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Stochastic description of gene regulatory mechanisms

This lecture discusses the stochastic simulation of gene regulatory mechanisms using the Gillespie algorithm. It covers topics such as the chemical master equation, reaction probability density function, autoregulatory network motif, model reduction, stochastic simulation, and fluctuation-driven transitions between fixed points.

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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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