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 algorithmPart I: Simulation of stochastic chemical systems with the Gillespie algorithm Chemical master equation (CME)Chemical master equation (CME) Reaction probability density function ) Gillespie algorithmReaction probability density function ) Gillespie algorithm Part II: Application to gene regulatory mechanismsPart II: Application to gene regulatory mechanisms Bistable autoregulatory network motifBistable autoregulatory network motif Deterministic description by ODE‘sDeterministic description by ODE‘s Model reductionModel reduction Fixedpoint analysisFixedpoint analysis Stochastic simulationStochastic simulation Glance at the C-codeGlance at the C-code Timeseries: fluctuation-driven transitions between ‚fixedpoints‘Timeseries: fluctuation-driven transitions between ‚fixedpoints‘ SummarySummary

3. Chemical master equation M reactions R , N reactants S i with molecule numbers X iM reactions R , N reactants S i with molecule numbers X i Well stirred system, no spacial effects consideredWell stirred system, no spacial effects considered c  dt: prob. of one reaction  in dt, given one reactant combinationc  dt: prob. of one reaction  in dt, given one reactant combination h  : number of distinct molecular reactant combinations, e.g. h 1 =X 1 X 2h  : number of distinct molecular reactant combinations, e.g. h 1 =X 1 X 2 a  dt := h  c  dt: prob. that any reaction of the type R  will occur in (t, t+dt)a  dt := h  c  dt: prob. that any reaction of the type R  will occur in (t, t+dt)

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

5. The Gillespie algorithm*: Simulation of the reaction probability density function Known as the BKL (Bortz-Kalos-Lebowitz) algorithm in the physical literatureKnown as the BKL (Bortz-Kalos-Lebowitz) algorithm in the physical literature Equivalent to the chemical master equationEquivalent to the chemical master equation Basic idea: when will the next reaction occur, what kind of reaction is it?Basic idea: when will the next reaction occur, what kind of reaction is it? Described by the reaction probability density function P(  )Described by the reaction probability density function P(  ) P( ,  ) d  := prob. that, given the state (X 1,…,X N ) at time t, the next reaction will occur in (t+ ,t+  +d  ) and will be an R  reactionP( ,  ) d  := prob. that, given the state (X 1,…,X N ) 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(  )Goal: determine P(  ) P 0 (  ) ´ prob. that no reaction occurs in (t, t+  )P 0 (  ) ´ prob. that no reaction occurs in (t, t+  ) P 0 (  +d  ) = P 0 (  ) [1-   a  d  ]P 0 (  +d  ) = P 0 (  ) [1-   a  d  ] P( ,  ) d  = P 0 (  ) a  d P( ,  ) d  = P 0 (  ) a  d 

7. Simulation of P(  ) Generate a random pair ( ,  ) according toGenerate a random pair ( ,  ) according to Remember Wolfram‘s talk: generateRemember Wolfram‘s talk: generate r 1,r 2 2 UD(0,1) and compute

8. The Algorithm Step 0 (Initialization): set the reaction rates c 1,…,c M and the initial molecular population numbers X 1,…,X NStep 0 (Initialization): set the reaction rates c 1,…,c M and the initial molecular population numbers X 1,…,X N Step 1: calculate the propensities a 1 =h 1 ¢ c 1, …, a M =h M ¢ c M and the total propensity a 0Step 1: calculate the propensities a 1 =h 1 ¢ c 1, …, a M =h M ¢ c M and the total propensity a 0 Step 2: generate random numbers  and  according to P( ,  )Step 2: generate random numbers  and  according to P( ,  ) Step 3: increase time t by  and update molecule numbers according to reaction Step 3: increase time t by  and update molecule numbers according to reaction  if t < t int goto Step 1if t < t int goto Step 1

9. Part II: Application to autoregulatory genetic network motif transcription translation transcription factor 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: c 0 << c 1 burst factor b = c 2 /c 9 determines the number of proteins produced per mRNA

11. Deterministic approach: model reduction

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

13. Stochastic simulation Step 0 (Initialization): set the reaction rates c 1,…,c M and the initial molecular population numbers X 1,…,X NStep 0 (Initialization): set the reaction rates c 1,…,c M and the initial molecular population numbers X 1,…,X N Step 1: calculate the propensities a1=h1 ¢ c1, …, a M =h M ¢ c M and the total propensity a 0Step 1: calculate the propensities a1=h1 ¢ c1, …, a M =h M ¢ c M and the total propensity a 0 Step 2: generate random numbers  and  according to P( ,  )Step 2: generate random numbers  and  according to P( ,  ) Step 3: increase time t by  and update molecule numbers according to reaction Step 3: increase time t by  and update molecule numbers according to reaction  if t < t int goto Step 1if t < t int goto Step 1

14. Stochastic timeseries 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 fluctuation-driven transitions between ‚fixedpoints‘

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