Template:ETH Modeling Formulas
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k<sub>−1</sub> | k<sub>−1</sub> | ||
| - | d[S]/dt = −k<sub>+1</sub>[S][E] + k<sub>−1</sub>[E•S] − d<sub>S</sub>[S] | + | d[S]/dt = −k<sub>+1</sub>[S][E] + k<sub>−1</sub>[E•S] − d<sub>S</sub>[S] |
| - | d[E]/dt = −k<sub>+1</sub>[S][E] + k<sub>−1</sub>[E•S] + k<sub>2</sub>[S•E] − | + | 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>[E•S] − d<sub>P</sub>[P] | + | 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 | [S] : substrate concentration | ||
Revision as of 11:37, 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, 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, MIT Press.
