Stochastic models for gene expression

Ovidiu Radulescu, DIMNP UMR 5235, Univ. of Montpellier 2

Colloque franco-roumain mathématiques appliquées, Poitiers 29/08/10

Summary

Motivation : fluctuations in gene networks

Stochastic models for gene expression

Application to biology: fluctuome

1. Fluctuations in gene networks

From genes to gene networks

From genes to proteins Interactions between genes,proteins, metabolites

Gene networks, feed-back

Fluctuations in molecular networks

Epigenetic variability

Which is the origin of fluctuations?Is there some order/logics in randomness?

Questions to answer

Becksai et al EMBO J 2001: S.cerevisiae

Ozbudak et al Nature 2002: one promoter noise in B.subtilis

Cai et al. Nature 2006:beta-gal operon

Intermittent protein production

How to tame fluctuations? either all numbers large, or all small numbers fast

The central limit theorem and the law of large numbers: if the number of molecules is large for ALL molecular species, then the fluctuations are small and Gaussian; dynamics is close to deterministic

The averaging theorem: if a slow deterministic system receives rapid random fluctuations as input then the output has small fluctuations

PG

k1 k2Fast

Noise in multiscale networks with inverted time hierarchy

Hybrid noise : discrete variations of low numbers species, continuous variation punctuated by jumps or switching of large numbers species

Low and large numbers : a broad distribution of abundances from a few to 104 per cell

Fast and slow processes : from 10-3 to 104 s

Multiscale

Inverted time hierarchy: some processes involving low numbers are slow

PG

k1 k2Slow

2. Stochastic models for gene expression

Delbrück-Rényi-Bartolomay approachTwo main assumptions:

Reactions are independent; Transport is instantaneous.

Dynamics variables are numbers of molecules of different species All species evolve by discrete jumps separated by random waiting times

Which reaction?

Which time?Generate exponential variable of

parameter (k1A+k2B)

Gillespie’s algorithm

Disadvantages:

Time costly Analytical solutions of theChemical Master Equation are rarelyavailable

Continuous time Markov jump processes

nr

iXiXi

iiXqXqX

1

(.)])((.))([,.)(

are n chemical species

jump vectorrninZiii ,1,

nininini AAAA ...... 1111

nAA ,...,1

nr

iii XVXVX

1

)]()([)(

nr

jjjii XVXVXVXq

1

)]()([)/()(

intensity

distribution of jumps

biochemical reaction

jump probability

nZX is the state

Hybrid approach

The hybrid stochastic dynamics is a piecewise deterministic Markov process

Assumption:

Law of large numbers can be applied to continuous variablesGaussian noise neglected

2 types of dynamical variables: discrete and continuousDiscrete variables undergo random jumps, continuous variables follow ODE dynamics

Hybrid algorithm

Advantages:

Slight improvement of execution timeEmphasize the hybrid nature of fluctuations Analytical solutions more easily available

Continuous variable : ODE

Discrete variable : Gillespie dynamics

23 kPkdtdu Pk

dtdu

3

321

1

,***,kkk

k

PPGGGG

switching

Partial fluid approximation:partitionGiven a pure jump model, find its hybrid approximation

Species partition: discrete, continuous),( CD XXCDCD RRR Reaction partition

Discrete transitions

Contributions to continuous flow

coupling intrinsic

321

1

,***,kkk

k

PPGGGG

Coupling between discrete and continuous only if super-reactions of type 1: fast, change mode XC, rates depend on XD

switching

Partial fluid approximation : expansion)/,(

~),( VXxXXXX CCDCD Rescaling

1st order Taylor expansion in

) , , ( ) , ( ) ,~

( )~

( ) ,~

(t x X p x X V t X p X t Xtpc D

R i

c D iDi

D

Di

) , / , ( ) / , (t V x X p V x X VCi

DC

Cic D

R i

c D i

) , / , ( ) / (t V x X p V x VCi

C

Cic D

R i

c i

VCi/

) , , ( ) , (t x X p x Xfc D c D xc

V x V V x X V x Xfdtdx

C

Ci

DC

Ci

R i

c i

R i

c D i c Dc

)/ ( )/ , ( ) , (

Chemical Master Equation

Hybrid Master/Fokker-Planck equation

Switching

Drift

of a PDP with switching

Coupling if super-reactions of type 1

Partial fluid approximation : breakage

Breakage+drift part in the master/Fokker-Planck equation

V x V V x X V V x X V x XfC

Ci

DC

Ci

Ci

R i

c i

R i

c D i

S i

c D i c D)/ ( )/ , ( )/ , ( 1 ) , (2

super-reactions of type 2:very fast, act on XC, rates depend on XD

breakage

... ) , , ( ) , ( 1 ) ,~

( )~

( ) ,~

(

t x X p x X V t X p X t Xtpc D

DR i

c D iDi

Di

fast back to ground discrete reactions DR

DX

CX... )] ,

~( ) ( ) , , ( )[

~( )) ,

~( )

~( ( ) ,

~(

t X p dy y t y x X p X t X p X gx t

t X pD

breakage frequency breakage size distribution

Hierarchical model reduction

Reduction of stochastic models:Linear sub-networksNeeds multi-scalenessWorks both for deterministicand for stochastic models

Eliminate inessential details: pruningGroup together reactions: lumpingZoom in and out complexityDrastic decrease of simulation time

Dominance and pruning Un-broken cycle: averaging

A1

A2

Ai Aj

A

k2

k1

klim ki

k

k<<ki

Aj

A

kklim/ki

i

jj

ii

kk

k

k plim

1

1

Inverted time hierarchiesStochastic leaks of low mass un-broken cycles produces slow transitions of discrete variables, thus inverted time hierarchies

A un-broken cycle can be a source of hybrid noise

A1

A2

Ai Aj

A

k2

k1

klim ki

k

k<<ki

Aj

A

kklim/ki

kkk ki

lim

Low intensity shot noise is amplified to bursts by fast transitions of the continuous variables

AjA Bkklim/ki

kd

kpBurst amplification kp>>kd

Continuous variableDiscrete variables

Prot FoldedProt

D D.R

D.RNAP

TrRNAP

RBS

Rib.RBS

ElRib

Ø

k1=400

km1=1

km2=10 k2=6

k3=0.1

k4=0.3

k6=60km6=2.25

k7=0.5

k8=0.015

k10=1e-5k11=1e-5

k9=1.3e-4

Ø

Øk5=0.3

model1

Prot FoldedProt

D**

TrRNAP

RBS

Rib.RBS

ElRib

0.3

602.25

0.5

0.015

1e-5

1.3e-4

Ø

1.5e-4

Prot FoldedProt

RBS*

ElRib

D**

TrRNAP

0.5

0.0151e-5

1.3e-4

Ø

model2 model31.5e-4

0.3

Ø

Ø

0.3

1e-2

Hierarchical model reduction

Inverted time hierarchy

Continuous variables

Burst amplification of ElRib

Hierarchical model reduction: gain in computational power

Crudu et al BMC Systems Biology 2009

A simple model : solution of the hybrid master/Fokker-Planck equation

a=k1/2 : number of transcription initiations per protein lifetime b=k2/1 : number of proteins per burst

)] , ( ) / exp( 1) , ( )) , ( ( [ ) , (0

2t x ap dy b ybt y x p a t x xp

xt x

tp

x

) / exp( 1 ) (b x ax x p

Gamma steady state distribution

1 a 1a

3. Application to biology: fluctuome

Biological system: central carbon metabolism in B.subtilis

Bacteria are grown in two experimental conditions: glucose-rich and malate-rich medium

Experimental method: number and brightness

Poissonian field F map Histograms

Part of the fluctuome: CggR, gapB, CcpN

CggRgapB

CcpN

Glucose

Malate

Glucose

Malate

Glucose

Malate

Matthew Ferguson, CBS

10min

60min

MdGFP

D D.R

RBS

ElRib

k3

k7

k8

kdegØ

Øk5

k3’

TrRNAP

k4

k1on

k1off

k4’~0Ø

k6

RNAP

pRNAP-R

RNAP

k1off

k1on

CggR

CggR

TrRNAPpaused

D.R.RNAP

k6

RBS

k4’RNAP RNAP RNAP

RNAP

RNAP

RNAP

TrRNAPD.RNAP RBSbursting

RNAP RNAP

RNAPRNAP

CcpNk3 or k3’

gapBA

C

TrRNAPD.RNAPRBSbursting

CggR

k4

RNAP

RNAP

RNAP

RNAP RNAP

RNAPRNAP

B

D

pcggR

pgapB

RNAP

Take home message

The origin of noise in multiscale molecular network is the inverted time hierarchy; noise is hybrid

Hierarchical model reduction unravels functional structure: unbroken cycles, burst amplifiers, integrators

Noise carries information about interactions : fluctuome could be an important tool

Acknowledgements

Nathalie Declerck, Matthew Ferguson, Catherine RoyerCBS Montpellier

Alina Crudu, Arnaud Debussche, Aurelie MullerIRMAR Rennes, ENS Cachan

Alexander GorbanUniversity of Leicester