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Stochastic modeling of molecular reaction networks

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Stochastic modeling of molecular reaction networks

Daniel Forger

University of Michigan

numsites = total # of sites on a gene, G = # sites bound

M = mRNA, Po = unmodified protein, Pt = modified protein

Transcription trans or 0+M

Translation tl*M+Po

Protein Modification conv*Po-Po, +Pt

M degradation degM*M-M

Po degradationdegPo*Po-Po

Pt degradationdegPt*Pt-Pt

Binding to DNAbin(numsites - G)*Pt -Pt, +G

Unbinding to DNAunbin*G-G

- Let M, Po, Pt be # molecules
- First order rate constants (tl, unbin, conv, degM, degPo and degPt) have units 1/time and stay constant
- Zero order rate constant (trans) has units conc/time, so multiply it by volume
- 2nd order rate constant (bin) has units 1/(conc*time), so divide it by volume

numsites = total # of sites on a gene, G = # sites bound

M = mRNA, Po = unmodified protein, Pt = modified protein

V = Volume

Transcription trans*V or 0 +M

Translation tl*M +Po

Protein Modification conv*Po -Po, +Pt

M degradation degM*M -M

Po degradationdegPo*Po -Po

Pt degradationdegPt*Pt -Pt

Binding to DNAbin/V(numsites - G)*Pt -Pt, +G

Unbinding to DNAunbin*G -G

- Choose which reaction happens next
- Find next reaction
- Update species by stochiometry of next reaction
- Find time to this next reaction

- Choose randomly based on their reaction rates

trans*V

tl*M

degM*M

degPo*Po

degPt*Pt

conv*Po

unbin*G

bin/V(numsites - G)*Pt

Random #

- Po = Po - 1
- Pt = Pt + 1
- How much time has elapsed
- a0 = sum of reaction rates
- r0 = random # between 0 and 1

- Computational Biologists typically call this the Gillespie Method
- Gillespie also has another method

- Material Scientists typically call this Kinetic Monte Carlo

A

B

C

A

B

C

- Here a protein can be in 3 states, A, B or C
- We start the system with 100 molecules of A
- Assume all rates are 1, and that reactions occur without randomness (it takes one time unit to go from A to B, etc.)

- Mass action and stochastic simulations should agree when certain “limits” are obtained
- Mass action typically represents the expected concentrations of chemical species (more later)

- This is particularly important for reversible reactions
- By the central limit theorem, fluctuations dissapear like n-1/2
- There are almost always a very limited number of genes,
- Ok if fast binding and unbinding

- Chemical Langevin Equations
- Master Equations
- Fokker-Planck
- Moment descriptions

- Gaussian Random Variables are fully characterized by their mean and standard deviation
- We can write down odes for the mean and standard deviation of each variable
- However, for bimolecular reactions, we need to know the correlations between variables (potentially N2)

- Let’s divide the master equation by the mean m*.
- Although this equation described many states, we can smooth the states to make a probability distribution function

Note

If 1/m* is small, we can then derive a simplifed

Version of the Master equations

- If we don’t want the whole probability distribution, we can sometimes derive a stochastic differential equation to generate a sample

- Transcription Control
- Lac Operon
- Oscillations
- Accounting for diffusion

Saddle-Node on an

Invariant Circle

x2

SNIC

max

max

saddle

min

node

p1

SNIC Bifurcation

Invariant Circle

Limit Cycle

x2

max

slc

uss

sss

min

p1

Hopf Bifurcation

stable

limit cycle

Liu et al. Cell 129:605