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# Stochastic modeling of molecular reaction networks - PowerPoint PPT Presentation

Stochastic modeling of molecular reaction networks. Daniel Forger University of Michigan. Let’s begin with a simple genetic network. We can list the basic reaction rates and stochiometry. numsites = total # of sites on a gene, G = # sites bound

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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

Binding to DNA bin(numsites - G)*Pt -Pt, +G

Unbinding to DNA unbin*G -G

We normally track concentrationLet’s track # molecules instead

• 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

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

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

Unbinding to DNA unbin*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

Myth 1:“Mass Action Formulations do not account for Stochasticity”

A

B

C

A Cyanobacteria

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.)

Matlab simulation Cyanobacteria

• Mass action and stochastic simulations should agree when certain “limits” are obtained

• Mass action typically represents the expected concentrations of chemical species (more later)

Myth 2: CyanobacteriaStochastic and Mass Action Approaches agree only if there are enough molecules

What matters is the number of reactions Cyanobacteria

• 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

We will illustrate this with an example GillespieKepler and Elston Biophysical Journal 81:3116

Distribution of molecules each stateoften looks Gaussian

Moment Descriptions each state

• 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)

Towards Fokker Planck each state

• 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 each state

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

Version of the Master equations

Chemical Langevin Equations each state

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

Examples each state

• Transcription Control

• Lac Operon

• Oscillations

• Accounting for diffusion

Rossi et al. each stateMolecular Cell

Invariant Circle

x2

SNIC

max

max

min

node

p1

SNIC Bifurcation

Invariant Circle

Limit Cycle

x each state2

max

slc

uss

sss

min

p1

Hopf Bifurcation

stable

limit cycle

3-D Gillespie each statehttp://www.math.utah.edu/~isaacson/3dmodel.html