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

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

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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

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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

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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

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

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

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

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Ultradian clock at yeast Klevecz et al, 2004 5,329 expressed genes Reductive phase Respiratory phase Average peak-to-trough ratio ~2 Synchronized culture

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

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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

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Oscillations in gene regulation promoter gene RNAP DNA RNAP repressor mRNA

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

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Single gene autoregulation Fast Slow

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

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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

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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”

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Multiple-scale analysis Fast reactions

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

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Prefactor w/o prefactor with prefactor full model

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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

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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!

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

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

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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

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Regulatory delay in genetic circuits

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

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

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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

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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

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Stochastic simulations

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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)

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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

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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

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Autocorrelation function Linear equation!

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

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Power spectrum: two-state model 10 -4 10 -3 10 -2 10 10 0 1 2 S ()() =0.05 =-0.05

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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)

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Analytical results Correlation function: Result:

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

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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, 2506021 (2001). D.A. Bratsun, D. Volfson, L.S. Tsimring, and J. Hasty, PNAS, 102, 14593-12598 (2005). M. Bennett, D. Volfson, L. Tsimring, and J. Hasty, Biophys. J., 2007, in press.

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