Template:ETH Modeling Formulas
From 2006.igem.org
(Difference between revisions)
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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 <==> S•E --> P + E | |
k<sub>−1</sub> | k<sub>−1</sub> | ||
| - | d[ | + | d[S]/dt = −k<sub>+1</sub>[S][E] + k<sub>−1</sub>[S•E] − 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>[S•E] + k<sub>2</sub>[S•E] − d<sub>E</sub>[E] |
| - | d[P]/dt = <sub> </sub> + k<sub>2</sub>[ | + | d[P]/dt = <sub> </sub> + k<sub>2</sub>[S•E] − d<sub>P</sub>[P] |
| + | |||
| + | [S] : substrate concentration | ||
| + | [E] : enzyme conc. | ||
| + | [E•] : concentration of enzyme-substrate complex | ||
| + | [P] : product concentration | ||
kinetic constants: | kinetic constants: | ||
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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 konstant (transcription) | k<sub>tr</sub> : kinetic konstant (transcription) | ||
| + | u<sub> </sub> : system input, e.g. transcription rate (≅PoPS) | ||
d<sub>M </sub> : degradation konstant for mRNA | d<sub>M </sub> : degradation konstant for mRNA | ||
Revision as of 12:40, 30 October 2006
To get our models into a form which can be simulated, we needed to transform the wiring diagrams into a set of ODE's (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 <==> S•E --> P + E
k−1
d[S]/dt = −k+1[S][E] + k−1[S•E] − dS[S]
d[E]/dt = −k+1[S][E] + k−1[S•E] + k2[S•E] − dE[E]
d[P]/dt = + k2[S•E] − dP[P]
[S] : substrate concentration
[E] : enzyme conc.
[E•] : 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 konstant (transcription) u : system input, e.g. transcription rate (≅PoPS) dM : degradation konstant 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 konstant (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 konstant for mRNA
Finally, translation is usually modelled like this:
d[P]/dt = ktl[M] − dP[P] [P] : product (protein) concentration [M] : mRNA concentration ktl : kinetic konstant (translation) dP : degradation konstant 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].
