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Stochastic description of gene regulatory mechanisms

Stochastic description of gene regulatory mechanisms. 08.02.2006 Georg Fritz Statistical and Biological Physics Group LMU München Albert-Ludwigs Universität Freiburg. Outline. Part I: Simulation of stochastic chemical systems with the Gillespie algorithm Chemical master equation (CME)

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Stochastic description of gene regulatory mechanisms

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  1. Stochastic description of gene regulatory mechanisms 08.02.2006 Georg Fritz Statistical and Biological Physics Group LMU München Albert-Ludwigs Universität Freiburg

  2. Outline • Part I: Simulation of stochastic chemical systems with the Gillespie algorithm • Chemical master equation (CME) • Reaction probability density function ) Gillespie algorithm • Part II: Application to gene regulatory mechanisms • Bistable autoregulatory network motif • Deterministic description by ODE‘s • Model reduction • Fixedpoint analysis • Stochastic simulation • Glance at the C-code • Timeseries: fluctuation-driven transitions between ‚fixedpoints‘ • Summary

  3. Chemical master equation • M reactions R, N reactants Si with molecule numbers Xi • Well stirred system, no spacial effects considered • c dt: prob. of one reaction  in dt, given one reactant combination • h: number of distinct molecular reactant combinations, e.g. h1=X1 X2 • a dt := h c dt: prob. that any reaction of the type R will occur in (t, t+dt)

  4. Chemical master equation • Solution hard/impossible (for interesting problems) • Use CME to derive time evolution of the moments • Nonlinearities lead to involvement of higher moments • Alternative: Measure many realizations of the stochastic process and estimate the quantity of interest ) Gillespie algorithm

  5. The Gillespie algorithm*: Simulation of the reaction probability density function • Known as the BKL (Bortz-Kalos-Lebowitz) algorithm in the physical literature • Equivalent to the chemical master equation • Basic idea: when will the next reaction occur, what kind of reaction is it? • Described by the reaction probability density function P(,m) • P(,) d := prob. that, given the state (X1,…,XN) at time t, the next reaction will occur in (t+,t++d) and will be an R reaction *D. Gillespie, Exact stochastic simulation of coupled chemical reactions, J. Phys. Chem. 81, 1977

  6. The reaction probability density function • Goal: determine P(,m) • P0() ´ prob. that no reaction occurs in (t, t+) • P0(+d) = P0() [1-a d] • P(,) d = P0() a d

  7. Simulation of P(,m) • Generate a random pair (,) according to • Remember Wolfram‘s talk: generate r1,r22 UD(0,1) and compute

  8. The Algorithm • Step 0 (Initialization): set the reaction rates c1,…,cM and the initial molecular population numbers X1,…,XN • Step 1: calculate the propensities a1=h1¢c1, …, aM=hM¢cM and the total propensity a0 • Step 2: generate random numbers  and  according to P(,) • Step 3: increase time t by  and update molecule numbers according to reaction  • if t < tint goto Step 1

  9. Part II: Application to autoregulatory genetic network motif transcription factor translation transcription M. Ptashne and A. Gann, Imposing specificity by localization: mechanism and evolvability, Curr. Biol., 1998, 8:R812-R822

  10. Positive autoregulation • # RNA polymerases large ) subsumed into transcription rate • positive regulation: c0 << c1 • burst factor b = c2/c9 determines the number of proteins produced per mRNA

  11. Deterministic approach: model reduction

  12. Fixedpoint analysis for / < 2K both unstable for / > 2K one stable, one unstable stable slope determined by /

  13. Stochastic simulation • Step 0 (Initialization): set the reaction rates c1,…,cM and the initial molecular population numbers X1,…,XN • Step 1: calculate the propensities a1=h1¢c1, …, aM=hM¢ cM and the total propensity a0 • Step 2: generate random numbers  and  according to P(,) • Step 3: increase time t by  and update molecule numbers according to reaction  • if t < tint goto Step 1

  14. Stochastic timeseries fluctuation-driven transitions between ‚fixedpoints‘ burst factor b = 0.1 b = 1 b = 10 transcription rate was adjusted in order to keep the protein production rate  = b ¢ [transcription rate] = const

  15. Summary • Part I: The Gillespie algorithm • The Gillespie algorithm is an exact simulation of the master equation • Basic idea: when will the next reaction occur and what kind of reaction will it be? • Part II: Autoregulatory network motiv • Positive autoregulation + nonlinearity leads to bistable behavior • A high burst factor is one source of strong noise

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