Stochastic modeling of molecular reaction networks
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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

Stochastic modeling of molecular reaction networks

Daniel Forger

University of Michigan


Let s begin with a simple genetic network

Let’s begin with a simple genetic network


We can list the basic reaction rates and stochiometry

We can list the basic reaction rates and stochiometry

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


We normally track concentration let s track molecules instead

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


Stochastic modeling of molecular reaction networks

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


How would you simulate this

How would you simulate this?

  • Choose which reaction happens next

    • Find next reaction

    • Update species by stochiometry of next reaction

    • Find time to this next reaction


How to find the next reaction

How to find the 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 #


Now that we know the next reaction modifies the protein

Now that we know the next reaction modifies the protein

  • Po = Po - 1

  • Pt = Pt + 1

  • How much time has elapsed

    • a0 = sum of reaction rates

    • r0 = random # between 0 and 1


This method goes by many names

This method goes by many names

  • 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

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


Consider a simple model inspired by the circadian clock in cyanobacteria

Consider a simple model inspired by the circadian clock in Cyanobacteria

A

B

C


Stochastic modeling of molecular reaction networks

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 representation

Mass Action Representation


Matlab simulation

Matlab simulation


Mass action represents a limiting case of stochastics

Mass Action represents a limiting case of Stochastics

  • 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 stochastic and mass action approaches agree only if there are enough molecules

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


What matters is the number of reactions

What matters is the number of reactions

  • 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


There are several representations in between mass action and gillespie

There are several representations in between Mass Action and Gillespie

  • Chemical Langevin Equations

  • Master Equations

  • Fokker-Planck

  • Moment descriptions


We will illustrate this with an example kepler and elston biophysical journal 81 3116

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


Master equations describe how the probability of being in each state

Master Equations describe how the probability of being in each state


Sometimes we can solve for the mean and variance

Sometimes we can solve for the mean and variance


Distribution of molecules often looks gaussian

Distribution of molecules often looks Gaussian


Moment descriptions

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)


Towards fokker planck

Towards Fokker Planck

  • 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


Stochastic modeling of molecular reaction networks

Note

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

Version of the Master equations


Chemical langevin equations

Chemical Langevin Equations

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


Adalsteinsson et al bmc bioinformatics 5 24

Adalsteinsson et al. BMC Bioinformatics 5:24


Examples

Examples

  • Transcription Control

  • Lac Operon

  • Oscillations

  • Accounting for diffusion


Rossi et al molecular cell

Rossi et al. Molecular Cell


Ozbudak et al nature 427 737

Ozbudak et al. Nature 427:737


Guantes and poyatos plos computational biology 2 e30

Guantes and Poyatos PLoS Computational Biology 2:e30


Stochastic modeling of molecular reaction networks

Saddle-Node on an

Invariant Circle

x2

SNIC

max

max

saddle

min

node

p1

SNIC Bifurcation

Invariant Circle

Limit Cycle


Stochastic modeling of molecular reaction networks

x2

max

slc

uss

sss

min

p1

Hopf Bifurcation

stable

limit cycle


Noise induced oscillations

Noise Induced oscillations


Stochastic modeling of molecular reaction networks

Liu et al. Cell 129:605


3 d gillespie http www math utah edu isaacson 3dmodel html

3-D Gillespiehttp://www.math.utah.edu/~isaacson/3dmodel.html


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