Non-Markovian dynamics of small genetic circuits Lev Tsimring Institute for Nonlinear Science University of California, San Diego Le Houches, 9-20 April, 2007

Outline Deterministic and stochastic descriptions of genetic circuits with very different time scales Non-Markovian effects in gene regulation –transcriptional delay-induced stochastic oscillations

Gene regulatory networks Proteins affect rates of production of other proteins (or themselves) This leads to formation of networks of interacting genes/proteins Different reaction channels operate at vastly different time scales and number densities Sub-networks are non-Markovian, even if the whole system is Compound reactions are non-Markovian A B C DE A B D

Transients in gene regulation Genetic circuits are never at a fixed point: –Cell cycle; volume growth; division –External signaling –Intrinsic noise –Extrinsic “noise” –Circadian rhythms; ultradian rhythms

Interesting design (modeling) issues arise naturally Separation of timescales – multiple time-scale analysis Nonlinearity due to multimerization, cooperativity and feedback – bifurcation analysis Time delays Spatial compartments and cell signaling - spatial models Cell-to-cell variations are large In order to build gene circuits to perform cellular “tasks”, we need to understand the origins of the variability

External signaling: -Phage Life Cycle M.Ptashne, 2002

Engineered Toggle Switch Gardner, Cantor & Collins, Nature 403:339 (2001) Construction/experiments: Model

Circadian clock in Neurospora crassa WC-1 WC-2 WCC FRQ P.Ruoff

Ultradian clock at yeast Klevecz et al, ,329 expressed genes Reductive phase Respiratory phase Average peak-to-trough ratio ~2 Synchronized culture

The Repressilator Elowitz and Leibler, Nature 403:335 (2001) Model Construction/experiments:

RNAP Auto-repressor: A cartoon promoter gene RNAP DNA Binding/unbinding rate: <1 sec Transcription rate: ~10 3 basepairs/min Translation rate: ~10 2 aminoacids/min mRNA degradation rate ~3min Transport in/out nucleus 10+ min Protein degradation rate ~ 30min..hours protein mRNA

Oscillations in gene regulation promoter gene RNAP DNA RNAP repressor mRNA

Single gene autoregulation Fast Slow Binding/unbinding rate (k -1,k -2 ): ~1 sec -1 Transcription rate (k t ): ~1 min min -1 Protein degradation (k x ) ~0.01 min -1 

Single gene autoregulation  Fast Slow

Quasi-steady-state approximation (naïve approach) Fixed points – yes, Dynamics – no: x is a fast variable also! Correct?

Separation of scales (Correct projection) slow variable Prefactor is important if x 2 /x~1, i.e. lots of dimers Prefactor makes transients slower Kepler & Elston, 2001 Bundschuh et al, 2003 Bennett et al, 2007, in press

Genetic toggle switch [Gardner, Cantor, Collins, Nature 2000] Gene AonGene B offReporter GFP Protein A Gene A off Gene B on Reporter Protein B “On” “Off”

Multiple-scale analysis Fast reactions

Multiple-scale analysis (cont’d) slow variable constant Local equilibrium for fast reactions Nullspace of the adjoint linear operator [2 eigenvectors] From orthogonality conditions:

Prefactor w/o prefactor with prefactor full model

Stochastic gene expression: Master Equation approach Two reactions: production degradation Probability of having x molecules of X at time t, Dynamics of Continuum limit ( x >>1): Fokker-Planck equation

Stochastic gene expression: Langevin equation approach Two reactions: production degradation Deterministic equation: Each reaction is a noisy Poisson process, mean=variance Separately: Since reactions are uncorrelated, variances add: Langevin equations From Langevin equation to FPE (van Kampen, Stochastic Processes in Chemistry and Physics,1992): …or from FPE to Langevin!

Autoregulation: stochastic description Master equation for Projection: using n – total # of monomers; u – # of unbound dimers; b - # of bound dimers

Back to ODE In the continuum limit (large n ): Fokker-Planck equation Corresponding Langevin equation with (no prefactor) Fast reaction noise is filtered out

Multiscale stochastic simulations (turbo-charged Gillespie algorithm) The computational analog of the projection procedure: stochastic partial equilibrium (Cao, Petzold, Gillespie, 2005): – Identify slow and fast variables –Fast reactions at quasi-equilibrium –distribution for fixed is assumed known –Compute propensities for slow reactions Easy for zero- and first-order reactions, more tricky for higher order reactions

Regulatory delay in genetic circuits

Single gene autoregulation: transcriptional delay  Fast Slow Delayed After projection [cf. Santillán & Mackey, 2001]

Genetic oscillations: Hopf bifurcation Fixed point: Complex eigenvalues Instability ktTktT

Transcriptional delay: a non-Markovian process Markovian reactions [dimerization, degradation, binding]: exponential “next reaction” time distribution Non-Markovian channels [transcription, translation]: Gaussian time distribution which reaction to choose? Stochastic simulations (modified Gillespie algorithm) update

Scheme of numerical simulation: delay time steps Modified Direct Gillespie algorithm (Gillespie, 1977): 1.Input values for initial state, set t=0 2.Compute propensities 3.Generate random numbers 4.Compute time step until next reaction 5.Check if there has been a delayed reaction scheduled in a) if yes, then last steps 2,3,4 are ignored, time advances to, update in accordance with delayed reaction b) if not, go to the step 6 6.Find the channel of the next reaction: 7.Update time and

Stochastic simulations

Analytical results Reactions: Deterministic model No Hopf bifurcation! Stochastic model (Master equations) probability to have n monomers at time t given the state s at time t-  Approximation: (no dimerization)

Boolean model Transition probability if at time t depends on the state at t-T: positive feedback negative feedback For double-well quartic potential Two-state gene: 1

Master equations the probability of having value s(t) =  1 at time t; s =  1 to 1 s = 1 to  1 probability of transition from within ( t,t+dt) Delayed master equation

Autocorrelation function Linear equation!

Autocorrelation function T=1000, p 1 =0.1 p 2 =0.3 Stochastic oscillations!

Power spectrum: two-state model  S ()()  =0.05  =-0.05

Analytical results Reactions: Deterministic model No Hopf bifurcation! Stochastic model (Master equations) probability to have n monomers at time t given the state s at time t-  Approximation: (no dimerization)

Analytical results Correlation function: Result:

Standard deviation/mean Time delay increases noise level Effect of stochasticity and delay on regulation

Conclusions Fast binding-unbinding processes can be eliminated both in deterministic and stochastic modeling, however an accurate averaging procedure has to be used: leads to prefactors affecting transient times and noise distributions Multimerization increases time scales of genetic regulation Deterministic and stochastic description of regulatory delays developed, delays of transcription/translation of auto-repressor may lead to increased fluctuations levels and oscillations even when deterministic model shows no Hopf bifurcation Modified Gillespie algorithm is developed for simulating time- delayed reactions L.S. Tsimring and A. Pikovsky, Phys. Rev. Lett., 87, (2001). D.A. Bratsun, D. Volfson, L.S. Tsimring, and J. Hasty, PNAS, 102, (2005). M. Bennett, D. Volfson, L. Tsimring, and J. Hasty, Biophys. J., 2007, in press.