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 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 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 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? 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*Vtl*M conv*Po degM*MdegPo*PodegPt*Pt bin/V(numsites - G)*Pt unbin*G Random #

Now that we know the next reaction modifies the protein Po = Po - 1 Pt = Pt + 1 How much time has elapsed –a 0 = sum of reaction rates –r 0 = 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”

Consider a simple model inspired by the circadian clock in Cyanobacteria AB 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.) AB C

Mass Action Representation

Matlab simulation

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

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 Chemical Langevin Equations Master Equations Fokker-Planck Moment descriptions

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

Master Equations describe how the probability of being in each state

Sometimes we can solve for the mean and variance

Distribution of molecules often looks Gaussian

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 N 2 )

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

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

Examples Transcription Control Lac Operon Oscillations Accounting for diffusion

Rossi et al. Molecular Cell

Ozbudak et al. Nature 427:737

Guantes and Poyatos PLoS Computational Biology 2:e30

SNIC Bifurcation Invariant Circle Limit Cycle x2x2 p1p1 node saddle Saddle-Node on an Invariant Circle max min max SNIC

Hopf Bifurcation x2x2 p1p1 stable limit cycle sss uss slc max min

Noise Induced oscillations

Liu et al. Cell 129:605

3-D Gillespie