Stochastic description of gene Stochastic description of gene regulatory mechanismsregulatory mechanisms

08.02.200608.02.2006Georg FritzGeorg Fritz

Statistical and Biological Physics GroupStatistical and Biological Physics GroupLMU MünchenLMU München

Albert-Ludwigs Universität FreiburgAlbert-Ludwigs Universität Freiburg

OutlineOutline• Part I: Simulation of stochastic chemical systems with the Part I: Simulation of stochastic chemical systems with the

Gillespie algorithmGillespie algorithm• Chemical master equation (CME)Chemical master equation (CME)• Reaction probability density function Reaction probability density function ) ) Gillespie algorithm 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‘s Deterministic 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

Chemical master equationChemical master equation

• M reactions RM reactions R, N reactants S, N reactants Si i with molecule numbers Xwith molecule numbers Xii

• Well stirred system, no spacial effects consideredWell stirred system, no spacial effects considered• cc dt: prob. of one reaction dt: prob. of one reaction in dt, given one reactant in dt, given one reactant

combinationcombination• hh: number of distinct molecular reactant combinations, e.g. : number of distinct molecular reactant combinations, e.g.

hh11=X=X11 X X22

• aa dt := h dt := h c c dt: prob. that any reaction of the type R dt: prob. that any reaction of the type R will will occur in (t, t+dt)occur in (t, t+dt)

• 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: Alternative: Measure many realizationsMeasure many realizations of the of the stochastic process and stochastic process and estimate the quantity of interestestimate the quantity of interest

)) Gillespie algorithm Gillespie algorithm

Chemical master equationChemical master equation

The Gillespie algorithm*: Simulation of the The Gillespie algorithm*: Simulation of the reaction probability density functionreaction probability density function

• Known as the BKL (Bortz-Kalos-Lebowitz) Known as the BKL (Bortz-Kalos-Lebowitz) algorithm in the physical literaturealgorithm in the physical literature

• Equivalent to the chemical master equation Equivalent to the chemical master equation • Basic idea: Basic idea: whenwhen will the next reaction occur, will the next reaction occur, what what

kind of reaction is it?kind of reaction is it?• Described by the reaction probability density Described by the reaction probability density

function function P(P())• P(P(,,) d) d := prob. that, given the state (X := prob. that, given the state (X11,…,X,…,XNN) at ) at

time t, the next reaction will occur in (t+time t, the next reaction will occur in (t+,t+,t++d+d) ) andand will be an Rwill be an R reaction reaction

*D. Gillespie, Exact stochastic simulation of coupled chemical reactions, J. Phys. Chem. 81, 1977

The reaction probability density functionThe reaction probability density function

• Goal: determine P(Goal: determine P())

• PP00(() ) ´́ prob. that no reaction occurs in (t, t+ prob. that no reaction occurs in (t, t+))

• PP00((+d+d) = P) = P00(() [1-) [1-aa d d] ]

• P(P(,,) d) d = P = P00(() a) a d d

Simulation of P(Simulation of P())

• Generate a random pair (Generate a random pair (,,) according to) according to

• Remember Wolfram‘s talk: generate Remember Wolfram‘s talk: generate r1,r2 2 UD(0,1) and compute

The AlgorithmThe Algorithm

• Step 0 (Initialization): set the reaction rates cStep 0 (Initialization): set the reaction rates c11,,…,c…,cMM and the initial molecular population and the initial molecular population numbers Xnumbers X11,…,X,…,XNN

• Step 1: calculate the propensities aStep 1: calculate the propensities a11=h=h11¢¢cc11, …, , …, aaMM=h=hMM¢¢ccMM and the total propensity a and the total propensity a00

• Step 2: generate random numbers Step 2: generate random numbers and and according to P(according to P(,,))

• Step 3: increase time t by Step 3: increase time t by and update molecule and update molecule numbers according to reaction numbers according to reaction

• if t < tif t < tintint goto Step 1 goto Step 1

Part II: Application to autoregulatory Part II: Application to autoregulatory genetic network motifgenetic network motif

transcriptiontranscription

translationtranslation

transcription factortranscription factor

M. Ptashne and A. Gann, Imposing specificity by localization: mechanism and evolvability, Curr. Biol., 1998, 8:R812-R822

Positive autoregulationPositive 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

Deterministic approach: model reductionDeterministic approach: model reduction

Fixedpoint analysisFixedpoint analysis

for / < 2K both unstable for / > 2K one stable, one

unstable

slope determined by /

stable

Stochastic simulationStochastic simulation

• Step 0 (Initialization): set the reaction rates cStep 0 (Initialization): set the reaction rates c11,,…,c…,cMM and the initial molecular population and the initial molecular population numbers Xnumbers X11,…,X,…,XNN

• Step 1: calculate the propensities a1=h1Step 1: calculate the propensities a1=h1¢¢ c1, …, c1, …, aaMM=h=hMM¢¢ c cMM and the total propensity a and the total propensity a00

• Step 2: generate random numbers Step 2: generate random numbers and and according to P(according to P(,,))

• Step 3: increase time t by Step 3: increase time t by and update molecule and update molecule numbers according to reaction numbers according to reaction

• if t < tif t < tintint goto Step 1 goto Step 1

Stochastic timeseriesStochastic timeseries

burst factor b = 0.1burst factor b = 0.1 b = 1b = 1 b = 10b = 10

transcription rate was adjusted in order to keep the protein transcription rate was adjusted in order to keep the protein production rate production rate = b = b ¢¢ [transcription rate] = [transcription rate] = constconst

fluctuation-driven transitions between ‚fixedpoints‘fluctuation-driven transitions between ‚fixedpoints‘

SummarySummary

• Part I: The Gillespie algorithmPart I: The Gillespie algorithm• The Gillespie algorithm is an exact simulation The Gillespie algorithm is an exact simulation

of the master equationof the master equation• Basic idea: Basic idea: whenwhen will the next reaction occur will the next reaction occur

and and what kind of reactionwhat kind of reaction will it be? will it be?

• Part II: Autoregulatory network motivPart II: Autoregulatory network motiv• Positive autoregulation + nonlinearity leads to Positive autoregulation + nonlinearity leads to

bistable behaviorbistable behavior• A high burst factor is one source of strong noiseA high burst factor is one source of strong noise