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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. 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

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  1. Stochastic modeling of molecular reaction networks Daniel Forger University of Michigan

  2. Let’s begin with a simple genetic network

  3. 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

  4. 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

  5. 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

  6. 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

  7. 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 #

  8. 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

  9. 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

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

  11. Consider a simple model inspired by the circadian clock in Cyanobacteria A B C

  12. 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.)

  13. Mass Action Representation

  14. Matlab simulation

  15. 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)

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

  17. 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

  18. There are several representations in between Mass Action and Gillespie • Chemical Langevin Equations • Master Equations • Fokker-Planck • Moment descriptions

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

  20. Master Equations describe how the probability of being in each state

  21. Sometimes we can solve for the mean and variance

  22. Distribution of molecules often looks Gaussian

  23. 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)

  24. 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

  25. Note If 1/m* is small, we can then derive a simplifed Version of the Master equations

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

  27. Adalsteinsson et al. BMC Bioinformatics 5:24

  28. Examples • Transcription Control • Lac Operon • Oscillations • Accounting for diffusion

  29. Rossi et al. Molecular Cell

  30. Ozbudak et al. Nature 427:737

  31. Guantes and Poyatos PLoS Computational Biology 2:e30

  32. Saddle-Node on an Invariant Circle x2 SNIC max max saddle min node p1 SNIC Bifurcation Invariant Circle Limit Cycle

  33. x2 max slc uss sss min p1 Hopf Bifurcation stable limit cycle

  34. Noise Induced oscillations

  35. Liu et al. Cell 129:605

  36. 3-D Gillespiehttp://www.math.utah.edu/~isaacson/3dmodel.html

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