Stochastic modeling of molecular reaction networks
Download
1 / 44

Stochastic modeling of molecular reaction networks - PowerPoint PPT Presentation


  • 78 Views
  • Uploaded on

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

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about ' Stochastic modeling of molecular reaction networks' - theta


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
Stochastic modeling of molecular reaction networks

Stochastic modeling of molecular reaction networks

Daniel Forger

University of Michigan



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 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 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


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
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”



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
Matlab simulation Cyanobacteria


Mass action represents a limiting case of stochastics
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 stochastic and mass action approaches agree only if there are enough molecules
Myth 2: CyanobacteriaStochastic 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 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
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 example GillespieKepler and Elston Biophysical Journal 81:3116




Distribution of molecules often looks gaussian
Distribution of molecules each stateoften looks Gaussian


Moment descriptions
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
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
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
Examples each state

  • Transcription Control

  • Lac Operon

  • Oscillations

  • Accounting for diffusion


Rossi et al molecular cell
Rossi et al. each stateMolecular Cell




Saddle-Node on an each state

Invariant Circle

x2

SNIC

max

max

saddle

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


ad