Template:ETH Modeling Formulas

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         k<sub>&minus;1</sub>
         k<sub>&minus;1</sub>
   
   
-
  d[S]/dt = &minus;k<sub>+1</sub>[S][E] + k<sub>&minus;1</sub>[E&bull;S]          &minus; d<sub>S</sub>[S]
+
  d[S]/dt   = &minus;k<sub>+1</sub>[S][E] + k<sub>&minus;1</sub>[E&bull;S]          &minus; d<sub>S</sub>[S]
-
  d[E]/dt = &minus;k<sub>+1</sub>[S][E] + k<sub>&minus;1</sub>[E&bull;S] + k<sub>2</sub>[S&bull;E] &minus; d<sub>E</sub>[E]
+
  d[E]/dt   = &minus;k<sub>+1</sub>[S][E] + k<sub>&minus;1</sub>[E&bull;S] + k<sub>2</sub>[E&bull;S] &minus; d<sub>E</sub>[E]
-
  d[P]/dt =                  <sub>    </sub> + k<sub>2</sub>[E&bull;S] &minus; d<sub>P</sub>[P]
+
d[E&bull;S]/dt =  k<sub>+1</sub>[S][E] &minus; k<sub>&minus;1</sub>[E&bull;S] &minus; k<sub>2</sub>[E&bull;S] &minus; d<sub>ES</sub>[E&bull;S]
 +
  d[P]/dt   =                  <sub>    </sub> + k<sub>2</sub>[E&bull;S] &minus; 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, [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].
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