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Stochastic modeling of molecular reaction networksPowerPoint Presentation

Stochastic modeling of molecular reaction networks

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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 degradation degPo*Po -Po

Pt degradation degPt*Pt -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

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 degradation degPo*Po -Po

Pt degradation degPt*Pt -Pt

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

Unbinding to DNA unbin*G -G

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

- 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

- 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

- 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 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.)

Mass Action Representation Cyanobacteria

Matlab simulation Cyanobacteria

Mass Action represents a limiting case of Stochastics 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

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 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

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

Adalsteinsson et al. BMC Bioinformatics 5:24 each state

Examples each state

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

Rossi et al. each stateMolecular Cell

Ozbudak et al. Nature 427:737 each state

Saddle-Node on an each state

Invariant Circle

x2

SNIC

max

max

saddle

min

node

p1

SNIC Bifurcation

Invariant Circle

Limit Cycle

Noise Induced oscillations each state

Liu et al. Cell 129:605 each state

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

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