https://2006.igem.org/wiki/index.php?title=Special:Contributions/Jlohmuel&feed=atom&limit=50&target=Jlohmuel&year=&month=2006.igem.org - User contributions [en]2021-05-06T02:59:37ZFrom 2006.igem.orgMediaWiki 1.16.5https://2006.igem.org/wiki/index.php/Brown:Tri-Stable_toggle_switchBrown:Tri-Stable toggle switch2006-11-12T05:31:14Z<p>Jlohmuel: </p>
<hr />
<div>{{Brown navigation bar}}<br />
=A tri-stable toggle switch=<br />
<br />
Pursuant to the the 1999 paper "Construction of a genetic toggle switch in Escherichia coli," by Timothy S. Gardner, Charles R. Cantor and James J. Collins, we wondered if the bi-stable toggle switch could be generalized to an n-stable switch. To that end, we conceived and began construction of the tri-stable toggle switch. The general idea is that when a selected promoter is activated, it represses the other two. As such each of the three states of our network are self-stabilizing.<br />
<br />
[[Image:Tristable_Toggle.jpg|left|500px|A general tri-stable toggle]]<br />
<br />
<br />
Our implementation uses the pLac, pTet, and pBad/AraC promoters and their respective inhibiting proteins. We chose these because of the relative ease with which each promoter is induced: by adding IPTG for the pLac-promoted region, tetracycline for the PTet-promoted region, and arabinose for the pBad/Ara genes. By characterizing each of the three pathways without following genes and terminators, we enable a tri-stable switching network of any three biobricks to be constructed with minimal cloning.<br />
<br />
[[Image:Registry_tristable.jpg|Our implementation]]<br />
<br />
<br />
<br><br><br />
<br />
Link to tri-stable switch on the registry:[http://partsregistry.org/Part:BBa_J24672 Tri-stable Switch]<br />
<br />
<br />
<br />
<br />
<br />
=Modeling the tri-stable toggle switch=<br />
While the tri-stable switch is seemingly simple in design, just like any other system it is subject to potentially comprising factors such as promoter leakiness and other stochastic fluctuations. In an attempt to predict the behavior of the tri-stable switch, we created a deterministic model of the system taking into account these factors. The model structure is based on that described in "Prediction and measurement of an autoregulatory genetic module" by Farren Isaacs, et al. This paper features an excellent supplement that takes you hand-in-hand through the derivation of model equations. I will attempt to emulate this derivation with our model below. It is important to note that while we have some preliminary results, our model is very much a work in progress. In order to model the system accurately, many of the fundamental constants governing the model will need to be determined experimentally. For now we have used some constants from the literature and estimated some based on similar the values of similar constants also from the literature. <br />
<br />
====Derivation of the Model Equations====<br />
[[Derivation of the Model Equations]]<br />
<br />
====Table of preliminary model constants====<br />
[[Table of preliminary model constants]]<br />
<br />
==Preliminary modeling results==<br />
<br />
The preliminary modeling results predict things about of the the tri-stable switch. First of all, the model confirms certain obvious points such as the a shift to the proper steady expression state in the presence of inducers. This prediction attests to model validity more than anything else. Secondly, the model predicts that the switch will reach a natural steady state with one promoter dominating in the absence of inducers. This makes intuitive sense as the individual promoters will most likely have different strengths. <br />
<br />
[[Image:no_inducer.png]]<br />
<br />
Next we tested the stability of the model by plugging in final steady-state induced levels of protein into an inducer-less system. The model predicts that the system will not remain stable and will instead revert back to the natural steady state for the inducer-less system. Even after changing several model parameters, the switch would not remain stable.<br />
<br />
<br />
[[Image:stability_test1.png]]<br />
<br />
As a test of the model, models of the three potential bi-stable switches (already proven to be successful experimentally) using the same components and parameters were tested. In accordance with the literature, each of these constructs maintained stable expression of the induced state following the removal of the inducer. In order to probe the instability of the tri-stable switch further, tri-stable switches in which all of the components were effectively the same were created (ex:for all LacI-like, TetR and AraC were modeled as forming tetramers and having one operator.) Each of these systems reached a steady state in which all of the proteins evolved identically and were present at the same levels in the absence of inducer. The same stability tests were then performed. All of the constructs remained stable. Thus our model suggests that the similarity of the repressor-expression systems dominates the success or failure of the tri-stable switch. Furthermore, the bi-stable switch is less sensitive to this factor. <br />
<br />
It should be noted that this is only a preliminary model. The values obtained from the literature will most like differ in our system. Through future experimentation we will elucidate these values and thus create a more accurate model. We will also conduct parameter scans aimed to provide further insight into the necessary system requirements for "tri-stability."<br />
<br />
<br />
<br />
<br />
Link to MATLAB code for tri-stable model [[Media:tristable1.txt]] <br />
<br />
Note: copy this code into MATLAB and save as an M-file<br />
<br />
=What we achieved=<br />
<br />
A successfully ligated pBad/arac construct in addition to partially completed ligations for the pLacI and the pTetR constructs were the results of our summer and semesters' work. The tri-stable toggle switch system was also modeled and proved to yield significant findings.<br />
<br />
=Challenges=<br />
<br />
Despite many cloning and ligation attempts, we were unable to create each component of the switch. More specifically, the pLacI and the pTetR circuits presented many ligation difficulties and proved to yield unsuccessful cloning. However, the pBAD/araC circuit was successfully constructed (yay!).<br />
<br />
=Conclusions=<br />
<br />
With the bigger picture in mind,this project yielded many successful outcomes.Firstly, we were able to successfully construct one of the three constructs necessary for a fully functional tri-stable toggle switch. Secondly, we were able to add more working parts to the registry, both a fully functioning pBad/AraC construct and many parts that were successfully ligated in the process of assembling the other two constructs. These parts will be used in the future by our team and perhaps many others as new ideas and projects develop. Also, it is necessary to mention our most recent finding from the modeling results for the tri-stable toggle switch. Interestingly enough, our model suggests that it is biologically unfeasible to create a switch with more than two components. If more than two components are introduced, such as three, the system is unable to maintain any one state and it will revert back to expressing its most dominant promoter construct. We find such results quite meaningful in demonstrating the biological limits of a system under study.<br />
<br />
=Future work=<br />
<br />
In the coming months, it is hoped that the failure of the ligation strategies for the pLacI and the pTetR constructs can be further investigated. We feel confident that we can continue to build from the one successful construction,the pBad/Arac construct and complete the other 2 constructs. To this end, it will be necessary to further troubleshoot the possible reasons why the cloning of the ligations were unsuccessful. Also we plan to conduct various experiments to elucidate more accurate model parameters specific to our system. Furthermore we will perform a parameter scan to investigate the determinates of "tri-stability."</div>Jlohmuelhttps://2006.igem.org/wiki/index.php/Brown:Tri-Stable_toggle_switchBrown:Tri-Stable toggle switch2006-11-12T05:30:28Z<p>Jlohmuel: </p>
<hr />
<div>{{Brown navigation bar}}<br />
=A tri-stable toggle switch=<br />
<br />
Pursuant to the the 1999 paper "Construction of a genetic toggle switch in Escherichia coli," by Timothy S. Gardner, Charles R. Cantor and James J. Collins, we wondered if the bi-stable toggle switch could be generalized to an n-stable switch. To that end, we conceived and began construction of the tri-stable toggle switch. The general idea is that when a selected promoter is activated, it represses the other two. As such each of the three states of our network are self-stabilizing.<br />
<br />
[[Image:Tristable_Toggle.jpg|left|500px|A general tri-stable toggle]]<br />
<br />
<br />
Our implementation uses the pLac, pTet, and pBad/AraC promoters and their respective inhibiting proteins. We chose these because of the relative ease with which each promoter is induced: by adding IPTG for the pLac-promoted region, tetracycline for the PTet-promoted region, and arabinose for the pBad/Ara genes. By characterizing each of the three pathways without following genes and terminators, we enable a tri-stable switching network of any three biobricks to be constructed with minimal cloning.<br />
<br />
[[Image:Registry_tristable.jpg|Our implementation]]<br />
<br />
<br />
<br><br><br />
<br />
Link to tri-stable switch on the registry:[http://partsregistry.org/Part:BBa_J24672 Tri-stable Switch]<br />
<br />
<br />
=Modeling the tri-stable toggle switch=<br />
While the tri-stable switch is seemingly simple in design, just like any other system it is subject to potentially comprising factors such as promoter leakiness and other stochastic fluctuations. In an attempt to predict the behavior of the tri-stable switch, we created a deterministic model of the system taking into account these factors. The model structure is based on that described in "Prediction and measurement of an autoregulatory genetic module" by Farren Isaacs, et al. This paper features an excellent supplement that takes you hand-in-hand through the derivation of model equations. I will attempt to emulate this derivation with our model below. It is important to note that while we have some preliminary results, our model is very much a work in progress. In order to model the system accurately, many of the fundamental constants governing the model will need to be determined experimentally. For now we have used some constants from the literature and estimated some based on similar the values of similar constants also from the literature. <br />
<br />
====Derivation of the Model Equations====<br />
[[Derivation of the Model Equations]]<br />
<br />
====Table of preliminary model constants====<br />
[[Table of preliminary model constants]]<br />
<br />
==Preliminary modeling results==<br />
<br />
The preliminary modeling results predict things about of the the tri-stable switch. First of all, the model confirms certain obvious points such as the a shift to the proper steady expression state in the presence of inducers. This prediction attests to model validity more than anything else. Secondly, the model predicts that the switch will reach a natural steady state with one promoter dominating in the absence of inducers. This makes intuitive sense as the individual promoters will most likely have different strengths. <br />
<br />
[[Image:no_inducer.png]]<br />
<br />
Next we tested the stability of the model by plugging in final steady-state induced levels of protein into an inducer-less system. The model predicts that the system will not remain stable and will instead revert back to the natural steady state for the inducer-less system. Even after changing several model parameters, the switch would not remain stable.<br />
<br />
<br />
[[Image:stability_test1.png]]<br />
<br />
As a test of the model, models of the three potential bi-stable switches (already proven to be successful experimentally) using the same components and parameters were tested. In accordance with the literature, each of these constructs maintained stable expression of the induced state following the removal of the inducer. In order to probe the instability of the tri-stable switch further, tri-stable switches in which all of the components were effectively the same were created (ex:for all LacI-like, TetR and AraC were modeled as forming tetramers and having one operator.) Each of these systems reached a steady state in which all of the proteins evolved identically and were present at the same levels in the absence of inducer. The same stability tests were then performed. All of the constructs remained stable. Thus our model suggests that the similarity of the repressor-expression systems dominates the success or failure of the tri-stable switch. Furthermore, the bi-stable switch is less sensitive to this factor. <br />
<br />
It should be noted that this is only a preliminary model. The values obtained from the literature will most like differ in our system. Through future experimentation we will elucidate these values and thus create a more accurate model. We will also conduct parameter scans aimed to provide further insight into the necessary system requirements for "tri-stability."<br />
<br />
<br />
<br />
<br />
Link to MATLAB code for tri-stable model [[Media:tristable1.txt]] <br />
<br />
Note: copy this code into MATLAB and save as an M-file<br />
<br />
=What we achieved=<br />
<br />
A successfully ligated pBad/arac construct in addition to partially completed ligations for the pLacI and the pTetR constructs were the results of our summer and semesters' work. The tri-stable toggle switch system was also modeled and proved to yield significant findings.<br />
<br />
=Challenges=<br />
<br />
Despite many cloning and ligation attempts, we were unable to create each component of the switch. More specifically, the pLacI and the pTetR circuits presented many ligation difficulties and proved to yield unsuccessful cloning. However, the pBAD/araC circuit was successfully constructed (yay!).<br />
<br />
=Conclusions=<br />
<br />
With the bigger picture in mind,this project yielded many successful outcomes.Firstly, we were able to successfully construct one of the three constructs necessary for a fully functional tri-stable toggle switch. Secondly, we were able to add more working parts to the registry, both a fully functioning pBad/AraC construct and many parts that were successfully ligated in the process of assembling the other two constructs. These parts will be used in the future by our team and perhaps many others as new ideas and projects develop. Also, it is necessary to mention our most recent finding from the modeling results for the tri-stable toggle switch. Interestingly enough, our model suggests that it is biologically unfeasible to create a switch with more than two components. If more than two components are introduced, such as three, the system is unable to maintain any one state and it will revert back to expressing its most dominant promoter construct. We find such results quite meaningful in demonstrating the biological limits of a system under study.<br />
<br />
=Future work=<br />
<br />
In the coming months, it is hoped that the failure of the ligation strategies for the pLacI and the pTetR constructs can be further investigated. We feel confident that we can continue to build from the one successful construction,the pBad/Arac construct and complete the other 2 constructs. To this end, it will be necessary to further troubleshoot the possible reasons why the cloning of the ligations were unsuccessful. Also we plan to conduct various experiments to elucidate more accurate model parameters specific to our system. Furthermore we will perform a parameter scan to investigate the determinates of "tri-stability."</div>Jlohmuelhttps://2006.igem.org/wiki/index.php/Derivation_of_the_Model_EquationsDerivation of the Model Equations2006-11-03T06:54:58Z<p>Jlohmuel: /* Equations for the total number of molecules */</p>
<hr />
<div>{{Brown navigation bar}}<br />
=Derivation of the model equations=<br />
The chemical reactions describing the tri-stable switch can be divided into the two categories of fast and slow reactions. Fast reactions such as dimer formation and promoter-binding occur in the scale of seconds and are therefore modeled to be in equilibrium. Conversely, slow reactions including the likes of transcription, translation, and protein degradation occur on the scale of minutes and are thus modeled to be evolving with time. <br />
<br />
====Fast reversible reaction equations====<br />
The following equations (1) describe the fast reactions. The characters L,T,A denote molecules of LacI, TetR, and, AraC, respectively while the subscripts denote whether the molecule is a monomer (blank), dimer(2), tri-mer(3), etc. The k's denote reaction rates.<br />
<br />
[[Image:Fast_rxn_eqs1.png]]<br />
<br />
Note that the equations include volume explicitly. Cell volume is modeled in this way as it is a slowly evolving function of time.<br />
<br />
====Slow irreversible reaction equations====<br />
The following equations (2) describe the slow irreversible reactions of transcription and translation (both taken into account with the reaction rate kti) and protein degradation (reaction constant kdi). The coefficients eta-ij take into account the relative translation rates of proteins from the same transcripts. For eta-ij, "i" represents the promoter responsible for producing the molecule and "j" represents the molecule being translated [ex: eta-LA corresponds to the relative rate of AraC production from the trascript produced by the LacI promoter]. To establish a convention, eta's are relative to the translation rate of the first gene on a particular transcript. Thus eta for the first gene on the mRNA transcript = 1. The alpha-i coefficients represent relative transcription rates. In this case, "i" denotes the promoter from which the mRNA is transcribed.<br />
<br />
[[Image:slow_rxn_eqs.png]]<br />
====Equations governing cell volume====<br />
The following two equations (3) describing cellular growth and division are taken directly from the aforementioned paper by Isaacs et al. The first equation describes the volume increase from the time immediately following cell division to the time immediately before it. In this equation, V0 denotes the volume of the cell at the beginning of growth and T0 denotes the time of cell division. In our model at times T=q*t0 in which q is an integer, we have volume V and protein concentration n halve - thus modeling volume division and the resulting protein redistribution. The second equation describes the dimensionless equation in which t is measured in terms of fcell-division time and the cell volume changes between 1 and 2.<br />
<br />
[[Image:vol_eq.png]]<br />
====Equations for the total number of molecules====<br />
The following equations (4) describe the total number of inhibitor protein molecules zi. d#i represents the number, "#", of molecules bound to the "i" promoter. (ex: all of the LacI tetramers bound to pLacI promoters are accounted for by the 4d1L term.)<br />
<br />
[[Image:Tot_mol_eq.png]]<br />
<br />
====Equations describing the temporal evolution of the total number inhibitor molecules====<br />
The following equations (5) describe the temporal evolution of the total number of inhibitor molecules. Beta-i = the cell division time multiplied by the combined transcription and translation rates from a given promoter "i" (t0*kti), thus representing the total number of protein molecules maximally produced by a given promoter. Similarly, Gamma-i = the cell division time multiplied by the degradation rate of protein i (t0*kdi), thus representing the total number of protein i destabilized over one cell division time. The eta and alpha terms are described in the section above describing the equations for the slow reactions. Note that the equation describing the temporal evolution of AraC includes the term "Aend." This term represents the endogenous expression of AraC in dH5alpha's. Our preliminary work suggests that this endogenous AraC has very little effect on our pBad promoter.<br />
<br />
[[Image:Ev_tot_mol_eqs.png]]<br />
<br />
====Fast reaction equilibrium equations====<br />
Compared to the slowly evolving reactions(2) described by the above equations (5), the fast reactions (1) can be considered to be in equilibrium. Thus the following equilibrium relations (6) hold.<br />
[[Image:Fast_eq_eqs.png]]<br />
<br />
====Equations modeling the presence of the inducers====<br />
The following equations (7) model the effect of the chemical inducers on the system. The model regards the proteins bound to the inducers as having increased dissociation rates (ki-#) as specified below. The reaction rate with the apostrophe denotes the original reaction rate in the absence of inducer inclusion. In these equations "I" denotes IPTG, "a" denotes arabinose, and "Tc" denotes tetracycline (or analogue aTc). The concept for and the form of these equations are based on equations described in the supplementary Information for "A bottom-up approach to gene regulation" by Guido et al. Much like the aforementioned paper by Isaacs et al., 'Bottom-up' has an extensive and brilliantly articulate derivation supplement. <br />
<br />
[[Image:inducer_eqs.png]]<br />
<br />
====Equations relating plasmid copy number to operator sites====<br />
The following equations (8) relate the plasmid copy number, "m", to the number of promoters with bound and unbound operators. As each plasmid contains 1 of each promoter, m is constant for all three promoter types. <br />
<br />
[[Image:cop_num_eqs.png]]<br />
<br />
====Simplified fast reaction equilibrium equations====<br />
The equilibrium equations governing the fast reactions (6) can be simplified by defining dimensionless equilibrium constants in the form of cij=kij/(k(-ij)*V0*A) in which A is Avogadro's number. The cL's are further simplified by defining the constant cL=c1L*c2L*c3L*c4L. Additionally by plugging in successive terms we were able to write the operator equations in terms of d0L. <br />
<br />
[[Image:Simp_fast_eq_eqs.png]]<br />
<br />
====Equations for the number of unbound operators====<br />
Combining equations (8) and the above simplified equilibrium equations(9), we are able to solve for the number of unbound operators d0i. <br />
<br />
[[Image:Unbound_eqs.png]]<br />
<br />
====Equations for the number of bound operators====<br />
By plugging the d0i's (10) back into the equations for bound operators (9), we can solve for these equations in terms of monomer concentrations.<br />
<br />
<br />
[[Image:bound_eqs.png]]<br />
<br />
====Equation manipulations to determine change in protein monomers per time====<br />
Next we can rewrite the equations describing the temporal evolution of the total number inhibitor molecules (5) in terms of equations (10) and (11). This yields the somewhat complex equations (12) which are solely in terms of monomer protein concentration. <br />
<br />
[[Image:dzdt_eqs.png]]<br />
<br />
====Equations for the change in protein monomers per time====<br />
By applying the simple mathematical manipulation (13) to equations (12) and the X derivative of equations (9) ((9) can be written in terms of X by plugging in equations (10) and (11)) we can solve for the following equations for the evolution of protein monomers as a function of time (14). I will spare you the dZi/dXi derivative equations of equation (9).<br />
<br />
[[Image:simp_mat_man_eq.png]]<br />
<br />
====Equations describing stochastic variation====<br />
<br />
====Equations for the evolution of fluorescent reporters====<br />
<br />
<br />
--[[User:Jlohmuel|Jlohmuel]] 03:59, 28 October 2006 (EDT)</div>Jlohmuelhttps://2006.igem.org/wiki/index.php/Brown:ProjectsBrown:Projects2006-11-02T03:04:42Z<p>Jlohmuel: </p>
<hr />
<div>{{Brown navigation bar}}<br />
{| cellspacing="2px" cellpadding="0" border="0" style="padding: 0; width: 750px; color: #000000; background-color: #ffffff;"<br />
| width=717px style="padding: 5px; background-color: #ffffff; border: 2px solid #993300;" |<br />
<br />
<h3>Brown 2006 iGEM entry </h3><br />
<br />
As the race to the finish draws near for iGEM 2006, our team has compiled a detailed summary of our summers work. This includes our initial ideas, designs, models and the parts that we have added to the registry. We began the summer with grand plans to design a game of bacterial freeze tag, modeled after the popular children's game where one player tries to catch the other. Our efforts were then switched to the extension of the bi-stable toggle switch and the design of a tri-stable toggle switch. Our projects highlight the team's creative and innovative approaches to designing and engineering working biological systems.<br />
<br />
Click here to see our BioBricks! [http://partsregistry.org/cgi/partsdb/pgroup.cgi?pgroup=iGEM2006partsregistry.org/cgi/partsdb/pgroup.cgi?pgroup=iGEM2006&group=Brown Brown iGEM 2006 Parts]<br />
<br />
{| cellspacing="2px" cellpadding="0" border="0" style="padding: 0px; width: 750px; color: #000000; background-color: #ffffff;"<br />
|-valign="top"<br />
|width=179.25px style="padding: 5px; background-color: #ffffff; border: 2px solid #993300;" |<br />
<br />
<h3>[[Brown:Bacterial Freeze Tag | Bacterial Freeze Tag]] </h3><br />
<br />
*[[Brown:Bacterial_Freeze_Tag#Overview |Overview]]<br />
*[[Brown:Bacterial_Freeze_Tag#Sender cell | Sender cell ]]<br />
*[[Brown:Bacterial_Freeze_Tag#Receiver cell | Receiver cell ]]<br />
*[[Brown:Bacterial_Freeze_Tag#Challenges| Challenges ]]<br />
<br />
|width=179.25px style="padding: 5px; background-color: #ffffff; border: 2px solid #993300;" |<br />
<br />
<h3>[[Brown:Tri-Stable toggle switch|Tri-Stable toggle switch]]</h3><br />
*[[Brown:Tri-Stable_toggle_switch|Introduction to the project]]<br />
*[[Brown:Tri-Stable toggle switch#Modeling the tri-stable toggle switch | Modeling the tri-stable switch]]<br />
*[[Brown:Tri-Stable toggle switch#What we achieved| What we achieved]]<br />
*[[Brown:Tri-Stable toggle switch#Challenges | Challenges ]]<br />
*[[Brown:Tri-Stable toggle switch#Challenges | Future work ]]<br />
<br />
|width=179.25px style="padding: 5px; background-color: #ffffff; border: 2px solid #993300;" |<br />
<br />
<h3>[[Brown:Team logistics | Team Logistics ]] </h3><br />
<br />
*[[Brown:Team logistics|How we got here]]<br />
*[[Brown:Team_logistics|Challenges]]<br />
*[[Brown:Team_logistics|Future work]]<br />
<br />
<br />
<br />
__NOTOC__</div>Jlohmuelhttps://2006.igem.org/wiki/index.php/Brown:Tri-Stable_toggle_switchBrown:Tri-Stable toggle switch2006-11-02T02:31:52Z<p>Jlohmuel: /* A tri-stable toggle switch */</p>
<hr />
<div>{{Brown navigation bar}}<br />
=A tri-stable toggle switch=<br />
<br />
Pursuant to the the 1999 paper "Construction of a genetic toggle switch in Escherichia coli," by Timothy S. Gardner, Charles R. Cantor and James J. Collins, we wondered if the bi-stable toggle switch could be generalized to an n-stable switch. To that end, we conceived and began construction of the tri-stable toggle switch. The general idea is that when a selected promoter is activated, it represses the other two. As such each of the three states of our network are self-stabilizing.<br />
<br />
[[Image:Tristable_Toggle.jpg|left|500px|A general tri-stable toggle]]<br />
<br />
<br />
Our implementation uses the pLac, pTet, and pBad/AraC promoters and their respective inhibiting proteins. We chose these because of the relative ease with which each promoter is induced: by adding IPTG for the pLac-promoted region, tetracycline for the PTet-promoted region, and arabinose for the pBad/Ara genes. By characterizing each of the three pathways without following genes and terminators, we enable a tri-stable switching network of any three biobricks to be constructed with minimal cloning.<br />
<br />
[[Image:Registry_tristable.jpg|Our implementation]]<br />
<br />
<br />
<br><br><br />
<br />
Link to tri-stable switch on the registry:[http://partsregistry.org/Part:BBa_J24672 Tri-stable Switch]<br />
<br />
=Modeling the tri-stable toggle switch=<br />
While the tri-stable switch is seemingly simple in design, just like any other system it is subject to potentially comprising factors such as promoter leakiness and other stochastic fluctuations. In an attempt to predict the behavior of the tri-stable switch, we created a deterministic model of the system taking into account these factors. The model structure is based on that described in "Prediction and measurement of an autoregulatory genetic module" by Farren Isaacs, et al. This paper features an excellent supplement that takes you hand-in-hand through the derivation of model equations. I will attempt to emulate this derivation with our model below. It is important to note that while we have some preliminary results, our model is very much a work in progress. In order to model the system accurately, many of the fundamental constants governing the model will need to be determined experimentally. For now we have used some constants from the literature and estimated some based on similar the values of similar constants also from the literature. <br />
<br />
====Derivation of the Model Equations====<br />
[[Derivation of the Model Equations]]<br />
<br />
====Table of preliminary model constants====<br />
[[Table of preliminary model constants]]<br />
<br />
==Preliminary modeling results==<br />
<br />
The preliminary modeling results predict things about of the the tri-stable switch. First of all, the model confirms certain obvious points such as the a shift to the proper steady expression state in the presence of inducers. This prediction attests to model validity more than anything else. Secondly, the model predicts that the switch will reach a natural steady state with one promoter dominating in the absence of inducers. This makes intuitive sense as the individual promoters will most likely have different strengths. <br />
<br />
[[Image:no_inducer.png]]<br />
<br />
Next we tested the stability of the model by plugging in final steady-state induced levels of protein into an inducer-less system. The model predicts that the system will not remain stable and will instead revert back to the natural steady state for the inducer-less system. Even after changing several model parameters, the switch would not remain stable.<br />
<br />
<br />
[[Image:stability_test1.png]]<br />
<br />
As a test of the model, models of the three potential bi-stable switches (already proven to be successful experimentally) using the same components and parameters were tested. In accordance with the literature, each of these constructs maintained stable expression of the induced state following the removal of the inducer. In order to probe the instability of the tri-stable switch further, tri-stable switches in which all of the components were effectively the same were created (ex:for all LacI-like, TetR and AraC were modeled as forming tetramers and having one operator.) Each of these systems reached a steady state in which all of the proteins evolved identically and were present at the same levels in the absence of inducer. The same stability tests were then performed. All of the constructs remained stable. Thus our model suggests that the similarity of the repressor-expression systems dominates the success or failure of the tri-stable switch. Furthermore, the bi-stable switch is less sensitive to this factor. <br />
<br />
It should be noted that this is only a preliminary model. The values obtained from the literature will most like differ in our system. Through future experimentation we will elucidate these values and thus create a more accurate model. We will also conduct parameter scans aimed to provide further insight into the necessary system requirements for "tri-stability."<br />
<br />
<br />
<br />
<br />
Link to MATLAB code for tri-stable model [[Media:tristable1.txt]] <br />
<br />
Note: copy this code into MATLAB and save as an M-file<br />
<br />
=What we achieved=<br />
<br />
A successfully ligated pBad/arac construct in addition to partially completed ligations for the pLacI and the pTetR constructs were the results of our summer and semesters' work. The tri-stable toggle switch system was also modeled and proved to yield significant findings.<br />
<br />
=Challenges=<br />
<br />
Despite many cloning and ligation attempts, we were unable to create each component of the switch. More specifically, the pLacI and the pTetR circuits presented many ligation difficulties and proved to yield unsuccessful cloning. However, the pBAD/araC circuit was successfully constructed (yay!).<br />
<br />
=Conclusions=<br />
<br />
With the bigger picture in mind,this project yielded many successful outcomes.Firstly, we were able to successfully construct one of the three constructs necessary for a fully functional tri-stable toggle switch. Secondly, we were able to add more working parts to the registry, both a fully functioning pBad/AraC construct and many parts that were successfully ligated in the process of assembling the other two constructs. These parts will be used in the future by our team and perhaps many others as new ideas and projects develop. Also, it is necessary to mention our most recent finding from the modeling results for the tri-stable toggle switch. Interestingly enough, our model suggests that it is biologically unfeasible to create a switch with more than two components. If more than two components are introduced, such as three, the system is unable to maintain any one state and it will revert back to expressing its most dominant promoter construct. We find such results quite meaningful in demonstrating the biological limits of a system under study.<br />
<br />
=Future work=<br />
<br />
In the coming months, it is hoped that the failure of the ligation strategies for the pLacI and the pTetR constructs can be further investigated. We feel confident that we can continue to build from the one successful construction,the pBad/Arac construct and complete the other 2 constructs. To this end, it will be necessary to further troubleshoot the possible reasons why the cloning of the ligations were unsuccessful. Also we plan to conduct various experiments to elucidate more accurate model parameters specific to our system. Furthermore we will perform a parameter scan to investigate the determinates of "tri-stability."</div>Jlohmuelhttps://2006.igem.org/wiki/index.php/Brown:Tri-Stable_toggle_switchBrown:Tri-Stable toggle switch2006-11-02T02:30:05Z<p>Jlohmuel: /* A tri-stable toggle switch */</p>
<hr />
<div>{{Brown navigation bar}}<br />
=A tri-stable toggle switch=<br />
<br />
Pursuant to the the 1999 paper "Construction of a genetic toggle switch in Escherichia coli," by Timothy S. Gardner, Charles R. Cantor and James J. Collins, we wondered if the bi-stable toggle switch could be generalized to an n-stable switch. To that end, we conceived and began construction of the tri-stable toggle switch. The general idea is that when a selected promoter is activated, it represses the other two. As such each of the three states of our network are self-stabilizing.<br />
<br />
[[Image:Tristable_Toggle.jpg|left|500px|A general tri-stable toggle]]<br />
<br />
Our implementation uses the pLac, pTet, and pBad/AraC promoters and their respective inhibiting proteins. We chose these because of the relative ease with which each promoter is induced: by adding IPTG for the pLac-promoted region, tetracycline for the PTet-promoted region, and arabinose for the pBad/Ara genes. By characterizing each of the three pathways without following genes and terminators, we enable a tri-stable switching network of any three biobricks to be constructed with minimal cloning.<br />
<br />
[[Image:Registry_tristable.jpg|Our implementation]]<br />
<br />
<br><br />
<br />
<br />
<br><br><br />
<br />
Link to tri-stable switch on the registry:[http://partsregistry.org/Part:BBa_J24672 Tri-stable Switch]<br />
<br />
=Modeling the tri-stable toggle switch=<br />
While the tri-stable switch is seemingly simple in design, just like any other system it is subject to potentially comprising factors such as promoter leakiness and other stochastic fluctuations. In an attempt to predict the behavior of the tri-stable switch, we created a deterministic model of the system taking into account these factors. The model structure is based on that described in "Prediction and measurement of an autoregulatory genetic module" by Farren Isaacs, et al. This paper features an excellent supplement that takes you hand-in-hand through the derivation of model equations. I will attempt to emulate this derivation with our model below. It is important to note that while we have some preliminary results, our model is very much a work in progress. In order to model the system accurately, many of the fundamental constants governing the model will need to be determined experimentally. For now we have used some constants from the literature and estimated some based on similar the values of similar constants also from the literature. <br />
<br />
====Derivation of the Model Equations====<br />
[[Derivation of the Model Equations]]<br />
<br />
====Table of preliminary model constants====<br />
[[Table of preliminary model constants]]<br />
<br />
==Preliminary modeling results==<br />
<br />
The preliminary modeling results predict things about of the the tri-stable switch. First of all, the model confirms certain obvious points such as the a shift to the proper steady expression state in the presence of inducers. This prediction attests to model validity more than anything else. Secondly, the model predicts that the switch will reach a natural steady state with one promoter dominating in the absence of inducers. This makes intuitive sense as the individual promoters will most likely have different strengths. <br />
<br />
[[Image:no_inducer.png]]<br />
<br />
Next we tested the stability of the model by plugging in final steady-state induced levels of protein into an inducer-less system. The model predicts that the system will not remain stable and will instead revert back to the natural steady state for the inducer-less system. Even after changing several model parameters, the switch would not remain stable.<br />
<br />
<br />
[[Image:stability_test1.png]]<br />
<br />
As a test of the model, models of the three potential bi-stable switches (already proven to be successful experimentally) using the same components and parameters were tested. In accordance with the literature, each of these constructs maintained stable expression of the induced state following the removal of the inducer. In order to probe the instability of the tri-stable switch further, tri-stable switches in which all of the components were effectively the same were created (ex:for all LacI-like, TetR and AraC were modeled as forming tetramers and having one operator.) Each of these systems reached a steady state in which all of the proteins evolved identically and were present at the same levels in the absence of inducer. The same stability tests were then performed. All of the constructs remained stable. Thus our model suggests that the similarity of the repressor-expression systems dominates the success or failure of the tri-stable switch. Furthermore, the bi-stable switch is less sensitive to this factor. <br />
<br />
It should be noted that this is only a preliminary model. The values obtained from the literature will most like differ in our system. Through future experimentation we will elucidate these values and thus create a more accurate model. We will also conduct parameter scans aimed to provide further insight into the necessary system requirements for "tri-stability."<br />
<br />
<br />
<br />
<br />
Link to MATLAB code for tri-stable model [[Media:tristable1.txt]] <br />
<br />
Note: copy this code into MATLAB and save as an M-file<br />
<br />
=What we achieved=<br />
<br />
A successfully ligated pBad/arac construct in addition to partially completed ligations for the pLacI and the pTetR constructs were the results of our summer and semesters' work. The tri-stable toggle switch system was also modeled and proved to yield significant findings.<br />
<br />
=Challenges=<br />
<br />
Despite many cloning and ligation attempts, we were unable to create each component of the switch. More specifically, the pLacI and the pTetR circuits presented many ligation difficulties and proved to yield unsuccessful cloning. However, the pBAD/araC circuit was successfully constructed (yay!).<br />
<br />
=Conclusions=<br />
<br />
With the bigger picture in mind,this project yielded many successful outcomes.Firstly, we were able to successfully construct one of the three constructs necessary for a fully functional tri-stable toggle switch. Secondly, we were able to add more working parts to the registry, both a fully functioning pBad/AraC construct and many parts that were successfully ligated in the process of assembling the other two constructs. These parts will be used in the future by our team and perhaps many others as new ideas and projects develop. Also, it is necessary to mention our most recent finding from the modeling results for the tri-stable toggle switch. Interestingly enough, our model suggests that it is biologically unfeasible to create a switch with more than two components. If more than two components are introduced, such as three, the system is unable to maintain any one state and it will revert back to expressing its most dominant promoter construct. We find such results quite meaningful in demonstrating the biological limits of a system under study.<br />
<br />
=Future work=<br />
<br />
In the coming months, it is hoped that the failure of the ligation strategies for the pLacI and the pTetR constructs can be further investigated. We feel confident that we can continue to build from the one successful construction,the pBad/Arac construct and complete the other 2 constructs. To this end, it will be necessary to further troubleshoot the possible reasons why the cloning of the ligations were unsuccessful. Also we plan to conduct various experiments to elucidate more accurate model parameters specific to our system. Furthermore we will perform a parameter scan to investigate the determinates of "tri-stability."</div>Jlohmuelhttps://2006.igem.org/wiki/index.php/Brown:Tri-Stable_toggle_switchBrown:Tri-Stable toggle switch2006-11-02T02:28:37Z<p>Jlohmuel: /* A tri-stable toggle switch */</p>
<hr />
<div>{{Brown navigation bar}}<br />
=A tri-stable toggle switch=<br />
<br />
Pursuant to the the 1999 paper "Construction of a genetic toggle switch in Escherichia coli," by Timothy S. Gardner, Charles R. Cantor and James J. Collins, we wondered if the bi-stable toggle switch could be generalized to an n-stable switch. To that end, we conceived and began construction of the tri-stable toggle switch. The general idea is that when a selected promoter is activated, it represses the other two. As such each of the three states of our network are self-stabilizing.<br />
<br />
[[Image:Tristable_Toggle.jpg|left|500px|A general tri-stable toggle]]<br />
<br />
Our implementation uses the pLac, pTet, and pBad/AraC promoters and their respective inhibiting proteins. We chose these because of the relative ease with which each promoter is induced: by adding IPTG for the pLac-promoted region, tetracycline for the PTet-promoted region, and arabinose for the pBad/Ara genes. By characterizing each of the three pathways without following genes and terminators, we enable a tri-stable switching network of any three biobricks to be constructed with minimal cloning.<br />
<br />
[[Image:Registry_tristable.jpg|Our implementation]]<br />
<br />
<br><br />
<br />
<br />
<br><br><br />
<br />
<br><br />
<br><br />
Link to Tri-stable Switch on the registry:[http://partsregistry.org/Part:BBa_J24672 Tri-stable Switch]<br />
<br />
=Modeling the tri-stable toggle switch=<br />
While the tri-stable switch is seemingly simple in design, just like any other system it is subject to potentially comprising factors such as promoter leakiness and other stochastic fluctuations. In an attempt to predict the behavior of the tri-stable switch, we created a deterministic model of the system taking into account these factors. The model structure is based on that described in "Prediction and measurement of an autoregulatory genetic module" by Farren Isaacs, et al. This paper features an excellent supplement that takes you hand-in-hand through the derivation of model equations. I will attempt to emulate this derivation with our model below. It is important to note that while we have some preliminary results, our model is very much a work in progress. In order to model the system accurately, many of the fundamental constants governing the model will need to be determined experimentally. For now we have used some constants from the literature and estimated some based on similar the values of similar constants also from the literature. <br />
<br />
====Derivation of the Model Equations====<br />
[[Derivation of the Model Equations]]<br />
<br />
====Table of preliminary model constants====<br />
[[Table of preliminary model constants]]<br />
<br />
==Preliminary modeling results==<br />
<br />
The preliminary modeling results predict things about of the the tri-stable switch. First of all, the model confirms certain obvious points such as the a shift to the proper steady expression state in the presence of inducers. This prediction attests to model validity more than anything else. Secondly, the model predicts that the switch will reach a natural steady state with one promoter dominating in the absence of inducers. This makes intuitive sense as the individual promoters will most likely have different strengths. <br />
<br />
[[Image:no_inducer.png]]<br />
<br />
Next we tested the stability of the model by plugging in final steady-state induced levels of protein into an inducer-less system. The model predicts that the system will not remain stable and will instead revert back to the natural steady state for the inducer-less system. Even after changing several model parameters, the switch would not remain stable.<br />
<br />
<br />
[[Image:stability_test1.png]]<br />
<br />
As a test of the model, models of the three potential bi-stable switches (already proven to be successful experimentally) using the same components and parameters were tested. In accordance with the literature, each of these constructs maintained stable expression of the induced state following the removal of the inducer. In order to probe the instability of the tri-stable switch further, tri-stable switches in which all of the components were effectively the same were created (ex:for all LacI-like, TetR and AraC were modeled as forming tetramers and having one operator.) Each of these systems reached a steady state in which all of the proteins evolved identically and were present at the same levels in the absence of inducer. The same stability tests were then performed. All of the constructs remained stable. Thus our model suggests that the similarity of the repressor-expression systems dominates the success or failure of the tri-stable switch. Furthermore, the bi-stable switch is less sensitive to this factor. <br />
<br />
It should be noted that this is only a preliminary model. The values obtained from the literature will most like differ in our system. Through future experimentation we will elucidate these values and thus create a more accurate model. We will also conduct parameter scans aimed to provide further insight into the necessary system requirements for "tri-stability."<br />
<br />
<br />
<br />
<br />
Link to MATLAB code for tri-stable model [[Media:tristable1.txt]] <br />
<br />
Note: copy this code into MATLAB and save as an M-file<br />
<br />
=What we achieved=<br />
<br />
A successfully ligated pBad/arac construct in addition to partially completed ligations for the pLacI and the pTetR constructs were the results of our summer and semesters' work. The tri-stable toggle switch system was also modeled and proved to yield significant findings.<br />
<br />
=Challenges=<br />
<br />
Despite many cloning and ligation attempts, we were unable to create each component of the switch. More specifically, the pLacI and the pTetR circuits presented many ligation difficulties and proved to yield unsuccessful cloning. However, the pBAD/araC circuit was successfully constructed (yay!).<br />
<br />
=Conclusions=<br />
<br />
With the bigger picture in mind,this project yielded many successful outcomes.Firstly, we were able to successfully construct one of the three constructs necessary for a fully functional tri-stable toggle switch. Secondly, we were able to add more working parts to the registry, both a fully functioning pBad/AraC construct and many parts that were successfully ligated in the process of assembling the other two constructs. These parts will be used in the future by our team and perhaps many others as new ideas and projects develop. Also, it is necessary to mention our most recent finding from the modeling results for the tri-stable toggle switch. Interestingly enough, our model suggests that it is biologically unfeasible to create a switch with more than two components. If more than two components are introduced, such as three, the system is unable to maintain any one state and it will revert back to expressing its most dominant promoter construct. We find such results quite meaningful in demonstrating the biological limits of a system under study.<br />
<br />
=Future work=<br />
<br />
In the coming months, it is hoped that the failure of the ligation strategies for the pLacI and the pTetR constructs can be further investigated. We feel confident that we can continue to build from the one successful construction,the pBad/Arac construct and complete the other 2 constructs. To this end, it will be necessary to further troubleshoot the possible reasons why the cloning of the ligations were unsuccessful. Also we plan to conduct various experiments to elucidate more accurate model parameters specific to our system. Furthermore we will perform a parameter scan to investigate the determinates of "tri-stability."</div>Jlohmuelhttps://2006.igem.org/wiki/index.php/Brown:Tri-Stable_toggle_switchBrown:Tri-Stable toggle switch2006-11-02T02:28:17Z<p>Jlohmuel: /* A tri-stable toggle switch */</p>
<hr />
<div>{{Brown navigation bar}}<br />
=A tri-stable toggle switch=<br />
<br />
Pursuant to the the 1999 paper "Construction of a genetic toggle switch in Escherichia coli," by Timothy S. Gardner, Charles R. Cantor and James J. Collins, we wondered if the bi-stable toggle switch could be generalized to an n-stable switch. To that end, we conceived and began construction of the tri-stable toggle switch. The general idea is that when a selected promoter is activated, it represses the other two. As such each of the three states of our network are self-stabilizing.<br />
<br />
[[Image:Tristable_Toggle.jpg|left|500px|A general tri-stable toggle]]<br />
<br />
Our implementation uses the pLac, pTet, and pBad/AraC promoters and their respective inhibiting proteins. We chose these because of the relative ease with which each promoter is induced: by adding IPTG for the pLac-promoted region, tetracycline for the PTet-promoted region, and arabinose for the pBad/Ara genes. By characterizing each of the three pathways without following genes and terminators, we enable a tri-stable switching network of any three biobricks to be constructed with minimal cloning.<br />
<br />
[[Image:Registry_tristable.jpg|Our implementation]]<br />
<br />
<br><br />
Link to Tri-stable Switch on the registry:[http://partsregistry.org/Part:BBa_J24672 Tri-stable Switch]<br />
<br />
<br><br><br />
<br />
<br><br />
<br><br />
<br />
=Modeling the tri-stable toggle switch=<br />
While the tri-stable switch is seemingly simple in design, just like any other system it is subject to potentially comprising factors such as promoter leakiness and other stochastic fluctuations. In an attempt to predict the behavior of the tri-stable switch, we created a deterministic model of the system taking into account these factors. The model structure is based on that described in "Prediction and measurement of an autoregulatory genetic module" by Farren Isaacs, et al. This paper features an excellent supplement that takes you hand-in-hand through the derivation of model equations. I will attempt to emulate this derivation with our model below. It is important to note that while we have some preliminary results, our model is very much a work in progress. In order to model the system accurately, many of the fundamental constants governing the model will need to be determined experimentally. For now we have used some constants from the literature and estimated some based on similar the values of similar constants also from the literature. <br />
<br />
====Derivation of the Model Equations====<br />
[[Derivation of the Model Equations]]<br />
<br />
====Table of preliminary model constants====<br />
[[Table of preliminary model constants]]<br />
<br />
==Preliminary modeling results==<br />
<br />
The preliminary modeling results predict things about of the the tri-stable switch. First of all, the model confirms certain obvious points such as the a shift to the proper steady expression state in the presence of inducers. This prediction attests to model validity more than anything else. Secondly, the model predicts that the switch will reach a natural steady state with one promoter dominating in the absence of inducers. This makes intuitive sense as the individual promoters will most likely have different strengths. <br />
<br />
[[Image:no_inducer.png]]<br />
<br />
Next we tested the stability of the model by plugging in final steady-state induced levels of protein into an inducer-less system. The model predicts that the system will not remain stable and will instead revert back to the natural steady state for the inducer-less system. Even after changing several model parameters, the switch would not remain stable.<br />
<br />
<br />
[[Image:stability_test1.png]]<br />
<br />
As a test of the model, models of the three potential bi-stable switches (already proven to be successful experimentally) using the same components and parameters were tested. In accordance with the literature, each of these constructs maintained stable expression of the induced state following the removal of the inducer. In order to probe the instability of the tri-stable switch further, tri-stable switches in which all of the components were effectively the same were created (ex:for all LacI-like, TetR and AraC were modeled as forming tetramers and having one operator.) Each of these systems reached a steady state in which all of the proteins evolved identically and were present at the same levels in the absence of inducer. The same stability tests were then performed. All of the constructs remained stable. Thus our model suggests that the similarity of the repressor-expression systems dominates the success or failure of the tri-stable switch. Furthermore, the bi-stable switch is less sensitive to this factor. <br />
<br />
It should be noted that this is only a preliminary model. The values obtained from the literature will most like differ in our system. Through future experimentation we will elucidate these values and thus create a more accurate model. We will also conduct parameter scans aimed to provide further insight into the necessary system requirements for "tri-stability."<br />
<br />
<br />
<br />
<br />
Link to MATLAB code for tri-stable model [[Media:tristable1.txt]] <br />
<br />
Note: copy this code into MATLAB and save as an M-file<br />
<br />
=What we achieved=<br />
<br />
A successfully ligated pBad/arac construct in addition to partially completed ligations for the pLacI and the pTetR constructs were the results of our summer and semesters' work. The tri-stable toggle switch system was also modeled and proved to yield significant findings.<br />
<br />
=Challenges=<br />
<br />
Despite many cloning and ligation attempts, we were unable to create each component of the switch. More specifically, the pLacI and the pTetR circuits presented many ligation difficulties and proved to yield unsuccessful cloning. However, the pBAD/araC circuit was successfully constructed (yay!).<br />
<br />
=Conclusions=<br />
<br />
With the bigger picture in mind,this project yielded many successful outcomes.Firstly, we were able to successfully construct one of the three constructs necessary for a fully functional tri-stable toggle switch. Secondly, we were able to add more working parts to the registry, both a fully functioning pBad/AraC construct and many parts that were successfully ligated in the process of assembling the other two constructs. These parts will be used in the future by our team and perhaps many others as new ideas and projects develop. Also, it is necessary to mention our most recent finding from the modeling results for the tri-stable toggle switch. Interestingly enough, our model suggests that it is biologically unfeasible to create a switch with more than two components. If more than two components are introduced, such as three, the system is unable to maintain any one state and it will revert back to expressing its most dominant promoter construct. We find such results quite meaningful in demonstrating the biological limits of a system under study.<br />
<br />
=Future work=<br />
<br />
In the coming months, it is hoped that the failure of the ligation strategies for the pLacI and the pTetR constructs can be further investigated. We feel confident that we can continue to build from the one successful construction,the pBad/Arac construct and complete the other 2 constructs. To this end, it will be necessary to further troubleshoot the possible reasons why the cloning of the ligations were unsuccessful. Also we plan to conduct various experiments to elucidate more accurate model parameters specific to our system. Furthermore we will perform a parameter scan to investigate the determinates of "tri-stability."</div>Jlohmuelhttps://2006.igem.org/wiki/index.php/Brown:ProjectsBrown:Projects2006-11-02T02:27:24Z<p>Jlohmuel: </p>
<hr />
<div>{{Brown navigation bar}}<br />
{| cellspacing="2px" cellpadding="0" border="0" style="padding: 0; width: 750px; color: #000000; background-color: #ffffff;"<br />
| width=717px style="padding: 5px; background-color: #ffffff; border: 2px solid #993300;" |<br />
<br />
<h3>Brown 2006 iGEM entry </h3><br />
<br />
As the race to the finish draws near for iGEM 2006, our team has compiled a detailed summary of our summers work. This includes our initial ideas, designs, models and the parts that we have added to the registry. We began the summer with grand plans to design a game of bacterial freeze tag, modeled after the popular children's game where one player tries to catch the other. Our efforts were then switched to the extension of the bi-stable toggle switch and the design of a tri-stable toggle switch. Our projects highlight the team's creative and innovative approaches to designing and engineering working biological systems.<br />
<br />
Click here to see our completed BioBricks! [http://partsregistry.org/cgi/partsdb/pgroup.cgi?pgroup=iGEM2006partsregistry.org/cgi/partsdb/pgroup.cgi?pgroup=iGEM2006&group=Brown Brown iGEM 2006 Parts]<br />
<br />
{| cellspacing="2px" cellpadding="0" border="0" style="padding: 0px; width: 750px; color: #000000; background-color: #ffffff;"<br />
|-valign="top"<br />
|width=179.25px style="padding: 5px; background-color: #ffffff; border: 2px solid #993300;" |<br />
<br />
<h3>[[Brown:Bacterial Freeze Tag | Bacterial Freeze Tag]] </h3><br />
<br />
*[[Brown:Bacterial_Freeze_Tag#Overview |Overview]]<br />
*[[Brown:Bacterial_Freeze_Tag#Sender cell | Sender cell ]]<br />
*[[Brown:Bacterial_Freeze_Tag#Receiver cell | Receiver cell ]]<br />
*[[Brown:Bacterial_Freeze_Tag#Challenges| Challenges ]]<br />
<br />
|width=179.25px style="padding: 5px; background-color: #ffffff; border: 2px solid #993300;" |<br />
<br />
<h3>[[Brown:Tri-Stable toggle switch|Tri-Stable toggle switch]]</h3><br />
*[[Brown:Tri-Stable_toggle_switch|Introduction to the project]]<br />
*[[Brown:Tri-Stable toggle switch#Modeling the tri-stable toggle switch | Modeling the tri-stable switch]]<br />
*[[Brown:Tri-Stable toggle switch#What we achieved| What we achieved]]<br />
*[[Brown:Tri-Stable toggle switch#Challenges | Challenges ]]<br />
*[[Brown:Tri-Stable toggle switch#Challenges | Future work ]]<br />
<br />
|width=179.25px style="padding: 5px; background-color: #ffffff; border: 2px solid #993300;" |<br />
<br />
<h3>[[Brown:Team logistics | Team Logistics ]] </h3><br />
<br />
*[[Brown:Team logistics|How we got here]]<br />
*[[Brown:Team_logistics|Challenges]]<br />
*[[Brown:Team_logistics|Future work]]<br />
<br />
<br />
<br />
__NOTOC__</div>Jlohmuelhttps://2006.igem.org/wiki/index.php/Brown:ProjectsBrown:Projects2006-11-02T02:26:53Z<p>Jlohmuel: </p>
<hr />
<div>{{Brown navigation bar}}<br />
{| cellspacing="2px" cellpadding="0" border="0" style="padding: 0; width: 750px; color: #000000; background-color: #ffffff;"<br />
| width=717px style="padding: 5px; background-color: #ffffff; border: 2px solid #993300;" |<br />
<br />
<h3>Brown 2006 iGEM entry </h3><br />
<br />
As the race to the finish draws near for iGEM 2006, our team has compiled a detailed summary of our summers work. This includes our initial ideas, designs, models and the parts that we have added to the registry. We began the summer with grand plans to design a game of bacterial freeze tag, modeled after the popular children's game where one player tries to catch the other. Our efforts were then switched to the extension of the bi-stable toggle switch and the design of a tri-stable toggle switch. Our projects highlight the team's creative and innovative approaches to designing and engineering working biological systems.<br />
<br />
Click here to see our completed BioBricks! [http://partsregistry.org/cgi/partsdb/pgroup.cgi?pgroup=iGEM2006partsregistry.org/cgi/partsdb/pgroup.cgi?pgroup=iGEM2006&group=Brown Parts]<br />
<br />
{| cellspacing="2px" cellpadding="0" border="0" style="padding: 0px; width: 750px; color: #000000; background-color: #ffffff;"<br />
|-valign="top"<br />
|width=179.25px style="padding: 5px; background-color: #ffffff; border: 2px solid #993300;" |<br />
<br />
<h3>[[Brown:Bacterial Freeze Tag | Bacterial Freeze Tag]] </h3><br />
<br />
*[[Brown:Bacterial_Freeze_Tag#Overview |Overview]]<br />
*[[Brown:Bacterial_Freeze_Tag#Sender cell | Sender cell ]]<br />
*[[Brown:Bacterial_Freeze_Tag#Receiver cell | Receiver cell ]]<br />
*[[Brown:Bacterial_Freeze_Tag#Challenges| Challenges ]]<br />
<br />
|width=179.25px style="padding: 5px; background-color: #ffffff; border: 2px solid #993300;" |<br />
<br />
<h3>[[Brown:Tri-Stable toggle switch|Tri-Stable toggle switch]]</h3><br />
*[[Brown:Tri-Stable_toggle_switch|Introduction to the project]]<br />
*[[Brown:Tri-Stable toggle switch#Modeling the tri-stable toggle switch | Modeling the tri-stable switch]]<br />
*[[Brown:Tri-Stable toggle switch#What we achieved| What we achieved]]<br />
*[[Brown:Tri-Stable toggle switch#Challenges | Challenges ]]<br />
*[[Brown:Tri-Stable toggle switch#Challenges | Future work ]]<br />
<br />
|width=179.25px style="padding: 5px; background-color: #ffffff; border: 2px solid #993300;" |<br />
<br />
<h3>[[Brown:Team logistics | Team Logistics ]] </h3><br />
<br />
*[[Brown:Team logistics|How we got here]]<br />
*[[Brown:Team_logistics|Challenges]]<br />
*[[Brown:Team_logistics|Future work]]<br />
<br />
<br />
<br />
__NOTOC__</div>Jlohmuelhttps://2006.igem.org/wiki/index.php/Brown:ProjectsBrown:Projects2006-11-02T02:26:30Z<p>Jlohmuel: </p>
<hr />
<div>{{Brown navigation bar}}<br />
{| cellspacing="2px" cellpadding="0" border="0" style="padding: 0; width: 750px; color: #000000; background-color: #ffffff;"<br />
| width=717px style="padding: 5px; background-color: #ffffff; border: 2px solid #993300;" |<br />
<br />
<h3>Brown 2006 iGEM entry </h3><br />
<br />
As the race to the finish draws near for iGEM 2006, our team has compiled a detailed summary of our summers work. This includes our initial ideas, designs, models and the parts that we have added to the registry. We began the summer with grand plans to design a game of bacterial freeze tag, modeled after the popular children's game where one player tries to catch the other. Our efforts were then switched to the extension of the bi-stable toggle switch and the design of a tri-stable toggle switch. Our projects highlight the team's creative and innovative approaches to designing and engineering working biological systems.<br />
<br />
Click here to see our completed BioBricks! [http://partsregistry.org/cgi/partsdb/pgroup.cgi?pgroup=iGEM2006partsregistry.org/cgi/partsdb/pgroup.cgi?pgroup=iGEM2006&group=Brown| Parts]<br />
<br />
{| cellspacing="2px" cellpadding="0" border="0" style="padding: 0px; width: 750px; color: #000000; background-color: #ffffff;"<br />
|-valign="top"<br />
|width=179.25px style="padding: 5px; background-color: #ffffff; border: 2px solid #993300;" |<br />
<br />
<h3>[[Brown:Bacterial Freeze Tag | Bacterial Freeze Tag]] </h3><br />
<br />
*[[Brown:Bacterial_Freeze_Tag#Overview |Overview]]<br />
*[[Brown:Bacterial_Freeze_Tag#Sender cell | Sender cell ]]<br />
*[[Brown:Bacterial_Freeze_Tag#Receiver cell | Receiver cell ]]<br />
*[[Brown:Bacterial_Freeze_Tag#Challenges| Challenges ]]<br />
<br />
|width=179.25px style="padding: 5px; background-color: #ffffff; border: 2px solid #993300;" |<br />
<br />
<h3>[[Brown:Tri-Stable toggle switch|Tri-Stable toggle switch]]</h3><br />
*[[Brown:Tri-Stable_toggle_switch|Introduction to the project]]<br />
*[[Brown:Tri-Stable toggle switch#Modeling the tri-stable toggle switch | Modeling the tri-stable switch]]<br />
*[[Brown:Tri-Stable toggle switch#What we achieved| What we achieved]]<br />
*[[Brown:Tri-Stable toggle switch#Challenges | Challenges ]]<br />
*[[Brown:Tri-Stable toggle switch#Challenges | Future work ]]<br />
<br />
|width=179.25px style="padding: 5px; background-color: #ffffff; border: 2px solid #993300;" |<br />
<br />
<h3>[[Brown:Team logistics | Team Logistics ]] </h3><br />
<br />
*[[Brown:Team logistics|How we got here]]<br />
*[[Brown:Team_logistics|Challenges]]<br />
*[[Brown:Team_logistics|Future work]]<br />
<br />
<br />
<br />
__NOTOC__</div>Jlohmuelhttps://2006.igem.org/wiki/index.php/User:JasonlohmuellerUser:Jasonlohmueller2006-11-01T07:37:45Z<p>Jlohmuel: </p>
<hr />
<div>[[Image:jasonl.jpg]]<br />
<br />
My name is Jason Lohmueller. I am a senior Biomedical Engineering concentrator. Currently I am working in two labs. Here at Brown I am working in Jeffrey Morgan's lab doing a senior honors thesis investigating the role of cell motility on cell aggregate self-assembly. I am also doing population genetics research, specifically identifying positive selection in the human lineage, with Pardis Sabeti at the Broad Institute in Cambridge. My current work there is on the co-evolution of the two closely related immune response genes IKKalpha and IKKbeta in European populations. I am planning to study synthetic biology in graduate school. My non-science interests include playing piano and guitar, german, coffee, running, and getting huge in the gym.</div>Jlohmuelhttps://2006.igem.org/wiki/index.php/Brown:iGEM_portal_2006Brown:iGEM portal 20062006-11-01T07:35:18Z<p>Jlohmuel: </p>
<hr />
<div>{{Brown navigation bar}}<br />
[[Image:TeamPhotoBrown1.jpg|750px|Some of the Brown iGEM Team]]<br />
<br />
{| cellspacing="2px" cellpadding="0" border="0" style="padding: 0; width: 750px; color: #000000; background-color: #ffffff;"<br />
| width=717px style="padding: 5px; background-color: #ffffff; border: 2px solid #993300;" |<br />
<br />
<h3>[[Brown:Projects | See our 2006 iGEM Jamboree entry here]] </h3><br />
<br />
<h3>Welcome</h3><br />
Welcome to the Brown [http://icampus.mit.edu/projects/iGem.shtml International Genetically Engineered Machine (iGEM)] Team Wiki. Feel free to explore our website or [[Brown:Contact | contact us]], we are constantly updating it with the latest developments in the project.<br />
<br />
<h3>Team news </h3><br />
<br />
11/02/06 - We're presenting our work this Saturday, at the [http://www.igem2006.com iGEM Jamboree at MIT] <br><br />
11/02/06 - Get info on the [[Brown:2007 | Brown iGEM team 2007]] <br><br />
10/22/06 - Abstract accepted for poster presentation at the BioSysBio Conference in the UK! <br><br />
09/26/06 - The team recently visited Pfizer in Groton, CT <br><br />
7/18/06 - Brendan has added details and [[Brown:Projects|the freeze-tag project]] <br><br />
<br />
{| cellspacing="2px" cellpadding="0" border="0" style="padding: 0px; width: 750px; color: #000000; background-color: #ffffff;"<br />
|-valign="top"<br />
|width=179.25px style="padding: 5px; background-color: #ffffff; border: 2px solid #993300;" |<br />
<br />
<h3>Team roles</h3><br />
<br />
Team Leaders: [[user:jamielemon|Jamie L.]], [[user:anniegao|Annie]]<br><br />
Treasurer: [[user:anniegao|Annie]]<br><br />
Secretary: [[user:Victorialattanzi|Victoria]]<br><br />
Journal Club Coordinator: [[user:petergoldstein|Peter]] <br><br />
Faculty Liaison: [[user:azeemkaka| Azeem]]<br><br />
Industry Liaison: [[user:victorialattanzi|Victoria]]<br> <br />
Outreach: [[user: meganschmidt|Megan]]<br> <br />
Future Planning: [[user:Jasonlohmueller|Jason]] <br> <br />
Chemical DB Manager: [[user:petergoldstein|Peter]] <br><br />
Guest Speaker Organizer: [[user:johncumbers|John]] <br><br />
|width=179.25px style="padding: 5px; background-color: #ffffff; border: 2px solid #993300;" |<br />
<br />
<h3>[[Brown:Todo list |Todo list]]</h3><br />
<h3>[[Brown:Lab Protocols| Lab Protocols]] </h3><br />
<h3>[[Brown:Next Meeting Agenda |Next Meeting Agenda ]]</h3><br />
<br />
<h3>[[Brown:Weekly Meetings|Weekly Meetings]]</h3><br />
<br />
<h3>[[Brown:Equipment needed|Equipment needed]]</h3><br />
<br />
<h3>[[Brown:Summer workshops|Summer Workshops]]</h3><br />
<br />
<h3>[[Brown:Calendar pages|Calendar Pages]]</h3><br />
<br />
<h3>[[Brown:Projects|Projects]]</h3><br />
<br />
<h3>[[Brown:Journal_club:Synthetic_biology_journal_club|Journal Club]]</h3><br />
<br />
|width=179.25px style="padding: 5px; background-color: #ffffff; border: 2px solid #993300;" |<br />
<br />
<h3>About Us</h3><br />
#[[Brown:Lab Members | Faculty Advisors]]<br />
#[[Brown:Lab Members | Team Members]]<br />
#[[Brown:Industry Affiliates | Industry Affiliates]]<br />
<br />
Our lab is in J Walter Wilson, Rm 228, call 401 523 8190<br />
<br />
#[[Brown:2007 | Brown iGEM team 2007]]<br />
<br />
<br />
__NOTOC__</div>Jlohmuelhttps://2006.igem.org/wiki/index.php/Brown:iGEM_portal_2006Brown:iGEM portal 20062006-11-01T07:34:23Z<p>Jlohmuel: </p>
<hr />
<div>{{Brown navigation bar}}<br />
[[Image:TeamPhotoBrown1.jpg|750px|Some of the Brown iGEM Team]]<br />
<br />
{| cellspacing="2px" cellpadding="0" border="0" style="padding: 0; width: 750px; color: #000000; background-color: #ffffff;"<br />
| width=717px style="padding: 5px; background-color: #ffffff; border: 2px solid #993300;" |<br />
<br />
<h3>[[Brown:Projects | See our 2006 iGEM Jamboree entry here]] </h3><br />
<br />
<h3>Welcome</h3><br />
Welcome to the Brown [http://icampus.mit.edu/projects/iGem.shtml International Genetically Engineered Machine (iGEM)] Team Wiki. Feel free to explore our website or [[Brown:Contact | contact us]], we are constantly updating it with the latest developments in the project.<br />
<br />
<h3>Team news </h3><br />
<br />
11/02/06 - We're presenting our work this Saturday, at the [http://www.igem2006.com iGEM Jamboree at MIT] <br><br />
11/02/06 - Get info on the [[Brown:2007 | Brown iGEM team 2007]] <br><br />
10/22/06 - Abstract accepted for poster presentation at the BioSysBio Conference in the UK! <br><br />
09/26/06 - The team recently visited Pfizer in Groton, CT <br><br />
7/18/06 - Brendan has added details and [[Brown:Projects|diagrams of our main project]] <br><br />
<br />
{| cellspacing="2px" cellpadding="0" border="0" style="padding: 0px; width: 750px; color: #000000; background-color: #ffffff;"<br />
|-valign="top"<br />
|width=179.25px style="padding: 5px; background-color: #ffffff; border: 2px solid #993300;" |<br />
<br />
<h3>Team roles</h3><br />
<br />
Team Leaders: [[user:jamielemon|Jamie L.]], [[user:anniegao|Annie]]<br><br />
Treasurer: [[user:anniegao|Annie]]<br><br />
Secretary: [[user:Victorialattanzi|Victoria]]<br><br />
Journal Club Coordinator: [[user:petergoldstein|Peter]] <br><br />
Faculty Liaison: [[user:azeemkaka| Azeem]]<br><br />
Industry Liaison: [[user:victorialattanzi|Victoria]]<br> <br />
Outreach: [[user: meganschmidt|Megan]]<br> <br />
Future Planning: [[user:Jasonlohmueller|Jason]] <br> <br />
Chemical DB Manager: [[user:petergoldstein|Peter]] <br><br />
Guest Speaker Organizer: [[user:johncumbers|John]] <br><br />
|width=179.25px style="padding: 5px; background-color: #ffffff; border: 2px solid #993300;" |<br />
<br />
<h3>[[Brown:Todo list |Todo list]]</h3><br />
<h3>[[Brown:Lab Protocols| Lab Protocols]] </h3><br />
<h3>[[Brown:Next Meeting Agenda |Next Meeting Agenda ]]</h3><br />
<br />
<h3>[[Brown:Weekly Meetings|Weekly Meetings]]</h3><br />
<br />
<h3>[[Brown:Equipment needed|Equipment needed]]</h3><br />
<br />
<h3>[[Brown:Summer workshops|Summer Workshops]]</h3><br />
<br />
<h3>[[Brown:Calendar pages|Calendar Pages]]</h3><br />
<br />
<h3>[[Brown:Projects|Projects]]</h3><br />
<br />
<h3>[[Brown:Journal_club:Synthetic_biology_journal_club|Journal Club]]</h3><br />
<br />
|width=179.25px style="padding: 5px; background-color: #ffffff; border: 2px solid #993300;" |<br />
<br />
<h3>About Us</h3><br />
#[[Brown:Lab Members | Faculty Advisors]]<br />
#[[Brown:Lab Members | Team Members]]<br />
#[[Brown:Industry Affiliates | Industry Affiliates]]<br />
<br />
Our lab is in J Walter Wilson, Rm 228, call 401 523 8190<br />
<br />
#[[Brown:2007 | Brown iGEM team 2007]]<br />
<br />
<br />
__NOTOC__</div>Jlohmuelhttps://2006.igem.org/wiki/index.php/Brown:iGEM_portal_2006Brown:iGEM portal 20062006-11-01T07:34:06Z<p>Jlohmuel: </p>
<hr />
<div>{{Brown navigation bar}}<br />
[[Image:TeamPhotoBrown1.jpg|750px|Some of the Brown iGEM Team]]<br />
<br />
{| cellspacing="2px" cellpadding="0" border="0" style="padding: 0; width: 750px; color: #000000; background-color: #ffffff;"<br />
| width=717px style="padding: 5px; background-color: #ffffff; border: 2px solid #993300;" |<br />
<br />
<h3>[[Brown:Projects | See our 2006 iGEM Jamboree entry here]] </h3><br />
<br />
<h3>Welcome</h3><br />
Welcome to the Brown [http://icampus.mit.edu/projects/iGem.shtml International Genetically Engineered Machine (iGEM)] Team Wiki. Feel free to explore our website or [[Brown:Contact | contact us]], we are constantly updating it with the latest developments in the project.<br />
<br />
<h3>Team news </h3><br />
<br />
11/02/06 - We're presenting our work this Saturday, at the [http://www.igem2006.com iGEM Jamboree at MIT] <br><br />
11/02/06 - Get info on the [[Brown:2007 | Brown iGEM team 2007]] <br><br />
10/22/06 - Abstract accepted for poster presentation at the BioSysBio Conference in the UK <br><br />
09/26/06 - The team recently visited Pfizer in Groton, CT <br><br />
7/18/06 - Brendan has added details and [[Brown:Projects|diagrams of our main project]] <br><br />
<br />
{| cellspacing="2px" cellpadding="0" border="0" style="padding: 0px; width: 750px; color: #000000; background-color: #ffffff;"<br />
|-valign="top"<br />
|width=179.25px style="padding: 5px; background-color: #ffffff; border: 2px solid #993300;" |<br />
<br />
<h3>Team roles</h3><br />
<br />
Team Leaders: [[user:jamielemon|Jamie L.]], [[user:anniegao|Annie]]<br><br />
Treasurer: [[user:anniegao|Annie]]<br><br />
Secretary: [[user:Victorialattanzi|Victoria]]<br><br />
Journal Club Coordinator: [[user:petergoldstein|Peter]] <br><br />
Faculty Liaison: [[user:azeemkaka| Azeem]]<br><br />
Industry Liaison: [[user:victorialattanzi|Victoria]]<br> <br />
Outreach: [[user: meganschmidt|Megan]]<br> <br />
Future Planning: [[user:Jasonlohmueller|Jason]] <br> <br />
Chemical DB Manager: [[user:petergoldstein|Peter]] <br><br />
Guest Speaker Organizer: [[user:johncumbers|John]] <br><br />
|width=179.25px style="padding: 5px; background-color: #ffffff; border: 2px solid #993300;" |<br />
<br />
<h3>[[Brown:Todo list |Todo list]]</h3><br />
<h3>[[Brown:Lab Protocols| Lab Protocols]] </h3><br />
<h3>[[Brown:Next Meeting Agenda |Next Meeting Agenda ]]</h3><br />
<br />
<h3>[[Brown:Weekly Meetings|Weekly Meetings]]</h3><br />
<br />
<h3>[[Brown:Equipment needed|Equipment needed]]</h3><br />
<br />
<h3>[[Brown:Summer workshops|Summer Workshops]]</h3><br />
<br />
<h3>[[Brown:Calendar pages|Calendar Pages]]</h3><br />
<br />
<h3>[[Brown:Projects|Projects]]</h3><br />
<br />
<h3>[[Brown:Journal_club:Synthetic_biology_journal_club|Journal Club]]</h3><br />
<br />
|width=179.25px style="padding: 5px; background-color: #ffffff; border: 2px solid #993300;" |<br />
<br />
<h3>About Us</h3><br />
#[[Brown:Lab Members | Faculty Advisors]]<br />
#[[Brown:Lab Members | Team Members]]<br />
#[[Brown:Industry Affiliates | Industry Affiliates]]<br />
<br />
Our lab is in J Walter Wilson, Rm 228, call 401 523 8190<br />
<br />
#[[Brown:2007 | Brown iGEM team 2007]]<br />
<br />
<br />
__NOTOC__</div>Jlohmuelhttps://2006.igem.org/wiki/index.php/Brown:iGEM_portal_2006Brown:iGEM portal 20062006-11-01T07:33:34Z<p>Jlohmuel: </p>
<hr />
<div>{{Brown navigation bar}}<br />
[[Image:TeamPhotoBrown1.jpg|750px|Some of the Brown iGEM Team]]<br />
<br />
{| cellspacing="2px" cellpadding="0" border="0" style="padding: 0; width: 750px; color: #000000; background-color: #ffffff;"<br />
| width=717px style="padding: 5px; background-color: #ffffff; border: 2px solid #993300;" |<br />
<br />
<h3>[[Brown:Projects | See our 2006 iGEM Jamboree entry here]] </h3><br />
<br />
<h3>Welcome</h3><br />
Welcome to the Brown [http://icampus.mit.edu/projects/iGem.shtml International Genetically Engineered Machine (iGEM)] Team Wiki. Feel free to explore our website or [[Brown:Contact | contact us]], we are constantly updating it with the latest developments in the project.<br />
<br />
<h3>Team news </h3><br />
<br />
11/02/06 - We're presenting our work this Saturday, at the [http://www.igem2006.com iGEM Jamboree at MIT] <br><br />
11/02/06 - Get info on the [[Brown:2007 | Brown iGEM team 2007]] <br><br />
10/22/06 - <br> Abstract accepted for poster presentation at the BioSysBio Conference in the UK<br />
09/26/06 - The team recently visited Pfizer in Groton, CT <br><br />
7/18/06 - Brendan has added details and [[Brown:Projects|diagrams of our main project]] <br><br />
<br />
{| cellspacing="2px" cellpadding="0" border="0" style="padding: 0px; width: 750px; color: #000000; background-color: #ffffff;"<br />
|-valign="top"<br />
|width=179.25px style="padding: 5px; background-color: #ffffff; border: 2px solid #993300;" |<br />
<br />
<h3>Team roles</h3><br />
<br />
Team Leaders: [[user:jamielemon|Jamie L.]], [[user:anniegao|Annie]]<br><br />
Treasurer: [[user:anniegao|Annie]]<br><br />
Secretary: [[user:Victorialattanzi|Victoria]]<br><br />
Journal Club Coordinator: [[user:petergoldstein|Peter]] <br><br />
Faculty Liaison: [[user:azeemkaka| Azeem]]<br><br />
Industry Liaison: [[user:victorialattanzi|Victoria]]<br> <br />
Outreach: [[user: meganschmidt|Megan]]<br> <br />
Future Planning: [[user:Jasonlohmueller|Jason]] <br> <br />
Chemical DB Manager: [[user:petergoldstein|Peter]] <br><br />
Guest Speaker Organizer: [[user:johncumbers|John]] <br><br />
|width=179.25px style="padding: 5px; background-color: #ffffff; border: 2px solid #993300;" |<br />
<br />
<h3>[[Brown:Todo list |Todo list]]</h3><br />
<h3>[[Brown:Lab Protocols| Lab Protocols]] </h3><br />
<h3>[[Brown:Next Meeting Agenda |Next Meeting Agenda ]]</h3><br />
<br />
<h3>[[Brown:Weekly Meetings|Weekly Meetings]]</h3><br />
<br />
<h3>[[Brown:Equipment needed|Equipment needed]]</h3><br />
<br />
<h3>[[Brown:Summer workshops|Summer Workshops]]</h3><br />
<br />
<h3>[[Brown:Calendar pages|Calendar Pages]]</h3><br />
<br />
<h3>[[Brown:Projects|Projects]]</h3><br />
<br />
<h3>[[Brown:Journal_club:Synthetic_biology_journal_club|Journal Club]]</h3><br />
<br />
|width=179.25px style="padding: 5px; background-color: #ffffff; border: 2px solid #993300;" |<br />
<br />
<h3>About Us</h3><br />
#[[Brown:Lab Members | Faculty Advisors]]<br />
#[[Brown:Lab Members | Team Members]]<br />
#[[Brown:Industry Affiliates | Industry Affiliates]]<br />
<br />
Our lab is in J Walter Wilson, Rm 228, call 401 523 8190<br />
<br />
#[[Brown:2007 | Brown iGEM team 2007]]<br />
<br />
<br />
__NOTOC__</div>Jlohmuelhttps://2006.igem.org/wiki/index.php/Table_of_preliminary_model_constantsTable of preliminary model constants2006-11-01T06:56:42Z<p>Jlohmuel: </p>
<hr />
<div>{{Brown navigation bar}}<br />
<br />
=Table of preliminary model constants=<br />
====General Parameters====<br />
<br />
{| border=1 cellspacing=0 cellpadding=5<br />
<br />
!align="left" width="50" | Parameter<br />
!width="50" | Value<br />
!width="300" | Meaning<br />
<br />
|-<br />
|m<br />
|20 Copies<br />
|Plasmid Copy Number<br />
|-<br />
|v<br />
|1um^3<br />
|cell volume<br />
|-<br />
<br />
|}<br />
<br />
====Inhibitor Protein Parameters====<br />
{| border=1 cellspacing=0 cellpadding=5<br />
<br />
!align="left" width="50" | Parameter<br />
!width="50" | Value<br />
!width="300" | Meaning<br />
<br />
|-<br />
|alphaT1<br />
|.01<br />
|Relative transcription level for repressed (1 TetR dimer bound) pTetR promoter<br />
|-<br />
|alphaT2<br />
|.01<br />
|Relative transcription level for doubly repressed (2 TetR dimers bound) pTetR promoter<br />
|-<br />
|alphaL<br />
|.01<br />
|Relative transcription level for repressed (1 LacI tetramer bound) pLacI promoter<br />
|-<br />
|alphaA<br />
|.01<br />
|Relative transcription level for repressed (two AraC monomers bound) pBAD promoter<br />
|-<br />
|betaA<br />
|1000<br />
|maximum number of proteins produced from the pBAD promoter per cell division<br />
|-<br />
|betaL<br />
|1000<br />
|maximum number of proteins produced from the pLacI promoter per cell division<br />
|-<br />
|betaT<br />
|1000<br />
|maximum number of proteins produced from the pTetR promoter per cell division<br />
|-<br />
|etaLT<br />
|1<br />
|Relative rate of TetR translation from pLacI promoter<br />
|-<br />
|etaLA<br />
|1<br />
|Relative rate of AraC translation from pLacI promoter<br />
|-<br />
|etaAL<br />
|1<br />
|Relative rate of LacI translation from pBAD promoter<br />
|-<br />
|etaAT<br />
|1<br />
|Relative rate of TetR translation from pBAD promoter<br />
|-<br />
|etaTL<br />
|1<br />
|Relative rate of LacI translation from pTetR promoter<br />
|-<br />
|etaTA<br />
|1<br />
|Relative rate of AraC translation from pTetR promoter<br />
|-<br />
|c1L<br />
|.05<br />
|Equilibrium constant for LacI dimerization<br />
|-<br />
|c2L<br />
|.05<br />
|Equilibrium constant for LacI trimerization<br />
|-<br />
|c3L<br />
|.05<br />
|Equilibrium constant for LacI tetramerization<br />
|-<br />
|c4L<br />
|.33<br />
|Equilibrium constant for LacI tetramer binding to Lac operator<br />
|-<br />
|c1T<br />
|.05<br />
|Equilibrium constant for TetR dimerization<br />
|-<br />
|c2T<br />
|.33<br />
|Equilibrium constant for TetR dimer binding to first operator<br />
|-<br />
|c3T<br />
|.33<br />
|Equilibrium constant for TetR dimer binding to second operator<br />
|-<br />
|c1A<br />
|.05<br />
|Equilibrium constant for AraC monomer binding to first operator<br />
|-<br />
|c2A<br />
|.33<br />
|Equilibrium constant for AraC monomer binding to second operator<br />
|-<br />
|}</div>Jlohmuelhttps://2006.igem.org/wiki/index.php/Brown:Tri-Stable_toggle_switchBrown:Tri-Stable toggle switch2006-11-01T06:55:40Z<p>Jlohmuel: /* Table of preliminary model constants */</p>
<hr />
<div>{{Brown navigation bar}}<br />
=A tri-stable toggle switch=<br />
<br />
Pursuant to the the 1999 paper "Construction of a genetic toggle switch in Escherichia coli," by Timothy S. Gardner, Charles R. Cantor and James J. Collins, we wondered if the bi-stable toggle switch could be generalized to an n-stable switch. To that end, we conceived and began construction of the tri-stable toggle switch. The general idea is that when a selected promoter is activated, it represses the other two. As such each of the three states of our network are self-stabilizing.<br />
<br />
[[Image:Tristable_Toggle.jpg|left|500px|A general tri-stable toggle]]<br />
<br />
Our implementation uses the pLac, pTet, and pBad/AraC promoters and their respective inhibiting proteins. We chose these because of the relative ease with which each promoter is induced: by adding IPTG for the pLac-promoted region, tetracycline for the PTet-promoted region, and arabinose for the pBad/Ara genes. By characterizing each of the three pathways without following genes and terminators, we enable a tri-stable switching network of any three biobricks to be constructed with minimal cloning.<br />
<br />
[[Image:Registry_tristable.jpg|Our implementation]]<br />
<br />
<br><br />
Link to Tri-stable Switch on the registry:[http://partsregistry.org/Part:BBa_J24672]<br />
<br />
<br><br><br />
<br />
<br><br />
<br><br />
<br />
=Modeling the tri-stable toggle switch=<br />
While the tri-stable switch is seemingly simple in design, just like any other system it is subject to potentially comprising factors such as promoter leakiness and other stochastic fluctuations. In an attempt to predict the behavior of the tri-stable switch, we created a deterministic model of the system taking into account these factors. The model structure is based on that described in "Prediction and measurement of an autoregulatory genetic module" by Farren Isaacs, et al. This paper features an excellent supplement that takes you hand-in-hand through the derivation of model equations. I will attempt to emulate this derivation with our model below. It is important to note that while we have some preliminary results, our model is very much a work in progress. In order to model the system accurately, many of the fundamental constants governing the model will need to be determined experimentally. For now we have used some constants from the literature and estimated some based on similar the values of similar constants also from the literature. <br />
<br />
====Derivation of the Model Equations====<br />
[[Derivation of the Model Equations]]<br />
<br />
====Table of preliminary model constants====<br />
[[Table of preliminary model constants]]<br />
<br />
==Preliminary modeling results==<br />
<br />
The preliminary modeling results predict things about of the the tri-stable switch. First of all, the model confirms certain obvious points such as the a shift to the proper steady expression state in the presence of inducers. This prediction attests to model validity more than anything else. Secondly, the model predicts that the switch will reach a natural steady state with one promoter dominating in the absence of inducers. This makes intuitive sense as the individual promoters will most likely have different strengths. <br />
<br />
[[Image:no_inducer.png]]<br />
<br />
Next we tested the stability of the model by plugging in final steady-state induced levels of protein into an inducer-less system. The model predicts that the system will not remain stable and will instead revert back to the natural steady state for the inducer-less system. Even after changing several model parameters, the switch would not remain stable.<br />
<br />
<br />
[[Image:stability_test1.png]]<br />
<br />
As a test of the model, models of the three potential bi-stable switches (already proven to be successful experimentally) using the same components and parameters were tested. In accordance with the literature, each of these constructs maintained stable expression of the induced state following the removal of the inducer. In order to probe the instability of the tri-stable switch further, tri-stable switches in which all of the components were effectively the same were created (ex:for all LacI-like, TetR and AraC were modeled as forming tetramers and having one operator.) Each of these systems reached a steady state in which all of the proteins evolved identically and were present at the same levels in the absence of inducer. The same stability tests were then performed. All of the constructs remained stable. Thus our model suggests that the similarity of the repressor-expression systems dominates the success or failure of the tri-stable switch. Furthermore, the bi-stable switch is less sensitive to this factor. <br />
<br />
It should be noted that this is only a preliminary model. The values obtained from the literature will most like differ in our system. Through future experimentation we will elucidate these values and thus create a more accurate model. We will also conduct parameter scans aimed to provide further insight into the necessary system requirements for "tri-stability."<br />
<br />
<br />
<br />
<br />
Link to MATLAB code for tri-stable model [[Media:tristable1.txt]] <br />
<br />
Note: copy this code into MATLAB and save as an M-file<br />
<br />
=What we achieved=<br />
<br />
A successfully ligated pBad/arac construct in addition to partially completed ligations for the pLacI and the pTetR constructs were the results of our summer and semesters' work. The tri-stable toggle switch system was also modeled and proved to yield significant findings.<br />
<br />
=Challenges=<br />
<br />
Despite many cloning and ligation attempts, we were unable to create each component of the switch. More specifically, the pLacI and the pTetR circuits presented many ligation difficulties and proved to yield unsuccessful cloning. However, the pBAD/araC circuit was successfully constructed (yay!).<br />
<br />
=Conclusions=<br />
<br />
With the bigger picture in mind,this project yielded many successful outcomes.Firstly, we were able to successfully construct one of the three constructs necessary for a fully functional tri-stable toggle switch. Secondly, we were able to add more working parts to the registry, both a fully functioning pBad/AraC construct and many parts that were successfully ligated in the process of assembling the other two constructs. These parts will be used in the future by our team and perhaps many others as new ideas and projects develop. Also, it is necessary to mention our most recent finding from the modeling results for the tri-stable toggle switch. Interestingly enough, our model suggests that it is biologically unfeasible to create a switch with more than two components. If more than two components are introduced, such as three, the system is unable to maintain any one state and it will revert back to expressing its most dominant promoter construct. We find such results quite meaningful in demonstrating the biological limits of a system under study.<br />
<br />
=Future work=<br />
<br />
In the coming months, it is hoped that the failure of the ligation strategies for the pLacI and the pTetR constructs can be further investigated. We feel confident that we can continue to build from the one successful construction,the pBad/Arac construct and complete the other 2 constructs. To this end, it will be necessary to further troubleshoot the possible reasons why the cloning of the ligations were unsuccessful. Also we plan to conduct various experiments to elucidate more accurate model parameters specific to our system. Furthermore we will perform a parameter scan to investigate the determinates of "tri-stability."</div>Jlohmuelhttps://2006.igem.org/wiki/index.php/Brown:Tri-Stable_toggle_switchBrown:Tri-Stable toggle switch2006-11-01T06:55:27Z<p>Jlohmuel: /* Derivation of the Model Equations */</p>
<hr />
<div>{{Brown navigation bar}}<br />
=A tri-stable toggle switch=<br />
<br />
Pursuant to the the 1999 paper "Construction of a genetic toggle switch in Escherichia coli," by Timothy S. Gardner, Charles R. Cantor and James J. Collins, we wondered if the bi-stable toggle switch could be generalized to an n-stable switch. To that end, we conceived and began construction of the tri-stable toggle switch. The general idea is that when a selected promoter is activated, it represses the other two. As such each of the three states of our network are self-stabilizing.<br />
<br />
[[Image:Tristable_Toggle.jpg|left|500px|A general tri-stable toggle]]<br />
<br />
Our implementation uses the pLac, pTet, and pBad/AraC promoters and their respective inhibiting proteins. We chose these because of the relative ease with which each promoter is induced: by adding IPTG for the pLac-promoted region, tetracycline for the PTet-promoted region, and arabinose for the pBad/Ara genes. By characterizing each of the three pathways without following genes and terminators, we enable a tri-stable switching network of any three biobricks to be constructed with minimal cloning.<br />
<br />
[[Image:Registry_tristable.jpg|Our implementation]]<br />
<br />
<br><br />
Link to Tri-stable Switch on the registry:[http://partsregistry.org/Part:BBa_J24672]<br />
<br />
<br><br><br />
<br />
<br><br />
<br><br />
<br />
=Modeling the tri-stable toggle switch=<br />
While the tri-stable switch is seemingly simple in design, just like any other system it is subject to potentially comprising factors such as promoter leakiness and other stochastic fluctuations. In an attempt to predict the behavior of the tri-stable switch, we created a deterministic model of the system taking into account these factors. The model structure is based on that described in "Prediction and measurement of an autoregulatory genetic module" by Farren Isaacs, et al. This paper features an excellent supplement that takes you hand-in-hand through the derivation of model equations. I will attempt to emulate this derivation with our model below. It is important to note that while we have some preliminary results, our model is very much a work in progress. In order to model the system accurately, many of the fundamental constants governing the model will need to be determined experimentally. For now we have used some constants from the literature and estimated some based on similar the values of similar constants also from the literature. <br />
<br />
====Derivation of the Model Equations====<br />
[[Derivation of the Model Equations]]<br />
<br />
==Table of preliminary model constants==<br />
[[Table of preliminary model constants]]<br />
<br />
==Preliminary modeling results==<br />
<br />
The preliminary modeling results predict things about of the the tri-stable switch. First of all, the model confirms certain obvious points such as the a shift to the proper steady expression state in the presence of inducers. This prediction attests to model validity more than anything else. Secondly, the model predicts that the switch will reach a natural steady state with one promoter dominating in the absence of inducers. This makes intuitive sense as the individual promoters will most likely have different strengths. <br />
<br />
[[Image:no_inducer.png]]<br />
<br />
Next we tested the stability of the model by plugging in final steady-state induced levels of protein into an inducer-less system. The model predicts that the system will not remain stable and will instead revert back to the natural steady state for the inducer-less system. Even after changing several model parameters, the switch would not remain stable.<br />
<br />
<br />
[[Image:stability_test1.png]]<br />
<br />
As a test of the model, models of the three potential bi-stable switches (already proven to be successful experimentally) using the same components and parameters were tested. In accordance with the literature, each of these constructs maintained stable expression of the induced state following the removal of the inducer. In order to probe the instability of the tri-stable switch further, tri-stable switches in which all of the components were effectively the same were created (ex:for all LacI-like, TetR and AraC were modeled as forming tetramers and having one operator.) Each of these systems reached a steady state in which all of the proteins evolved identically and were present at the same levels in the absence of inducer. The same stability tests were then performed. All of the constructs remained stable. Thus our model suggests that the similarity of the repressor-expression systems dominates the success or failure of the tri-stable switch. Furthermore, the bi-stable switch is less sensitive to this factor. <br />
<br />
It should be noted that this is only a preliminary model. The values obtained from the literature will most like differ in our system. Through future experimentation we will elucidate these values and thus create a more accurate model. We will also conduct parameter scans aimed to provide further insight into the necessary system requirements for "tri-stability."<br />
<br />
<br />
<br />
<br />
Link to MATLAB code for tri-stable model [[Media:tristable1.txt]] <br />
<br />
Note: copy this code into MATLAB and save as an M-file<br />
<br />
=What we achieved=<br />
<br />
A successfully ligated pBad/arac construct in addition to partially completed ligations for the pLacI and the pTetR constructs were the results of our summer and semesters' work. The tri-stable toggle switch system was also modeled and proved to yield significant findings.<br />
<br />
=Challenges=<br />
<br />
Despite many cloning and ligation attempts, we were unable to create each component of the switch. More specifically, the pLacI and the pTetR circuits presented many ligation difficulties and proved to yield unsuccessful cloning. However, the pBAD/araC circuit was successfully constructed (yay!).<br />
<br />
=Conclusions=<br />
<br />
With the bigger picture in mind,this project yielded many successful outcomes.Firstly, we were able to successfully construct one of the three constructs necessary for a fully functional tri-stable toggle switch. Secondly, we were able to add more working parts to the registry, both a fully functioning pBad/AraC construct and many parts that were successfully ligated in the process of assembling the other two constructs. These parts will be used in the future by our team and perhaps many others as new ideas and projects develop. Also, it is necessary to mention our most recent finding from the modeling results for the tri-stable toggle switch. Interestingly enough, our model suggests that it is biologically unfeasible to create a switch with more than two components. If more than two components are introduced, such as three, the system is unable to maintain any one state and it will revert back to expressing its most dominant promoter construct. We find such results quite meaningful in demonstrating the biological limits of a system under study.<br />
<br />
=Future work=<br />
<br />
In the coming months, it is hoped that the failure of the ligation strategies for the pLacI and the pTetR constructs can be further investigated. We feel confident that we can continue to build from the one successful construction,the pBad/Arac construct and complete the other 2 constructs. To this end, it will be necessary to further troubleshoot the possible reasons why the cloning of the ligations were unsuccessful. Also we plan to conduct various experiments to elucidate more accurate model parameters specific to our system. Furthermore we will perform a parameter scan to investigate the determinates of "tri-stability."</div>Jlohmuelhttps://2006.igem.org/wiki/index.php/Derivation_of_the_Model_EquationsDerivation of the Model Equations2006-11-01T06:53:42Z<p>Jlohmuel: </p>
<hr />
<div>{{Brown navigation bar}}<br />
=Derivation of the model equations=<br />
The chemical reactions describing the tri-stable switch can be divided into the two categories of fast and slow reactions. Fast reactions such as dimer formation and promoter-binding occur in the scale of seconds and are therefore modeled to be in equilibrium. Conversely, slow reactions including the likes of transcription, translation, and protein degradation occur on the scale of minutes and are thus modeled to be evolving with time. <br />
<br />
====Fast reversible reaction equations====<br />
The following equations (1) describe the fast reactions. The characters L,T,A denote molecules of LacI, TetR, and, AraC, respectively while the subscripts denote whether the molecule is a monomer (blank), dimer(2), tri-mer(3), etc. The k's denote reaction rates.<br />
<br />
[[Image:Fast_rxn_eqs1.png]]<br />
<br />
Note that the equations include volume explicitly. Cell volume is modeled in this way as it is a slowly evolving function of time.<br />
<br />
====Slow irreversible reaction equations====<br />
The following equations (2) describe the slow irreversible reactions of transcription and translation (both taken into account with the reaction rate kti) and protein degradation (reaction constant kdi). The coefficients eta-ij take into account the relative translation rates of proteins from the same transcripts. For eta-ij, "i" represents the promoter responsible for producing the molecule and "j" represents the molecule being translated [ex: eta-LA corresponds to the relative rate of AraC production from the trascript produced by the LacI promoter]. To establish a convention, eta's are relative to the translation rate of the first gene on a particular transcript. Thus eta for the first gene on the mRNA transcript = 1. The alpha-i coefficients represent relative transcription rates. In this case, "i" denotes the promoter from which the mRNA is transcribed.<br />
<br />
[[Image:slow_rxn_eqs.png]]<br />
====Equations governing cell volume====<br />
The following two equations (3) describing cellular growth and division are taken directly from the aforementioned paper by Isaacs et al. The first equation describes the volume increase from the time immediately following cell division to the time immediately before it. In this equation, V0 denotes the volume of the cell at the beginning of growth and T0 denotes the time of cell division. In our model at times T=q*t0 in which q is an integer, we have volume V and protein concentration n halve - thus modeling volume division and the resulting protein redistribution. The second equation describes the dimensionless equation in which t is measured in terms of fcell-division time and the cell volume changes between 1 and 2.<br />
<br />
[[Image:vol_eq.png]]<br />
====Equations for the total number of molecules====<br />
The following equations (4) describe the total number of inhibitor protein molecules zi. d#i represents the number, "#", of molecules bound to the "i" promoter. (ex: all of the LacI tetramers bound to pLacI promoters are accounted for by the 4d1L term.)<br />
[[Image:Tot_mol_eq.png]]<br />
<br />
====Equations describing the temporal evolution of the total number inhibitor molecules====<br />
The following equations (5) describe the temporal evolution of the total number of inhibitor molecules. Beta-i = the cell division time multiplied by the combined transcription and translation rates from a given promoter "i" (t0*kti), thus representing the total number of protein molecules maximally produced by a given promoter. Similarly, Gamma-i = the cell division time multiplied by the degradation rate of protein i (t0*kdi), thus representing the total number of protein i destabilized over one cell division time. The eta and alpha terms are described in the section above describing the equations for the slow reactions. Note that the equation describing the temporal evolution of AraC includes the term "Aend." This term represents the endogenous expression of AraC in dH5alpha's. Our preliminary work suggests that this endogenous AraC has very little effect on our pBad promoter.<br />
<br />
[[Image:Ev_tot_mol_eqs.png]]<br />
<br />
====Fast reaction equilibrium equations====<br />
Compared to the slowly evolving reactions(2) described by the above equations (5), the fast reactions (1) can be considered to be in equilibrium. Thus the following equilibrium relations (6) hold.<br />
[[Image:Fast_eq_eqs.png]]<br />
<br />
====Equations modeling the presence of the inducers====<br />
The following equations (7) model the effect of the chemical inducers on the system. The model regards the proteins bound to the inducers as having increased dissociation rates (ki-#) as specified below. The reaction rate with the apostrophe denotes the original reaction rate in the absence of inducer inclusion. In these equations "I" denotes IPTG, "a" denotes arabinose, and "Tc" denotes tetracycline (or analogue aTc). The concept for and the form of these equations are based on equations described in the supplementary Information for "A bottom-up approach to gene regulation" by Guido et al. Much like the aforementioned paper by Isaacs et al., 'Bottom-up' has an extensive and brilliantly articulate derivation supplement. <br />
<br />
[[Image:inducer_eqs.png]]<br />
<br />
====Equations relating plasmid copy number to operator sites====<br />
The following equations (8) relate the plasmid copy number, "m", to the number of promoters with bound and unbound operators. As each plasmid contains 1 of each promoter, m is constant for all three promoter types. <br />
<br />
[[Image:cop_num_eqs.png]]<br />
<br />
====Simplified fast reaction equilibrium equations====<br />
The equilibrium equations governing the fast reactions (6) can be simplified by defining dimensionless equilibrium constants in the form of cij=kij/(k(-ij)*V0*A) in which A is Avogadro's number. The cL's are further simplified by defining the constant cL=c1L*c2L*c3L*c4L. Additionally by plugging in successive terms we were able to write the operator equations in terms of d0L. <br />
<br />
[[Image:Simp_fast_eq_eqs.png]]<br />
<br />
====Equations for the number of unbound operators====<br />
Combining equations (8) and the above simplified equilibrium equations(9), we are able to solve for the number of unbound operators d0i. <br />
<br />
[[Image:Unbound_eqs.png]]<br />
<br />
====Equations for the number of bound operators====<br />
By plugging the d0i's (10) back into the equations for bound operators (9), we can solve for these equations in terms of monomer concentrations.<br />
<br />
<br />
[[Image:bound_eqs.png]]<br />
<br />
====Equation manipulations to determine change in protein monomers per time====<br />
Next we can rewrite the equations describing the temporal evolution of the total number inhibitor molecules (5) in terms of equations (10) and (11). This yields the somewhat complex equations (12) which are solely in terms of monomer protein concentration. <br />
<br />
[[Image:dzdt_eqs.png]]<br />
<br />
====Equations for the change in protein monomers per time====<br />
By applying the simple mathematical manipulation (13) to equations (12) and the X derivative of equations (9) ((9) can be written in terms of X by plugging in equations (10) and (11)) we can solve for the following equations for the evolution of protein monomers as a function of time (14). I will spare you the dZi/dXi derivative equations of equation (9).<br />
<br />
[[Image:simp_mat_man_eq.png]]<br />
<br />
====Equations describing stochastic variation====<br />
<br />
====Equations for the evolution of fluorescent reporters====<br />
<br />
<br />
--[[User:Jlohmuel|Jlohmuel]] 03:59, 28 October 2006 (EDT)</div>Jlohmuelhttps://2006.igem.org/wiki/index.php/Brown:Tri-Stable_toggle_switchBrown:Tri-Stable toggle switch2006-11-01T06:52:24Z<p>Jlohmuel: /* Preliminary modeling results */</p>
<hr />
<div>{{Brown navigation bar}}<br />
=A tri-stable toggle switch=<br />
<br />
Pursuant to the the 1999 paper "Construction of a genetic toggle switch in Escherichia coli," by Timothy S. Gardner, Charles R. Cantor and James J. Collins, we wondered if the bi-stable toggle switch could be generalized to an n-stable switch. To that end, we conceived and began construction of the tri-stable toggle switch. The general idea is that when a selected promoter is activated, it represses the other two. As such each of the three states of our network are self-stabilizing.<br />
<br />
[[Image:Tristable_Toggle.jpg|left|500px|A general tri-stable toggle]]<br />
<br />
Our implementation uses the pLac, pTet, and pBad/AraC promoters and their respective inhibiting proteins. We chose these because of the relative ease with which each promoter is induced: by adding IPTG for the pLac-promoted region, tetracycline for the PTet-promoted region, and arabinose for the pBad/Ara genes. By characterizing each of the three pathways without following genes and terminators, we enable a tri-stable switching network of any three biobricks to be constructed with minimal cloning.<br />
<br />
[[Image:Registry_tristable.jpg|Our implementation]]<br />
<br />
<br><br />
Link to Tri-stable Switch on the registry:[http://partsregistry.org/Part:BBa_J24672]<br />
<br />
<br><br><br />
<br />
<br><br />
<br><br />
<br />
=Modeling the tri-stable toggle switch=<br />
While the tri-stable switch is seemingly simple in design, just like any other system it is subject to potentially comprising factors such as promoter leakiness and other stochastic fluctuations. In an attempt to predict the behavior of the tri-stable switch, we created a deterministic model of the system taking into account these factors. The model structure is based on that described in "Prediction and measurement of an autoregulatory genetic module" by Farren Isaacs, et al. This paper features an excellent supplement that takes you hand-in-hand through the derivation of model equations. I will attempt to emulate this derivation with our model below. It is important to note that while we have some preliminary results, our model is very much a work in progress. In order to model the system accurately, many of the fundamental constants governing the model will need to be determined experimentally. For now we have used some constants from the literature and estimated some based on similar the values of similar constants also from the literature. <br />
<br />
==Derivation of the Model Equations==<br />
[[Derivation of the Model Equations]]<br />
<br />
==Table of preliminary model constants==<br />
[[Table of preliminary model constants]]<br />
<br />
==Preliminary modeling results==<br />
<br />
The preliminary modeling results predict things about of the the tri-stable switch. First of all, the model confirms certain obvious points such as the a shift to the proper steady expression state in the presence of inducers. This prediction attests to model validity more than anything else. Secondly, the model predicts that the switch will reach a natural steady state with one promoter dominating in the absence of inducers. This makes intuitive sense as the individual promoters will most likely have different strengths. <br />
<br />
[[Image:no_inducer.png]]<br />
<br />
Next we tested the stability of the model by plugging in final steady-state induced levels of protein into an inducer-less system. The model predicts that the system will not remain stable and will instead revert back to the natural steady state for the inducer-less system. Even after changing several model parameters, the switch would not remain stable.<br />
<br />
<br />
[[Image:stability_test1.png]]<br />
<br />
As a test of the model, models of the three potential bi-stable switches (already proven to be successful experimentally) using the same components and parameters were tested. In accordance with the literature, each of these constructs maintained stable expression of the induced state following the removal of the inducer. In order to probe the instability of the tri-stable switch further, tri-stable switches in which all of the components were effectively the same were created (ex:for all LacI-like, TetR and AraC were modeled as forming tetramers and having one operator.) Each of these systems reached a steady state in which all of the proteins evolved identically and were present at the same levels in the absence of inducer. The same stability tests were then performed. All of the constructs remained stable. Thus our model suggests that the similarity of the repressor-expression systems dominates the success or failure of the tri-stable switch. Furthermore, the bi-stable switch is less sensitive to this factor. <br />
<br />
It should be noted that this is only a preliminary model. The values obtained from the literature will most like differ in our system. Through future experimentation we will elucidate these values and thus create a more accurate model. We will also conduct parameter scans aimed to provide further insight into the necessary system requirements for "tri-stability."<br />
<br />
<br />
<br />
<br />
Link to MATLAB code for tri-stable model [[Media:tristable1.txt]] <br />
<br />
Note: copy this code into MATLAB and save as an M-file<br />
<br />
=What we achieved=<br />
<br />
A successfully ligated pBad/arac construct in addition to partially completed ligations for the pLacI and the pTetR constructs were the results of our summer and semesters' work. The tri-stable toggle switch system was also modeled and proved to yield significant findings.<br />
<br />
=Challenges=<br />
<br />
Despite many cloning and ligation attempts, we were unable to create each component of the switch. More specifically, the pLacI and the pTetR circuits presented many ligation difficulties and proved to yield unsuccessful cloning. However, the pBAD/araC circuit was successfully constructed (yay!).<br />
<br />
=Conclusions=<br />
<br />
With the bigger picture in mind,this project yielded many successful outcomes.Firstly, we were able to successfully construct one of the three constructs necessary for a fully functional tri-stable toggle switch. Secondly, we were able to add more working parts to the registry, both a fully functioning pBad/AraC construct and many parts that were successfully ligated in the process of assembling the other two constructs. These parts will be used in the future by our team and perhaps many others as new ideas and projects develop. Also, it is necessary to mention our most recent finding from the modeling results for the tri-stable toggle switch. Interestingly enough, our model suggests that it is biologically unfeasible to create a switch with more than two components. If more than two components are introduced, such as three, the system is unable to maintain any one state and it will revert back to expressing its most dominant promoter construct. We find such results quite meaningful in demonstrating the biological limits of a system under study.<br />
<br />
=Future work=<br />
<br />
In the coming months, it is hoped that the failure of the ligation strategies for the pLacI and the pTetR constructs can be further investigated. We feel confident that we can continue to build from the one successful construction,the pBad/Arac construct and complete the other 2 constructs. To this end, it will be necessary to further troubleshoot the possible reasons why the cloning of the ligations were unsuccessful. Also we plan to conduct various experiments to elucidate more accurate model parameters specific to our system. Furthermore we will perform a parameter scan to investigate the determinates of "tri-stability."</div>Jlohmuelhttps://2006.igem.org/wiki/index.php/Brown:Tri-Stable_toggle_switchBrown:Tri-Stable toggle switch2006-11-01T06:52:09Z<p>Jlohmuel: /* Preliminary modeling results */</p>
<hr />
<div>{{Brown navigation bar}}<br />
=A tri-stable toggle switch=<br />
<br />
Pursuant to the the 1999 paper "Construction of a genetic toggle switch in Escherichia coli," by Timothy S. Gardner, Charles R. Cantor and James J. Collins, we wondered if the bi-stable toggle switch could be generalized to an n-stable switch. To that end, we conceived and began construction of the tri-stable toggle switch. The general idea is that when a selected promoter is activated, it represses the other two. As such each of the three states of our network are self-stabilizing.<br />
<br />
[[Image:Tristable_Toggle.jpg|left|500px|A general tri-stable toggle]]<br />
<br />
Our implementation uses the pLac, pTet, and pBad/AraC promoters and their respective inhibiting proteins. We chose these because of the relative ease with which each promoter is induced: by adding IPTG for the pLac-promoted region, tetracycline for the PTet-promoted region, and arabinose for the pBad/Ara genes. By characterizing each of the three pathways without following genes and terminators, we enable a tri-stable switching network of any three biobricks to be constructed with minimal cloning.<br />
<br />
[[Image:Registry_tristable.jpg|Our implementation]]<br />
<br />
<br><br />
Link to Tri-stable Switch on the registry:[http://partsregistry.org/Part:BBa_J24672]<br />
<br />
<br><br><br />
<br />
<br><br />
<br><br />
<br />
=Modeling the tri-stable toggle switch=<br />
While the tri-stable switch is seemingly simple in design, just like any other system it is subject to potentially comprising factors such as promoter leakiness and other stochastic fluctuations. In an attempt to predict the behavior of the tri-stable switch, we created a deterministic model of the system taking into account these factors. The model structure is based on that described in "Prediction and measurement of an autoregulatory genetic module" by Farren Isaacs, et al. This paper features an excellent supplement that takes you hand-in-hand through the derivation of model equations. I will attempt to emulate this derivation with our model below. It is important to note that while we have some preliminary results, our model is very much a work in progress. In order to model the system accurately, many of the fundamental constants governing the model will need to be determined experimentally. For now we have used some constants from the literature and estimated some based on similar the values of similar constants also from the literature. <br />
<br />
==Derivation of the Model Equations==<br />
[[Derivation of the Model Equations]]<br />
<br />
==Table of preliminary model constants==<br />
[[Table of preliminary model constants]]<br />
<br />
==Preliminary modeling results==<br />
<br />
The preliminary modeling results predict things about of the the tri-stable switch. First of all, the model confirms certain obvious points such as the a shift to the proper steady expression state in the presence of inducers. This prediction attests to model validity more than anything else. Secondly, the model predicts that the switch will reach a natural steady state with one promoter dominating in the absence of inducers. This makes intuitive sense as the individual promoters will most likely have different strengths. <br />
<br />
[[Image:no_inducer.png]]<br />
<br />
Next we tested the stability of the model by plugging in final steady-state induced levels of protein into an inducer-less system. The model predicts that the system will not remain stable and will instead revert back to the natural steady state for the inducer-less system. Even after changing several model parameters, the switch would not remain stable.<br />
<br />
<br />
[[Image:stability_test1.png]]<br />
<br />
As a test of the model, models of the three potential bi-stable switches (already proven to be successful experimentally) using the same components and parameters were tested. In accordance with the literature, each of these constructs maintained stable expression of the induced state following the removal of the inducer. In order to probe the instability of the tri-stable switch further, tri-stable switches in which all of the components were effectively the same were created (ex:for all LacI-like, TetR and AraC were modeled as forming tetramers and having one operator.) Each of these systems reached a steady state in which all of the proteins evolved identically and were present at the same levels in the absence of inducer. The same stability tests were then performed. All of the constructs remained stable. Thus our model suggests that the similarity of the repressor-expression systems dominates the success or failure of the tri-stable switch. Furthermore, the bi-stable switch is less sensitive to this factor. <br />
<br />
It should be noted that this is only a preliminary model. The values obtained from the literature will most like differ in our system. Through future experimentation we will elucidate these values and thus create a more accurate model. We will also conduct parameter scans aimed to provide further insight into the necessary system requirements for "tri-stability."<br />
<br />
<br />
<br />
<br />
Link to MATLAB code for tri-stable model [[tristable1.txt]] <br />
<br />
Note: copy this code into MATLAB and save as an M-file<br />
<br />
=What we achieved=<br />
<br />
A successfully ligated pBad/arac construct in addition to partially completed ligations for the pLacI and the pTetR constructs were the results of our summer and semesters' work. The tri-stable toggle switch system was also modeled and proved to yield significant findings.<br />
<br />
=Challenges=<br />
<br />
Despite many cloning and ligation attempts, we were unable to create each component of the switch. More specifically, the pLacI and the pTetR circuits presented many ligation difficulties and proved to yield unsuccessful cloning. However, the pBAD/araC circuit was successfully constructed (yay!).<br />
<br />
=Conclusions=<br />
<br />
With the bigger picture in mind,this project yielded many successful outcomes.Firstly, we were able to successfully construct one of the three constructs necessary for a fully functional tri-stable toggle switch. Secondly, we were able to add more working parts to the registry, both a fully functioning pBad/AraC construct and many parts that were successfully ligated in the process of assembling the other two constructs. These parts will be used in the future by our team and perhaps many others as new ideas and projects develop. Also, it is necessary to mention our most recent finding from the modeling results for the tri-stable toggle switch. Interestingly enough, our model suggests that it is biologically unfeasible to create a switch with more than two components. If more than two components are introduced, such as three, the system is unable to maintain any one state and it will revert back to expressing its most dominant promoter construct. We find such results quite meaningful in demonstrating the biological limits of a system under study.<br />
<br />
=Future work=<br />
<br />
In the coming months, it is hoped that the failure of the ligation strategies for the pLacI and the pTetR constructs can be further investigated. We feel confident that we can continue to build from the one successful construction,the pBad/Arac construct and complete the other 2 constructs. To this end, it will be necessary to further troubleshoot the possible reasons why the cloning of the ligations were unsuccessful. Also we plan to conduct various experiments to elucidate more accurate model parameters specific to our system. Furthermore we will perform a parameter scan to investigate the determinates of "tri-stability."</div>Jlohmuelhttps://2006.igem.org/wiki/index.php/Brown:ProjectsBrown:Projects2006-11-01T06:51:34Z<p>Jlohmuel: </p>
<hr />
<div>{{Brown navigation bar}}<br />
{| cellspacing="2px" cellpadding="0" border="0" style="padding: 0; width: 750px; color: #000000; background-color: #ffffff;"<br />
| width=717px style="padding: 5px; background-color: #ffffff; border: 2px solid #993300;" |<br />
<br />
<h3>Brown 2006 iGEM entry </h3><br />
<br />
As the race to the finish draws near for iGEM 2006, our team has compiled a detailed summary of our summers work. This includes our initial ideas, designs, models and the parts that we have added to the registry. We began the summer with grand plans to design a game of bacterial freeze tag, modeled after the popular children's game where one player tries to catch the other. Our efforts were then switched to the extension of the bi-stable toggle switch and the design of a tri-stable toggle switch. Our projects highlight the team's creative and innovative approaches to designing and engineering working biological systems.<br />
{| cellspacing="2px" cellpadding="0" border="0" style="padding: 0px; width: 750px; color: #000000; background-color: #ffffff;"<br />
|-valign="top"<br />
|width=179.25px style="padding: 5px; background-color: #ffffff; border: 2px solid #993300;" |<br />
<br />
<h3>[[Brown:Bacterial Freeze Tag | Bacterial Freeze Tag]] </h3><br />
<br />
*[[Brown:Bacterial_Freeze_Tag#Overview |Overview]]<br />
*[[Brown:Bacterial_Freeze_Tag#Sender cell | Sender cell ]]<br />
*[[Brown:Bacterial_Freeze_Tag#Receiver cell | Receiver cell ]]<br />
*[[Brown:Bacterial_Freeze_Tag#Challenges| Challenges ]]<br />
<br />
|width=179.25px style="padding: 5px; background-color: #ffffff; border: 2px solid #993300;" |<br />
<br />
<h3>[[Brown:Tri-Stable toggle switch|Tri-Stable toggle switch]]</h3><br />
*[[Brown:Tri-Stable_toggle_switch|Introduction to the project]]<br />
*[[Brown:Tri-Stable toggle switch#Modeling the tri-stable toggle switch | Modeling the tri-stable switch]]<br />
*[[Brown:Tri-Stable toggle switch#What we achieved| What we achieved]]<br />
*[[Brown:Tri-Stable toggle switch#Challenges | Challenges ]]<br />
*[[Brown:Tri-Stable toggle switch#Challenges | Future work ]]<br />
<br />
|width=179.25px style="padding: 5px; background-color: #ffffff; border: 2px solid #993300;" |<br />
<br />
<h3>[[Brown:Team logistics | Team Logistics ]] </h3><br />
<br />
*[[Brown:Team logistics|How we got here]]<br />
*[[Brown:Team_logistics|Challenges]]<br />
*[[Brown:Team_logistics|Future work]]<br />
<br />
<br />
<br />
__NOTOC__</div>Jlohmuelhttps://2006.igem.org/wiki/index.php/Derivation_of_the_Model_EquationsDerivation of the Model Equations2006-11-01T06:50:34Z<p>Jlohmuel: </p>
<hr />
<div>{{Brown navigation bar}}<br />
====Derivation of the model equations====<br />
The chemical reactions describing the tri-stable switch can be divided into the two categories of fast and slow reactions. Fast reactions such as dimer formation and promoter-binding occur in the scale of seconds and are therefore modeled to be in equilibrium. Conversely, slow reactions including the likes of transcription, translation, and protein degradation occur on the scale of minutes and are thus modeled to be evolving with time. <br />
<br />
====Fast reversible reaction equations====<br />
The following equations (1) describe the fast reactions. The characters L,T,A denote molecules of LacI, TetR, and, AraC, respectively while the subscripts denote whether the molecule is a monomer (blank), dimer(2), tri-mer(3), etc. The k's denote reaction rates.<br />
<br />
[[Image:Fast_rxn_eqs1.png]]<br />
<br />
Note that the equations include volume explicitly. Cell volume is modeled in this way as it is a slowly evolving function of time.<br />
<br />
====Slow irreversible reaction equations====<br />
The following equations (2) describe the slow irreversible reactions of transcription and translation (both taken into account with the reaction rate kti) and protein degradation (reaction constant kdi). The coefficients eta-ij take into account the relative translation rates of proteins from the same transcripts. For eta-ij, "i" represents the promoter responsible for producing the molecule and "j" represents the molecule being translated [ex: eta-LA corresponds to the relative rate of AraC production from the trascript produced by the LacI promoter]. To establish a convention, eta's are relative to the translation rate of the first gene on a particular transcript. Thus eta for the first gene on the mRNA transcript = 1. The alpha-i coefficients represent relative transcription rates. In this case, "i" denotes the promoter from which the mRNA is transcribed.<br />
<br />
[[Image:slow_rxn_eqs.png]]<br />
====Equations governing cell volume====<br />
The following two equations (3) describing cellular growth and division are taken directly from the aforementioned paper by Isaacs et al. The first equation describes the volume increase from the time immediately following cell division to the time immediately before it. In this equation, V0 denotes the volume of the cell at the beginning of growth and T0 denotes the time of cell division. In our model at times T=q*t0 in which q is an integer, we have volume V and protein concentration n halve - thus modeling volume division and the resulting protein redistribution. The second equation describes the dimensionless equation in which t is measured in terms of fcell-division time and the cell volume changes between 1 and 2.<br />
<br />
[[Image:vol_eq.png]]<br />
====Equations for the total number of molecules====<br />
The following equations (4) describe the total number of inhibitor protein molecules zi. d#i represents the number, "#", of molecules bound to the "i" promoter. (ex: all of the LacI tetramers bound to pLacI promoters are accounted for by the 4d1L term.)<br />
[[Image:Tot_mol_eq.png]]<br />
<br />
====Equations describing the temporal evolution of the total number inhibitor molecules====<br />
The following equations (5) describe the temporal evolution of the total number of inhibitor molecules. Beta-i = the cell division time multiplied by the combined transcription and translation rates from a given promoter "i" (t0*kti), thus representing the total number of protein molecules maximally produced by a given promoter. Similarly, Gamma-i = the cell division time multiplied by the degradation rate of protein i (t0*kdi), thus representing the total number of protein i destabilized over one cell division time. The eta and alpha terms are described in the section above describing the equations for the slow reactions. Note that the equation describing the temporal evolution of AraC includes the term "Aend." This term represents the endogenous expression of AraC in dH5alpha's. Our preliminary work suggests that this endogenous AraC has very little effect on our pBad promoter.<br />
<br />
[[Image:Ev_tot_mol_eqs.png]]<br />
<br />
====Fast reaction equilibrium equations====<br />
Compared to the slowly evolving reactions(2) described by the above equations (5), the fast reactions (1) can be considered to be in equilibrium. Thus the following equilibrium relations (6) hold.<br />
[[Image:Fast_eq_eqs.png]]<br />
<br />
====Equations modeling the presence of the inducers====<br />
The following equations (7) model the effect of the chemical inducers on the system. The model regards the proteins bound to the inducers as having increased dissociation rates (ki-#) as specified below. The reaction rate with the apostrophe denotes the original reaction rate in the absence of inducer inclusion. In these equations "I" denotes IPTG, "a" denotes arabinose, and "Tc" denotes tetracycline (or analogue aTc). The concept for and the form of these equations are based on equations described in the supplementary Information for "A bottom-up approach to gene regulation" by Guido et al. Much like the aforementioned paper by Isaacs et al., 'Bottom-up' has an extensive and brilliantly articulate derivation supplement. <br />
<br />
[[Image:inducer_eqs.png]]<br />
<br />
====Equations relating plasmid copy number to operator sites====<br />
The following equations (8) relate the plasmid copy number, "m", to the number of promoters with bound and unbound operators. As each plasmid contains 1 of each promoter, m is constant for all three promoter types. <br />
<br />
[[Image:cop_num_eqs.png]]<br />
<br />
====Simplified fast reaction equilibrium equations====<br />
The equilibrium equations governing the fast reactions (6) can be simplified by defining dimensionless equilibrium constants in the form of cij=kij/(k(-ij)*V0*A) in which A is Avogadro's number. The cL's are further simplified by defining the constant cL=c1L*c2L*c3L*c4L. Additionally by plugging in successive terms we were able to write the operator equations in terms of d0L. <br />
<br />
[[Image:Simp_fast_eq_eqs.png]]<br />
<br />
====Equations for the number of unbound operators====<br />
Combining equations (8) and the above simplified equilibrium equations(9), we are able to solve for the number of unbound operators d0i. <br />
<br />
[[Image:Unbound_eqs.png]]<br />
<br />
====Equations for the number of bound operators====<br />
By plugging the d0i's (10) back into the equations for bound operators (9), we can solve for these equations in terms of monomer concentrations.<br />
<br />
<br />
[[Image:bound_eqs.png]]<br />
<br />
====Equation manipulations to determine change in protein monomers per time====<br />
Next we can rewrite the equations describing the temporal evolution of the total number inhibitor molecules (5) in terms of equations (10) and (11). This yields the somewhat complex equations (12) which are solely in terms of monomer protein concentration. <br />
<br />
[[Image:dzdt_eqs.png]]<br />
<br />
====Equations for the change in protein monomers per time====<br />
By applying the simple mathematical manipulation (13) to equations (12) and the X derivative of equations (9) ((9) can be written in terms of X by plugging in equations (10) and (11)) we can solve for the following equations for the evolution of protein monomers as a function of time (14). I will spare you the dZi/dXi derivative equations of equation (9).<br />
<br />
[[Image:simp_mat_man_eq.png]]<br />
<br />
====Equations describing stochastic variation====<br />
<br />
====Equations for the evolution of fluorescent reporters====<br />
<br />
<br />
--[[User:Jlohmuel|Jlohmuel]] 03:59, 28 October 2006 (EDT)</div>Jlohmuelhttps://2006.igem.org/wiki/index.php/Brown:Tri-Stable_toggle_switchBrown:Tri-Stable toggle switch2006-11-01T06:50:01Z<p>Jlohmuel: /* Future work */</p>
<hr />
<div>{{Brown navigation bar}}<br />
=A tri-stable toggle switch=<br />
<br />
Pursuant to the the 1999 paper "Construction of a genetic toggle switch in Escherichia coli," by Timothy S. Gardner, Charles R. Cantor and James J. Collins, we wondered if the bi-stable toggle switch could be generalized to an n-stable switch. To that end, we conceived and began construction of the tri-stable toggle switch. The general idea is that when a selected promoter is activated, it represses the other two. As such each of the three states of our network are self-stabilizing.<br />
<br />
[[Image:Tristable_Toggle.jpg|left|500px|A general tri-stable toggle]]<br />
<br />
Our implementation uses the pLac, pTet, and pBad/AraC promoters and their respective inhibiting proteins. We chose these because of the relative ease with which each promoter is induced: by adding IPTG for the pLac-promoted region, tetracycline for the PTet-promoted region, and arabinose for the pBad/Ara genes. By characterizing each of the three pathways without following genes and terminators, we enable a tri-stable switching network of any three biobricks to be constructed with minimal cloning.<br />
<br />
[[Image:Registry_tristable.jpg|Our implementation]]<br />
<br />
<br><br />
Link to Tri-stable Switch on the registry:[http://partsregistry.org/Part:BBa_J24672]<br />
<br />
<br><br><br />
<br />
<br><br />
<br><br />
<br />
=Modeling the tri-stable toggle switch=<br />
While the tri-stable switch is seemingly simple in design, just like any other system it is subject to potentially comprising factors such as promoter leakiness and other stochastic fluctuations. In an attempt to predict the behavior of the tri-stable switch, we created a deterministic model of the system taking into account these factors. The model structure is based on that described in "Prediction and measurement of an autoregulatory genetic module" by Farren Isaacs, et al. This paper features an excellent supplement that takes you hand-in-hand through the derivation of model equations. I will attempt to emulate this derivation with our model below. It is important to note that while we have some preliminary results, our model is very much a work in progress. In order to model the system accurately, many of the fundamental constants governing the model will need to be determined experimentally. For now we have used some constants from the literature and estimated some based on similar the values of similar constants also from the literature. <br />
<br />
==Derivation of the Model Equations==<br />
[[Derivation of the Model Equations]]<br />
<br />
==Table of preliminary model constants==<br />
[[Table of preliminary model constants]]<br />
<br />
==Preliminary modeling results==<br />
<br />
The preliminary modeling results predict things about of the the tri-stable switch. First of all, the model confirms certain obvious points such as the a shift to the proper steady expression state in the presence of inducers. This prediction attests to model validity more than anything else. Secondly, the model predicts that the switch will reach a natural steady state with one promoter dominating in the absence of inducers. This makes intuitive sense as the individual promoters will most likely have different strengths. <br />
<br />
[[Image:no_inducer.png]]<br />
<br />
Next we tested the stability of the model by plugging in final steady-state induced levels of protein into an inducer-less system. The model predicts that the system will not remain stable and will instead revert back to the natural steady state for the inducer-less system. Even after changing several model parameters, the switch would not remain stable.<br />
<br />
<br />
[[Image:stability_test1.png]]<br />
<br />
As a test of the model, models of the three potential bi-stable switches (already proven to be successful experimentally) using the same components and parameters were tested. In accordance with the literature, each of these constructs maintained stable expression of the induced state following the removal of the inducer. In order to probe the instability of the tri-stable switch further, tri-stable switches in which all of the components were effectively the same were created (ex:for all LacI-like, TetR and AraC were modeled as forming tetramers and having one operator.) Each of these systems reached a steady state in which all of the proteins evolved identically and were present at the same levels in the absence of inducer. The same stability tests were then performed. All of the constructs remained stable. Thus our model suggests that the similarity of the repressor-expression systems dominates the success or failure of the tri-stable switch. Furthermore, the bi-stable switch is less sensitive to this factor. <br />
<br />
It should be noted that this is only a preliminary model. The values obtained from the literature will most like differ in our system. Through future experimentation we will elucidate these values and thus create a more accurate model. We will also conduct parameter scans aimed to provide further insight into the necessary system requirements for "tri-stability."<br />
<br />
<br />
<br />
<br />
Link to MATLAB code for tri-stable model [[Media:tristable1.txt]] <br />
<br />
Note: copy this code into MATLAB and save as an M-file<br />
<br />
=What we achieved=<br />
<br />
A successfully ligated pBad/arac construct in addition to partially completed ligations for the pLacI and the pTetR constructs were the results of our summer and semesters' work. The tri-stable toggle switch system was also modeled and proved to yield significant findings.<br />
<br />
=Challenges=<br />
<br />
Despite many cloning and ligation attempts, we were unable to create each component of the switch. More specifically, the pLacI and the pTetR circuits presented many ligation difficulties and proved to yield unsuccessful cloning. However, the pBAD/araC circuit was successfully constructed (yay!).<br />
<br />
=Conclusions=<br />
<br />
With the bigger picture in mind,this project yielded many successful outcomes.Firstly, we were able to successfully construct one of the three constructs necessary for a fully functional tri-stable toggle switch. Secondly, we were able to add more working parts to the registry, both a fully functioning pBad/AraC construct and many parts that were successfully ligated in the process of assembling the other two constructs. These parts will be used in the future by our team and perhaps many others as new ideas and projects develop. Also, it is necessary to mention our most recent finding from the modeling results for the tri-stable toggle switch. Interestingly enough, our model suggests that it is biologically unfeasible to create a switch with more than two components. If more than two components are introduced, such as three, the system is unable to maintain any one state and it will revert back to expressing its most dominant promoter construct. We find such results quite meaningful in demonstrating the biological limits of a system under study.<br />
<br />
=Future work=<br />
<br />
In the coming months, it is hoped that the failure of the ligation strategies for the pLacI and the pTetR constructs can be further investigated. We feel confident that we can continue to build from the one successful construction,the pBad/Arac construct and complete the other 2 constructs. To this end, it will be necessary to further troubleshoot the possible reasons why the cloning of the ligations were unsuccessful. Also we plan to conduct various experiments to elucidate more accurate model parameters specific to our system. Furthermore we will perform a parameter scan to investigate the determinates of "tri-stability."</div>Jlohmuelhttps://2006.igem.org/wiki/index.php/Brown:Tri-Stable_toggle_switchBrown:Tri-Stable toggle switch2006-11-01T06:49:52Z<p>Jlohmuel: /* Conclusions */</p>
<hr />
<div>{{Brown navigation bar}}<br />
=A tri-stable toggle switch=<br />
<br />
Pursuant to the the 1999 paper "Construction of a genetic toggle switch in Escherichia coli," by Timothy S. Gardner, Charles R. Cantor and James J. Collins, we wondered if the bi-stable toggle switch could be generalized to an n-stable switch. To that end, we conceived and began construction of the tri-stable toggle switch. The general idea is that when a selected promoter is activated, it represses the other two. As such each of the three states of our network are self-stabilizing.<br />
<br />
[[Image:Tristable_Toggle.jpg|left|500px|A general tri-stable toggle]]<br />
<br />
Our implementation uses the pLac, pTet, and pBad/AraC promoters and their respective inhibiting proteins. We chose these because of the relative ease with which each promoter is induced: by adding IPTG for the pLac-promoted region, tetracycline for the PTet-promoted region, and arabinose for the pBad/Ara genes. By characterizing each of the three pathways without following genes and terminators, we enable a tri-stable switching network of any three biobricks to be constructed with minimal cloning.<br />
<br />
[[Image:Registry_tristable.jpg|Our implementation]]<br />
<br />
<br><br />
Link to Tri-stable Switch on the registry:[http://partsregistry.org/Part:BBa_J24672]<br />
<br />
<br><br><br />
<br />
<br><br />
<br><br />
<br />
=Modeling the tri-stable toggle switch=<br />
While the tri-stable switch is seemingly simple in design, just like any other system it is subject to potentially comprising factors such as promoter leakiness and other stochastic fluctuations. In an attempt to predict the behavior of the tri-stable switch, we created a deterministic model of the system taking into account these factors. The model structure is based on that described in "Prediction and measurement of an autoregulatory genetic module" by Farren Isaacs, et al. This paper features an excellent supplement that takes you hand-in-hand through the derivation of model equations. I will attempt to emulate this derivation with our model below. It is important to note that while we have some preliminary results, our model is very much a work in progress. In order to model the system accurately, many of the fundamental constants governing the model will need to be determined experimentally. For now we have used some constants from the literature and estimated some based on similar the values of similar constants also from the literature. <br />
<br />
==Derivation of the Model Equations==<br />
[[Derivation of the Model Equations]]<br />
<br />
==Table of preliminary model constants==<br />
[[Table of preliminary model constants]]<br />
<br />
==Preliminary modeling results==<br />
<br />
The preliminary modeling results predict things about of the the tri-stable switch. First of all, the model confirms certain obvious points such as the a shift to the proper steady expression state in the presence of inducers. This prediction attests to model validity more than anything else. Secondly, the model predicts that the switch will reach a natural steady state with one promoter dominating in the absence of inducers. This makes intuitive sense as the individual promoters will most likely have different strengths. <br />
<br />
[[Image:no_inducer.png]]<br />
<br />
Next we tested the stability of the model by plugging in final steady-state induced levels of protein into an inducer-less system. The model predicts that the system will not remain stable and will instead revert back to the natural steady state for the inducer-less system. Even after changing several model parameters, the switch would not remain stable.<br />
<br />
<br />
[[Image:stability_test1.png]]<br />
<br />
As a test of the model, models of the three potential bi-stable switches (already proven to be successful experimentally) using the same components and parameters were tested. In accordance with the literature, each of these constructs maintained stable expression of the induced state following the removal of the inducer. In order to probe the instability of the tri-stable switch further, tri-stable switches in which all of the components were effectively the same were created (ex:for all LacI-like, TetR and AraC were modeled as forming tetramers and having one operator.) Each of these systems reached a steady state in which all of the proteins evolved identically and were present at the same levels in the absence of inducer. The same stability tests were then performed. All of the constructs remained stable. Thus our model suggests that the similarity of the repressor-expression systems dominates the success or failure of the tri-stable switch. Furthermore, the bi-stable switch is less sensitive to this factor. <br />
<br />
It should be noted that this is only a preliminary model. The values obtained from the literature will most like differ in our system. Through future experimentation we will elucidate these values and thus create a more accurate model. We will also conduct parameter scans aimed to provide further insight into the necessary system requirements for "tri-stability."<br />
<br />
<br />
<br />
<br />
Link to MATLAB code for tri-stable model [[Media:tristable1.txt]] <br />
<br />
Note: copy this code into MATLAB and save as an M-file<br />
<br />
=What we achieved=<br />
<br />
A successfully ligated pBad/arac construct in addition to partially completed ligations for the pLacI and the pTetR constructs were the results of our summer and semesters' work. The tri-stable toggle switch system was also modeled and proved to yield significant findings.<br />
<br />
=Challenges=<br />
<br />
Despite many cloning and ligation attempts, we were unable to create each component of the switch. More specifically, the pLacI and the pTetR circuits presented many ligation difficulties and proved to yield unsuccessful cloning. However, the pBAD/araC circuit was successfully constructed (yay!).<br />
<br />
=Conclusions=<br />
<br />
With the bigger picture in mind,this project yielded many successful outcomes.Firstly, we were able to successfully construct one of the three constructs necessary for a fully functional tri-stable toggle switch. Secondly, we were able to add more working parts to the registry, both a fully functioning pBad/AraC construct and many parts that were successfully ligated in the process of assembling the other two constructs. These parts will be used in the future by our team and perhaps many others as new ideas and projects develop. Also, it is necessary to mention our most recent finding from the modeling results for the tri-stable toggle switch. Interestingly enough, our model suggests that it is biologically unfeasible to create a switch with more than two components. If more than two components are introduced, such as three, the system is unable to maintain any one state and it will revert back to expressing its most dominant promoter construct. We find such results quite meaningful in demonstrating the biological limits of a system under study.<br />
<br />
==Future work==<br />
<br />
In the coming months, it is hoped that the failure of the ligation strategies for the pLacI and the pTetR constructs can be further investigated. We feel confident that we can continue to build from the one successful construction,the pBad/Arac construct and complete the other 2 constructs. To this end, it will be necessary to further troubleshoot the possible reasons why the cloning of the ligations were unsuccessful. Also we plan to conduct various experiments to elucidate more accurate model parameters specific to our system. Furthermore we will perform a parameter scan to investigate the determinates of "tri-stability."</div>Jlohmuelhttps://2006.igem.org/wiki/index.php/Brown:Tri-Stable_toggle_switchBrown:Tri-Stable toggle switch2006-11-01T06:49:43Z<p>Jlohmuel: /* Challenges */</p>
<hr />
<div>{{Brown navigation bar}}<br />
=A tri-stable toggle switch=<br />
<br />
Pursuant to the the 1999 paper "Construction of a genetic toggle switch in Escherichia coli," by Timothy S. Gardner, Charles R. Cantor and James J. Collins, we wondered if the bi-stable toggle switch could be generalized to an n-stable switch. To that end, we conceived and began construction of the tri-stable toggle switch. The general idea is that when a selected promoter is activated, it represses the other two. As such each of the three states of our network are self-stabilizing.<br />
<br />
[[Image:Tristable_Toggle.jpg|left|500px|A general tri-stable toggle]]<br />
<br />
Our implementation uses the pLac, pTet, and pBad/AraC promoters and their respective inhibiting proteins. We chose these because of the relative ease with which each promoter is induced: by adding IPTG for the pLac-promoted region, tetracycline for the PTet-promoted region, and arabinose for the pBad/Ara genes. By characterizing each of the three pathways without following genes and terminators, we enable a tri-stable switching network of any three biobricks to be constructed with minimal cloning.<br />
<br />
[[Image:Registry_tristable.jpg|Our implementation]]<br />
<br />
<br><br />
Link to Tri-stable Switch on the registry:[http://partsregistry.org/Part:BBa_J24672]<br />
<br />
<br><br><br />
<br />
<br><br />
<br><br />
<br />
=Modeling the tri-stable toggle switch=<br />
While the tri-stable switch is seemingly simple in design, just like any other system it is subject to potentially comprising factors such as promoter leakiness and other stochastic fluctuations. In an attempt to predict the behavior of the tri-stable switch, we created a deterministic model of the system taking into account these factors. The model structure is based on that described in "Prediction and measurement of an autoregulatory genetic module" by Farren Isaacs, et al. This paper features an excellent supplement that takes you hand-in-hand through the derivation of model equations. I will attempt to emulate this derivation with our model below. It is important to note that while we have some preliminary results, our model is very much a work in progress. In order to model the system accurately, many of the fundamental constants governing the model will need to be determined experimentally. For now we have used some constants from the literature and estimated some based on similar the values of similar constants also from the literature. <br />
<br />
==Derivation of the Model Equations==<br />
[[Derivation of the Model Equations]]<br />
<br />
==Table of preliminary model constants==<br />
[[Table of preliminary model constants]]<br />
<br />
==Preliminary modeling results==<br />
<br />
The preliminary modeling results predict things about of the the tri-stable switch. First of all, the model confirms certain obvious points such as the a shift to the proper steady expression state in the presence of inducers. This prediction attests to model validity more than anything else. Secondly, the model predicts that the switch will reach a natural steady state with one promoter dominating in the absence of inducers. This makes intuitive sense as the individual promoters will most likely have different strengths. <br />
<br />
[[Image:no_inducer.png]]<br />
<br />
Next we tested the stability of the model by plugging in final steady-state induced levels of protein into an inducer-less system. The model predicts that the system will not remain stable and will instead revert back to the natural steady state for the inducer-less system. Even after changing several model parameters, the switch would not remain stable.<br />
<br />
<br />
[[Image:stability_test1.png]]<br />
<br />
As a test of the model, models of the three potential bi-stable switches (already proven to be successful experimentally) using the same components and parameters were tested. In accordance with the literature, each of these constructs maintained stable expression of the induced state following the removal of the inducer. In order to probe the instability of the tri-stable switch further, tri-stable switches in which all of the components were effectively the same were created (ex:for all LacI-like, TetR and AraC were modeled as forming tetramers and having one operator.) Each of these systems reached a steady state in which all of the proteins evolved identically and were present at the same levels in the absence of inducer. The same stability tests were then performed. All of the constructs remained stable. Thus our model suggests that the similarity of the repressor-expression systems dominates the success or failure of the tri-stable switch. Furthermore, the bi-stable switch is less sensitive to this factor. <br />
<br />
It should be noted that this is only a preliminary model. The values obtained from the literature will most like differ in our system. Through future experimentation we will elucidate these values and thus create a more accurate model. We will also conduct parameter scans aimed to provide further insight into the necessary system requirements for "tri-stability."<br />
<br />
<br />
<br />
<br />
Link to MATLAB code for tri-stable model [[Media:tristable1.txt]] <br />
<br />
Note: copy this code into MATLAB and save as an M-file<br />
<br />
=What we achieved=<br />
<br />
A successfully ligated pBad/arac construct in addition to partially completed ligations for the pLacI and the pTetR constructs were the results of our summer and semesters' work. The tri-stable toggle switch system was also modeled and proved to yield significant findings.<br />
<br />
=Challenges=<br />
<br />
Despite many cloning and ligation attempts, we were unable to create each component of the switch. More specifically, the pLacI and the pTetR circuits presented many ligation difficulties and proved to yield unsuccessful cloning. However, the pBAD/araC circuit was successfully constructed (yay!).<br />
<br />
==Conclusions==<br />
<br />
With the bigger picture in mind,this project yielded many successful outcomes.Firstly, we were able to successfully construct one of the three constructs necessary for a fully functional tri-stable toggle switch. Secondly, we were able to add more working parts to the registry, both a fully functioning pBad/AraC construct and many parts that were successfully ligated in the process of assembling the other two constructs. These parts will be used in the future by our team and perhaps many others as new ideas and projects develop. Also, it is necessary to mention our most recent finding from the modeling results for the tri-stable toggle switch. Interestingly enough, our model suggests that it is biologically unfeasible to create a switch with more than two components. If more than two components are introduced, such as three, the system is unable to maintain any one state and it will revert back to expressing its most dominant promoter construct. We find such results quite meaningful in demonstrating the biological limits of a system under study.<br />
<br />
==Future work==<br />
<br />
In the coming months, it is hoped that the failure of the ligation strategies for the pLacI and the pTetR constructs can be further investigated. We feel confident that we can continue to build from the one successful construction,the pBad/Arac construct and complete the other 2 constructs. To this end, it will be necessary to further troubleshoot the possible reasons why the cloning of the ligations were unsuccessful. Also we plan to conduct various experiments to elucidate more accurate model parameters specific to our system. Furthermore we will perform a parameter scan to investigate the determinates of "tri-stability."</div>Jlohmuelhttps://2006.igem.org/wiki/index.php/Brown:Tri-Stable_toggle_switchBrown:Tri-Stable toggle switch2006-11-01T06:49:33Z<p>Jlohmuel: /* What we achieved */</p>
<hr />
<div>{{Brown navigation bar}}<br />
=A tri-stable toggle switch=<br />
<br />
Pursuant to the the 1999 paper "Construction of a genetic toggle switch in Escherichia coli," by Timothy S. Gardner, Charles R. Cantor and James J. Collins, we wondered if the bi-stable toggle switch could be generalized to an n-stable switch. To that end, we conceived and began construction of the tri-stable toggle switch. The general idea is that when a selected promoter is activated, it represses the other two. As such each of the three states of our network are self-stabilizing.<br />
<br />
[[Image:Tristable_Toggle.jpg|left|500px|A general tri-stable toggle]]<br />
<br />
Our implementation uses the pLac, pTet, and pBad/AraC promoters and their respective inhibiting proteins. We chose these because of the relative ease with which each promoter is induced: by adding IPTG for the pLac-promoted region, tetracycline for the PTet-promoted region, and arabinose for the pBad/Ara genes. By characterizing each of the three pathways without following genes and terminators, we enable a tri-stable switching network of any three biobricks to be constructed with minimal cloning.<br />
<br />
[[Image:Registry_tristable.jpg|Our implementation]]<br />
<br />
<br><br />
Link to Tri-stable Switch on the registry:[http://partsregistry.org/Part:BBa_J24672]<br />
<br />
<br><br><br />
<br />
<br><br />
<br><br />
<br />
=Modeling the tri-stable toggle switch=<br />
While the tri-stable switch is seemingly simple in design, just like any other system it is subject to potentially comprising factors such as promoter leakiness and other stochastic fluctuations. In an attempt to predict the behavior of the tri-stable switch, we created a deterministic model of the system taking into account these factors. The model structure is based on that described in "Prediction and measurement of an autoregulatory genetic module" by Farren Isaacs, et al. This paper features an excellent supplement that takes you hand-in-hand through the derivation of model equations. I will attempt to emulate this derivation with our model below. It is important to note that while we have some preliminary results, our model is very much a work in progress. In order to model the system accurately, many of the fundamental constants governing the model will need to be determined experimentally. For now we have used some constants from the literature and estimated some based on similar the values of similar constants also from the literature. <br />
<br />
==Derivation of the Model Equations==<br />
[[Derivation of the Model Equations]]<br />
<br />
==Table of preliminary model constants==<br />
[[Table of preliminary model constants]]<br />
<br />
==Preliminary modeling results==<br />
<br />
The preliminary modeling results predict things about of the the tri-stable switch. First of all, the model confirms certain obvious points such as the a shift to the proper steady expression state in the presence of inducers. This prediction attests to model validity more than anything else. Secondly, the model predicts that the switch will reach a natural steady state with one promoter dominating in the absence of inducers. This makes intuitive sense as the individual promoters will most likely have different strengths. <br />
<br />
[[Image:no_inducer.png]]<br />
<br />
Next we tested the stability of the model by plugging in final steady-state induced levels of protein into an inducer-less system. The model predicts that the system will not remain stable and will instead revert back to the natural steady state for the inducer-less system. Even after changing several model parameters, the switch would not remain stable.<br />
<br />
<br />
[[Image:stability_test1.png]]<br />
<br />
As a test of the model, models of the three potential bi-stable switches (already proven to be successful experimentally) using the same components and parameters were tested. In accordance with the literature, each of these constructs maintained stable expression of the induced state following the removal of the inducer. In order to probe the instability of the tri-stable switch further, tri-stable switches in which all of the components were effectively the same were created (ex:for all LacI-like, TetR and AraC were modeled as forming tetramers and having one operator.) Each of these systems reached a steady state in which all of the proteins evolved identically and were present at the same levels in the absence of inducer. The same stability tests were then performed. All of the constructs remained stable. Thus our model suggests that the similarity of the repressor-expression systems dominates the success or failure of the tri-stable switch. Furthermore, the bi-stable switch is less sensitive to this factor. <br />
<br />
It should be noted that this is only a preliminary model. The values obtained from the literature will most like differ in our system. Through future experimentation we will elucidate these values and thus create a more accurate model. We will also conduct parameter scans aimed to provide further insight into the necessary system requirements for "tri-stability."<br />
<br />
<br />
<br />
<br />
Link to MATLAB code for tri-stable model [[Media:tristable1.txt]] <br />
<br />
Note: copy this code into MATLAB and save as an M-file<br />
<br />
=What we achieved=<br />
<br />
A successfully ligated pBad/arac construct in addition to partially completed ligations for the pLacI and the pTetR constructs were the results of our summer and semesters' work. The tri-stable toggle switch system was also modeled and proved to yield significant findings.<br />
<br />
==Challenges==<br />
<br />
Despite many cloning and ligation attempts, we were unable to create each component of the switch. More specifically, the pLacI and the pTetR circuits presented many ligation difficulties and proved to yield unsuccessful cloning. However, the pBAD/araC circuit was successfully constructed (yay!).<br />
<br />
==Conclusions==<br />
<br />
With the bigger picture in mind,this project yielded many successful outcomes.Firstly, we were able to successfully construct one of the three constructs necessary for a fully functional tri-stable toggle switch. Secondly, we were able to add more working parts to the registry, both a fully functioning pBad/AraC construct and many parts that were successfully ligated in the process of assembling the other two constructs. These parts will be used in the future by our team and perhaps many others as new ideas and projects develop. Also, it is necessary to mention our most recent finding from the modeling results for the tri-stable toggle switch. Interestingly enough, our model suggests that it is biologically unfeasible to create a switch with more than two components. If more than two components are introduced, such as three, the system is unable to maintain any one state and it will revert back to expressing its most dominant promoter construct. We find such results quite meaningful in demonstrating the biological limits of a system under study.<br />
<br />
==Future work==<br />
<br />
In the coming months, it is hoped that the failure of the ligation strategies for the pLacI and the pTetR constructs can be further investigated. We feel confident that we can continue to build from the one successful construction,the pBad/Arac construct and complete the other 2 constructs. To this end, it will be necessary to further troubleshoot the possible reasons why the cloning of the ligations were unsuccessful. Also we plan to conduct various experiments to elucidate more accurate model parameters specific to our system. Furthermore we will perform a parameter scan to investigate the determinates of "tri-stability."</div>Jlohmuelhttps://2006.igem.org/wiki/index.php/Brown:Tri-Stable_toggle_switchBrown:Tri-Stable toggle switch2006-11-01T06:48:52Z<p>Jlohmuel: /* Preliminary modeling results */</p>
<hr />
<div>{{Brown navigation bar}}<br />
=A tri-stable toggle switch=<br />
<br />
Pursuant to the the 1999 paper "Construction of a genetic toggle switch in Escherichia coli," by Timothy S. Gardner, Charles R. Cantor and James J. Collins, we wondered if the bi-stable toggle switch could be generalized to an n-stable switch. To that end, we conceived and began construction of the tri-stable toggle switch. The general idea is that when a selected promoter is activated, it represses the other two. As such each of the three states of our network are self-stabilizing.<br />
<br />
[[Image:Tristable_Toggle.jpg|left|500px|A general tri-stable toggle]]<br />
<br />
Our implementation uses the pLac, pTet, and pBad/AraC promoters and their respective inhibiting proteins. We chose these because of the relative ease with which each promoter is induced: by adding IPTG for the pLac-promoted region, tetracycline for the PTet-promoted region, and arabinose for the pBad/Ara genes. By characterizing each of the three pathways without following genes and terminators, we enable a tri-stable switching network of any three biobricks to be constructed with minimal cloning.<br />
<br />
[[Image:Registry_tristable.jpg|Our implementation]]<br />
<br />
<br><br />
Link to Tri-stable Switch on the registry:[http://partsregistry.org/Part:BBa_J24672]<br />
<br />
<br><br><br />
<br />
<br><br />
<br><br />
<br />
=Modeling the tri-stable toggle switch=<br />
While the tri-stable switch is seemingly simple in design, just like any other system it is subject to potentially comprising factors such as promoter leakiness and other stochastic fluctuations. In an attempt to predict the behavior of the tri-stable switch, we created a deterministic model of the system taking into account these factors. The model structure is based on that described in "Prediction and measurement of an autoregulatory genetic module" by Farren Isaacs, et al. This paper features an excellent supplement that takes you hand-in-hand through the derivation of model equations. I will attempt to emulate this derivation with our model below. It is important to note that while we have some preliminary results, our model is very much a work in progress. In order to model the system accurately, many of the fundamental constants governing the model will need to be determined experimentally. For now we have used some constants from the literature and estimated some based on similar the values of similar constants also from the literature. <br />
<br />
==Derivation of the Model Equations==<br />
[[Derivation of the Model Equations]]<br />
<br />
==Table of preliminary model constants==<br />
[[Table of preliminary model constants]]<br />
<br />
==Preliminary modeling results==<br />
<br />
The preliminary modeling results predict things about of the the tri-stable switch. First of all, the model confirms certain obvious points such as the a shift to the proper steady expression state in the presence of inducers. This prediction attests to model validity more than anything else. Secondly, the model predicts that the switch will reach a natural steady state with one promoter dominating in the absence of inducers. This makes intuitive sense as the individual promoters will most likely have different strengths. <br />
<br />
[[Image:no_inducer.png]]<br />
<br />
Next we tested the stability of the model by plugging in final steady-state induced levels of protein into an inducer-less system. The model predicts that the system will not remain stable and will instead revert back to the natural steady state for the inducer-less system. Even after changing several model parameters, the switch would not remain stable.<br />
<br />
<br />
[[Image:stability_test1.png]]<br />
<br />
As a test of the model, models of the three potential bi-stable switches (already proven to be successful experimentally) using the same components and parameters were tested. In accordance with the literature, each of these constructs maintained stable expression of the induced state following the removal of the inducer. In order to probe the instability of the tri-stable switch further, tri-stable switches in which all of the components were effectively the same were created (ex:for all LacI-like, TetR and AraC were modeled as forming tetramers and having one operator.) Each of these systems reached a steady state in which all of the proteins evolved identically and were present at the same levels in the absence of inducer. The same stability tests were then performed. All of the constructs remained stable. Thus our model suggests that the similarity of the repressor-expression systems dominates the success or failure of the tri-stable switch. Furthermore, the bi-stable switch is less sensitive to this factor. <br />
<br />
It should be noted that this is only a preliminary model. The values obtained from the literature will most like differ in our system. Through future experimentation we will elucidate these values and thus create a more accurate model. We will also conduct parameter scans aimed to provide further insight into the necessary system requirements for "tri-stability."<br />
<br />
<br />
<br />
<br />
Link to MATLAB code for tri-stable model [[Media:tristable1.txt]] <br />
<br />
Note: copy this code into MATLAB and save as an M-file<br />
<br />
==What we achieved==<br />
<br />
A successfully ligated pBad/arac construct in addition to partially completed ligations for the pLacI and the pTetR constructs were the results of our summer and semesters' work. The tri-stable toggle switch system was also modeled and proved to yield significant findings.<br />
<br />
==Challenges==<br />
<br />
Despite many cloning and ligation attempts, we were unable to create each component of the switch. More specifically, the pLacI and the pTetR circuits presented many ligation difficulties and proved to yield unsuccessful cloning. However, the pBAD/araC circuit was successfully constructed (yay!).<br />
<br />
==Conclusions==<br />
<br />
With the bigger picture in mind,this project yielded many successful outcomes.Firstly, we were able to successfully construct one of the three constructs necessary for a fully functional tri-stable toggle switch. Secondly, we were able to add more working parts to the registry, both a fully functioning pBad/AraC construct and many parts that were successfully ligated in the process of assembling the other two constructs. These parts will be used in the future by our team and perhaps many others as new ideas and projects develop. Also, it is necessary to mention our most recent finding from the modeling results for the tri-stable toggle switch. Interestingly enough, our model suggests that it is biologically unfeasible to create a switch with more than two components. If more than two components are introduced, such as three, the system is unable to maintain any one state and it will revert back to expressing its most dominant promoter construct. We find such results quite meaningful in demonstrating the biological limits of a system under study.<br />
<br />
==Future work==<br />
<br />
In the coming months, it is hoped that the failure of the ligation strategies for the pLacI and the pTetR constructs can be further investigated. We feel confident that we can continue to build from the one successful construction,the pBad/Arac construct and complete the other 2 constructs. To this end, it will be necessary to further troubleshoot the possible reasons why the cloning of the ligations were unsuccessful. Also we plan to conduct various experiments to elucidate more accurate model parameters specific to our system. Furthermore we will perform a parameter scan to investigate the determinates of "tri-stability."</div>Jlohmuelhttps://2006.igem.org/wiki/index.php/Brown:Tri-Stable_toggle_switchBrown:Tri-Stable toggle switch2006-11-01T06:48:25Z<p>Jlohmuel: /* Future work */</p>
<hr />
<div>{{Brown navigation bar}}<br />
=A tri-stable toggle switch=<br />
<br />
Pursuant to the the 1999 paper "Construction of a genetic toggle switch in Escherichia coli," by Timothy S. Gardner, Charles R. Cantor and James J. Collins, we wondered if the bi-stable toggle switch could be generalized to an n-stable switch. To that end, we conceived and began construction of the tri-stable toggle switch. The general idea is that when a selected promoter is activated, it represses the other two. As such each of the three states of our network are self-stabilizing.<br />
<br />
[[Image:Tristable_Toggle.jpg|left|500px|A general tri-stable toggle]]<br />
<br />
Our implementation uses the pLac, pTet, and pBad/AraC promoters and their respective inhibiting proteins. We chose these because of the relative ease with which each promoter is induced: by adding IPTG for the pLac-promoted region, tetracycline for the PTet-promoted region, and arabinose for the pBad/Ara genes. By characterizing each of the three pathways without following genes and terminators, we enable a tri-stable switching network of any three biobricks to be constructed with minimal cloning.<br />
<br />
[[Image:Registry_tristable.jpg|Our implementation]]<br />
<br />
<br><br />
Link to Tri-stable Switch on the registry:[http://partsregistry.org/Part:BBa_J24672]<br />
<br />
<br><br><br />
<br />
<br><br />
<br><br />
<br />
=Modeling the tri-stable toggle switch=<br />
While the tri-stable switch is seemingly simple in design, just like any other system it is subject to potentially comprising factors such as promoter leakiness and other stochastic fluctuations. In an attempt to predict the behavior of the tri-stable switch, we created a deterministic model of the system taking into account these factors. The model structure is based on that described in "Prediction and measurement of an autoregulatory genetic module" by Farren Isaacs, et al. This paper features an excellent supplement that takes you hand-in-hand through the derivation of model equations. I will attempt to emulate this derivation with our model below. It is important to note that while we have some preliminary results, our model is very much a work in progress. In order to model the system accurately, many of the fundamental constants governing the model will need to be determined experimentally. For now we have used some constants from the literature and estimated some based on similar the values of similar constants also from the literature. <br />
<br />
==Derivation of the Model Equations==<br />
[[Derivation of the Model Equations]]<br />
<br />
==Table of preliminary model constants==<br />
[[Table of preliminary model constants]]<br />
<br />
==Preliminary modeling results==<br />
<br />
The preliminary modeling results predict things about of the the tri-stable switch. First of all, the model confirms certain obvious points such as the a shift to the proper steady expression state in the presence of inducers. This prediction attests to model validity more than anything else. Secondly, the model predicts that the switch will reach a natural steady state with one promoter dominating in the absence of inducers. This makes intuitive sense as the individual promoters will most likely have different strengths. <br />
<br />
[[Image:no_inducer.png]]<br />
<br />
Next we tested the stability of the model by plugging in final steady-state induced levels of protein into an inducer-less system. The model predicts that the system will not remain stable and will instead revert back to the natural steady state for the inducer-less system. Even after changing several model parameters, the switch would not remain stable.<br />
<br />
<br />
[[Image:stability_test1.png]]<br />
<br />
As a test of the model, models of the three potential bi-stable switches (already proven to be successful experimentally) using the same components and parameters were tested. In accordance with the literature, each of these constructs maintained stable expression of the induced state following the removal of the inducer. In order to probe the instability of the tri-stable switch further, tri-stable switches in which all of the components were effectively the same were created (ex:for all LacI-like, TetR and AraC were modeled as forming tetramers and having one operator.) Each of these systems reached a steady state in which all of the proteins evolved identically and were present at the same levels in the absence of inducer. The same stability tests were then performed. All of the constructs remained stable. Thus our model suggests that the similarity of the repressor-expression systems dominates the success or failure of the tri-stable switch. Furthermore, the bi-stable switch is less sensitive to this factor. <br />
<br />
It should be noted that this is only a preliminary model. The values obtained from the literature will most like differ in our system. Through future experimentation we will elucidate these values and thus create a more accurate model. We will also conduct parameter scans aimed to provide further insight into the necessary system requirements for "tri-stability."<br />
<br />
<br />
<br />
<br />
Link to MATLAB code for tri-stable model [[Media:tristable1.txt]] <br />
<br />
Note: copy this code into MATLAB and save as an M-file<br />
<br />
--[[User:Jlohmuel|Jlohmuel]] 03:59, 28 October 2006 (EDT)<br />
<br />
==What we achieved==<br />
<br />
A successfully ligated pBad/arac construct in addition to partially completed ligations for the pLacI and the pTetR constructs were the results of our summer and semesters' work. The tri-stable toggle switch system was also modeled and proved to yield significant findings.<br />
<br />
==Challenges==<br />
<br />
Despite many cloning and ligation attempts, we were unable to create each component of the switch. More specifically, the pLacI and the pTetR circuits presented many ligation difficulties and proved to yield unsuccessful cloning. However, the pBAD/araC circuit was successfully constructed (yay!).<br />
<br />
==Conclusions==<br />
<br />
With the bigger picture in mind,this project yielded many successful outcomes.Firstly, we were able to successfully construct one of the three constructs necessary for a fully functional tri-stable toggle switch. Secondly, we were able to add more working parts to the registry, both a fully functioning pBad/AraC construct and many parts that were successfully ligated in the process of assembling the other two constructs. These parts will be used in the future by our team and perhaps many others as new ideas and projects develop. Also, it is necessary to mention our most recent finding from the modeling results for the tri-stable toggle switch. Interestingly enough, our model suggests that it is biologically unfeasible to create a switch with more than two components. If more than two components are introduced, such as three, the system is unable to maintain any one state and it will revert back to expressing its most dominant promoter construct. We find such results quite meaningful in demonstrating the biological limits of a system under study.<br />
<br />
==Future work==<br />
<br />
In the coming months, it is hoped that the failure of the ligation strategies for the pLacI and the pTetR constructs can be further investigated. We feel confident that we can continue to build from the one successful construction,the pBad/Arac construct and complete the other 2 constructs. To this end, it will be necessary to further troubleshoot the possible reasons why the cloning of the ligations were unsuccessful. Also we plan to conduct various experiments to elucidate more accurate model parameters specific to our system. Furthermore we will perform a parameter scan to investigate the determinates of "tri-stability."</div>Jlohmuelhttps://2006.igem.org/wiki/index.php/Derivation_of_the_Model_EquationsDerivation of the Model Equations2006-11-01T06:45:25Z<p>Jlohmuel: </p>
<hr />
<div>{{Brown navigation bar}}<br />
====Derivation of the model equations====<br />
The chemical reactions describing the tri-stable switch can be divided into the two categories of fast and slow reactions. Fast reactions such as dimer formation and promoter-binding occur in the scale of seconds and are therefore modeled to be in equilibrium. Conversely, slow reactions including the likes of transcription, translation, and protein degradation occur on the scale of minutes and are thus modeled to be evolving with time. <br />
<br />
====Fast reversible reaction equations====<br />
The following equations (1) describe the fast reactions. The characters L,T,A denote molecules of LacI, TetR, and, AraC, respectively while the subscripts denote whether the molecule is a monomer (blank), dimer(2), tri-mer(3), etc. The k's denote reaction rates.<br />
<br />
[[Image:Fast_rxn_eqs1.png]]<br />
<br />
Note that the equations include volume explicitly. Cell volume is modeled in this way as it is a slowly evolving function of time.<br />
<br />
====Slow irreversible reaction equations====<br />
The following equations (2) describe the slow irreversible reactions of transcription and translation (both taken into account with the reaction rate kti) and protein degradation (reaction constant kdi). The coefficients eta-ij take into account the relative translation rates of proteins from the same transcripts. For eta-ij, "i" represents the promoter responsible for producing the molecule and "j" represents the molecule being translated [ex: eta-LA corresponds to the relative rate of AraC production from the trascript produced by the LacI promoter]. To establish a convention, eta's are relative to the translation rate of the first gene on a particular transcript. Thus eta for the first gene on the mRNA transcript = 1. The alpha-i coefficients represent relative transcription rates. In this case, "i" denotes the promoter from which the mRNA is transcribed.<br />
<br />
[[Image:slow_rxn_eqs.png]]<br />
====Equations governing cell volume====<br />
The following two equations (3) describing cellular growth and division are taken directly from the aforementioned paper by Isaacs et al. The first equation describes the volume increase from the time immediately following cell division to the time immediately before it. In this equation, V0 denotes the volume of the cell at the beginning of growth and T0 denotes the time of cell division. In our model at times T=q*t0 in which q is an integer, we have volume V and protein concentration n halve - thus modeling volume division and the resulting protein redistribution. The second equation describes the dimensionless equation in which t is measured in terms of fcell-division time and the cell volume changes between 1 and 2.<br />
<br />
[[Image:vol_eq.png]]<br />
====Equations for the total number of molecules====<br />
The following equations (4) describe the total number of inhibitor protein molecules zi. d#i represents the number, "#", of molecules bound to the "i" promoter. (ex: all of the LacI tetramers bound to pLacI promoters are accounted for by the 4d1L term.)<br />
[[Image:Tot_mol_eq.png]]<br />
<br />
====Equations describing the temporal evolution of the total number inhibitor molecules====<br />
The following equations (5) describe the temporal evolution of the total number of inhibitor molecules. Beta-i = the cell division time multiplied by the combined transcription and translation rates from a given promoter "i" (t0*kti), thus representing the total number of protein molecules maximally produced by a given promoter. Similarly, Gamma-i = the cell division time multiplied by the degradation rate of protein i (t0*kdi), thus representing the total number of protein i destabilized over one cell division time. The eta and alpha terms are described in the section above describing the equations for the slow reactions. Note that the equation describing the temporal evolution of AraC includes the term "Aend." This term represents the endogenous expression of AraC in dH5alpha's. Our preliminary work suggests that this endogenous AraC has very little effect on our pBad promoter.<br />
<br />
[[Image:Ev_tot_mol_eqs.png]]<br />
<br />
====Fast reaction equilibrium equations====<br />
Compared to the slowly evolving reactions(2) described by the above equations (5), the fast reactions (1) can be considered to be in equilibrium. Thus the following equilibrium relations (6) hold.<br />
[[Image:Fast_eq_eqs.png]]<br />
<br />
====Equations modeling the presence of the inducers====<br />
The following equations (7) model the effect of the chemical inducers on the system. The model regards the proteins bound to the inducers as having increased dissociation rates (ki-#) as specified below. The reaction rate with the apostrophe denotes the original reaction rate in the absence of inducer inclusion. In these equations "I" denotes IPTG, "a" denotes arabinose, and "Tc" denotes tetracycline (or analogue aTc). The concept for and the form of these equations are based on equations described in the supplementary Information for "A bottom-up approach to gene regulation" by Guido et al. Much like the aforementioned paper by Isaacs et al., 'Bottom-up' has an extensive and brilliantly articulate derivation supplement. <br />
<br />
[[Image:inducer_eqs.png]]<br />
<br />
====Equations relating plasmid copy number to operator sites====<br />
The following equations (8) relate the plasmid copy number, "m", to the number of promoters with bound and unbound operators. As each plasmid contains 1 of each promoter, m is constant for all three promoter types. <br />
<br />
[[Image:cop_num_eqs.png]]<br />
<br />
====Simplified fast reaction equilibrium equations====<br />
The equilibrium equations governing the fast reactions (6) can be simplified by defining dimensionless equilibrium constants in the form of cij=kij/(k(-ij)*V0*A) in which A is Avogadro's number. The cL's are further simplified by defining the constant cL=c1L*c2L*c3L*c4L. Additionally by plugging in successive terms we were able to write the operator equations in terms of d0L. <br />
<br />
[[Image:Simp_fast_eq_eqs.png]]<br />
<br />
====Equations for the number of unbound operators====<br />
Combining equations (8) and the above simplified equilibrium equations(9), we are able to solve for the number of unbound operators d0i. <br />
<br />
[[Image:Unbound_eqs.png]]<br />
<br />
====Equations for the number of bound operators====<br />
By plugging the d0i's (10) back into the equations for bound operators (9), we can solve for these equations in terms of monomer concentrations.<br />
<br />
<br />
[[Image:bound_eqs.png]]<br />
<br />
====Equation manipulations to determine change in protein monomers per time====<br />
Next we can rewrite the equations describing the temporal evolution of the total number inhibitor molecules (5) in terms of equations (10) and (11). This yields the somewhat complex equations (12) which are solely in terms of monomer protein concentration. <br />
<br />
[[Image:dzdt_eqs.png]]<br />
<br />
====Equations for the change in protein monomers per time====<br />
By applying the simple mathematical manipulation (13) to equations (12) and the X derivative of equations (9) ((9) can be written in terms of X by plugging in equations (10) and (11)) we can solve for the following equations for the evolution of protein monomers as a function of time (14). I will spare you the dZi/dXi derivative equations of equation (9).<br />
<br />
[[Image:simp_mat_man_eq.png]]<br />
<br />
====Equations describing stochastic variation====<br />
<br />
====Equations for the evolution of fluorescent reporters====</div>Jlohmuelhttps://2006.igem.org/wiki/index.php/Derivation_of_the_Model_EquationsDerivation of the Model Equations2006-11-01T06:44:12Z<p>Jlohmuel: </p>
<hr />
<div>{{Brown navigation bar}}<br />
<br />
The chemical reactions describing the tri-stable switch can be divided into the two categories of fast and slow reactions. Fast reactions such as dimer formation and promoter-binding occur in the scale of seconds and are therefore modeled to be in equilibrium. Conversely, slow reactions including the likes of transcription, translation, and protein degradation occur on the scale of minutes and are thus modeled to be evolving with time. <br />
<br />
====Fast reversible reaction equations====<br />
The following equations (1) describe the fast reactions. The characters L,T,A denote molecules of LacI, TetR, and, AraC, respectively while the subscripts denote whether the molecule is a monomer (blank), dimer(2), tri-mer(3), etc. The k's denote reaction rates.<br />
<br />
[[Image:Fast_rxn_eqs1.png]]<br />
<br />
Note that the equations include volume explicitly. Cell volume is modeled in this way as it is a slowly evolving function of time.<br />
<br />
====Slow irreversible reaction equations====<br />
The following equations (2) describe the slow irreversible reactions of transcription and translation (both taken into account with the reaction rate kti) and protein degradation (reaction constant kdi). The coefficients eta-ij take into account the relative translation rates of proteins from the same transcripts. For eta-ij, "i" represents the promoter responsible for producing the molecule and "j" represents the molecule being translated [ex: eta-LA corresponds to the relative rate of AraC production from the trascript produced by the LacI promoter]. To establish a convention, eta's are relative to the translation rate of the first gene on a particular transcript. Thus eta for the first gene on the mRNA transcript = 1. The alpha-i coefficients represent relative transcription rates. In this case, "i" denotes the promoter from which the mRNA is transcribed.<br />
<br />
[[Image:slow_rxn_eqs.png]]<br />
====Equations governing cell volume====<br />
The following two equations (3) describing cellular growth and division are taken directly from the aforementioned paper by Isaacs et al. The first equation describes the volume increase from the time immediately following cell division to the time immediately before it. In this equation, V0 denotes the volume of the cell at the beginning of growth and T0 denotes the time of cell division. In our model at times T=q*t0 in which q is an integer, we have volume V and protein concentration n halve - thus modeling volume division and the resulting protein redistribution. The second equation describes the dimensionless equation in which t is measured in terms of fcell-division time and the cell volume changes between 1 and 2.<br />
<br />
[[Image:vol_eq.png]]<br />
====Equations for the total number of molecules====<br />
The following equations (4) describe the total number of inhibitor protein molecules zi. d#i represents the number, "#", of molecules bound to the "i" promoter. (ex: all of the LacI tetramers bound to pLacI promoters are accounted for by the 4d1L term.)<br />
[[Image:Tot_mol_eq.png]]<br />
<br />
====Equations describing the temporal evolution of the total number inhibitor molecules====<br />
The following equations (5) describe the temporal evolution of the total number of inhibitor molecules. Beta-i = the cell division time multiplied by the combined transcription and translation rates from a given promoter "i" (t0*kti), thus representing the total number of protein molecules maximally produced by a given promoter. Similarly, Gamma-i = the cell division time multiplied by the degradation rate of protein i (t0*kdi), thus representing the total number of protein i destabilized over one cell division time. The eta and alpha terms are described in the section above describing the equations for the slow reactions. Note that the equation describing the temporal evolution of AraC includes the term "Aend." This term represents the endogenous expression of AraC in dH5alpha's. Our preliminary work suggests that this endogenous AraC has very little effect on our pBad promoter.<br />
<br />
[[Image:Ev_tot_mol_eqs.png]]<br />
<br />
====Fast reaction equilibrium equations====<br />
Compared to the slowly evolving reactions(2) described by the above equations (5), the fast reactions (1) can be considered to be in equilibrium. Thus the following equilibrium relations (6) hold.<br />
[[Image:Fast_eq_eqs.png]]<br />
<br />
====Equations modeling the presence of the inducers====<br />
The following equations (7) model the effect of the chemical inducers on the system. The model regards the proteins bound to the inducers as having increased dissociation rates (ki-#) as specified below. The reaction rate with the apostrophe denotes the original reaction rate in the absence of inducer inclusion. In these equations "I" denotes IPTG, "a" denotes arabinose, and "Tc" denotes tetracycline (or analogue aTc). The concept for and the form of these equations are based on equations described in the supplementary Information for "A bottom-up approach to gene regulation" by Guido et al. Much like the aforementioned paper by Isaacs et al., 'Bottom-up' has an extensive and brilliantly articulate derivation supplement. <br />
<br />
[[Image:inducer_eqs.png]]<br />
<br />
====Equations relating plasmid copy number to operator sites====<br />
The following equations (8) relate the plasmid copy number, "m", to the number of promoters with bound and unbound operators. As each plasmid contains 1 of each promoter, m is constant for all three promoter types. <br />
<br />
[[Image:cop_num_eqs.png]]<br />
<br />
====Simplified fast reaction equilibrium equations====<br />
The equilibrium equations governing the fast reactions (6) can be simplified by defining dimensionless equilibrium constants in the form of cij=kij/(k(-ij)*V0*A) in which A is Avogadro's number. The cL's are further simplified by defining the constant cL=c1L*c2L*c3L*c4L. Additionally by plugging in successive terms we were able to write the operator equations in terms of d0L. <br />
<br />
[[Image:Simp_fast_eq_eqs.png]]<br />
<br />
====Equations for the number of unbound operators====<br />
Combining equations (8) and the above simplified equilibrium equations(9), we are able to solve for the number of unbound operators d0i. <br />
<br />
[[Image:Unbound_eqs.png]]<br />
<br />
====Equations for the number of bound operators====<br />
By plugging the d0i's (10) back into the equations for bound operators (9), we can solve for these equations in terms of monomer concentrations.<br />
<br />
<br />
[[Image:bound_eqs.png]]<br />
<br />
====Equation manipulations to determine change in protein monomers per time====<br />
Next we can rewrite the equations describing the temporal evolution of the total number inhibitor molecules (5) in terms of equations (10) and (11). This yields the somewhat complex equations (12) which are solely in terms of monomer protein concentration. <br />
<br />
[[Image:dzdt_eqs.png]]<br />
<br />
====Equations for the change in protein monomers per time====<br />
By applying the simple mathematical manipulation (13) to equations (12) and the X derivative of equations (9) ((9) can be written in terms of X by plugging in equations (10) and (11)) we can solve for the following equations for the evolution of protein monomers as a function of time (14). I will spare you the dZi/dXi derivative equations of equation (9).<br />
<br />
[[Image:simp_mat_man_eq.png]]<br />
<br />
====Equations describing stochastic variation====<br />
<br />
====Equations for the evolution of fluorescent reporters====</div>Jlohmuelhttps://2006.igem.org/wiki/index.php/Derivation_of_the_Model_EquationsDerivation of the Model Equations2006-11-01T06:43:17Z<p>Jlohmuel: </p>
<hr />
<div>The chemical reactions describing the tri-stable switch can be divided into the two categories of fast and slow reactions. Fast reactions such as dimer formation and promoter-binding occur in the scale of seconds and are therefore modeled to be in equilibrium. Conversely, slow reactions including the likes of transcription, translation, and protein degradation occur on the scale of minutes and are thus modeled to be evolving with time. <br />
<br />
====Fast reversible reaction equations====<br />
The following equations (1) describe the fast reactions. The characters L,T,A denote molecules of LacI, TetR, and, AraC, respectively while the subscripts denote whether the molecule is a monomer (blank), dimer(2), tri-mer(3), etc. The k's denote reaction rates.<br />
<br />
[[Image:Fast_rxn_eqs1.png]]<br />
<br />
Note that the equations include volume explicitly. Cell volume is modeled in this way as it is a slowly evolving function of time.<br />
<br />
====Slow irreversible reaction equations====<br />
The following equations (2) describe the slow irreversible reactions of transcription and translation (both taken into account with the reaction rate kti) and protein degradation (reaction constant kdi). The coefficients eta-ij take into account the relative translation rates of proteins from the same transcripts. For eta-ij, "i" represents the promoter responsible for producing the molecule and "j" represents the molecule being translated [ex: eta-LA corresponds to the relative rate of AraC production from the trascript produced by the LacI promoter]. To establish a convention, eta's are relative to the translation rate of the first gene on a particular transcript. Thus eta for the first gene on the mRNA transcript = 1. The alpha-i coefficients represent relative transcription rates. In this case, "i" denotes the promoter from which the mRNA is transcribed.<br />
<br />
[[Image:slow_rxn_eqs.png]]<br />
====Equations governing cell volume====<br />
The following two equations (3) describing cellular growth and division are taken directly from the aforementioned paper by Isaacs et al. The first equation describes the volume increase from the time immediately following cell division to the time immediately before it. In this equation, V0 denotes the volume of the cell at the beginning of growth and T0 denotes the time of cell division. In our model at times T=q*t0 in which q is an integer, we have volume V and protein concentration n halve - thus modeling volume division and the resulting protein redistribution. The second equation describes the dimensionless equation in which t is measured in terms of fcell-division time and the cell volume changes between 1 and 2.<br />
<br />
[[Image:vol_eq.png]]<br />
====Equations for the total number of molecules====<br />
The following equations (4) describe the total number of inhibitor protein molecules zi. d#i represents the number, "#", of molecules bound to the "i" promoter. (ex: all of the LacI tetramers bound to pLacI promoters are accounted for by the 4d1L term.)<br />
[[Image:Tot_mol_eq.png]]<br />
<br />
====Equations describing the temporal evolution of the total number inhibitor molecules====<br />
The following equations (5) describe the temporal evolution of the total number of inhibitor molecules. Beta-i = the cell division time multiplied by the combined transcription and translation rates from a given promoter "i" (t0*kti), thus representing the total number of protein molecules maximally produced by a given promoter. Similarly, Gamma-i = the cell division time multiplied by the degradation rate of protein i (t0*kdi), thus representing the total number of protein i destabilized over one cell division time. The eta and alpha terms are described in the section above describing the equations for the slow reactions. Note that the equation describing the temporal evolution of AraC includes the term "Aend." This term represents the endogenous expression of AraC in dH5alpha's. Our preliminary work suggests that this endogenous AraC has very little effect on our pBad promoter.<br />
<br />
[[Image:Ev_tot_mol_eqs.png]]<br />
<br />
====Fast reaction equilibrium equations====<br />
Compared to the slowly evolving reactions(2) described by the above equations (5), the fast reactions (1) can be considered to be in equilibrium. Thus the following equilibrium relations (6) hold.<br />
[[Image:Fast_eq_eqs.png]]<br />
<br />
====Equations modeling the presence of the inducers====<br />
The following equations (7) model the effect of the chemical inducers on the system. The model regards the proteins bound to the inducers as having increased dissociation rates (ki-#) as specified below. The reaction rate with the apostrophe denotes the original reaction rate in the absence of inducer inclusion. In these equations "I" denotes IPTG, "a" denotes arabinose, and "Tc" denotes tetracycline (or analogue aTc). The concept for and the form of these equations are based on equations described in the supplementary Information for "A bottom-up approach to gene regulation" by Guido et al. Much like the aforementioned paper by Isaacs et al., 'Bottom-up' has an extensive and brilliantly articulate derivation supplement. <br />
<br />
[[Image:inducer_eqs.png]]<br />
<br />
====Equations relating plasmid copy number to operator sites====<br />
The following equations (8) relate the plasmid copy number, "m", to the number of promoters with bound and unbound operators. As each plasmid contains 1 of each promoter, m is constant for all three promoter types. <br />
<br />
[[Image:cop_num_eqs.png]]<br />
<br />
====Simplified fast reaction equilibrium equations====<br />
The equilibrium equations governing the fast reactions (6) can be simplified by defining dimensionless equilibrium constants in the form of cij=kij/(k(-ij)*V0*A) in which A is Avogadro's number. The cL's are further simplified by defining the constant cL=c1L*c2L*c3L*c4L. Additionally by plugging in successive terms we were able to write the operator equations in terms of d0L. <br />
<br />
[[Image:Simp_fast_eq_eqs.png]]<br />
<br />
====Equations for the number of unbound operators====<br />
Combining equations (8) and the above simplified equilibrium equations(9), we are able to solve for the number of unbound operators d0i. <br />
<br />
[[Image:Unbound_eqs.png]]<br />
<br />
====Equations for the number of bound operators====<br />
By plugging the d0i's (10) back into the equations for bound operators (9), we can solve for these equations in terms of monomer concentrations.<br />
<br />
<br />
[[Image:bound_eqs.png]]<br />
<br />
====Equation manipulations to determine change in protein monomers per time====<br />
Next we can rewrite the equations describing the temporal evolution of the total number inhibitor molecules (5) in terms of equations (10) and (11). This yields the somewhat complex equations (12) which are solely in terms of monomer protein concentration. <br />
<br />
[[Image:dzdt_eqs.png]]<br />
<br />
====Equations for the change in protein monomers per time====<br />
By applying the simple mathematical manipulation (13) to equations (12) and the X derivative of equations (9) ((9) can be written in terms of X by plugging in equations (10) and (11)) we can solve for the following equations for the evolution of protein monomers as a function of time (14). I will spare you the dZi/dXi derivative equations of equation (9).<br />
<br />
[[Image:simp_mat_man_eq.png]]<br />
<br />
====Equations describing stochastic variation====<br />
<br />
====Equations for the evolution of fluorescent reporters====</div>Jlohmuelhttps://2006.igem.org/wiki/index.php/Brown:Tri-Stable_toggle_switchBrown:Tri-Stable toggle switch2006-11-01T06:42:55Z<p>Jlohmuel: /* Derivation of the Model Equations */</p>
<hr />
<div>{{Brown navigation bar}}<br />
=A tri-stable toggle switch=<br />
<br />
Pursuant to the the 1999 paper "Construction of a genetic toggle switch in Escherichia coli," by Timothy S. Gardner, Charles R. Cantor and James J. Collins, we wondered if the bi-stable toggle switch could be generalized to an n-stable switch. To that end, we conceived and began construction of the tri-stable toggle switch. The general idea is that when a selected promoter is activated, it represses the other two. As such each of the three states of our network are self-stabilizing.<br />
<br />
[[Image:Tristable_Toggle.jpg|left|500px|A general tri-stable toggle]]<br />
<br />
Our implementation uses the pLac, pTet, and pBad/AraC promoters and their respective inhibiting proteins. We chose these because of the relative ease with which each promoter is induced: by adding IPTG for the pLac-promoted region, tetracycline for the PTet-promoted region, and arabinose for the pBad/Ara genes. By characterizing each of the three pathways without following genes and terminators, we enable a tri-stable switching network of any three biobricks to be constructed with minimal cloning.<br />
<br />
[[Image:Registry_tristable.jpg|Our implementation]]<br />
<br />
<br><br />
Link to Tri-stable Switch on the registry:[http://partsregistry.org/Part:BBa_J24672]<br />
<br />
<br><br><br />
<br />
<br><br />
<br><br />
<br />
=Modeling the tri-stable toggle switch=<br />
While the tri-stable switch is seemingly simple in design, just like any other system it is subject to potentially comprising factors such as promoter leakiness and other stochastic fluctuations. In an attempt to predict the behavior of the tri-stable switch, we created a deterministic model of the system taking into account these factors. The model structure is based on that described in "Prediction and measurement of an autoregulatory genetic module" by Farren Isaacs, et al. This paper features an excellent supplement that takes you hand-in-hand through the derivation of model equations. I will attempt to emulate this derivation with our model below. It is important to note that while we have some preliminary results, our model is very much a work in progress. In order to model the system accurately, many of the fundamental constants governing the model will need to be determined experimentally. For now we have used some constants from the literature and estimated some based on similar the values of similar constants also from the literature. <br />
<br />
==Derivation of the Model Equations==<br />
[[Derivation of the Model Equations]]<br />
<br />
==Table of preliminary model constants==<br />
[[Table of preliminary model constants]]<br />
<br />
==Preliminary modeling results==<br />
<br />
The preliminary modeling results predict things about of the the tri-stable switch. First of all, the model confirms certain obvious points such as the a shift to the proper steady expression state in the presence of inducers. This prediction attests to model validity more than anything else. Secondly, the model predicts that the switch will reach a natural steady state with one promoter dominating in the absence of inducers. This makes intuitive sense as the individual promoters will most likely have different strengths. <br />
<br />
[[Image:no_inducer.png]]<br />
<br />
Next we tested the stability of the model by plugging in final steady-state induced levels of protein into an inducer-less system. The model predicts that the system will not remain stable and will instead revert back to the natural steady state for the inducer-less system. Even after changing several model parameters, the switch would not remain stable.<br />
<br />
<br />
[[Image:stability_test1.png]]<br />
<br />
As a test of the model, models of the three potential bi-stable switches (already proven to be successful experimentally) using the same components and parameters were tested. In accordance with the literature, each of these constructs maintained stable expression of the induced state following the removal of the inducer. In order to probe the instability of the tri-stable switch further, tri-stable switches in which all of the components were effectively the same were created (ex:for all LacI-like, TetR and AraC were modeled as forming tetramers and having one operator.) Each of these systems reached a steady state in which all of the proteins evolved identically and were present at the same levels in the absence of inducer. The same stability tests were then performed. All of the constructs remained stable. Thus our model suggests that the similarity of the repressor-expression systems dominates the success or failure of the tri-stable switch. Furthermore, the bi-stable switch is less sensitive to this factor. <br />
<br />
It should be noted that this is only a preliminary model. The values obtained from the literature will most like differ in our system. Through future experimentation we will elucidate these values and thus create a more accurate model. We will also conduct parameter scans aimed to provide further insight into the necessary system requirements for "tri-stability."<br />
<br />
<br />
<br />
<br />
Link to MATLAB code for tri-stable model [[Media:tristable1.txt]] <br />
<br />
Note: copy this code into MATLAB and save as an M-file<br />
<br />
--[[User:Jlohmuel|Jlohmuel]] 03:59, 28 October 2006 (EDT)<br />
<br />
==What we achieved==<br />
<br />
A successfully ligated pBad/arac construct in addition to partially completed ligations for the pLacI and the pTetR constructs were the results of our summer and semesters' work. The tri-stable toggle switch system was also modeled and proved to yield significant findings.<br />
<br />
==Challenges==<br />
<br />
Despite many cloning and ligation attempts, we were unable to create each component of the switch. More specifically, the pLacI and the pTetR circuits presented many ligation difficulties and proved to yield unsuccessful cloning. However, the pBAD/araC circuit was successfully constructed (yay!).<br />
<br />
==Conclusions==<br />
<br />
With the bigger picture in mind,this project yielded many successful outcomes.Firstly, we were able to successfully construct one of the three constructs necessary for a fully functional tri-stable toggle switch. Secondly, we were able to add more working parts to the registry, both a fully functioning pBad/AraC construct and many parts that were successfully ligated in the process of assembling the other two constructs. These parts will be used in the future by our team and perhaps many others as new ideas and projects develop. Also, it is necessary to mention our most recent finding from the modeling results for the tri-stable toggle switch. Interestingly enough, our model suggests that it is biologically unfeasible to create a switch with more than two components. If more than two components are introduced, such as three, the system is unable to maintain any one state and it will revert back to expressing its most dominant promoter construct. We find such results quite meaningful in demonstrating the biological limits of a system under study.<br />
<br />
==Future work==<br />
<br />
In the coming months, it is hoped that the failure of the ligation strategies for the pLacI and the pTetR constructs can be further investigated. We feel confident that we can continue to build from the one successful construction,the pBad/Arac construct and complete the other 2 constructs. To this end, it will be necessary to further troubleshoot the possible reasons why the cloning of the ligations were unsuccessful.</div>Jlohmuelhttps://2006.igem.org/wiki/index.php/Table_of_preliminary_model_constantsTable of preliminary model constants2006-11-01T06:41:36Z<p>Jlohmuel: </p>
<hr />
<div>====General Parameters====<br />
<br />
{| border=1 cellspacing=0 cellpadding=5<br />
<br />
!align="left" width="50" | Parameter<br />
!width="50" | Value<br />
!width="300" | Meaning<br />
<br />
|-<br />
|m<br />
|20 Copies<br />
|Plasmid Copy Number<br />
|-<br />
|v<br />
|1um^3<br />
|cell volume<br />
|-<br />
<br />
|}<br />
<br />
====Inhibitor Protein Parameters====<br />
{| border=1 cellspacing=0 cellpadding=5<br />
<br />
!align="left" width="50" | Parameter<br />
!width="50" | Value<br />
!width="300" | Meaning<br />
<br />
|-<br />
|alphaT1<br />
|.01<br />
|Relative transcription level for repressed (1 TetR dimer bound) pTetR promoter<br />
|-<br />
|alphaT2<br />
|.01<br />
|Relative transcription level for doubly repressed (2 TetR dimers bound) pTetR promoter<br />
|-<br />
|alphaL<br />
|.01<br />
|Relative transcription level for repressed (1 LacI tetramer bound) pLacI promoter<br />
|-<br />
|alphaA<br />
|.01<br />
|Relative transcription level for repressed (two AraC monomers bound) pBAD promoter<br />
|-<br />
|betaA<br />
|1000<br />
|maximum number of proteins produced from the pBAD promoter per cell division<br />
|-<br />
|betaL<br />
|1000<br />
|maximum number of proteins produced from the pLacI promoter per cell division<br />
|-<br />
|betaT<br />
|1000<br />
|maximum number of proteins produced from the pTetR promoter per cell division<br />
|-<br />
|etaLT<br />
|1<br />
|Relative rate of TetR translation from pLacI promoter<br />
|-<br />
|etaLA<br />
|1<br />
|Relative rate of AraC translation from pLacI promoter<br />
|-<br />
|etaAL<br />
|1<br />
|Relative rate of LacI translation from pBAD promoter<br />
|-<br />
|etaAT<br />
|1<br />
|Relative rate of TetR translation from pBAD promoter<br />
|-<br />
|etaTL<br />
|1<br />
|Relative rate of LacI translation from pTetR promoter<br />
|-<br />
|etaTA<br />
|1<br />
|Relative rate of AraC translation from pTetR promoter<br />
|-<br />
|c1L<br />
|.05<br />
|Equilibrium constant for LacI dimerization<br />
|-<br />
|c2L<br />
|.05<br />
|Equilibrium constant for LacI trimerization<br />
|-<br />
|c3L<br />
|.05<br />
|Equilibrium constant for LacI tetramerization<br />
|-<br />
|c4L<br />
|.33<br />
|Equilibrium constant for LacI tetramer binding to Lac operator<br />
|-<br />
|c1T<br />
|.05<br />
|Equilibrium constant for TetR dimerization<br />
|-<br />
|c2T<br />
|.33<br />
|Equilibrium constant for TetR dimer binding to first operator<br />
|-<br />
|c3T<br />
|.33<br />
|Equilibrium constant for TetR dimer binding to second operator<br />
|-<br />
|c1A<br />
|.05<br />
|Equilibrium constant for AraC monomer binding to first operator<br />
|-<br />
|c2A<br />
|.33<br />
|Equilibrium constant for AraC monomer binding to second operator<br />
|-<br />
|}<br />
<br />
<br />
<br />
{{Brown navigation bar}}</div>Jlohmuelhttps://2006.igem.org/wiki/index.php/Table_of_preliminary_model_constantsTable of preliminary model constants2006-11-01T06:41:04Z<p>Jlohmuel: </p>
<hr />
<div>====General Parameters====<br />
<br />
{| border=1 cellspacing=0 cellpadding=5<br />
<br />
!align="left" width="50" | Parameter<br />
!width="50" | Value<br />
!width="300" | Meaning<br />
<br />
|-<br />
|m<br />
|20 Copies<br />
|Plasmid Copy Number<br />
|-<br />
|v<br />
|1um^3<br />
|cell volume<br />
|-<br />
<br />
|}<br />
<br />
====Inhibitor Protein Parameters====<br />
{| border=1 cellspacing=0 cellpadding=5<br />
<br />
!align="left" width="50" | Parameter<br />
!width="50" | Value<br />
!width="300" | Meaning<br />
<br />
|-<br />
|alphaT1<br />
|.01<br />
|Relative transcription level for repressed (1 TetR dimer bound) pTetR promoter<br />
|-<br />
|alphaT2<br />
|.01<br />
|Relative transcription level for doubly repressed (2 TetR dimers bound) pTetR promoter<br />
|-<br />
|alphaL<br />
|.01<br />
|Relative transcription level for repressed (1 LacI tetramer bound) pLacI promoter<br />
|-<br />
|alphaA<br />
|.01<br />
|Relative transcription level for repressed (two AraC monomers bound) pBAD promoter<br />
|-<br />
|betaA<br />
|1000<br />
|maximum number of proteins produced from the pBAD promoter per cell division<br />
|-<br />
|betaL<br />
|1000<br />
|maximum number of proteins produced from the pLacI promoter per cell division<br />
|-<br />
|betaT<br />
|1000<br />
|maximum number of proteins produced from the pTetR promoter per cell division<br />
|-<br />
|etaLT<br />
|1<br />
|Relative rate of TetR translation from pLacI promoter<br />
|-<br />
|etaLA<br />
|1<br />
|Relative rate of AraC translation from pLacI promoter<br />
|-<br />
|etaAL<br />
|1<br />
|Relative rate of LacI translation from pBAD promoter<br />
|-<br />
|etaAT<br />
|1<br />
|Relative rate of TetR translation from pBAD promoter<br />
|-<br />
|etaTL<br />
|1<br />
|Relative rate of LacI translation from pTetR promoter<br />
|-<br />
|etaTA<br />
|1<br />
|Relative rate of AraC translation from pTetR promoter<br />
|-<br />
|c1L<br />
|.05<br />
|Equilibrium constant for LacI dimerization<br />
|-<br />
|c2L<br />
|.05<br />
|Equilibrium constant for LacI trimerization<br />
|-<br />
|c3L<br />
|.05<br />
|Equilibrium constant for LacI tetramerization<br />
|-<br />
|c4L<br />
|.33<br />
|Equilibrium constant for LacI tetramer binding to Lac operator<br />
|-<br />
|c1T<br />
|.05<br />
|Equilibrium constant for TetR dimerization<br />
|-<br />
|c2T<br />
|.33<br />
|Equilibrium constant for TetR dimer binding to first operator<br />
|-<br />
|c3T<br />
|.33<br />
|Equilibrium constant for TetR dimer binding to second operator<br />
|-<br />
|c1A<br />
|.05<br />
|Equilibrium constant for AraC monomer binding to first operator<br />
|-<br />
|c2A<br />
|.33<br />
|Equilibrium constant for AraC monomer binding to second operator<br />
|-<br />
|}<br />
<br />
{{Brown navigation bar}}</div>Jlohmuelhttps://2006.igem.org/wiki/index.php/Table_of_preliminary_model_constantsTable of preliminary model constants2006-11-01T06:40:46Z<p>Jlohmuel: /* General Parameters */</p>
<hr />
<div>====General Parameters====<br />
<br />
{| border=1 cellspacing=0 cellpadding=5<br />
<br />
!align="left" width="50" | Parameter<br />
!width="50" | Value<br />
!width="300" | Meaning<br />
<br />
|-<br />
|m<br />
|20 Copies<br />
|Plasmid Copy Number<br />
|-<br />
|v<br />
|1um^3<br />
|cell volume<br />
|-<br />
<br />
|}<br />
<br />
====Inhibitor Protein Parameters====<br />
{| border=1 cellspacing=0 cellpadding=5<br />
<br />
!align="left" width="50" | Parameter<br />
!width="50" | Value<br />
!width="300" | Meaning<br />
<br />
|-<br />
|alphaT1<br />
|.01<br />
|Relative transcription level for repressed (1 TetR dimer bound) pTetR promoter<br />
|-<br />
|alphaT2<br />
|.01<br />
|Relative transcription level for doubly repressed (2 TetR dimers bound) pTetR promoter<br />
|-<br />
|alphaL<br />
|.01<br />
|Relative transcription level for repressed (1 LacI tetramer bound) pLacI promoter<br />
|-<br />
|alphaA<br />
|.01<br />
|Relative transcription level for repressed (two AraC monomers bound) pBAD promoter<br />
|-<br />
|betaA<br />
|1000<br />
|maximum number of proteins produced from the pBAD promoter per cell division<br />
|-<br />
|betaL<br />
|1000<br />
|maximum number of proteins produced from the pLacI promoter per cell division<br />
|-<br />
|betaT<br />
|1000<br />
|maximum number of proteins produced from the pTetR promoter per cell division<br />
|-<br />
|etaLT<br />
|1<br />
|Relative rate of TetR translation from pLacI promoter<br />
|-<br />
|etaLA<br />
|1<br />
|Relative rate of AraC translation from pLacI promoter<br />
|-<br />
|etaAL<br />
|1<br />
|Relative rate of LacI translation from pBAD promoter<br />
|-<br />
|etaAT<br />
|1<br />
|Relative rate of TetR translation from pBAD promoter<br />
|-<br />
|etaTL<br />
|1<br />
|Relative rate of LacI translation from pTetR promoter<br />
|-<br />
|etaTA<br />
|1<br />
|Relative rate of AraC translation from pTetR promoter<br />
|-<br />
|c1L<br />
|.05<br />
|Equilibrium constant for LacI dimerization<br />
|-<br />
|c2L<br />
|.05<br />
|Equilibrium constant for LacI trimerization<br />
|-<br />
|c3L<br />
|.05<br />
|Equilibrium constant for LacI tetramerization<br />
|-<br />
|c4L<br />
|.33<br />
|Equilibrium constant for LacI tetramer binding to Lac operator<br />
|-<br />
|c1T<br />
|.05<br />
|Equilibrium constant for TetR dimerization<br />
|-<br />
|c2T<br />
|.33<br />
|Equilibrium constant for TetR dimer binding to first operator<br />
|-<br />
|c3T<br />
|.33<br />
|Equilibrium constant for TetR dimer binding to second operator<br />
|-<br />
|c1A<br />
|.05<br />
|Equilibrium constant for AraC monomer binding to first operator<br />
|-<br />
|c2A<br />
|.33<br />
|Equilibrium constant for AraC monomer binding to second operator<br />
|-<br />
|}</div>Jlohmuelhttps://2006.igem.org/wiki/index.php/Table_of_preliminary_model_constantsTable of preliminary model constants2006-11-01T06:40:23Z<p>Jlohmuel: /* General Parameters */</p>
<hr />
<div>====General Parameters====<br />
<br />
{| border=1 cellspacing=0 cellpadding=5<br />
<br />
!align="left" width="50" | Parameter<br />
!width="50" | Value<br />
!width="300" | Meaning<br />
<br />
|-<br />
|m<br />
|20 Copies<br />
|Plasmid Copy Number<br />
|-<br />
|v<br />
|1um^3<br />
|cell volume<br />
|-<br />
<br />
|}<br />
{{Brown navigation bar}}<br />
<br />
====Inhibitor Protein Parameters====<br />
{| border=1 cellspacing=0 cellpadding=5<br />
<br />
!align="left" width="50" | Parameter<br />
!width="50" | Value<br />
!width="300" | Meaning<br />
<br />
|-<br />
|alphaT1<br />
|.01<br />
|Relative transcription level for repressed (1 TetR dimer bound) pTetR promoter<br />
|-<br />
|alphaT2<br />
|.01<br />
|Relative transcription level for doubly repressed (2 TetR dimers bound) pTetR promoter<br />
|-<br />
|alphaL<br />
|.01<br />
|Relative transcription level for repressed (1 LacI tetramer bound) pLacI promoter<br />
|-<br />
|alphaA<br />
|.01<br />
|Relative transcription level for repressed (two AraC monomers bound) pBAD promoter<br />
|-<br />
|betaA<br />
|1000<br />
|maximum number of proteins produced from the pBAD promoter per cell division<br />
|-<br />
|betaL<br />
|1000<br />
|maximum number of proteins produced from the pLacI promoter per cell division<br />
|-<br />
|betaT<br />
|1000<br />
|maximum number of proteins produced from the pTetR promoter per cell division<br />
|-<br />
|etaLT<br />
|1<br />
|Relative rate of TetR translation from pLacI promoter<br />
|-<br />
|etaLA<br />
|1<br />
|Relative rate of AraC translation from pLacI promoter<br />
|-<br />
|etaAL<br />
|1<br />
|Relative rate of LacI translation from pBAD promoter<br />
|-<br />
|etaAT<br />
|1<br />
|Relative rate of TetR translation from pBAD promoter<br />
|-<br />
|etaTL<br />
|1<br />
|Relative rate of LacI translation from pTetR promoter<br />
|-<br />
|etaTA<br />
|1<br />
|Relative rate of AraC translation from pTetR promoter<br />
|-<br />
|c1L<br />
|.05<br />
|Equilibrium constant for LacI dimerization<br />
|-<br />
|c2L<br />
|.05<br />
|Equilibrium constant for LacI trimerization<br />
|-<br />
|c3L<br />
|.05<br />
|Equilibrium constant for LacI tetramerization<br />
|-<br />
|c4L<br />
|.33<br />
|Equilibrium constant for LacI tetramer binding to Lac operator<br />
|-<br />
|c1T<br />
|.05<br />
|Equilibrium constant for TetR dimerization<br />
|-<br />
|c2T<br />
|.33<br />
|Equilibrium constant for TetR dimer binding to first operator<br />
|-<br />
|c3T<br />
|.33<br />
|Equilibrium constant for TetR dimer binding to second operator<br />
|-<br />
|c1A<br />
|.05<br />
|Equilibrium constant for AraC monomer binding to first operator<br />
|-<br />
|c2A<br />
|.33<br />
|Equilibrium constant for AraC monomer binding to second operator<br />
|-<br />
|}</div>Jlohmuelhttps://2006.igem.org/wiki/index.php/Table_of_preliminary_model_constantsTable of preliminary model constants2006-11-01T06:39:54Z<p>Jlohmuel: /* General Parameters */</p>
<hr />
<div>{{Brown navigation bar}}<br />
====General Parameters====<br />
<br />
{| border=1 cellspacing=0 cellpadding=5<br />
<br />
!align="left" width="50" | Parameter<br />
!width="50" | Value<br />
!width="300" | Meaning<br />
<br />
|-<br />
|m<br />
|20 Copies<br />
|Plasmid Copy Number<br />
|-<br />
|v<br />
|1um^3<br />
|cell volume<br />
|-<br />
<br />
|}<br />
<br />
====Inhibitor Protein Parameters====<br />
{| border=1 cellspacing=0 cellpadding=5<br />
<br />
!align="left" width="50" | Parameter<br />
!width="50" | Value<br />
!width="300" | Meaning<br />
<br />
|-<br />
|alphaT1<br />
|.01<br />
|Relative transcription level for repressed (1 TetR dimer bound) pTetR promoter<br />
|-<br />
|alphaT2<br />
|.01<br />
|Relative transcription level for doubly repressed (2 TetR dimers bound) pTetR promoter<br />
|-<br />
|alphaL<br />
|.01<br />
|Relative transcription level for repressed (1 LacI tetramer bound) pLacI promoter<br />
|-<br />
|alphaA<br />
|.01<br />
|Relative transcription level for repressed (two AraC monomers bound) pBAD promoter<br />
|-<br />
|betaA<br />
|1000<br />
|maximum number of proteins produced from the pBAD promoter per cell division<br />
|-<br />
|betaL<br />
|1000<br />
|maximum number of proteins produced from the pLacI promoter per cell division<br />
|-<br />
|betaT<br />
|1000<br />
|maximum number of proteins produced from the pTetR promoter per cell division<br />
|-<br />
|etaLT<br />
|1<br />
|Relative rate of TetR translation from pLacI promoter<br />
|-<br />
|etaLA<br />
|1<br />
|Relative rate of AraC translation from pLacI promoter<br />
|-<br />
|etaAL<br />
|1<br />
|Relative rate of LacI translation from pBAD promoter<br />
|-<br />
|etaAT<br />
|1<br />
|Relative rate of TetR translation from pBAD promoter<br />
|-<br />
|etaTL<br />
|1<br />
|Relative rate of LacI translation from pTetR promoter<br />
|-<br />
|etaTA<br />
|1<br />
|Relative rate of AraC translation from pTetR promoter<br />
|-<br />
|c1L<br />
|.05<br />
|Equilibrium constant for LacI dimerization<br />
|-<br />
|c2L<br />
|.05<br />
|Equilibrium constant for LacI trimerization<br />
|-<br />
|c3L<br />
|.05<br />
|Equilibrium constant for LacI tetramerization<br />
|-<br />
|c4L<br />
|.33<br />
|Equilibrium constant for LacI tetramer binding to Lac operator<br />
|-<br />
|c1T<br />
|.05<br />
|Equilibrium constant for TetR dimerization<br />
|-<br />
|c2T<br />
|.33<br />
|Equilibrium constant for TetR dimer binding to first operator<br />
|-<br />
|c3T<br />
|.33<br />
|Equilibrium constant for TetR dimer binding to second operator<br />
|-<br />
|c1A<br />
|.05<br />
|Equilibrium constant for AraC monomer binding to first operator<br />
|-<br />
|c2A<br />
|.33<br />
|Equilibrium constant for AraC monomer binding to second operator<br />
|-<br />
|}</div>Jlohmuelhttps://2006.igem.org/wiki/index.php/Table_of_preliminary_model_constantsTable of preliminary model constants2006-11-01T06:38:22Z<p>Jlohmuel: /* General Parameters */</p>
<hr />
<div>====General Parameters====<br />
{| border=1 cellspacing=0 cellpadding=5<br />
<br />
!align="left" width="50" | Parameter<br />
!width="50" | Value<br />
!width="300" | Meaning<br />
<br />
|-<br />
|m<br />
|20 Copies<br />
|Plasmid Copy Number<br />
|-<br />
|v<br />
|1um^3<br />
|cell volume<br />
|-<br />
<br />
|}<br />
<br />
====Inhibitor Protein Parameters====<br />
{| border=1 cellspacing=0 cellpadding=5<br />
<br />
!align="left" width="50" | Parameter<br />
!width="50" | Value<br />
!width="300" | Meaning<br />
<br />
|-<br />
|alphaT1<br />
|.01<br />
|Relative transcription level for repressed (1 TetR dimer bound) pTetR promoter<br />
|-<br />
|alphaT2<br />
|.01<br />
|Relative transcription level for doubly repressed (2 TetR dimers bound) pTetR promoter<br />
|-<br />
|alphaL<br />
|.01<br />
|Relative transcription level for repressed (1 LacI tetramer bound) pLacI promoter<br />
|-<br />
|alphaA<br />
|.01<br />
|Relative transcription level for repressed (two AraC monomers bound) pBAD promoter<br />
|-<br />
|betaA<br />
|1000<br />
|maximum number of proteins produced from the pBAD promoter per cell division<br />
|-<br />
|betaL<br />
|1000<br />
|maximum number of proteins produced from the pLacI promoter per cell division<br />
|-<br />
|betaT<br />
|1000<br />
|maximum number of proteins produced from the pTetR promoter per cell division<br />
|-<br />
|etaLT<br />
|1<br />
|Relative rate of TetR translation from pLacI promoter<br />
|-<br />
|etaLA<br />
|1<br />
|Relative rate of AraC translation from pLacI promoter<br />
|-<br />
|etaAL<br />
|1<br />
|Relative rate of LacI translation from pBAD promoter<br />
|-<br />
|etaAT<br />
|1<br />
|Relative rate of TetR translation from pBAD promoter<br />
|-<br />
|etaTL<br />
|1<br />
|Relative rate of LacI translation from pTetR promoter<br />
|-<br />
|etaTA<br />
|1<br />
|Relative rate of AraC translation from pTetR promoter<br />
|-<br />
|c1L<br />
|.05<br />
|Equilibrium constant for LacI dimerization<br />
|-<br />
|c2L<br />
|.05<br />
|Equilibrium constant for LacI trimerization<br />
|-<br />
|c3L<br />
|.05<br />
|Equilibrium constant for LacI tetramerization<br />
|-<br />
|c4L<br />
|.33<br />
|Equilibrium constant for LacI tetramer binding to Lac operator<br />
|-<br />
|c1T<br />
|.05<br />
|Equilibrium constant for TetR dimerization<br />
|-<br />
|c2T<br />
|.33<br />
|Equilibrium constant for TetR dimer binding to first operator<br />
|-<br />
|c3T<br />
|.33<br />
|Equilibrium constant for TetR dimer binding to second operator<br />
|-<br />
|c1A<br />
|.05<br />
|Equilibrium constant for AraC monomer binding to first operator<br />
|-<br />
|c2A<br />
|.33<br />
|Equilibrium constant for AraC monomer binding to second operator<br />
|-<br />
|}</div>Jlohmuelhttps://2006.igem.org/wiki/index.php/Table_of_preliminary_model_constantsTable of preliminary model constants2006-11-01T06:36:51Z<p>Jlohmuel: /* Inhibitor Protein Parameters */</p>
<hr />
<div>====General Parameters====<br />
{| border=1 cellspacing=0 cellpadding=5<br />
<br />
!align="left" width="50" | Parameter<br />
!width="50" | Value<br />
!width="300" | Meaning<br />
<br />
|-<br />
|m<br />
|20 Copies<br />
|Plasmid Copy Number<br />
|-<br />
|}<br />
<br />
====Inhibitor Protein Parameters====<br />
{| border=1 cellspacing=0 cellpadding=5<br />
<br />
!align="left" width="50" | Parameter<br />
!width="50" | Value<br />
!width="300" | Meaning<br />
<br />
|-<br />
|alphaT1<br />
|.01<br />
|Relative transcription level for repressed (1 TetR dimer bound) pTetR promoter<br />
|-<br />
|alphaT2<br />
|.01<br />
|Relative transcription level for doubly repressed (2 TetR dimers bound) pTetR promoter<br />
|-<br />
|alphaL<br />
|.01<br />
|Relative transcription level for repressed (1 LacI tetramer bound) pLacI promoter<br />
|-<br />
|alphaA<br />
|.01<br />
|Relative transcription level for repressed (two AraC monomers bound) pBAD promoter<br />
|-<br />
|betaA<br />
|1000<br />
|maximum number of proteins produced from the pBAD promoter per cell division<br />
|-<br />
|betaL<br />
|1000<br />
|maximum number of proteins produced from the pLacI promoter per cell division<br />
|-<br />
|betaT<br />
|1000<br />
|maximum number of proteins produced from the pTetR promoter per cell division<br />
|-<br />
|etaLT<br />
|1<br />
|Relative rate of TetR translation from pLacI promoter<br />
|-<br />
|etaLA<br />
|1<br />
|Relative rate of AraC translation from pLacI promoter<br />
|-<br />
|etaAL<br />
|1<br />
|Relative rate of LacI translation from pBAD promoter<br />
|-<br />
|etaAT<br />
|1<br />
|Relative rate of TetR translation from pBAD promoter<br />
|-<br />
|etaTL<br />
|1<br />
|Relative rate of LacI translation from pTetR promoter<br />
|-<br />
|etaTA<br />
|1<br />
|Relative rate of AraC translation from pTetR promoter<br />
|-<br />
|c1L<br />
|.05<br />
|Equilibrium constant for LacI dimerization<br />
|-<br />
|c2L<br />
|.05<br />
|Equilibrium constant for LacI trimerization<br />
|-<br />
|c3L<br />
|.05<br />
|Equilibrium constant for LacI tetramerization<br />
|-<br />
|c4L<br />
|.33<br />
|Equilibrium constant for LacI tetramer binding to Lac operator<br />
|-<br />
|c1T<br />
|.05<br />
|Equilibrium constant for TetR dimerization<br />
|-<br />
|c2T<br />
|.33<br />
|Equilibrium constant for TetR dimer binding to first operator<br />
|-<br />
|c3T<br />
|.33<br />
|Equilibrium constant for TetR dimer binding to second operator<br />
|-<br />
|c1A<br />
|.05<br />
|Equilibrium constant for AraC monomer binding to first operator<br />
|-<br />
|c2A<br />
|.33<br />
|Equilibrium constant for AraC monomer binding to second operator<br />
|-<br />
|}</div>Jlohmuelhttps://2006.igem.org/wiki/index.php/Table_of_preliminary_model_constantsTable of preliminary model constants2006-11-01T06:32:16Z<p>Jlohmuel: </p>
<hr />
<div>====General Parameters====<br />
{| border=1 cellspacing=0 cellpadding=5<br />
<br />
!align="left" width="50" | Parameter<br />
!width="50" | Value<br />
!width="300" | Meaning<br />
<br />
|-<br />
|m<br />
|20 Copies<br />
|Plasmid Copy Number<br />
|-<br />
|}<br />
<br />
====Inhibitor Protein Parameters====<br />
{| border=1 cellspacing=0 cellpadding=5<br />
<br />
!align="left" width="50" | Parameter<br />
!width="50" | Value<br />
!width="300" | Meaning<br />
<br />
|-<br />
|alphaT1<br />
|.01<br />
|Relative transcription level for repressed (1 TetR dimer bound) pTetR promoter<br />
|-<br />
|alphaT2<br />
|.01<br />
|Relative transcription level for doubly repressed (2 TetR dimers bound) pTetR promoter<br />
|-<br />
|alphaL<br />
|.01<br />
|Relative transcription level for repressed (1 LacI tetramer bound) pLacI promoter<br />
|-<br />
|alphaA<br />
|.01<br />
|Relative transcription level for repressed (two AraC monomers bound) pBAD promoter<br />
|-<br />
|betaA<br />
|1000<br />
|maximum number of proteins produced from the pBAD promoter per cell division<br />
|-<br />
|betaL<br />
|1000<br />
|maximum number of proteins produced from the pLacI promoter per cell division<br />
|-<br />
|betaT<br />
|1000<br />
|maximum number of proteins produced from the pTetR promoter per cell division<br />
|-<br />
|etaLT<br />
|1<br />
|Relative rate of TetR translation from pLacI promoter<br />
|-<br />
|etaLA<br />
|1<br />
|Relative rate of AraC translation from pLacI promoter<br />
|-<br />
|etaAL<br />
|1<br />
|Relative rate of LacI translation from pBAD promoter<br />
|-<br />
|etaAT<br />
|1<br />
|Relative rate of TetR translation from pBAD promoter<br />
|-<br />
|etaTL<br />
|1<br />
|Relative rate of LacI translation from pTetR promoter<br />
|-<br />
|etaTA<br />
|1<br />
|Relative rate of AraC translation from pTetR promoter<br />
|-<br />
|c1L<br />
|.05<br />
|Equilibrium constant for LacI dimerization<br />
|-<br />
|c2L<br />
|.05<br />
|Equilibrium constant for LacI trimerization<br />
|-<br />
|c3L<br />
|.05<br />
|Equilibrium constant for LacI tetramerization<br />
|-<br />
|c4L<br />
|.33<br />
|Equilibrium constant for LacI binding<br />
|-<br />
|}</div>Jlohmuelhttps://2006.igem.org/wiki/index.php/Brown:Tri-Stable_toggle_switchBrown:Tri-Stable toggle switch2006-11-01T06:32:06Z<p>Jlohmuel: /* Table of preliminary model constants */</p>
<hr />
<div>{{Brown navigation bar}}<br />
=A tri-stable toggle switch=<br />
<br />
Pursuant to the the 1999 paper "Construction of a genetic toggle switch in Escherichia coli," by Timothy S. Gardner, Charles R. Cantor and James J. Collins, we wondered if the bi-stable toggle switch could be generalized to an n-stable switch. To that end, we conceived and began construction of the tri-stable toggle switch. The general idea is that when a selected promoter is activated, it represses the other two. As such each of the three states of our network are self-stabilizing.<br />
<br />
[[Image:Tristable_Toggle.jpg|left|500px|A general tri-stable toggle]]<br />
<br />
Our implementation uses the pLac, pTet, and pBad/AraC promoters and their respective inhibiting proteins. We chose these because of the relative ease with which each promoter is induced: by adding IPTG for the pLac-promoted region, tetracycline for the PTet-promoted region, and arabinose for the pBad/Ara genes. By characterizing each of the three pathways without following genes and terminators, we enable a tri-stable switching network of any three biobricks to be constructed with minimal cloning.<br />
<br />
[[Image:Registry_tristable.jpg|Our implementation]]<br />
<br />
<br><br />
Link to Tri-stable Switch on the registry:[http://partsregistry.org/Part:BBa_J24672]<br />
<br />
<br><br><br />
<br />
<br><br />
<br><br />
<br />
=Modeling the tri-stable toggle switch=<br />
While the tri-stable switch is seemingly simple in design, just like any other system it is subject to potentially comprising factors such as promoter leakiness and other stochastic fluctuations. In an attempt to predict the behavior of the tri-stable switch, we created a deterministic model of the system taking into account these factors. The model structure is based on that described in "Prediction and measurement of an autoregulatory genetic module" by Farren Isaacs, et al. This paper features an excellent supplement that takes you hand-in-hand through the derivation of model equations. I will attempt to emulate this derivation with our model below. It is important to note that while we have some preliminary results, our model is very much a work in progress. In order to model the system accurately, many of the fundamental constants governing the model will need to be determined experimentally. For now we have used some constants from the literature and estimated some based on similar the values of similar constants also from the literature. <br />
<br />
==Derivation of the Model Equations==<br />
The chemical reactions describing the tri-stable switch can be divided into the two categories of fast and slow reactions. Fast reactions such as dimer formation and promoter-binding occur in the scale of seconds and are therefore modeled to be in equilibrium. Conversely, slow reactions including the likes of transcription, translation, and protein degradation occur on the scale of minutes and are thus modeled to be evolving with time. <br />
<br />
====Fast reversible reaction equations====<br />
The following equations (1) describe the fast reactions. The characters L,T,A denote molecules of LacI, TetR, and, AraC, respectively while the subscripts denote whether the molecule is a monomer (blank), dimer(2), tri-mer(3), etc. The k's denote reaction rates.<br />
<br />
[[Image:Fast_rxn_eqs1.png]]<br />
<br />
Note that the equations include volume explicitly. Cell volume is modeled in this way as it is a slowly evolving function of time.<br />
<br />
====Slow irreversible reaction equations====<br />
The following equations (2) describe the slow irreversible reactions of transcription and translation (both taken into account with the reaction rate kti) and protein degradation (reaction constant kdi). The coefficients eta-ij take into account the relative translation rates of proteins from the same transcripts. For eta-ij, "i" represents the promoter responsible for producing the molecule and "j" represents the molecule being translated [ex: eta-LA corresponds to the relative rate of AraC production from the trascript produced by the LacI promoter]. To establish a convention, eta's are relative to the translation rate of the first gene on a particular transcript. Thus eta for the first gene on the mRNA transcript = 1. The alpha-i coefficients represent relative transcription rates. In this case, "i" denotes the promoter from which the mRNA is transcribed.<br />
<br />
[[Image:slow_rxn_eqs.png]]<br />
====Equations governing cell volume====<br />
The following two equations (3) describing cellular growth and division are taken directly from the aforementioned paper by Isaacs et al. The first equation describes the volume increase from the time immediately following cell division to the time immediately before it. In this equation, V0 denotes the volume of the cell at the beginning of growth and T0 denotes the time of cell division. In our model at times T=q*t0 in which q is an integer, we have volume V and protein concentration n halve - thus modeling volume division and the resulting protein redistribution. The second equation describes the dimensionless equation in which t is measured in terms of fcell-division time and the cell volume changes between 1 and 2.<br />
<br />
[[Image:vol_eq.png]]<br />
====Equations for the total number of molecules====<br />
The following equations (4) describe the total number of inhibitor protein molecules zi. d#i represents the number, "#", of molecules bound to the "i" promoter. (ex: all of the LacI tetramers bound to pLacI promoters are accounted for by the 4d1L term.)<br />
[[Image:Tot_mol_eq.png]]<br />
<br />
====Equations describing the temporal evolution of the total number inhibitor molecules====<br />
The following equations (5) describe the temporal evolution of the total number of inhibitor molecules. Beta-i = the cell division time multiplied by the combined transcription and translation rates from a given promoter "i" (t0*kti), thus representing the total number of protein molecules maximally produced by a given promoter. Similarly, Gamma-i = the cell division time multiplied by the degradation rate of protein i (t0*kdi), thus representing the total number of protein i destabilized over one cell division time. The eta and alpha terms are described in the section above describing the equations for the slow reactions. Note that the equation describing the temporal evolution of AraC includes the term "Aend." This term represents the endogenous expression of AraC in dH5alpha's. Our preliminary work suggests that this endogenous AraC has very little effect on our pBad promoter.<br />
<br />
[[Image:Ev_tot_mol_eqs.png]]<br />
<br />
====Fast reaction equilibrium equations====<br />
Compared to the slowly evolving reactions(2) described by the above equations (5), the fast reactions (1) can be considered to be in equilibrium. Thus the following equilibrium relations (6) hold.<br />
[[Image:Fast_eq_eqs.png]]<br />
<br />
====Equations modeling the presence of the inducers====<br />
The following equations (7) model the effect of the chemical inducers on the system. The model regards the proteins bound to the inducers as having increased dissociation rates (ki-#) as specified below. The reaction rate with the apostrophe denotes the original reaction rate in the absence of inducer inclusion. In these equations "I" denotes IPTG, "a" denotes arabinose, and "Tc" denotes tetracycline (or analogue aTc). The concept for and the form of these equations are based on equations described in the supplementary Information for "A bottom-up approach to gene regulation" by Guido et al. Much like the aforementioned paper by Isaacs et al., 'Bottom-up' has an extensive and brilliantly articulate derivation supplement. <br />
<br />
[[Image:inducer_eqs.png]]<br />
<br />
====Equations relating plasmid copy number to operator sites====<br />
The following equations (8) relate the plasmid copy number, "m", to the number of promoters with bound and unbound operators. As each plasmid contains 1 of each promoter, m is constant for all three promoter types. <br />
<br />
[[Image:cop_num_eqs.png]]<br />
<br />
====Simplified fast reaction equilibrium equations====<br />
The equilibrium equations governing the fast reactions (6) can be simplified by defining dimensionless equilibrium constants in the form of cij=kij/(k(-ij)*V0*A) in which A is Avogadro's number. The cL's are further simplified by defining the constant cL=c1L*c2L*c3L*c4L. Additionally by plugging in successive terms we were able to write the operator equations in terms of d0L. <br />
<br />
[[Image:Simp_fast_eq_eqs.png]]<br />
<br />
====Equations for the number of unbound operators====<br />
Combining equations (8) and the above simplified equilibrium equations(9), we are able to solve for the number of unbound operators d0i. <br />
<br />
[[Image:Unbound_eqs.png]]<br />
<br />
====Equations for the number of bound operators====<br />
By plugging the d0i's (10) back into the equations for bound operators (9), we can solve for these equations in terms of monomer concentrations.<br />
<br />
<br />
[[Image:bound_eqs.png]]<br />
<br />
====Equation manipulations to determine change in protein monomers per time====<br />
Next we can rewrite the equations describing the temporal evolution of the total number inhibitor molecules (5) in terms of equations (10) and (11). This yields the somewhat complex equations (12) which are solely in terms of monomer protein concentration. <br />
<br />
[[Image:dzdt_eqs.png]]<br />
<br />
====Equations for the change in protein monomers per time====<br />
By applying the simple mathematical manipulation (13) to equations (12) and the X derivative of equations (9) ((9) can be written in terms of X by plugging in equations (10) and (11)) we can solve for the following equations for the evolution of protein monomers as a function of time (14). I will spare you the dZi/dXi derivative equations of equation (9).<br />
<br />
[[Image:simp_mat_man_eq.png]]<br />
<br />
====Equations describing stochastic variation====<br />
<br />
====Equations for the evolution of fluorescent reporters====<br />
<br />
==Table of preliminary model constants==<br />
[[Table of preliminary model constants]]<br />
<br />
==Preliminary modeling results==<br />
<br />
The preliminary modeling results predict things about of the the tri-stable switch. First of all, the model confirms certain obvious points such as the a shift to the proper steady expression state in the presence of inducers. This prediction attests to model validity more than anything else. Secondly, the model predicts that the switch will reach a natural steady state with one promoter dominating in the absence of inducers. This makes intuitive sense as the individual promoters will most likely have different strengths. <br />
<br />
[[Image:no_inducer.png]]<br />
<br />
Next we tested the stability of the model by plugging in final steady-state induced levels of protein into an inducer-less system. The model predicts that the system will not remain stable and will instead revert back to the natural steady state for the inducer-less system. Even after changing several model parameters, the switch would not remain stable.<br />
<br />
<br />
[[Image:stability_test1.png]]<br />
<br />
As a test of the model, models of the three potential bi-stable switches (already proven to be successful experimentally) using the same components and parameters were tested. In accordance with the literature, each of these constructs maintained stable expression of the induced state following the removal of the inducer. In order to probe the instability of the tri-stable switch further, tri-stable switches in which all of the components were effectively the same were created (ex:for all LacI-like, TetR and AraC were modeled as forming tetramers and having one operator.) Each of these systems reached a steady state in which all of the proteins evolved identically and were present at the same levels in the absence of inducer. The same stability tests were then performed. All of the constructs remained stable. Thus our model suggests that the similarity of the repressor-expression systems dominates the success or failure of the tri-stable switch. Furthermore, the bi-stable switch is less sensitive to this factor. <br />
<br />
It should be noted that this is only a preliminary model. The values obtained from the literature will most like differ in our system. Through future experimentation we will elucidate these values and thus create a more accurate model. We will also conduct parameter scans aimed to provide further insight into the necessary system requirements for "tri-stability."<br />
<br />
<br />
<br />
<br />
Link to MATLAB code for tri-stable model [[Media:tristable1.txt]] <br />
<br />
Note: copy this code into MATLAB and save as an M-file<br />
<br />
--[[User:Jlohmuel|Jlohmuel]] 03:59, 28 October 2006 (EDT)<br />
<br />
==What we achieved==<br />
<br />
A successfully ligated pBad/arac construct in addition to partially completed ligations for the pLacI and the pTetR constructs were the results of our summer and semesters' work. The tri-stable toggle switch system was also modeled and proved to yield significant findings.<br />
<br />
==Challenges==<br />
<br />
Despite many cloning and ligation attempts, we were unable to create each component of the switch. More specifically, the pLacI and the pTetR circuits presented many ligation difficulties and proved to yield unsuccessful cloning. However, the pBAD/araC circuit was successfully constructed (yay!).<br />
<br />
==Conclusions==<br />
<br />
With the bigger picture in mind,this project yielded many successful outcomes.Firstly, we were able to successfully construct one of the three constructs necessary for a fully functional tri-stable toggle switch. Secondly, we were able to add more working parts to the registry, both a fully functioning pBad/AraC construct and many parts that were successfully ligated in the process of assembling the other two constructs. These parts will be used in the future by our team and perhaps many others as new ideas and projects develop. Also, it is necessary to mention our most recent finding from the modeling results for the tri-stable toggle switch. Interestingly enough, our model suggests that it is biologically unfeasible to create a switch with more than two components. If more than two components are introduced, such as three, the system is unable to maintain any one state and it will revert back to expressing its most dominant promoter construct. We find such results quite meaningful in demonstrating the biological limits of a system under study.<br />
<br />
==Future work==<br />
<br />
In the coming months, it is hoped that the failure of the ligation strategies for the pLacI and the pTetR constructs can be further investigated. We feel confident that we can continue to build from the one successful construction,the pBad/Arac construct and complete the other 2 constructs. To this end, it will be necessary to further troubleshoot the possible reasons why the cloning of the ligations were unsuccessful.</div>Jlohmuelhttps://2006.igem.org/wiki/index.php/Brown:Tri-Stable_toggle_switchBrown:Tri-Stable toggle switch2006-11-01T06:30:45Z<p>Jlohmuel: /* Preliminary modeling results */</p>
<hr />
<div>{{Brown navigation bar}}<br />
=A tri-stable toggle switch=<br />
<br />
Pursuant to the the 1999 paper "Construction of a genetic toggle switch in Escherichia coli," by Timothy S. Gardner, Charles R. Cantor and James J. Collins, we wondered if the bi-stable toggle switch could be generalized to an n-stable switch. To that end, we conceived and began construction of the tri-stable toggle switch. The general idea is that when a selected promoter is activated, it represses the other two. As such each of the three states of our network are self-stabilizing.<br />
<br />
[[Image:Tristable_Toggle.jpg|left|500px|A general tri-stable toggle]]<br />
<br />
Our implementation uses the pLac, pTet, and pBad/AraC promoters and their respective inhibiting proteins. We chose these because of the relative ease with which each promoter is induced: by adding IPTG for the pLac-promoted region, tetracycline for the PTet-promoted region, and arabinose for the pBad/Ara genes. By characterizing each of the three pathways without following genes and terminators, we enable a tri-stable switching network of any three biobricks to be constructed with minimal cloning.<br />
<br />
[[Image:Registry_tristable.jpg|Our implementation]]<br />
<br />
<br><br />
Link to Tri-stable Switch on the registry:[http://partsregistry.org/Part:BBa_J24672]<br />
<br />
<br><br><br />
<br />
<br><br />
<br><br />
<br />
=Modeling the tri-stable toggle switch=<br />
While the tri-stable switch is seemingly simple in design, just like any other system it is subject to potentially comprising factors such as promoter leakiness and other stochastic fluctuations. In an attempt to predict the behavior of the tri-stable switch, we created a deterministic model of the system taking into account these factors. The model structure is based on that described in "Prediction and measurement of an autoregulatory genetic module" by Farren Isaacs, et al. This paper features an excellent supplement that takes you hand-in-hand through the derivation of model equations. I will attempt to emulate this derivation with our model below. It is important to note that while we have some preliminary results, our model is very much a work in progress. In order to model the system accurately, many of the fundamental constants governing the model will need to be determined experimentally. For now we have used some constants from the literature and estimated some based on similar the values of similar constants also from the literature. <br />
<br />
==Derivation of the Model Equations==<br />
The chemical reactions describing the tri-stable switch can be divided into the two categories of fast and slow reactions. Fast reactions such as dimer formation and promoter-binding occur in the scale of seconds and are therefore modeled to be in equilibrium. Conversely, slow reactions including the likes of transcription, translation, and protein degradation occur on the scale of minutes and are thus modeled to be evolving with time. <br />
<br />
====Fast reversible reaction equations====<br />
The following equations (1) describe the fast reactions. The characters L,T,A denote molecules of LacI, TetR, and, AraC, respectively while the subscripts denote whether the molecule is a monomer (blank), dimer(2), tri-mer(3), etc. The k's denote reaction rates.<br />
<br />
[[Image:Fast_rxn_eqs1.png]]<br />
<br />
Note that the equations include volume explicitly. Cell volume is modeled in this way as it is a slowly evolving function of time.<br />
<br />
====Slow irreversible reaction equations====<br />
The following equations (2) describe the slow irreversible reactions of transcription and translation (both taken into account with the reaction rate kti) and protein degradation (reaction constant kdi). The coefficients eta-ij take into account the relative translation rates of proteins from the same transcripts. For eta-ij, "i" represents the promoter responsible for producing the molecule and "j" represents the molecule being translated [ex: eta-LA corresponds to the relative rate of AraC production from the trascript produced by the LacI promoter]. To establish a convention, eta's are relative to the translation rate of the first gene on a particular transcript. Thus eta for the first gene on the mRNA transcript = 1. The alpha-i coefficients represent relative transcription rates. In this case, "i" denotes the promoter from which the mRNA is transcribed.<br />
<br />
[[Image:slow_rxn_eqs.png]]<br />
====Equations governing cell volume====<br />
The following two equations (3) describing cellular growth and division are taken directly from the aforementioned paper by Isaacs et al. The first equation describes the volume increase from the time immediately following cell division to the time immediately before it. In this equation, V0 denotes the volume of the cell at the beginning of growth and T0 denotes the time of cell division. In our model at times T=q*t0 in which q is an integer, we have volume V and protein concentration n halve - thus modeling volume division and the resulting protein redistribution. The second equation describes the dimensionless equation in which t is measured in terms of fcell-division time and the cell volume changes between 1 and 2.<br />
<br />
[[Image:vol_eq.png]]<br />
====Equations for the total number of molecules====<br />
The following equations (4) describe the total number of inhibitor protein molecules zi. d#i represents the number, "#", of molecules bound to the "i" promoter. (ex: all of the LacI tetramers bound to pLacI promoters are accounted for by the 4d1L term.)<br />
[[Image:Tot_mol_eq.png]]<br />
<br />
====Equations describing the temporal evolution of the total number inhibitor molecules====<br />
The following equations (5) describe the temporal evolution of the total number of inhibitor molecules. Beta-i = the cell division time multiplied by the combined transcription and translation rates from a given promoter "i" (t0*kti), thus representing the total number of protein molecules maximally produced by a given promoter. Similarly, Gamma-i = the cell division time multiplied by the degradation rate of protein i (t0*kdi), thus representing the total number of protein i destabilized over one cell division time. The eta and alpha terms are described in the section above describing the equations for the slow reactions. Note that the equation describing the temporal evolution of AraC includes the term "Aend." This term represents the endogenous expression of AraC in dH5alpha's. Our preliminary work suggests that this endogenous AraC has very little effect on our pBad promoter.<br />
<br />
[[Image:Ev_tot_mol_eqs.png]]<br />
<br />
====Fast reaction equilibrium equations====<br />
Compared to the slowly evolving reactions(2) described by the above equations (5), the fast reactions (1) can be considered to be in equilibrium. Thus the following equilibrium relations (6) hold.<br />
[[Image:Fast_eq_eqs.png]]<br />
<br />
====Equations modeling the presence of the inducers====<br />
The following equations (7) model the effect of the chemical inducers on the system. The model regards the proteins bound to the inducers as having increased dissociation rates (ki-#) as specified below. The reaction rate with the apostrophe denotes the original reaction rate in the absence of inducer inclusion. In these equations "I" denotes IPTG, "a" denotes arabinose, and "Tc" denotes tetracycline (or analogue aTc). The concept for and the form of these equations are based on equations described in the supplementary Information for "A bottom-up approach to gene regulation" by Guido et al. Much like the aforementioned paper by Isaacs et al., 'Bottom-up' has an extensive and brilliantly articulate derivation supplement. <br />
<br />
[[Image:inducer_eqs.png]]<br />
<br />
====Equations relating plasmid copy number to operator sites====<br />
The following equations (8) relate the plasmid copy number, "m", to the number of promoters with bound and unbound operators. As each plasmid contains 1 of each promoter, m is constant for all three promoter types. <br />
<br />
[[Image:cop_num_eqs.png]]<br />
<br />
====Simplified fast reaction equilibrium equations====<br />
The equilibrium equations governing the fast reactions (6) can be simplified by defining dimensionless equilibrium constants in the form of cij=kij/(k(-ij)*V0*A) in which A is Avogadro's number. The cL's are further simplified by defining the constant cL=c1L*c2L*c3L*c4L. Additionally by plugging in successive terms we were able to write the operator equations in terms of d0L. <br />
<br />
[[Image:Simp_fast_eq_eqs.png]]<br />
<br />
====Equations for the number of unbound operators====<br />
Combining equations (8) and the above simplified equilibrium equations(9), we are able to solve for the number of unbound operators d0i. <br />
<br />
[[Image:Unbound_eqs.png]]<br />
<br />
====Equations for the number of bound operators====<br />
By plugging the d0i's (10) back into the equations for bound operators (9), we can solve for these equations in terms of monomer concentrations.<br />
<br />
<br />
[[Image:bound_eqs.png]]<br />
<br />
====Equation manipulations to determine change in protein monomers per time====<br />
Next we can rewrite the equations describing the temporal evolution of the total number inhibitor molecules (5) in terms of equations (10) and (11). This yields the somewhat complex equations (12) which are solely in terms of monomer protein concentration. <br />
<br />
[[Image:dzdt_eqs.png]]<br />
<br />
====Equations for the change in protein monomers per time====<br />
By applying the simple mathematical manipulation (13) to equations (12) and the X derivative of equations (9) ((9) can be written in terms of X by plugging in equations (10) and (11)) we can solve for the following equations for the evolution of protein monomers as a function of time (14). I will spare you the dZi/dXi derivative equations of equation (9).<br />
<br />
[[Image:simp_mat_man_eq.png]]<br />
<br />
====Equations describing stochastic variation====<br />
<br />
====Equations for the evolution of fluorescent reporters====<br />
<br />
==Table of preliminary model constants==<br />
====General Parameters====<br />
{| border=1 cellspacing=0 cellpadding=5<br />
<br />
!align="left" width="50" | Parameter<br />
!width="50" | Value<br />
!width="300" | Meaning<br />
<br />
|-<br />
|m<br />
|20 Copies<br />
|Plasmid Copy Number<br />
|-<br />
|}<br />
<br />
====Inhibitor Protein Parameters====<br />
{| border=1 cellspacing=0 cellpadding=5<br />
<br />
!align="left" width="50" | Parameter<br />
!width="50" | Value<br />
!width="300" | Meaning<br />
<br />
|-<br />
|alphaT1<br />
|.01<br />
|Relative transcription level for repressed (1 TetR dimer bound) pTetR promoter<br />
|-<br />
|alphaT2<br />
|.01<br />
|Relative transcription level for doubly repressed (2 TetR dimers bound) pTetR promoter<br />
|-<br />
|alphaL<br />
|.01<br />
|Relative transcription level for repressed (1 LacI tetramer bound) pLacI promoter<br />
|-<br />
|alphaA<br />
|.01<br />
|Relative transcription level for repressed (two AraC monomers bound) pBAD promoter<br />
|-<br />
|betaA<br />
|1000<br />
|maximum number of proteins produced from the pBAD promoter per cell division<br />
|-<br />
|betaL<br />
|1000<br />
|maximum number of proteins produced from the pLacI promoter per cell division<br />
|-<br />
|betaT<br />
|1000<br />
|maximum number of proteins produced from the pTetR promoter per cell division<br />
|-<br />
|etaLT<br />
|1<br />
|Relative rate of TetR translation from pLacI promoter<br />
|-<br />
|etaLA<br />
|1<br />
|Relative rate of AraC translation from pLacI promoter<br />
|-<br />
|etaAL<br />
|1<br />
|Relative rate of LacI translation from pBAD promoter<br />
|-<br />
|etaAT<br />
|1<br />
|Relative rate of TetR translation from pBAD promoter<br />
|-<br />
|etaTL<br />
|1<br />
|Relative rate of LacI translation from pTetR promoter<br />
|-<br />
|etaTA<br />
|1<br />
|Relative rate of AraC translation from pTetR promoter<br />
|-<br />
|c1L<br />
|.05<br />
|Equilibrium constant for LacI dimerization<br />
|-<br />
|c2L<br />
|.05<br />
|Equilibrium constant for LacI trimerization<br />
|-<br />
|c3L<br />
|.05<br />
|Equilibrium constant for LacI tetramerization<br />
|-<br />
|c4L<br />
|.33<br />
|Equilibrium constant for LacI binding<br />
|-<br />
|}<br />
<br />
==Preliminary modeling results==<br />
<br />
The preliminary modeling results predict things about of the the tri-stable switch. First of all, the model confirms certain obvious points such as the a shift to the proper steady expression state in the presence of inducers. This prediction attests to model validity more than anything else. Secondly, the model predicts that the switch will reach a natural steady state with one promoter dominating in the absence of inducers. This makes intuitive sense as the individual promoters will most likely have different strengths. <br />
<br />
[[Image:no_inducer.png]]<br />
<br />
Next we tested the stability of the model by plugging in final steady-state induced levels of protein into an inducer-less system. The model predicts that the system will not remain stable and will instead revert back to the natural steady state for the inducer-less system. Even after changing several model parameters, the switch would not remain stable.<br />
<br />
<br />
[[Image:stability_test1.png]]<br />
<br />
As a test of the model, models of the three potential bi-stable switches (already proven to be successful experimentally) using the same components and parameters were tested. In accordance with the literature, each of these constructs maintained stable expression of the induced state following the removal of the inducer. In order to probe the instability of the tri-stable switch further, tri-stable switches in which all of the components were effectively the same were created (ex:for all LacI-like, TetR and AraC were modeled as forming tetramers and having one operator.) Each of these systems reached a steady state in which all of the proteins evolved identically and were present at the same levels in the absence of inducer. The same stability tests were then performed. All of the constructs remained stable. Thus our model suggests that the similarity of the repressor-expression systems dominates the success or failure of the tri-stable switch. Furthermore, the bi-stable switch is less sensitive to this factor. <br />
<br />
It should be noted that this is only a preliminary model. The values obtained from the literature will most like differ in our system. Through future experimentation we will elucidate these values and thus create a more accurate model. We will also conduct parameter scans aimed to provide further insight into the necessary system requirements for "tri-stability."<br />
<br />
<br />
<br />
<br />
Link to MATLAB code for tri-stable model [[Media:tristable1.txt]] <br />
<br />
Note: copy this code into MATLAB and save as an M-file<br />
<br />
--[[User:Jlohmuel|Jlohmuel]] 03:59, 28 October 2006 (EDT)<br />
<br />
==What we achieved==<br />
<br />
A successfully ligated pBad/arac construct in addition to partially completed ligations for the pLacI and the pTetR constructs were the results of our summer and semesters' work. The tri-stable toggle switch system was also modeled and proved to yield significant findings.<br />
<br />
==Challenges==<br />
<br />
Despite many cloning and ligation attempts, we were unable to create each component of the switch. More specifically, the pLacI and the pTetR circuits presented many ligation difficulties and proved to yield unsuccessful cloning. However, the pBAD/araC circuit was successfully constructed (yay!).<br />
<br />
==Conclusions==<br />
<br />
With the bigger picture in mind,this project yielded many successful outcomes.Firstly, we were able to successfully construct one of the three constructs necessary for a fully functional tri-stable toggle switch. Secondly, we were able to add more working parts to the registry, both a fully functioning pBad/AraC construct and many parts that were successfully ligated in the process of assembling the other two constructs. These parts will be used in the future by our team and perhaps many others as new ideas and projects develop. Also, it is necessary to mention our most recent finding from the modeling results for the tri-stable toggle switch. Interestingly enough, our model suggests that it is biologically unfeasible to create a switch with more than two components. If more than two components are introduced, such as three, the system is unable to maintain any one state and it will revert back to expressing its most dominant promoter construct. We find such results quite meaningful in demonstrating the biological limits of a system under study.<br />
<br />
==Future work==<br />
<br />
In the coming months, it is hoped that the failure of the ligation strategies for the pLacI and the pTetR constructs can be further investigated. We feel confident that we can continue to build from the one successful construction,the pBad/Arac construct and complete the other 2 constructs. To this end, it will be necessary to further troubleshoot the possible reasons why the cloning of the ligations were unsuccessful.</div>Jlohmuelhttps://2006.igem.org/wiki/index.php/Brown:Tri-Stable_toggle_switchBrown:Tri-Stable toggle switch2006-11-01T06:28:53Z<p>Jlohmuel: /* Preliminary modeling results */</p>
<hr />
<div>{{Brown navigation bar}}<br />
=A tri-stable toggle switch=<br />
<br />
Pursuant to the the 1999 paper "Construction of a genetic toggle switch in Escherichia coli," by Timothy S. Gardner, Charles R. Cantor and James J. Collins, we wondered if the bi-stable toggle switch could be generalized to an n-stable switch. To that end, we conceived and began construction of the tri-stable toggle switch. The general idea is that when a selected promoter is activated, it represses the other two. As such each of the three states of our network are self-stabilizing.<br />
<br />
[[Image:Tristable_Toggle.jpg|left|500px|A general tri-stable toggle]]<br />
<br />
Our implementation uses the pLac, pTet, and pBad/AraC promoters and their respective inhibiting proteins. We chose these because of the relative ease with which each promoter is induced: by adding IPTG for the pLac-promoted region, tetracycline for the PTet-promoted region, and arabinose for the pBad/Ara genes. By characterizing each of the three pathways without following genes and terminators, we enable a tri-stable switching network of any three biobricks to be constructed with minimal cloning.<br />
<br />
[[Image:Registry_tristable.jpg|Our implementation]]<br />
<br />
<br><br />
Link to Tri-stable Switch on the registry:[http://partsregistry.org/Part:BBa_J24672]<br />
<br />
<br><br><br />
<br />
<br><br />
<br><br />
<br />
=Modeling the tri-stable toggle switch=<br />
While the tri-stable switch is seemingly simple in design, just like any other system it is subject to potentially comprising factors such as promoter leakiness and other stochastic fluctuations. In an attempt to predict the behavior of the tri-stable switch, we created a deterministic model of the system taking into account these factors. The model structure is based on that described in "Prediction and measurement of an autoregulatory genetic module" by Farren Isaacs, et al. This paper features an excellent supplement that takes you hand-in-hand through the derivation of model equations. I will attempt to emulate this derivation with our model below. It is important to note that while we have some preliminary results, our model is very much a work in progress. In order to model the system accurately, many of the fundamental constants governing the model will need to be determined experimentally. For now we have used some constants from the literature and estimated some based on similar the values of similar constants also from the literature. <br />
<br />
==Derivation of the Model Equations==<br />
The chemical reactions describing the tri-stable switch can be divided into the two categories of fast and slow reactions. Fast reactions such as dimer formation and promoter-binding occur in the scale of seconds and are therefore modeled to be in equilibrium. Conversely, slow reactions including the likes of transcription, translation, and protein degradation occur on the scale of minutes and are thus modeled to be evolving with time. <br />
<br />
====Fast reversible reaction equations====<br />
The following equations (1) describe the fast reactions. The characters L,T,A denote molecules of LacI, TetR, and, AraC, respectively while the subscripts denote whether the molecule is a monomer (blank), dimer(2), tri-mer(3), etc. The k's denote reaction rates.<br />
<br />
[[Image:Fast_rxn_eqs1.png]]<br />
<br />
Note that the equations include volume explicitly. Cell volume is modeled in this way as it is a slowly evolving function of time.<br />
<br />
====Slow irreversible reaction equations====<br />
The following equations (2) describe the slow irreversible reactions of transcription and translation (both taken into account with the reaction rate kti) and protein degradation (reaction constant kdi). The coefficients eta-ij take into account the relative translation rates of proteins from the same transcripts. For eta-ij, "i" represents the promoter responsible for producing the molecule and "j" represents the molecule being translated [ex: eta-LA corresponds to the relative rate of AraC production from the trascript produced by the LacI promoter]. To establish a convention, eta's are relative to the translation rate of the first gene on a particular transcript. Thus eta for the first gene on the mRNA transcript = 1. The alpha-i coefficients represent relative transcription rates. In this case, "i" denotes the promoter from which the mRNA is transcribed.<br />
<br />
[[Image:slow_rxn_eqs.png]]<br />
====Equations governing cell volume====<br />
The following two equations (3) describing cellular growth and division are taken directly from the aforementioned paper by Isaacs et al. The first equation describes the volume increase from the time immediately following cell division to the time immediately before it. In this equation, V0 denotes the volume of the cell at the beginning of growth and T0 denotes the time of cell division. In our model at times T=q*t0 in which q is an integer, we have volume V and protein concentration n halve - thus modeling volume division and the resulting protein redistribution. The second equation describes the dimensionless equation in which t is measured in terms of fcell-division time and the cell volume changes between 1 and 2.<br />
<br />
[[Image:vol_eq.png]]<br />
====Equations for the total number of molecules====<br />
The following equations (4) describe the total number of inhibitor protein molecules zi. d#i represents the number, "#", of molecules bound to the "i" promoter. (ex: all of the LacI tetramers bound to pLacI promoters are accounted for by the 4d1L term.)<br />
[[Image:Tot_mol_eq.png]]<br />
<br />
====Equations describing the temporal evolution of the total number inhibitor molecules====<br />
The following equations (5) describe the temporal evolution of the total number of inhibitor molecules. Beta-i = the cell division time multiplied by the combined transcription and translation rates from a given promoter "i" (t0*kti), thus representing the total number of protein molecules maximally produced by a given promoter. Similarly, Gamma-i = the cell division time multiplied by the degradation rate of protein i (t0*kdi), thus representing the total number of protein i destabilized over one cell division time. The eta and alpha terms are described in the section above describing the equations for the slow reactions. Note that the equation describing the temporal evolution of AraC includes the term "Aend." This term represents the endogenous expression of AraC in dH5alpha's. Our preliminary work suggests that this endogenous AraC has very little effect on our pBad promoter.<br />
<br />
[[Image:Ev_tot_mol_eqs.png]]<br />
<br />
====Fast reaction equilibrium equations====<br />
Compared to the slowly evolving reactions(2) described by the above equations (5), the fast reactions (1) can be considered to be in equilibrium. Thus the following equilibrium relations (6) hold.<br />
[[Image:Fast_eq_eqs.png]]<br />
<br />
====Equations modeling the presence of the inducers====<br />
The following equations (7) model the effect of the chemical inducers on the system. The model regards the proteins bound to the inducers as having increased dissociation rates (ki-#) as specified below. The reaction rate with the apostrophe denotes the original reaction rate in the absence of inducer inclusion. In these equations "I" denotes IPTG, "a" denotes arabinose, and "Tc" denotes tetracycline (or analogue aTc). The concept for and the form of these equations are based on equations described in the supplementary Information for "A bottom-up approach to gene regulation" by Guido et al. Much like the aforementioned paper by Isaacs et al., 'Bottom-up' has an extensive and brilliantly articulate derivation supplement. <br />
<br />
[[Image:inducer_eqs.png]]<br />
<br />
====Equations relating plasmid copy number to operator sites====<br />
The following equations (8) relate the plasmid copy number, "m", to the number of promoters with bound and unbound operators. As each plasmid contains 1 of each promoter, m is constant for all three promoter types. <br />
<br />
[[Image:cop_num_eqs.png]]<br />
<br />
====Simplified fast reaction equilibrium equations====<br />
The equilibrium equations governing the fast reactions (6) can be simplified by defining dimensionless equilibrium constants in the form of cij=kij/(k(-ij)*V0*A) in which A is Avogadro's number. The cL's are further simplified by defining the constant cL=c1L*c2L*c3L*c4L. Additionally by plugging in successive terms we were able to write the operator equations in terms of d0L. <br />
<br />
[[Image:Simp_fast_eq_eqs.png]]<br />
<br />
====Equations for the number of unbound operators====<br />
Combining equations (8) and the above simplified equilibrium equations(9), we are able to solve for the number of unbound operators d0i. <br />
<br />
[[Image:Unbound_eqs.png]]<br />
<br />
====Equations for the number of bound operators====<br />
By plugging the d0i's (10) back into the equations for bound operators (9), we can solve for these equations in terms of monomer concentrations.<br />
<br />
<br />
[[Image:bound_eqs.png]]<br />
<br />
====Equation manipulations to determine change in protein monomers per time====<br />
Next we can rewrite the equations describing the temporal evolution of the total number inhibitor molecules (5) in terms of equations (10) and (11). This yields the somewhat complex equations (12) which are solely in terms of monomer protein concentration. <br />
<br />
[[Image:dzdt_eqs.png]]<br />
<br />
====Equations for the change in protein monomers per time====<br />
By applying the simple mathematical manipulation (13) to equations (12) and the X derivative of equations (9) ((9) can be written in terms of X by plugging in equations (10) and (11)) we can solve for the following equations for the evolution of protein monomers as a function of time (14). I will spare you the dZi/dXi derivative equations of equation (9).<br />
<br />
[[Image:simp_mat_man_eq.png]]<br />
<br />
====Equations describing stochastic variation====<br />
<br />
====Equations for the evolution of fluorescent reporters====<br />
<br />
==Table of preliminary model constants==<br />
====General Parameters====<br />
{| border=1 cellspacing=0 cellpadding=5<br />
<br />
!align="left" width="50" | Parameter<br />
!width="50" | Value<br />
!width="300" | Meaning<br />
<br />
|-<br />
|m<br />
|20 Copies<br />
|Plasmid Copy Number<br />
|-<br />
|}<br />
<br />
====Inhibitor Protein Parameters====<br />
{| border=1 cellspacing=0 cellpadding=5<br />
<br />
!align="left" width="50" | Parameter<br />
!width="50" | Value<br />
!width="300" | Meaning<br />
<br />
|-<br />
|alphaT1<br />
|.01<br />
|Relative transcription level for repressed (1 TetR dimer bound) pTetR promoter<br />
|-<br />
|alphaT2<br />
|.01<br />
|Relative transcription level for doubly repressed (2 TetR dimers bound) pTetR promoter<br />
|-<br />
|alphaL<br />
|.01<br />
|Relative transcription level for repressed (1 LacI tetramer bound) pLacI promoter<br />
|-<br />
|alphaA<br />
|.01<br />
|Relative transcription level for repressed (two AraC monomers bound) pBAD promoter<br />
|-<br />
|betaA<br />
|1000<br />
|maximum number of proteins produced from the pBAD promoter per cell division<br />
|-<br />
|betaL<br />
|1000<br />
|maximum number of proteins produced from the pLacI promoter per cell division<br />
|-<br />
|betaT<br />
|1000<br />
|maximum number of proteins produced from the pTetR promoter per cell division<br />
|-<br />
|etaLT<br />
|1<br />
|Relative rate of TetR translation from pLacI promoter<br />
|-<br />
|etaLA<br />
|1<br />
|Relative rate of AraC translation from pLacI promoter<br />
|-<br />
|etaAL<br />
|1<br />
|Relative rate of LacI translation from pBAD promoter<br />
|-<br />
|etaAT<br />
|1<br />
|Relative rate of TetR translation from pBAD promoter<br />
|-<br />
|etaTL<br />
|1<br />
|Relative rate of LacI translation from pTetR promoter<br />
|-<br />
|etaTA<br />
|1<br />
|Relative rate of AraC translation from pTetR promoter<br />
|-<br />
|c1L<br />
|.05<br />
|Equilibrium constant for LacI dimerization<br />
|-<br />
|c2L<br />
|.05<br />
|Equilibrium constant for LacI trimerization<br />
|-<br />
|c3L<br />
|.05<br />
|Equilibrium constant for LacI tetramerization<br />
|-<br />
|c4L<br />
|.33<br />
|Equilibrium constant for LacI binding<br />
|-<br />
|}<br />
<br />
==Preliminary modeling results==<br />
<br />
The preliminary modeling results predict things about of the the tri-stable switch. First of all, the model confirms certain obvious points such as the a shift to the proper steady expression state in the presence of inducers. This prediction attests to model validity more than anything else. Secondly, the model predicts that the switch will reach a natural steady state with one promoter dominating in the absence of inducers. This makes intuitive sense as the individual promoters will most likely have different strengths. <br />
<br />
[[Image:no_inducer.png]]<br />
<br />
Next we tested the stability of the model by plugging in final steady-state induced levels of protein into an inducer-less system. The model predicts that the system will not remain stable and will instead revert back to the natural steady state for the inducer-less system. Even after changing several model parameters, the switch would not remain stable.<br />
<br />
<br />
[[Image:stability_test1.png]]<br />
<br />
As a test of the model, models of the three potential bi-stable switches (already proven to be successful experimentally) using the same components and parameters were tested. In accordance with the literature, each of these constructs maintained stable expression of the induced state following the removal of the inducer. In order to probe the instability of the tri-stable switch further, tri-stable switches in which all of the components were effectively the same were created (ex:for all LacI-like, TetR and AraC were modeled as forming tetramers and having one operator.) Each of these systems reached a steady state in which all of the proteins evolved identically and were present at the same levels in the absence of inducer. The same stability tests were then performed. All of the constructs remained stable. Thus our model suggests that the similarity of the repressor-expression systems dominates the success or failure of the tri-stable switch. Furthermore, the bi-stable switch is less sensitive to this factor. <br />
<br />
It should be noted that this is only a preliminary model. The values obtained from the literature will most like differ in our system. Through future experimentation we will elucidate these values and thus create a more accurate model. We will also conduct parameter scans aimed to provide further insight into the necessary system requirements for "tri-stability"<br />
<br />
<br />
<br />
<br />
Link to MATLAB code for tri-stable model [[Media:tristable1.txt]] <br />
<br />
Note: copy this code into MATLAB and save as an M-file<br />
<br />
--[[User:Jlohmuel|Jlohmuel]] 03:59, 28 October 2006 (EDT)<br />
<br />
==What we achieved==<br />
<br />
A successfully ligated pBad/arac construct in addition to partially completed ligations for the pLacI and the pTetR constructs were the results of our summer and semesters' work. The tri-stable toggle switch system was also modeled and proved to yield significant findings.<br />
<br />
==Challenges==<br />
<br />
Despite many cloning and ligation attempts, we were unable to create each component of the switch. More specifically, the pLacI and the pTetR circuits presented many ligation difficulties and proved to yield unsuccessful cloning. However, the pBAD/araC circuit was successfully constructed (yay!).<br />
<br />
==Conclusions==<br />
<br />
With the bigger picture in mind,this project yielded many successful outcomes.Firstly, we were able to successfully construct one of the three constructs necessary for a fully functional tri-stable toggle switch. Secondly, we were able to add more working parts to the registry, both a fully functioning pBad/AraC construct and many parts that were successfully ligated in the process of assembling the other two constructs. These parts will be used in the future by our team and perhaps many others as new ideas and projects develop. Also, it is necessary to mention our most recent finding from the modeling results for the tri-stable toggle switch. Interestingly enough, our model suggests that it is biologically unfeasible to create a switch with more than two components. If more than two components are introduced, such as three, the system is unable to maintain any one state and it will revert back to expressing its most dominant promoter construct. We find such results quite meaningful in demonstrating the biological limits of a system under study.<br />
<br />
==Future work==<br />
<br />
In the coming months, it is hoped that the failure of the ligation strategies for the pLacI and the pTetR constructs can be further investigated. We feel confident that we can continue to build from the one successful construction,the pBad/Arac construct and complete the other 2 constructs. To this end, it will be necessary to further troubleshoot the possible reasons why the cloning of the ligations were unsuccessful.</div>Jlohmuelhttps://2006.igem.org/wiki/index.php/Brown:Tri-Stable_toggle_switchBrown:Tri-Stable toggle switch2006-11-01T06:23:38Z<p>Jlohmuel: /* Preliminary modeling results */</p>
<hr />
<div>{{Brown navigation bar}}<br />
=A tri-stable toggle switch=<br />
<br />
Pursuant to the the 1999 paper "Construction of a genetic toggle switch in Escherichia coli," by Timothy S. Gardner, Charles R. Cantor and James J. Collins, we wondered if the bi-stable toggle switch could be generalized to an n-stable switch. To that end, we conceived and began construction of the tri-stable toggle switch. The general idea is that when a selected promoter is activated, it represses the other two. As such each of the three states of our network are self-stabilizing.<br />
<br />
[[Image:Tristable_Toggle.jpg|left|500px|A general tri-stable toggle]]<br />
<br />
Our implementation uses the pLac, pTet, and pBad/AraC promoters and their respective inhibiting proteins. We chose these because of the relative ease with which each promoter is induced: by adding IPTG for the pLac-promoted region, tetracycline for the PTet-promoted region, and arabinose for the pBad/Ara genes. By characterizing each of the three pathways without following genes and terminators, we enable a tri-stable switching network of any three biobricks to be constructed with minimal cloning.<br />
<br />
[[Image:Registry_tristable.jpg|Our implementation]]<br />
<br />
<br><br />
Link to Tri-stable Switch on the registry:[http://partsregistry.org/Part:BBa_J24672]<br />
<br />
<br><br><br />
<br />
<br><br />
<br><br />
<br />
=Modeling the tri-stable toggle switch=<br />
While the tri-stable switch is seemingly simple in design, just like any other system it is subject to potentially comprising factors such as promoter leakiness and other stochastic fluctuations. In an attempt to predict the behavior of the tri-stable switch, we created a deterministic model of the system taking into account these factors. The model structure is based on that described in "Prediction and measurement of an autoregulatory genetic module" by Farren Isaacs, et al. This paper features an excellent supplement that takes you hand-in-hand through the derivation of model equations. I will attempt to emulate this derivation with our model below. It is important to note that while we have some preliminary results, our model is very much a work in progress. In order to model the system accurately, many of the fundamental constants governing the model will need to be determined experimentally. For now we have used some constants from the literature and estimated some based on similar the values of similar constants also from the literature. <br />
<br />
==Derivation of the Model Equations==<br />
The chemical reactions describing the tri-stable switch can be divided into the two categories of fast and slow reactions. Fast reactions such as dimer formation and promoter-binding occur in the scale of seconds and are therefore modeled to be in equilibrium. Conversely, slow reactions including the likes of transcription, translation, and protein degradation occur on the scale of minutes and are thus modeled to be evolving with time. <br />
<br />
====Fast reversible reaction equations====<br />
The following equations (1) describe the fast reactions. The characters L,T,A denote molecules of LacI, TetR, and, AraC, respectively while the subscripts denote whether the molecule is a monomer (blank), dimer(2), tri-mer(3), etc. The k's denote reaction rates.<br />
<br />
[[Image:Fast_rxn_eqs1.png]]<br />
<br />
Note that the equations include volume explicitly. Cell volume is modeled in this way as it is a slowly evolving function of time.<br />
<br />
====Slow irreversible reaction equations====<br />
The following equations (2) describe the slow irreversible reactions of transcription and translation (both taken into account with the reaction rate kti) and protein degradation (reaction constant kdi). The coefficients eta-ij take into account the relative translation rates of proteins from the same transcripts. For eta-ij, "i" represents the promoter responsible for producing the molecule and "j" represents the molecule being translated [ex: eta-LA corresponds to the relative rate of AraC production from the trascript produced by the LacI promoter]. To establish a convention, eta's are relative to the translation rate of the first gene on a particular transcript. Thus eta for the first gene on the mRNA transcript = 1. The alpha-i coefficients represent relative transcription rates. In this case, "i" denotes the promoter from which the mRNA is transcribed.<br />
<br />
[[Image:slow_rxn_eqs.png]]<br />
====Equations governing cell volume====<br />
The following two equations (3) describing cellular growth and division are taken directly from the aforementioned paper by Isaacs et al. The first equation describes the volume increase from the time immediately following cell division to the time immediately before it. In this equation, V0 denotes the volume of the cell at the beginning of growth and T0 denotes the time of cell division. In our model at times T=q*t0 in which q is an integer, we have volume V and protein concentration n halve - thus modeling volume division and the resulting protein redistribution. The second equation describes the dimensionless equation in which t is measured in terms of fcell-division time and the cell volume changes between 1 and 2.<br />
<br />
[[Image:vol_eq.png]]<br />
====Equations for the total number of molecules====<br />
The following equations (4) describe the total number of inhibitor protein molecules zi. d#i represents the number, "#", of molecules bound to the "i" promoter. (ex: all of the LacI tetramers bound to pLacI promoters are accounted for by the 4d1L term.)<br />
[[Image:Tot_mol_eq.png]]<br />
<br />
====Equations describing the temporal evolution of the total number inhibitor molecules====<br />
The following equations (5) describe the temporal evolution of the total number of inhibitor molecules. Beta-i = the cell division time multiplied by the combined transcription and translation rates from a given promoter "i" (t0*kti), thus representing the total number of protein molecules maximally produced by a given promoter. Similarly, Gamma-i = the cell division time multiplied by the degradation rate of protein i (t0*kdi), thus representing the total number of protein i destabilized over one cell division time. The eta and alpha terms are described in the section above describing the equations for the slow reactions. Note that the equation describing the temporal evolution of AraC includes the term "Aend." This term represents the endogenous expression of AraC in dH5alpha's. Our preliminary work suggests that this endogenous AraC has very little effect on our pBad promoter.<br />
<br />
[[Image:Ev_tot_mol_eqs.png]]<br />
<br />
====Fast reaction equilibrium equations====<br />
Compared to the slowly evolving reactions(2) described by the above equations (5), the fast reactions (1) can be considered to be in equilibrium. Thus the following equilibrium relations (6) hold.<br />
[[Image:Fast_eq_eqs.png]]<br />
<br />
====Equations modeling the presence of the inducers====<br />
The following equations (7) model the effect of the chemical inducers on the system. The model regards the proteins bound to the inducers as having increased dissociation rates (ki-#) as specified below. The reaction rate with the apostrophe denotes the original reaction rate in the absence of inducer inclusion. In these equations "I" denotes IPTG, "a" denotes arabinose, and "Tc" denotes tetracycline (or analogue aTc). The concept for and the form of these equations are based on equations described in the supplementary Information for "A bottom-up approach to gene regulation" by Guido et al. Much like the aforementioned paper by Isaacs et al., 'Bottom-up' has an extensive and brilliantly articulate derivation supplement. <br />
<br />
[[Image:inducer_eqs.png]]<br />
<br />
====Equations relating plasmid copy number to operator sites====<br />
The following equations (8) relate the plasmid copy number, "m", to the number of promoters with bound and unbound operators. As each plasmid contains 1 of each promoter, m is constant for all three promoter types. <br />
<br />
[[Image:cop_num_eqs.png]]<br />
<br />
====Simplified fast reaction equilibrium equations====<br />
The equilibrium equations governing the fast reactions (6) can be simplified by defining dimensionless equilibrium constants in the form of cij=kij/(k(-ij)*V0*A) in which A is Avogadro's number. The cL's are further simplified by defining the constant cL=c1L*c2L*c3L*c4L. Additionally by plugging in successive terms we were able to write the operator equations in terms of d0L. <br />
<br />
[[Image:Simp_fast_eq_eqs.png]]<br />
<br />
====Equations for the number of unbound operators====<br />
Combining equations (8) and the above simplified equilibrium equations(9), we are able to solve for the number of unbound operators d0i. <br />
<br />
[[Image:Unbound_eqs.png]]<br />
<br />
====Equations for the number of bound operators====<br />
By plugging the d0i's (10) back into the equations for bound operators (9), we can solve for these equations in terms of monomer concentrations.<br />
<br />
<br />
[[Image:bound_eqs.png]]<br />
<br />
====Equation manipulations to determine change in protein monomers per time====<br />
Next we can rewrite the equations describing the temporal evolution of the total number inhibitor molecules (5) in terms of equations (10) and (11). This yields the somewhat complex equations (12) which are solely in terms of monomer protein concentration. <br />
<br />
[[Image:dzdt_eqs.png]]<br />
<br />
====Equations for the change in protein monomers per time====<br />
By applying the simple mathematical manipulation (13) to equations (12) and the X derivative of equations (9) ((9) can be written in terms of X by plugging in equations (10) and (11)) we can solve for the following equations for the evolution of protein monomers as a function of time (14). I will spare you the dZi/dXi derivative equations of equation (9).<br />
<br />
[[Image:simp_mat_man_eq.png]]<br />
<br />
====Equations describing stochastic variation====<br />
<br />
====Equations for the evolution of fluorescent reporters====<br />
<br />
==Table of preliminary model constants==<br />
====General Parameters====<br />
{| border=1 cellspacing=0 cellpadding=5<br />
<br />
!align="left" width="50" | Parameter<br />
!width="50" | Value<br />
!width="300" | Meaning<br />
<br />
|-<br />
|m<br />
|20 Copies<br />
|Plasmid Copy Number<br />
|-<br />
|}<br />
<br />
====Inhibitor Protein Parameters====<br />
{| border=1 cellspacing=0 cellpadding=5<br />
<br />
!align="left" width="50" | Parameter<br />
!width="50" | Value<br />
!width="300" | Meaning<br />
<br />
|-<br />
|alphaT1<br />
|.01<br />
|Relative transcription level for repressed (1 TetR dimer bound) pTetR promoter<br />
|-<br />
|alphaT2<br />
|.01<br />
|Relative transcription level for doubly repressed (2 TetR dimers bound) pTetR promoter<br />
|-<br />
|alphaL<br />
|.01<br />
|Relative transcription level for repressed (1 LacI tetramer bound) pLacI promoter<br />
|-<br />
|alphaA<br />
|.01<br />
|Relative transcription level for repressed (two AraC monomers bound) pBAD promoter<br />
|-<br />
|betaA<br />
|1000<br />
|maximum number of proteins produced from the pBAD promoter per cell division<br />
|-<br />
|betaL<br />
|1000<br />
|maximum number of proteins produced from the pLacI promoter per cell division<br />
|-<br />
|betaT<br />
|1000<br />
|maximum number of proteins produced from the pTetR promoter per cell division<br />
|-<br />
|etaLT<br />
|1<br />
|Relative rate of TetR translation from pLacI promoter<br />
|-<br />
|etaLA<br />
|1<br />
|Relative rate of AraC translation from pLacI promoter<br />
|-<br />
|etaAL<br />
|1<br />
|Relative rate of LacI translation from pBAD promoter<br />
|-<br />
|etaAT<br />
|1<br />
|Relative rate of TetR translation from pBAD promoter<br />
|-<br />
|etaTL<br />
|1<br />
|Relative rate of LacI translation from pTetR promoter<br />
|-<br />
|etaTA<br />
|1<br />
|Relative rate of AraC translation from pTetR promoter<br />
|-<br />
|c1L<br />
|.05<br />
|Equilibrium constant for LacI dimerization<br />
|-<br />
|c2L<br />
|.05<br />
|Equilibrium constant for LacI trimerization<br />
|-<br />
|c3L<br />
|.05<br />
|Equilibrium constant for LacI tetramerization<br />
|-<br />
|c4L<br />
|.33<br />
|Equilibrium constant for LacI binding<br />
|-<br />
|}<br />
<br />
==Preliminary modeling results==<br />
<br />
The preliminary modeling results predict things about of the the tri-stable switch. First of all, the model confirms certain obvious points such as the a shift to the proper steady expression state in the presence of inducers. This prediction attests to model validity more than anything else. Secondly, the model predicts that the switch will reach a natural steady state with one promoter dominating in the absence of inducers. This makes intuitive sense as the individual promoters will most likely have different strengths. <br />
<br />
[[Image:no_inducer.png]]<br />
<br />
Next we tested the stability of the model by plugging in final steady-state induced levels of protein into an inducer-less system. The model predicts that the system will not remain stable and will instead revert back to the natural steady state for the inducer-less system. <br />
<br />
<br />
[[Image:stability_test1.png]]<br />
<br />
As a test of the model, models of the three potential bi-stable switches (already proven to be successful experimentally) using the same components and parameters were tested. Each of these constructs remained stable. As a further means of testing the model, switches in which all of the components were effectively the same were created (ex:for all LacI-like, TetR and AraC were modeled as forming tetramers and having one operator. In each of these systems a steady state was reached in which all of the proteins evolved and were present at the same levels. The same stability tests were performed and all constructs remained stable. Thus our model suggests that the similarity of the repressor-expression systems dominates the success or failure of the tri-stable switch. Furthermore, the bi-stable switch is less sensitive to this factor. <br />
<br />
It should be noted that this is only a preliminary model. The values obtained from the literature will most like differ in our system. Through future experimentation we will elucidate these values and thus create a more accurate model. We will also conduct parameter scans aimed to provide further insight into the necessary system requirements for "tri-stability"<br />
<br />
<br />
<br />
<br />
Link to MATLAB code for tri-stable model [[Media:tristable1.txt]] <br />
<br />
Note: copy this code into MATLAB and save as an M-file<br />
<br />
--[[User:Jlohmuel|Jlohmuel]] 03:59, 28 October 2006 (EDT)<br />
<br />
==What we achieved==<br />
<br />
A successfully ligated pBad/arac construct in addition to partially completed ligations for the pLacI and the pTetR constructs were the results of our summer and semesters' work. The tri-stable toggle switch system was also modeled and proved to yield significant findings.<br />
<br />
==Challenges==<br />
<br />
Despite many cloning and ligation attempts, we were unable to create each component of the switch. More specifically, the pLacI and the pTetR circuits presented many ligation difficulties and proved to yield unsuccessful cloning. However, the pBAD/araC circuit was successfully constructed (yay!).<br />
<br />
==Conclusions==<br />
<br />
With the bigger picture in mind,this project yielded many successful outcomes.Firstly, we were able to successfully construct one of the three constructs necessary for a fully functional tri-stable toggle switch. Secondly, we were able to add more working parts to the registry, both a fully functioning pBad/AraC construct and many parts that were successfully ligated in the process of assembling the other two constructs. These parts will be used in the future by our team and perhaps many others as new ideas and projects develop. Also, it is necessary to mention our most recent finding from the modeling results for the tri-stable toggle switch. Interestingly enough, our model suggests that it is biologically unfeasible to create a switch with more than two components. If more than two components are introduced, such as three, the system is unable to maintain any one state and it will revert back to expressing its most dominant promoter construct. We find such results quite meaningful in demonstrating the biological limits of a system under study.<br />
<br />
==Future work==<br />
<br />
In the coming months, it is hoped that the failure of the ligation strategies for the pLacI and the pTetR constructs can be further investigated. We feel confident that we can continue to build from the one successful construction,the pBad/Arac construct and complete the other 2 constructs. To this end, it will be necessary to further troubleshoot the possible reasons why the cloning of the ligations were unsuccessful.</div>Jlohmuelhttps://2006.igem.org/wiki/index.php/Brown:Tri-Stable_toggle_switchBrown:Tri-Stable toggle switch2006-11-01T05:54:39Z<p>Jlohmuel: /* Preliminary modeling results */</p>
<hr />
<div>{{Brown navigation bar}}<br />
=A tri-stable toggle switch=<br />
<br />
Pursuant to the the 1999 paper "Construction of a genetic toggle switch in Escherichia coli," by Timothy S. Gardner, Charles R. Cantor and James J. Collins, we wondered if the bi-stable toggle switch could be generalized to an n-stable switch. To that end, we conceived and began construction of the tri-stable toggle switch. The general idea is that when a selected promoter is activated, it represses the other two. As such each of the three states of our network are self-stabilizing.<br />
<br />
[[Image:Tristable_Toggle.jpg|left|500px|A general tri-stable toggle]]<br />
<br />
Our implementation uses the pLac, pTet, and pBad/AraC promoters and their respective inhibiting proteins. We chose these because of the relative ease with which each promoter is induced: by adding IPTG for the pLac-promoted region, tetracycline for the PTet-promoted region, and arabinose for the pBad/Ara genes. By characterizing each of the three pathways without following genes and terminators, we enable a tri-stable switching network of any three biobricks to be constructed with minimal cloning.<br />
<br />
[[Image:Registry_tristable.jpg|Our implementation]]<br />
<br />
<br><br />
Link to Tri-stable Switch on the registry:[http://partsregistry.org/Part:BBa_J24672]<br />
<br />
<br><br><br />
<br />
<br><br />
<br><br />
<br />
=Modeling the tri-stable toggle switch=<br />
While the tri-stable switch is seemingly simple in design, just like any other system it is subject to potentially comprising factors such as promoter leakiness and other stochastic fluctuations. In an attempt to predict the behavior of the tri-stable switch, we created a deterministic model of the system taking into account these factors. The model structure is based on that described in "Prediction and measurement of an autoregulatory genetic module" by Farren Isaacs, et al. This paper features an excellent supplement that takes you hand-in-hand through the derivation of model equations. I will attempt to emulate this derivation with our model below. It is important to note that while we have some preliminary results, our model is very much a work in progress. In order to model the system accurately, many of the fundamental constants governing the model will need to be determined experimentally. For now we have used some constants from the literature and estimated some based on similar the values of similar constants also from the literature. <br />
<br />
==Derivation of the Model Equations==<br />
The chemical reactions describing the tri-stable switch can be divided into the two categories of fast and slow reactions. Fast reactions such as dimer formation and promoter-binding occur in the scale of seconds and are therefore modeled to be in equilibrium. Conversely, slow reactions including the likes of transcription, translation, and protein degradation occur on the scale of minutes and are thus modeled to be evolving with time. <br />
<br />
====Fast reversible reaction equations====<br />
The following equations (1) describe the fast reactions. The characters L,T,A denote molecules of LacI, TetR, and, AraC, respectively while the subscripts denote whether the molecule is a monomer (blank), dimer(2), tri-mer(3), etc. The k's denote reaction rates.<br />
<br />
[[Image:Fast_rxn_eqs1.png]]<br />
<br />
Note that the equations include volume explicitly. Cell volume is modeled in this way as it is a slowly evolving function of time.<br />
<br />
====Slow irreversible reaction equations====<br />
The following equations (2) describe the slow irreversible reactions of transcription and translation (both taken into account with the reaction rate kti) and protein degradation (reaction constant kdi). The coefficients eta-ij take into account the relative translation rates of proteins from the same transcripts. For eta-ij, "i" represents the promoter responsible for producing the molecule and "j" represents the molecule being translated [ex: eta-LA corresponds to the relative rate of AraC production from the trascript produced by the LacI promoter]. To establish a convention, eta's are relative to the translation rate of the first gene on a particular transcript. Thus eta for the first gene on the mRNA transcript = 1. The alpha-i coefficients represent relative transcription rates. In this case, "i" denotes the promoter from which the mRNA is transcribed.<br />
<br />
[[Image:slow_rxn_eqs.png]]<br />
====Equations governing cell volume====<br />
The following two equations (3) describing cellular growth and division are taken directly from the aforementioned paper by Isaacs et al. The first equation describes the volume increase from the time immediately following cell division to the time immediately before it. In this equation, V0 denotes the volume of the cell at the beginning of growth and T0 denotes the time of cell division. In our model at times T=q*t0 in which q is an integer, we have volume V and protein concentration n halve - thus modeling volume division and the resulting protein redistribution. The second equation describes the dimensionless equation in which t is measured in terms of fcell-division time and the cell volume changes between 1 and 2.<br />
<br />
[[Image:vol_eq.png]]<br />
====Equations for the total number of molecules====<br />
The following equations (4) describe the total number of inhibitor protein molecules zi. d#i represents the number, "#", of molecules bound to the "i" promoter. (ex: all of the LacI tetramers bound to pLacI promoters are accounted for by the 4d1L term.)<br />
[[Image:Tot_mol_eq.png]]<br />
<br />
====Equations describing the temporal evolution of the total number inhibitor molecules====<br />
The following equations (5) describe the temporal evolution of the total number of inhibitor molecules. Beta-i = the cell division time multiplied by the combined transcription and translation rates from a given promoter "i" (t0*kti), thus representing the total number of protein molecules maximally produced by a given promoter. Similarly, Gamma-i = the cell division time multiplied by the degradation rate of protein i (t0*kdi), thus representing the total number of protein i destabilized over one cell division time. The eta and alpha terms are described in the section above describing the equations for the slow reactions. Note that the equation describing the temporal evolution of AraC includes the term "Aend." This term represents the endogenous expression of AraC in dH5alpha's. Our preliminary work suggests that this endogenous AraC has very little effect on our pBad promoter.<br />
<br />
[[Image:Ev_tot_mol_eqs.png]]<br />
<br />
====Fast reaction equilibrium equations====<br />
Compared to the slowly evolving reactions(2) described by the above equations (5), the fast reactions (1) can be considered to be in equilibrium. Thus the following equilibrium relations (6) hold.<br />
[[Image:Fast_eq_eqs.png]]<br />
<br />
====Equations modeling the presence of the inducers====<br />
The following equations (7) model the effect of the chemical inducers on the system. The model regards the proteins bound to the inducers as having increased dissociation rates (ki-#) as specified below. The reaction rate with the apostrophe denotes the original reaction rate in the absence of inducer inclusion. In these equations "I" denotes IPTG, "a" denotes arabinose, and "Tc" denotes tetracycline (or analogue aTc). The concept for and the form of these equations are based on equations described in the supplementary Information for "A bottom-up approach to gene regulation" by Guido et al. Much like the aforementioned paper by Isaacs et al., 'Bottom-up' has an extensive and brilliantly articulate derivation supplement. <br />
<br />
[[Image:inducer_eqs.png]]<br />
<br />
====Equations relating plasmid copy number to operator sites====<br />
The following equations (8) relate the plasmid copy number, "m", to the number of promoters with bound and unbound operators. As each plasmid contains 1 of each promoter, m is constant for all three promoter types. <br />
<br />
[[Image:cop_num_eqs.png]]<br />
<br />
====Simplified fast reaction equilibrium equations====<br />
The equilibrium equations governing the fast reactions (6) can be simplified by defining dimensionless equilibrium constants in the form of cij=kij/(k(-ij)*V0*A) in which A is Avogadro's number. The cL's are further simplified by defining the constant cL=c1L*c2L*c3L*c4L. Additionally by plugging in successive terms we were able to write the operator equations in terms of d0L. <br />
<br />
[[Image:Simp_fast_eq_eqs.png]]<br />
<br />
====Equations for the number of unbound operators====<br />
Combining equations (8) and the above simplified equilibrium equations(9), we are able to solve for the number of unbound operators d0i. <br />
<br />
[[Image:Unbound_eqs.png]]<br />
<br />
====Equations for the number of bound operators====<br />
By plugging the d0i's (10) back into the equations for bound operators (9), we can solve for these equations in terms of monomer concentrations.<br />
<br />
<br />
[[Image:bound_eqs.png]]<br />
<br />
====Equation manipulations to determine change in protein monomers per time====<br />
Next we can rewrite the equations describing the temporal evolution of the total number inhibitor molecules (5) in terms of equations (10) and (11). This yields the somewhat complex equations (12) which are solely in terms of monomer protein concentration. <br />
<br />
[[Image:dzdt_eqs.png]]<br />
<br />
====Equations for the change in protein monomers per time====<br />
By applying the simple mathematical manipulation (13) to equations (12) and the X derivative of equations (9) ((9) can be written in terms of X by plugging in equations (10) and (11)) we can solve for the following equations for the evolution of protein monomers as a function of time (14). I will spare you the dZi/dXi derivative equations of equation (9).<br />
<br />
[[Image:simp_mat_man_eq.png]]<br />
<br />
====Equations describing stochastic variation====<br />
<br />
====Equations for the evolution of fluorescent reporters====<br />
<br />
==Table of preliminary model constants==<br />
====General Parameters====<br />
{| border=1 cellspacing=0 cellpadding=5<br />
<br />
!align="left" width="50" | Parameter<br />
!width="50" | Value<br />
!width="300" | Meaning<br />
<br />
|-<br />
|m<br />
|20 Copies<br />
|Plasmid Copy Number<br />
|-<br />
|}<br />
<br />
====Inhibitor Protein Parameters====<br />
{| border=1 cellspacing=0 cellpadding=5<br />
<br />
!align="left" width="50" | Parameter<br />
!width="50" | Value<br />
!width="300" | Meaning<br />
<br />
|-<br />
|alphaT1<br />
|.01<br />
|Relative transcription level for repressed (1 TetR dimer bound) pTetR promoter<br />
|-<br />
|alphaT2<br />
|.01<br />
|Relative transcription level for doubly repressed (2 TetR dimers bound) pTetR promoter<br />
|-<br />
|alphaL<br />
|.01<br />
|Relative transcription level for repressed (1 LacI tetramer bound) pLacI promoter<br />
|-<br />
|alphaA<br />
|.01<br />
|Relative transcription level for repressed (two AraC monomers bound) pBAD promoter<br />
|-<br />
|betaA<br />
|1000<br />
|maximum number of proteins produced from the pBAD promoter per cell division<br />
|-<br />
|betaL<br />
|1000<br />
|maximum number of proteins produced from the pLacI promoter per cell division<br />
|-<br />
|betaT<br />
|1000<br />
|maximum number of proteins produced from the pTetR promoter per cell division<br />
|-<br />
|etaLT<br />
|1<br />
|Relative rate of TetR translation from pLacI promoter<br />
|-<br />
|etaLA<br />
|1<br />
|Relative rate of AraC translation from pLacI promoter<br />
|-<br />
|etaAL<br />
|1<br />
|Relative rate of LacI translation from pBAD promoter<br />
|-<br />
|etaAT<br />
|1<br />
|Relative rate of TetR translation from pBAD promoter<br />
|-<br />
|etaTL<br />
|1<br />
|Relative rate of LacI translation from pTetR promoter<br />
|-<br />
|etaTA<br />
|1<br />
|Relative rate of AraC translation from pTetR promoter<br />
|-<br />
|c1L<br />
|.05<br />
|Equilibrium constant for LacI dimerization<br />
|-<br />
|c2L<br />
|.05<br />
|Equilibrium constant for LacI trimerization<br />
|-<br />
|c3L<br />
|.05<br />
|Equilibrium constant for LacI tetramerization<br />
|-<br />
|c4L<br />
|.33<br />
|Equilibrium constant for LacI binding<br />
|-<br />
|}<br />
<br />
==Preliminary modeling results==<br />
<br />
The preliminary modeling results predict that the tri-stable switch will reach a natural steady state in the absence of inducers. not remain stable in the absence of the inducer. As a means of testing the validity of the model, the fundamental equations were slightly modified to code for the three combinations of bi-stable switches. However, This suggests that Full parameter scans will have to be run to test for <br />
With all of this said the modeling parameters are likely to be. Only future experiments <br />
<br />
[[Image:no_inducer.png]]<br />
<br />
[[Image:stability_test1.png]]<br />
<br />
<br />
Link to MATLAB code for tri-stable model [[Media:tristable1.txt]] <br />
<br />
Note: copy this code into MATLAB and save as an M-file<br />
<br />
--[[User:Jlohmuel|Jlohmuel]] 03:59, 28 October 2006 (EDT)<br />
<br />
==What we achieved==<br />
<br />
A successfully ligated pBad/arac construct in addition to partially completed ligations for the pLacI and the pTetR constructs were the results of our summer and semesters' work. The tri-stable toggle switch system was also modeled and proved to yield significant findings.<br />
<br />
==Challenges==<br />
<br />
Despite many cloning and ligation attempts, we were unable to create each component of the switch. More specifically, the pLacI and the pTetR circuits presented many ligation difficulties and proved to yield unsuccessful cloning. However, the pBAD/araC circuit was successfully constructed (yay!).<br />
<br />
==Conclusions==<br />
<br />
With the bigger picture in mind,this project yielded many successful outcomes.Firstly, we were able to successfully construct one of the three constructs necessary for a fully functional tri-stable toggle switch. Secondly, we were able to add more working parts to the registry, both a fully functioning pBad/AraC construct and many parts that were successfully ligated in the process of assembling the other two constructs. These parts will be used in the future by our team and perhaps many others as new ideas and projects develop. Also, it is necessary to mention our most recent finding from the modeling results for the tri-stable toggle switch. Interestingly enough, our model suggests that it is biologically unfeasible to create a switch with more than two components. If more than two components are introduced, such as three, the system is unable to maintain any one state and it will revert back to expressing its most dominant promoter construct. We find such results quite meaningful in demonstrating the biological limits of a system under study.<br />
<br />
==Future work==<br />
<br />
In the coming months, it is hoped that the failure of the ligation strategies for the pLacI and the pTetR constructs can be further investigated. We feel confident that we can continue to build from the one successful construction,the pBad/Arac construct and complete the other 2 constructs. To this end, it will be necessary to further troubleshoot the possible reasons why the cloning of the ligations were unsuccessful.</div>Jlohmuelhttps://2006.igem.org/wiki/index.php/File:Stability_test1.pngFile:Stability test1.png2006-11-01T05:54:15Z<p>Jlohmuel: </p>
<hr />
<div></div>Jlohmuelhttps://2006.igem.org/wiki/index.php/Brown:Tri-Stable_toggle_switchBrown:Tri-Stable toggle switch2006-11-01T05:51:29Z<p>Jlohmuel: /* Preliminary modeling results */</p>
<hr />
<div>{{Brown navigation bar}}<br />
=A tri-stable toggle switch=<br />
<br />
Pursuant to the the 1999 paper "Construction of a genetic toggle switch in Escherichia coli," by Timothy S. Gardner, Charles R. Cantor and James J. Collins, we wondered if the bi-stable toggle switch could be generalized to an n-stable switch. To that end, we conceived and began construction of the tri-stable toggle switch. The general idea is that when a selected promoter is activated, it represses the other two. As such each of the three states of our network are self-stabilizing.<br />
<br />
[[Image:Tristable_Toggle.jpg|left|500px|A general tri-stable toggle]]<br />
<br />
Our implementation uses the pLac, pTet, and pBad/AraC promoters and their respective inhibiting proteins. We chose these because of the relative ease with which each promoter is induced: by adding IPTG for the pLac-promoted region, tetracycline for the PTet-promoted region, and arabinose for the pBad/Ara genes. By characterizing each of the three pathways without following genes and terminators, we enable a tri-stable switching network of any three biobricks to be constructed with minimal cloning.<br />
<br />
[[Image:Registry_tristable.jpg|Our implementation]]<br />
<br />
<br><br />
Link to Tri-stable Switch on the registry:[http://partsregistry.org/Part:BBa_J24672]<br />
<br />
<br><br><br />
<br />
<br><br />
<br><br />
<br />
=Modeling the tri-stable toggle switch=<br />
While the tri-stable switch is seemingly simple in design, just like any other system it is subject to potentially comprising factors such as promoter leakiness and other stochastic fluctuations. In an attempt to predict the behavior of the tri-stable switch, we created a deterministic model of the system taking into account these factors. The model structure is based on that described in "Prediction and measurement of an autoregulatory genetic module" by Farren Isaacs, et al. This paper features an excellent supplement that takes you hand-in-hand through the derivation of model equations. I will attempt to emulate this derivation with our model below. It is important to note that while we have some preliminary results, our model is very much a work in progress. In order to model the system accurately, many of the fundamental constants governing the model will need to be determined experimentally. For now we have used some constants from the literature and estimated some based on similar the values of similar constants also from the literature. <br />
<br />
==Derivation of the Model Equations==<br />
The chemical reactions describing the tri-stable switch can be divided into the two categories of fast and slow reactions. Fast reactions such as dimer formation and promoter-binding occur in the scale of seconds and are therefore modeled to be in equilibrium. Conversely, slow reactions including the likes of transcription, translation, and protein degradation occur on the scale of minutes and are thus modeled to be evolving with time. <br />
<br />
====Fast reversible reaction equations====<br />
The following equations (1) describe the fast reactions. The characters L,T,A denote molecules of LacI, TetR, and, AraC, respectively while the subscripts denote whether the molecule is a monomer (blank), dimer(2), tri-mer(3), etc. The k's denote reaction rates.<br />
<br />
[[Image:Fast_rxn_eqs1.png]]<br />
<br />
Note that the equations include volume explicitly. Cell volume is modeled in this way as it is a slowly evolving function of time.<br />
<br />
====Slow irreversible reaction equations====<br />
The following equations (2) describe the slow irreversible reactions of transcription and translation (both taken into account with the reaction rate kti) and protein degradation (reaction constant kdi). The coefficients eta-ij take into account the relative translation rates of proteins from the same transcripts. For eta-ij, "i" represents the promoter responsible for producing the molecule and "j" represents the molecule being translated [ex: eta-LA corresponds to the relative rate of AraC production from the trascript produced by the LacI promoter]. To establish a convention, eta's are relative to the translation rate of the first gene on a particular transcript. Thus eta for the first gene on the mRNA transcript = 1. The alpha-i coefficients represent relative transcription rates. In this case, "i" denotes the promoter from which the mRNA is transcribed.<br />
<br />
[[Image:slow_rxn_eqs.png]]<br />
====Equations governing cell volume====<br />
The following two equations (3) describing cellular growth and division are taken directly from the aforementioned paper by Isaacs et al. The first equation describes the volume increase from the time immediately following cell division to the time immediately before it. In this equation, V0 denotes the volume of the cell at the beginning of growth and T0 denotes the time of cell division. In our model at times T=q*t0 in which q is an integer, we have volume V and protein concentration n halve - thus modeling volume division and the resulting protein redistribution. The second equation describes the dimensionless equation in which t is measured in terms of fcell-division time and the cell volume changes between 1 and 2.<br />
<br />
[[Image:vol_eq.png]]<br />
====Equations for the total number of molecules====<br />
The following equations (4) describe the total number of inhibitor protein molecules zi. d#i represents the number, "#", of molecules bound to the "i" promoter. (ex: all of the LacI tetramers bound to pLacI promoters are accounted for by the 4d1L term.)<br />
[[Image:Tot_mol_eq.png]]<br />
<br />
====Equations describing the temporal evolution of the total number inhibitor molecules====<br />
The following equations (5) describe the temporal evolution of the total number of inhibitor molecules. Beta-i = the cell division time multiplied by the combined transcription and translation rates from a given promoter "i" (t0*kti), thus representing the total number of protein molecules maximally produced by a given promoter. Similarly, Gamma-i = the cell division time multiplied by the degradation rate of protein i (t0*kdi), thus representing the total number of protein i destabilized over one cell division time. The eta and alpha terms are described in the section above describing the equations for the slow reactions. Note that the equation describing the temporal evolution of AraC includes the term "Aend." This term represents the endogenous expression of AraC in dH5alpha's. Our preliminary work suggests that this endogenous AraC has very little effect on our pBad promoter.<br />
<br />
[[Image:Ev_tot_mol_eqs.png]]<br />
<br />
====Fast reaction equilibrium equations====<br />
Compared to the slowly evolving reactions(2) described by the above equations (5), the fast reactions (1) can be considered to be in equilibrium. Thus the following equilibrium relations (6) hold.<br />
[[Image:Fast_eq_eqs.png]]<br />
<br />
====Equations modeling the presence of the inducers====<br />
The following equations (7) model the effect of the chemical inducers on the system. The model regards the proteins bound to the inducers as having increased dissociation rates (ki-#) as specified below. The reaction rate with the apostrophe denotes the original reaction rate in the absence of inducer inclusion. In these equations "I" denotes IPTG, "a" denotes arabinose, and "Tc" denotes tetracycline (or analogue aTc). The concept for and the form of these equations are based on equations described in the supplementary Information for "A bottom-up approach to gene regulation" by Guido et al. Much like the aforementioned paper by Isaacs et al., 'Bottom-up' has an extensive and brilliantly articulate derivation supplement. <br />
<br />
[[Image:inducer_eqs.png]]<br />
<br />
====Equations relating plasmid copy number to operator sites====<br />
The following equations (8) relate the plasmid copy number, "m", to the number of promoters with bound and unbound operators. As each plasmid contains 1 of each promoter, m is constant for all three promoter types. <br />
<br />
[[Image:cop_num_eqs.png]]<br />
<br />
====Simplified fast reaction equilibrium equations====<br />
The equilibrium equations governing the fast reactions (6) can be simplified by defining dimensionless equilibrium constants in the form of cij=kij/(k(-ij)*V0*A) in which A is Avogadro's number. The cL's are further simplified by defining the constant cL=c1L*c2L*c3L*c4L. Additionally by plugging in successive terms we were able to write the operator equations in terms of d0L. <br />
<br />
[[Image:Simp_fast_eq_eqs.png]]<br />
<br />
====Equations for the number of unbound operators====<br />
Combining equations (8) and the above simplified equilibrium equations(9), we are able to solve for the number of unbound operators d0i. <br />
<br />
[[Image:Unbound_eqs.png]]<br />
<br />
====Equations for the number of bound operators====<br />
By plugging the d0i's (10) back into the equations for bound operators (9), we can solve for these equations in terms of monomer concentrations.<br />
<br />
<br />
[[Image:bound_eqs.png]]<br />
<br />
====Equation manipulations to determine change in protein monomers per time====<br />
Next we can rewrite the equations describing the temporal evolution of the total number inhibitor molecules (5) in terms of equations (10) and (11). This yields the somewhat complex equations (12) which are solely in terms of monomer protein concentration. <br />
<br />
[[Image:dzdt_eqs.png]]<br />
<br />
====Equations for the change in protein monomers per time====<br />
By applying the simple mathematical manipulation (13) to equations (12) and the X derivative of equations (9) ((9) can be written in terms of X by plugging in equations (10) and (11)) we can solve for the following equations for the evolution of protein monomers as a function of time (14). I will spare you the dZi/dXi derivative equations of equation (9).<br />
<br />
[[Image:simp_mat_man_eq.png]]<br />
<br />
====Equations describing stochastic variation====<br />
<br />
====Equations for the evolution of fluorescent reporters====<br />
<br />
==Table of preliminary model constants==<br />
====General Parameters====<br />
{| border=1 cellspacing=0 cellpadding=5<br />
<br />
!align="left" width="50" | Parameter<br />
!width="50" | Value<br />
!width="300" | Meaning<br />
<br />
|-<br />
|m<br />
|20 Copies<br />
|Plasmid Copy Number<br />
|-<br />
|}<br />
<br />
====Inhibitor Protein Parameters====<br />
{| border=1 cellspacing=0 cellpadding=5<br />
<br />
!align="left" width="50" | Parameter<br />
!width="50" | Value<br />
!width="300" | Meaning<br />
<br />
|-<br />
|alphaT1<br />
|.01<br />
|Relative transcription level for repressed (1 TetR dimer bound) pTetR promoter<br />
|-<br />
|alphaT2<br />
|.01<br />
|Relative transcription level for doubly repressed (2 TetR dimers bound) pTetR promoter<br />
|-<br />
|alphaL<br />
|.01<br />
|Relative transcription level for repressed (1 LacI tetramer bound) pLacI promoter<br />
|-<br />
|alphaA<br />
|.01<br />
|Relative transcription level for repressed (two AraC monomers bound) pBAD promoter<br />
|-<br />
|betaA<br />
|1000<br />
|maximum number of proteins produced from the pBAD promoter per cell division<br />
|-<br />
|betaL<br />
|1000<br />
|maximum number of proteins produced from the pLacI promoter per cell division<br />
|-<br />
|betaT<br />
|1000<br />
|maximum number of proteins produced from the pTetR promoter per cell division<br />
|-<br />
|etaLT<br />
|1<br />
|Relative rate of TetR translation from pLacI promoter<br />
|-<br />
|etaLA<br />
|1<br />
|Relative rate of AraC translation from pLacI promoter<br />
|-<br />
|etaAL<br />
|1<br />
|Relative rate of LacI translation from pBAD promoter<br />
|-<br />
|etaAT<br />
|1<br />
|Relative rate of TetR translation from pBAD promoter<br />
|-<br />
|etaTL<br />
|1<br />
|Relative rate of LacI translation from pTetR promoter<br />
|-<br />
|etaTA<br />
|1<br />
|Relative rate of AraC translation from pTetR promoter<br />
|-<br />
|c1L<br />
|.05<br />
|Equilibrium constant for LacI dimerization<br />
|-<br />
|c2L<br />
|.05<br />
|Equilibrium constant for LacI trimerization<br />
|-<br />
|c3L<br />
|.05<br />
|Equilibrium constant for LacI tetramerization<br />
|-<br />
|c4L<br />
|.33<br />
|Equilibrium constant for LacI binding<br />
|-<br />
|}<br />
<br />
==Preliminary modeling results==<br />
<br />
The preliminary modeling results predict that the tri-stable switch will reach a natural steady state in the absence of inducers. not remain stable in the absence of the inducer. As a means of testing the validity of the model, the fundamental equations were slightly modified to code for the three combinations of bi-stable switches. However, This suggests that Full parameter scans will have to be run to test for <br />
With all of this said the modeling parameters are likely to be. Only future experiments <br />
<br />
[[Image:no_inducer.png]]<br />
<br />
[[Image:stability_test.png]]<br />
<br />
<br />
Link to MATLAB code for tri-stable model [[Media:tristable1.txt]] <br />
<br />
Note: copy this code into MATLAB and save as an M-file<br />
<br />
--[[User:Jlohmuel|Jlohmuel]] 03:59, 28 October 2006 (EDT)<br />
<br />
==What we achieved==<br />
<br />
A successfully ligated pBad/arac construct in addition to partially completed ligations for the pLacI and the pTetR constructs were the results of our summer and semesters' work. The tri-stable toggle switch system was also modeled and proved to yield significant findings.<br />
<br />
==Challenges==<br />
<br />
Despite many cloning and ligation attempts, we were unable to create each component of the switch. More specifically, the pLacI and the pTetR circuits presented many ligation difficulties and proved to yield unsuccessful cloning. However, the pBAD/araC circuit was successfully constructed (yay!).<br />
<br />
==Conclusions==<br />
<br />
With the bigger picture in mind,this project yielded many successful outcomes.Firstly, we were able to successfully construct one of the three constructs necessary for a fully functional tri-stable toggle switch. Secondly, we were able to add more working parts to the registry, both a fully functioning pBad/AraC construct and many parts that were successfully ligated in the process of assembling the other two constructs. These parts will be used in the future by our team and perhaps many others as new ideas and projects develop. Also, it is necessary to mention our most recent finding from the modeling results for the tri-stable toggle switch. Interestingly enough, our model suggests that it is biologically unfeasible to create a switch with more than two components. If more than two components are introduced, such as three, the system is unable to maintain any one state and it will revert back to expressing its most dominant promoter construct. We find such results quite meaningful in demonstrating the biological limits of a system under study.<br />
<br />
==Future work==<br />
<br />
In the coming months, it is hoped that the failure of the ligation strategies for the pLacI and the pTetR constructs can be further investigated. We feel confident that we can continue to build from the one successful construction,the pBad/Arac construct and complete the other 2 constructs. To this end, it will be necessary to further troubleshoot the possible reasons why the cloning of the ligations were unsuccessful.</div>Jlohmuelhttps://2006.igem.org/wiki/index.php/File:Stability_test.pngFile:Stability test.png2006-11-01T05:50:47Z<p>Jlohmuel: </p>
<hr />
<div></div>Jlohmuelhttps://2006.igem.org/wiki/index.php/Brown:Tri-Stable_toggle_switchBrown:Tri-Stable toggle switch2006-11-01T05:46:45Z<p>Jlohmuel: /* Preliminary modeling results */</p>
<hr />
<div>{{Brown navigation bar}}<br />
=A tri-stable toggle switch=<br />
<br />
Pursuant to the the 1999 paper "Construction of a genetic toggle switch in Escherichia coli," by Timothy S. Gardner, Charles R. Cantor and James J. Collins, we wondered if the bi-stable toggle switch could be generalized to an n-stable switch. To that end, we conceived and began construction of the tri-stable toggle switch. The general idea is that when a selected promoter is activated, it represses the other two. As such each of the three states of our network are self-stabilizing.<br />
<br />
[[Image:Tristable_Toggle.jpg|left|500px|A general tri-stable toggle]]<br />
<br />
Our implementation uses the pLac, pTet, and pBad/AraC promoters and their respective inhibiting proteins. We chose these because of the relative ease with which each promoter is induced: by adding IPTG for the pLac-promoted region, tetracycline for the PTet-promoted region, and arabinose for the pBad/Ara genes. By characterizing each of the three pathways without following genes and terminators, we enable a tri-stable switching network of any three biobricks to be constructed with minimal cloning.<br />
<br />
[[Image:Registry_tristable.jpg|Our implementation]]<br />
<br />
<br><br />
Link to Tri-stable Switch on the registry:[http://partsregistry.org/Part:BBa_J24672]<br />
<br />
<br><br><br />
<br />
<br><br />
<br><br />
<br />
=Modeling the tri-stable toggle switch=<br />
While the tri-stable switch is seemingly simple in design, just like any other system it is subject to potentially comprising factors such as promoter leakiness and other stochastic fluctuations. In an attempt to predict the behavior of the tri-stable switch, we created a deterministic model of the system taking into account these factors. The model structure is based on that described in "Prediction and measurement of an autoregulatory genetic module" by Farren Isaacs, et al. This paper features an excellent supplement that takes you hand-in-hand through the derivation of model equations. I will attempt to emulate this derivation with our model below. It is important to note that while we have some preliminary results, our model is very much a work in progress. In order to model the system accurately, many of the fundamental constants governing the model will need to be determined experimentally. For now we have used some constants from the literature and estimated some based on similar the values of similar constants also from the literature. <br />
<br />
==Derivation of the Model Equations==<br />
The chemical reactions describing the tri-stable switch can be divided into the two categories of fast and slow reactions. Fast reactions such as dimer formation and promoter-binding occur in the scale of seconds and are therefore modeled to be in equilibrium. Conversely, slow reactions including the likes of transcription, translation, and protein degradation occur on the scale of minutes and are thus modeled to be evolving with time. <br />
<br />
====Fast reversible reaction equations====<br />
The following equations (1) describe the fast reactions. The characters L,T,A denote molecules of LacI, TetR, and, AraC, respectively while the subscripts denote whether the molecule is a monomer (blank), dimer(2), tri-mer(3), etc. The k's denote reaction rates.<br />
<br />
[[Image:Fast_rxn_eqs1.png]]<br />
<br />
Note that the equations include volume explicitly. Cell volume is modeled in this way as it is a slowly evolving function of time.<br />
<br />
====Slow irreversible reaction equations====<br />
The following equations (2) describe the slow irreversible reactions of transcription and translation (both taken into account with the reaction rate kti) and protein degradation (reaction constant kdi). The coefficients eta-ij take into account the relative translation rates of proteins from the same transcripts. For eta-ij, "i" represents the promoter responsible for producing the molecule and "j" represents the molecule being translated [ex: eta-LA corresponds to the relative rate of AraC production from the trascript produced by the LacI promoter]. To establish a convention, eta's are relative to the translation rate of the first gene on a particular transcript. Thus eta for the first gene on the mRNA transcript = 1. The alpha-i coefficients represent relative transcription rates. In this case, "i" denotes the promoter from which the mRNA is transcribed.<br />
<br />
[[Image:slow_rxn_eqs.png]]<br />
====Equations governing cell volume====<br />
The following two equations (3) describing cellular growth and division are taken directly from the aforementioned paper by Isaacs et al. The first equation describes the volume increase from the time immediately following cell division to the time immediately before it. In this equation, V0 denotes the volume of the cell at the beginning of growth and T0 denotes the time of cell division. In our model at times T=q*t0 in which q is an integer, we have volume V and protein concentration n halve - thus modeling volume division and the resulting protein redistribution. The second equation describes the dimensionless equation in which t is measured in terms of fcell-division time and the cell volume changes between 1 and 2.<br />
<br />
[[Image:vol_eq.png]]<br />
====Equations for the total number of molecules====<br />
The following equations (4) describe the total number of inhibitor protein molecules zi. d#i represents the number, "#", of molecules bound to the "i" promoter. (ex: all of the LacI tetramers bound to pLacI promoters are accounted for by the 4d1L term.)<br />
[[Image:Tot_mol_eq.png]]<br />
<br />
====Equations describing the temporal evolution of the total number inhibitor molecules====<br />
The following equations (5) describe the temporal evolution of the total number of inhibitor molecules. Beta-i = the cell division time multiplied by the combined transcription and translation rates from a given promoter "i" (t0*kti), thus representing the total number of protein molecules maximally produced by a given promoter. Similarly, Gamma-i = the cell division time multiplied by the degradation rate of protein i (t0*kdi), thus representing the total number of protein i destabilized over one cell division time. The eta and alpha terms are described in the section above describing the equations for the slow reactions. Note that the equation describing the temporal evolution of AraC includes the term "Aend." This term represents the endogenous expression of AraC in dH5alpha's. Our preliminary work suggests that this endogenous AraC has very little effect on our pBad promoter.<br />
<br />
[[Image:Ev_tot_mol_eqs.png]]<br />
<br />
====Fast reaction equilibrium equations====<br />
Compared to the slowly evolving reactions(2) described by the above equations (5), the fast reactions (1) can be considered to be in equilibrium. Thus the following equilibrium relations (6) hold.<br />
[[Image:Fast_eq_eqs.png]]<br />
<br />
====Equations modeling the presence of the inducers====<br />
The following equations (7) model the effect of the chemical inducers on the system. The model regards the proteins bound to the inducers as having increased dissociation rates (ki-#) as specified below. The reaction rate with the apostrophe denotes the original reaction rate in the absence of inducer inclusion. In these equations "I" denotes IPTG, "a" denotes arabinose, and "Tc" denotes tetracycline (or analogue aTc). The concept for and the form of these equations are based on equations described in the supplementary Information for "A bottom-up approach to gene regulation" by Guido et al. Much like the aforementioned paper by Isaacs et al., 'Bottom-up' has an extensive and brilliantly articulate derivation supplement. <br />
<br />
[[Image:inducer_eqs.png]]<br />
<br />
====Equations relating plasmid copy number to operator sites====<br />
The following equations (8) relate the plasmid copy number, "m", to the number of promoters with bound and unbound operators. As each plasmid contains 1 of each promoter, m is constant for all three promoter types. <br />
<br />
[[Image:cop_num_eqs.png]]<br />
<br />
====Simplified fast reaction equilibrium equations====<br />
The equilibrium equations governing the fast reactions (6) can be simplified by defining dimensionless equilibrium constants in the form of cij=kij/(k(-ij)*V0*A) in which A is Avogadro's number. The cL's are further simplified by defining the constant cL=c1L*c2L*c3L*c4L. Additionally by plugging in successive terms we were able to write the operator equations in terms of d0L. <br />
<br />
[[Image:Simp_fast_eq_eqs.png]]<br />
<br />
====Equations for the number of unbound operators====<br />
Combining equations (8) and the above simplified equilibrium equations(9), we are able to solve for the number of unbound operators d0i. <br />
<br />
[[Image:Unbound_eqs.png]]<br />
<br />
====Equations for the number of bound operators====<br />
By plugging the d0i's (10) back into the equations for bound operators (9), we can solve for these equations in terms of monomer concentrations.<br />
<br />
<br />
[[Image:bound_eqs.png]]<br />
<br />
====Equation manipulations to determine change in protein monomers per time====<br />
Next we can rewrite the equations describing the temporal evolution of the total number inhibitor molecules (5) in terms of equations (10) and (11). This yields the somewhat complex equations (12) which are solely in terms of monomer protein concentration. <br />
<br />
[[Image:dzdt_eqs.png]]<br />
<br />
====Equations for the change in protein monomers per time====<br />
By applying the simple mathematical manipulation (13) to equations (12) and the X derivative of equations (9) ((9) can be written in terms of X by plugging in equations (10) and (11)) we can solve for the following equations for the evolution of protein monomers as a function of time (14). I will spare you the dZi/dXi derivative equations of equation (9).<br />
<br />
[[Image:simp_mat_man_eq.png]]<br />
<br />
====Equations describing stochastic variation====<br />
<br />
====Equations for the evolution of fluorescent reporters====<br />
<br />
==Table of preliminary model constants==<br />
====General Parameters====<br />
{| border=1 cellspacing=0 cellpadding=5<br />
<br />
!align="left" width="50" | Parameter<br />
!width="50" | Value<br />
!width="300" | Meaning<br />
<br />
|-<br />
|m<br />
|20 Copies<br />
|Plasmid Copy Number<br />
|-<br />
|}<br />
<br />
====Inhibitor Protein Parameters====<br />
{| border=1 cellspacing=0 cellpadding=5<br />
<br />
!align="left" width="50" | Parameter<br />
!width="50" | Value<br />
!width="300" | Meaning<br />
<br />
|-<br />
|alphaT1<br />
|.01<br />
|Relative transcription level for repressed (1 TetR dimer bound) pTetR promoter<br />
|-<br />
|alphaT2<br />
|.01<br />
|Relative transcription level for doubly repressed (2 TetR dimers bound) pTetR promoter<br />
|-<br />
|alphaL<br />
|.01<br />
|Relative transcription level for repressed (1 LacI tetramer bound) pLacI promoter<br />
|-<br />
|alphaA<br />
|.01<br />
|Relative transcription level for repressed (two AraC monomers bound) pBAD promoter<br />
|-<br />
|betaA<br />
|1000<br />
|maximum number of proteins produced from the pBAD promoter per cell division<br />
|-<br />
|betaL<br />
|1000<br />
|maximum number of proteins produced from the pLacI promoter per cell division<br />
|-<br />
|betaT<br />
|1000<br />
|maximum number of proteins produced from the pTetR promoter per cell division<br />
|-<br />
|etaLT<br />
|1<br />
|Relative rate of TetR translation from pLacI promoter<br />
|-<br />
|etaLA<br />
|1<br />
|Relative rate of AraC translation from pLacI promoter<br />
|-<br />
|etaAL<br />
|1<br />
|Relative rate of LacI translation from pBAD promoter<br />
|-<br />
|etaAT<br />
|1<br />
|Relative rate of TetR translation from pBAD promoter<br />
|-<br />
|etaTL<br />
|1<br />
|Relative rate of LacI translation from pTetR promoter<br />
|-<br />
|etaTA<br />
|1<br />
|Relative rate of AraC translation from pTetR promoter<br />
|-<br />
|c1L<br />
|.05<br />
|Equilibrium constant for LacI dimerization<br />
|-<br />
|c2L<br />
|.05<br />
|Equilibrium constant for LacI trimerization<br />
|-<br />
|c3L<br />
|.05<br />
|Equilibrium constant for LacI tetramerization<br />
|-<br />
|c4L<br />
|.33<br />
|Equilibrium constant for LacI binding<br />
|-<br />
|}<br />
<br />
==Preliminary modeling results==<br />
<br />
The preliminary modeling results predict that the tri-stable switch will reach a natural steady state in the absence of inducers. not remain stable in the absence of the inducer. As a means of testing the validity of the model, the fundamental equations were slightly modified to code for the three combinations of bi-stable switches. However, This suggests that Full parameter scans will have to be run to test for <br />
With all of this said the modeling parameters are likely to be. Only future experiments <br />
<br />
[[Image:no_inducer.png]]<br />
<br />
Link to MATLAB code for tri-stable model [[Media:tristable1.txt]] <br />
<br />
Note: copy this code into MATLAB and save as an M-file<br />
<br />
--[[User:Jlohmuel|Jlohmuel]] 03:59, 28 October 2006 (EDT)<br />
<br />
==What we achieved==<br />
<br />
A successfully ligated pBad/arac construct in addition to partially completed ligations for the pLacI and the pTetR constructs were the results of our summer and semesters' work. The tri-stable toggle switch system was also modeled and proved to yield significant findings.<br />
<br />
==Challenges==<br />
<br />
Despite many cloning and ligation attempts, we were unable to create each component of the switch. More specifically, the pLacI and the pTetR circuits presented many ligation difficulties and proved to yield unsuccessful cloning. However, the pBAD/araC circuit was successfully constructed (yay!).<br />
<br />
==Conclusions==<br />
<br />
With the bigger picture in mind,this project yielded many successful outcomes.Firstly, we were able to successfully construct one of the three constructs necessary for a fully functional tri-stable toggle switch. Secondly, we were able to add more working parts to the registry, both a fully functioning pBad/AraC construct and many parts that were successfully ligated in the process of assembling the other two constructs. These parts will be used in the future by our team and perhaps many others as new ideas and projects develop. Also, it is necessary to mention our most recent finding from the modeling results for the tri-stable toggle switch. Interestingly enough, our model suggests that it is biologically unfeasible to create a switch with more than two components. If more than two components are introduced, such as three, the system is unable to maintain any one state and it will revert back to expressing its most dominant promoter construct. We find such results quite meaningful in demonstrating the biological limits of a system under study.<br />
<br />
==Future work==<br />
<br />
In the coming months, it is hoped that the failure of the ligation strategies for the pLacI and the pTetR constructs can be further investigated. We feel confident that we can continue to build from the one successful construction,the pBad/Arac construct and complete the other 2 constructs. To this end, it will be necessary to further troubleshoot the possible reasons why the cloning of the ligations were unsuccessful.</div>Jlohmuel