Template:ETH Modeling Formulas

From 2006.igem.org

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].