Template:ETH Modeling Formulas
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| - | To get our models into a form which can be simulated, we needed to transform the ''wiring diagrams'' into a set of | + | To get our models into a form which can be simulated, we needed to transform the ''wiring diagrams'' into a set of ODEs (ordinary differential equations), which in our case will be non-linear. |
For every concerned species <tt>X</tt>, we write | For every concerned species <tt>X</tt>, we write | ||
d[X]/dt = production - consumption | d[X]/dt = production - consumption | ||
| - | For enzymatic transformation of substrate | + | For enzymatic transformation of substrate S into product P (catalyzed by enzyme E), we write |
k<sub>+1</sub> k<sub>2</sub> | k<sub>+1</sub> k<sub>2</sub> | ||
| - | + | S + E <==> E•S --> P + E | |
k<sub>−1</sub> | k<sub>−1</sub> | ||
| - | d[ | + | d[S]/dt = −k<sub>+1</sub>[S][E] + k<sub>−1</sub>[E•S] <sub> </sub> − d<sub>S</sub>[S] |
| - | d[E]/dt = −k<sub>+1</sub>[ | + | d[E]/dt = −k<sub>+1</sub>[S][E] + k<sub>−1</sub>[E•S] + k<sub>2</sub>[E•S] − d<sub>E</sub>[E] |
| - | d[P]/dt = <sub> </sub> + k<sub>2</sub>[ | + | d[E•S]/dt = k<sub>+1</sub>[S][E] − k<sub>−1</sub>[E•S] − k<sub>2</sub>[E•S] − d<sub>ES</sub>[E•S] |
| + | d[P]/dt = <sub> </sub> + k<sub>2</sub>[E•S] − d<sub>P</sub>[P] | ||
| + | |||
| + | [S] : substrate concentration | ||
| + | [E] : enzyme conc. | ||
| + | [E•S] : concentration of enzyme-substrate complex | ||
| + | [P] : product concentration | ||
kinetic constants: | kinetic constants: | ||
| Line 21: | Line 27: | ||
For constitutive transcription, we have constant production rate and simply write | For constitutive transcription, we have constant production rate and simply write | ||
| - | d[M]/dt = k<sub>tr</sub> − d<sub>M</sub>[M] | + | d[M]/dt = k<sub>tr</sub>•u − d<sub>M</sub>[M] |
[M]<sub> </sub>: mRNA concentration | [M]<sub> </sub>: mRNA concentration | ||
| - | k<sub>tr</sub> : kinetic | + | k<sub>tr</sub> : kinetic constant (transcription) |
| - | d<sub>M </sub> : degradation | + | u<sub> </sub> : system input, e.g. transcription rate (≅PoPS) |
| + | d<sub>M </sub> : degradation constant for mRNA | ||
A transcriptional regulatory module can be described by and ODE of the following form: | A transcriptional regulatory module can be described by and ODE of the following form: | ||
| Line 33: | Line 40: | ||
[M]<sub> </sub>: mRNA concentration | [M]<sub> </sub>: mRNA concentration | ||
| - | k<sub>tr</sub> : kinetic | + | k<sub>tr</sub> : kinetic constant (transcription) |
a<sub> </sub> : constitutive portion, 0 ≤ a < 1 | a<sub> </sub> : constitutive portion, 0 ≤ a < 1 | ||
[S]<sub> </sub>: inducer (α=+1) / repressor (α=−1) concentration | [S]<sub> </sub>: inducer (α=+1) / repressor (α=−1) concentration | ||
| Line 39: | Line 46: | ||
n<sub> </sub> : hill coefficient | n<sub> </sub> : hill coefficient | ||
α<sub> </sub> : α=+1 for induction, α=−1 for repression | α<sub> </sub> : α=+1 for induction, α=−1 for repression | ||
| - | d<sub>M </sub> : degradation | + | d<sub>M </sub> : degradation constant for mRNA |
| - | Finally, translation is usually | + | Finally, translation is usually modeled like this: |
d[P]/dt = k<sub>tl</sub>[M] − d<sub>P</sub>[P] | d[P]/dt = k<sub>tl</sub>[M] − d<sub>P</sub>[P] | ||
[P]<sub> </sub>: product (protein) concentration | [P]<sub> </sub>: product (protein) concentration | ||
[M]<sub> </sub>: mRNA concentration | [M]<sub> </sub>: mRNA concentration | ||
| - | k<sub>tl</sub> : kinetic | + | k<sub>tl</sub> : kinetic constant (translation) |
| - | d<sub>P </sub> : degradation | + | d<sub>P </sub> : degradation constant for protein P |
References: | References: | ||
* ''Modeling Molecular Interaction Networks with Nonlinear Ordinary Differential Equations''. Emery D. Conrad and John J. Tyson<br/>in ''System Modeling in Cellular Biology. From Concepts to Nuts and Bolts.''<br/>Editors: Zoltan Szallasi, Jorg Stelling and Vipul Periwal, [http://mitpress.mit.edu/catalog/item/default.asp?ttype=2&tid=10923 MIT Press]. | * ''Modeling Molecular Interaction Networks with Nonlinear Ordinary Differential Equations''. Emery D. Conrad and John J. Tyson<br/>in ''System Modeling in Cellular Biology. From Concepts to Nuts and Bolts.''<br/>Editors: Zoltan Szallasi, Jorg Stelling and Vipul Periwal, [http://mitpress.mit.edu/catalog/item/default.asp?ttype=2&tid=10923 MIT Press]. | ||
* ''Synthetic Gene Regulatory Systems''. Mads Kaern and Ron Weiss<br/>in ''System Modeling in Cellular Biology. From Concepts to Nuts and Bolts.''<br/>Editors: Zoltan Szallasi, Jorg Stelling and Vipul Periwal, [http://mitpress.mit.edu/catalog/item/default.asp?ttype=2&tid=10923 MIT Press]. | * ''Synthetic Gene Regulatory Systems''. Mads Kaern and Ron Weiss<br/>in ''System Modeling in Cellular Biology. From Concepts to Nuts and Bolts.''<br/>Editors: Zoltan Szallasi, Jorg Stelling and Vipul Periwal, [http://mitpress.mit.edu/catalog/item/default.asp?ttype=2&tid=10923 MIT Press]. | ||
Latest revision as of 11:41, 20 November 2006
To get our models into a form which can be simulated, we needed to transform the wiring diagrams into a set of ODEs (ordinary differential equations), which in our case will be non-linear.
For every concerned species X, we write
d[X]/dt = production - consumption
For enzymatic transformation of substrate S into product P (catalyzed by enzyme E), we write
k+1 k2
S + E <==> E•S --> P + E
k−1
d[S]/dt = −k+1[S][E] + k−1[E•S] − dS[S]
d[E]/dt = −k+1[S][E] + k−1[E•S] + k2[E•S] − dE[E]
d[E•S]/dt = k+1[S][E] − k−1[E•S] − k2[E•S] − dES[E•S]
d[P]/dt = + k2[E•S] − dP[P]
[S] : substrate concentration
[E] : enzyme conc.
[E•S] : concentration of enzyme-substrate complex
[P] : product concentration
kinetic constants:
kk+1 : building enzyme-substrate complex (forward)
kk−1 : resolving enzyme-substrate complex (backward)
k2 : product formation
dXXX : degradation constants
For constitutive transcription, we have constant production rate and simply write
d[M]/dt = ktr•u − dM[M] [M] : mRNA concentration ktr : kinetic constant (transcription) u : system input, e.g. transcription rate (≅PoPS) dM : degradation constant for mRNA
A transcriptional regulatory module can be described by and ODE of the following form:
1
d[M]/dt = ktr ( a + −−−−−−−−−−−−−− ) − dM[M]
1 + (K/[S])α•n
[M] : mRNA concentration
ktr : kinetic constant (transcription)
a : constitutive portion, 0 ≤ a < 1
[S] : inducer (α=+1) / repressor (α=−1) concentration
K : hill constant
n : hill coefficient
α : α=+1 for induction, α=−1 for repression
dM : degradation constant for mRNA
Finally, translation is usually modeled like this:
d[P]/dt = ktl[M] − dP[P] [P] : product (protein) concentration [M] : mRNA concentration ktl : kinetic constant (translation) dP : degradation constant for protein P
References:
- Modeling Molecular Interaction Networks with Nonlinear Ordinary Differential Equations. Emery D. Conrad and John J. Tyson
in System Modeling in Cellular Biology. From Concepts to Nuts and Bolts.
Editors: Zoltan Szallasi, Jorg Stelling and Vipul Periwal, [http://mitpress.mit.edu/catalog/item/default.asp?ttype=2&tid=10923 MIT Press]. - Synthetic Gene Regulatory Systems. Mads Kaern and Ron Weiss
in System Modeling in Cellular Biology. From Concepts to Nuts and Bolts.
Editors: Zoltan Szallasi, Jorg Stelling and Vipul Periwal, [http://mitpress.mit.edu/catalog/item/default.asp?ttype=2&tid=10923 MIT Press].
