ComputationalSystemsBiology Paola Quaglia University of Trento and - - PowerPoint PPT Presentation

ComputationalSystemsBiology

Paola Quaglia

University of Trento and CoSBi

• Deterministicchemicalkinetics
• Stochasticchemicalkinetics
• Simulation:Gillespie’sdirectmethod

Agenda

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• Simulation:Gillespie’sdirectmethod
• BlenX:alanguageformodellingsystemdynamicswitha

stochasticrun$timesupportforsimulation

Chemicalkinetics:reactions

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n1 Y1 +...+nj Yj m1 X1 +...+mk Xk

Reactions:terminology

Reactants

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n1 Y1 +...+nj Yj m1 X1 +...+mk Xk

Reactions:terminology

Reactants Products

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n1 Y1 +...+nj Yj m1 X1 +...+mk Xk

Reactions:terminology

Reactants Products

PlantBioinformatics,SystemsandSyntheticBiologySummerSchool,Nottingham,July2009

n1 Y1 +...+nj Yj m1 X1 +...+mk Xk

Stoichiometries

Assumewehave:

• awell$stirredandfixedvolumeVinthermalequilibrium;
• Nchemicalspecies,eachwithaninitialnumberof

molecules;

• Mreactionsthroughwhichthespeciescaninteract.

Generalquestion:

Deterministicapproach

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Generalquestion: Whichwillbethepopulationlevelsofspeciesafteraperiodof time? Thedeterministicapproachassumesthatthenumberof moleculesofthei$thspeciesattimetcanberepresented byacontinuousfunctionXi(t): dXi/dt=f(X1(t),...,XN(t))

Lotka*Volterraprey*predatoreco*system

Example

Y 2Y Y+R 2R R Ø

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YrepresentsthepreY,andRthepredatoR

• 1. preyreproduction
• 2. predatorreproduction,favouredbyfeedingonpreys
• 3. predatornaturaldeath

Deterministicformulation: Timeevolutionisawhollypredictableprocess,governedbya setofcoupledODEs. Inmanycasestimeevolutioncanbetreatedasa deterministic andcontinuous processwithanacceptable

Deterministicformulation

PlantBioinformatics,SystemsandSyntheticBiologySummerSchool,Nottingham,July2009

degreeofaccuracy,however:

• Deterministicmodellingofabiologicalsystemrequiresthe

preciseknowledgeofmoleculardynamics(preciseposition andvelocityofeachmolecule).Athigherlevel(whenless detailsareknown),theevolutionisintrinsicallystochastic.

• Timeevolutionisnotreallyacontinuousprocess:

populationlevelscanchangeonlyinadiscreteway.

Stochasticformulation: Timeevolutionisarandom*walk process,governedbya singlestochasticdifferentialequation(). Thestochasticformulationhasafirmerphysicalkineticbasis

Stochasticformulation

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thanthedeterministicformulation,andisespeciallyrelevant whendealingwithlowconcentrations. Thestochasticmasterequation,though,isveryoften mathematicallyintractable.

Itisacomputationalmethod(analgorithm)whichtakes explicitaccountofthefactthattimeevolutionofspatially homogeneoussystemsisadiscrete (vs.continuous) stochastic (vs.deterministic)processandoffersan applicablealternativetothesolutionofthemasterequation.

Gillespie’sDirectMethod

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References:

• D.T.Gillespie,J.Comput.Phys.,vol.22,1976.
• D.T.Gillespie,J.PhysicalChemistry,vol.81,1977.

Themethodisimplementedtoanswerthe Generalquestion: IfNspeciescaninteractthroughoneofMreactionsinafixed volume,whichwillbethepopulationlevelsofspeciesaftera

Gillespie’sDirectMethod(ctd.)

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periodoftime? Thealgorithmgeneratesatrajectoryoftheevolutionofthe systems:itcalculateswhich reactionwilloccurnextand whenitwilloccur.

Underlyingphysics:

• reactionsarecollisions
• moleculesarerandomlyanduniformlydistributedinthe

volume(assumingthesystembeinthermalequilibrium) FromthisGillespiearguesthat,althoughonecannotrigorously computethenumberofcollisionsoccurringinVbetween

Gillespie’sDirectMethod(ctd.)

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computethenumberofcollisionsoccurringinVbetween moleculesoftwogivenspecies,itispossibletoprecisely computetheprobabilityofsuchcollisionoccurringinany infinitesimaltimeinterval. Thenthekeypointofthemethodis: using insteadof

GivenMreactionsR1,...,RM,thereexistMconstants,which

• nlydependonthephysicalpropertiesoftheinvolved

moleculesandonthetemperatureofthesystem,suchthat: cj dt=average probabilitythataparticularcombination

• fRjreactantswillreactinthenextinfinitesimal

timeintervaldt.

Gillespie’sDirectMethod(ctd.)

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Why“average”? hj =numberofdistinctRj molecularreactantcombinationsinV attimet. cjhjdt istheprobabilitythatanRj reactionwilloccurin thenextinfinitesimaltimeinterval(t,t+dt).

Computinghj isnothard:

• Y...

h=|Y|

Gillespie’sDirectMethod(ctd.)

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Y... h=|Y|

• X+Y...

h=|X||Y| 2X... h=|X|(|X|$1)/2

AttimeT,whatweneedtoknowtoimplementthenext simulationstepis:

• whenthenextreactionwilloccur,
• whichkindofreactionitwillbe.

Thisisaprobabilisticinformationgivenby:

Gillespie’sDirectMethod(ctd.)

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P(t,j)dt= probabilitythatattimetthenextreactionwill beaRjreactionandwilloccurinthe infinitesimalinterval(T+t,T+t+dt) =P(t,j)dt=aj exp($a0 t)(t≥0) where aj =cjhj anda0 =Σ j=1..M aj

Simulation algorithm Initialization (set the values cj and the population levels) Compute a0 = Σ j=1..M aj Generate two random numbers n1,n2 in [0,1] and compute

• t = (1/a0) ln (1/n1)
• j such that Σk=1..j-1 aj < n2 a0 ≤ Σ k=1..j aj

Adjust population levels according to Rj and set T=T+t then iterate from step 2

Stochasticprocesscalculi: formallanguagesforinteractingprocesses Basicingredients:

• 1. asetofelementaryactionswithassociatedratevalues

(meaningthatthedelayofthecorrespondingactivityisa

ApplyingGillespie’smethodinprocesscalculi

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(meaningthatthedelayofthecorrespondingactivityisa randomvariablewithanexponentialdistribution)

• 2. alimitedsetofoperatorstospecify(atleast):

$ thetemporalorderingofactions $ possiblecoordination/interactionbetweenactions

Theseformalismsare:

• scalable(todescribephenomenafrombiochemistryupto

populationsofcells);

• amenabletocomputerexecution(analysisand/or

simulation)

Applyingthemethodinprocesscalculi(ctd)

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Averygoodpoint: Thesecalculicomewithanoperationalsemanticsthat easetherepresentationofprocessbehavioursasgraphs.

Example:biochemicalstochasticpi*calculus (Priami,Regev,Silverman,Shapiro,2001)

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moreexamples: BioAmbients,Brane Calculi,CoreFormalBiology,Beta$binders,Bio$PEPA,...

• Deterministicchemicalkinetics
• Stochasticchemicalkinetics
• Simulation:Gillespie’sdirectmethod

Agenda

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• Simulation:Gillespie’sdirectmethod
• BlenX:alanguageformodellingsystemdynamicswitha

stochasticrun$timesupportforsimulation

BlenXisthekernelofaprogramminglanguagebasedon Beta$binders(PriamiandQuaglia,2004). Inturn,BlenXisthecoreofCoSBiLab.

BlenX

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http://www.cosbi.eu

Compiler Public Data Bases Literature

BlenX program

BlenX VL

CoSBiLab

supportsboth

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BWB plotter MC SBML Run-time environment Sim CTMC React

supportsboth Gillespies’ssimulationsand spatialdiffusion visual,Markovchain,and SBMLexport

Compiler

P-Systems Simulator

Concentration Time courses

Kinetic Inference

Public Data Bases Literature

BlenX program

BlenX VL

CoSBiLab

inferenceof quantitative parameters

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BWB plotter MC SBML Run-time environment Sim CTMC React

Simulator

Compiler

P-Systems Simulator

Concentration Time courses

Kinetic Inference

Public Data Bases Literature

BlenX program

BlenX VL

CoSBiLab

statistical analysisand visualization

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BWB plotter MC SBML Run-time environment Sim CTMC React

Simulator Statistical Analysis

Internal Representation

Graphical Network Inspector

Boxeswithtypedinteractionsites

MainingredientsofBlenX

P

x,A y,B z,C

Interfaces

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Internal process

P

• interaction between two boxes is allowed over “affine” interfaces,

and is based on a race condition

• complexation of two boxes is driven by the affinity of the relevant

sites

Biologicalentities (mRNA,protein, ...) Boxes Interactioncapabilities (protein domains,...) Boxinteractionsites & Communication Interactionpotentials Affinityofinteractionsites

Biologicalinteractions

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Interactionpotentials Affinityofinteractionsites Complexation Linking boxestogetherinto graphs Decomplexation Removingedgesfromgraphs

[steps=150000] letY:bproc=#(y,DY) [nil]; letR:bproc=#(r,DR) [nil];

Asimpleprogram

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letYR:bproc=#(yr,DYR) [nil]; when(Y::10)split(Y,Y); when(Y,R::0.01)join(YR); when(YR::inf)split(R,R); when(R::10)delete; run1000Y||1000R||0YR

[steps=150000] letY:bproc=#(y,DY) [nil]; letR:bproc=#(r,DR) [nil];

Asimpleprogram:structureoffile.prog

Preamble Declarations

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letYR:bproc=#(yr,DYR) [nil]; when(Y::10)split(Y,Y); when(Y,R::0.01)join(YR); when(YR::inf)split(R,R); when(R::10)delete; run1000Y||1000R||0YR

Declarations Directives

[steps=150000] letY:bproc=#(y,DY) [nil]; letR:bproc=#(r,DR) [nil];

Asimpleprogram:preamble

Simulationinformation [STEPS=10000] [TIME=70] [STEPS=7,DELTA=10] Globalstochasticrates

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letYR:bproc=#(yr,DYR) [nil]; when(Y::10)split(Y,Y); when(Y,R::0.01)join(YR); when(YR::inf)split(R,R); when(R::10)delete; run1000Y||1000R||0YR

<< BASERATE:inf, >>

[steps=150000] letY:bproc=#(y,DY) [nil]; letR:bproc=#(r,DR) [nil];

Asimpleprogram:boxdeclaration

nil nil

y,DY y,DY

nil nil

r,DR r,DR

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letYR:bproc=#(yr,DYR) [nil]; when(Y::10)split(Y,Y); when(Y,R::0.01)join(YR); when(YR::inf)split(R,R); when(R::10)delete; run1000Y||1000R||0YR

nil nil

yr,DYR yr,DYR

nil isthesimplestinternalprocess

[steps=150000] letY:bproc=#(y,DY) [nil]; letR:bproc=#(r,DR) [nil];

Asimpleprogram:eventsdeclaration

Events (split,join,delete)

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letYR:bproc=#(yr,DYR) [nil]; when(Y::10)split(Y,Y); when(Y,R::0.01)join(YR); when(YR::inf)split(R,R); when(R::10)delete; run1000Y||1000R||0YR

Events (split,join,delete) withassociatedrates

Lotka*Volterra,computationally

when(Y::10)split(Y,Y); when(Y,R::0.01)join(YR); when(YR::inf)split(R,R); when(R::10)delete;

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Lotka*Volterra,computationally

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Simulationrun

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Communicationprimitivesforboth

• interactionsbetweenboxes,and
• interactionbetweenparallelsub$processeswithinthesame

box Bindingandunbindingofboxes

Morethanevents

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Foreachmodel: √file.prog file.types

[steps=150000] letY:bproc=#(y,DY) [nil]; letR:bproc=#(r,DR) [nil];

Bindingandunbinding:file.types

{ DY, DR, DYR } %% { (DY,DR,0,0,0)

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letYR:bproc=#(yr,DYR) [nil]; when(Y::10)split(Y,Y); when(Y,R::0.01)join(YR); when(YR::inf)split(R,R); when(R::10)delete; run1000Y||1000R||0YR (DY,DR,0,0,0) }

nobinding(norsubsequent unbinding)isallowed betweenYandR

Bindingandunbinding:simpleexample

[steps=150000] letY:bproc =#(y,DY) [nil]; letR:bproc =#(r,DR) [nil]; { DY, DR, DYR } %% { (DY,DR,3,2,0) }

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nil nil

y,DY y,DY

nil nil

r,DR r,DR letYR:bproc =#(yr,DYR) [nil]; when(Y::10)split(Y,Y); when(Y,R::0.01)join(YR); when(YR::inf)split(R,R); when(R::10)delete; run 1000Y||1000R||0YR }

√ Events √Boxes Declarationsinfile.prog

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Internalprocesses

Interfacemanagement:

Internalprocesses

x,DX x,DX y,DY y,DY z,DZ z,DZ

Interactionmanagement:

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Interfacemanagement: change(x,DX) expose(w,DW) hide(y) Interactionmanagement: x!<value>.P z?(parameter).P u!<value>.P u?(parameter).P P|Q P+Q if thenP

Value*passing

y!<3>.nil y!<3>.nil

y,DY y,DY

r?(p).ifp>1thenP r?(p).ifp>1thenP

r,DR r,DR

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nil nil

y,DY y,DY

if3>1thenP if3>1thenP

r,DR r,DR

Filamentsaregeneratedfromaninitialfeed. FilamentscanbranchbycomplexationwithARPmolecules. Aminimumdistancebetweenadjacentbranchesisalways grantedbyaspecificinteractionprotocol.

Actinpolymerization

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by Roberto Larcher

Seeds: Otherelements:

Actinpolymerization

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Protocoltocontrolproximityofbranches

Actinpolymerization

distanceis:

1 2 1

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1 2 1

Polymerizationcomputationally

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PlantBioinformatics,SystemsandSyntheticBiologySummerSchool,Nottingham,July2009

Thanks!