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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