for reaction–diffusion systems A hybrid stochastic method with adaptive time step control Journal of Computational Physics

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Journal of Computational Physics

www.elsevier.com/locate/jcp

A hybrid stochastic method with adaptive time step control for reaction–diffusion systems

Wing-Cheong Lo

^∗

, Shaokun Mao

DepartmentofMathematics,CityUniversityofHongKong,Kowloon,HongKongSpecialAdministrativeRegion

a rt i c l e i n f o a b s t ra c t

Articlehistory:

Received15August2018

Receivedinrevisedform29November2018 Accepted30November2018

Availableonline12December2018

Keywords:

Reaction–diffusionsystems Stochasticsimulation Hybridmethod Biologicalpatterning

Randomnessoftenplaysanimportantroleinthespatialandtemporaldynamicsofbiolog- icalsystems.General stochasticsimulationmethods mayleadtoexcessivecomputational costforasysteminwhichalargenumber ofmoleculesinvolved.Therefore,multi-scale hybridsimulationmethodsbecomeimportantforstochasticsimulations.Herewebuilda spatiallyhybridmethodwhichcouplestwoapproaches:discretestochasticsimulationand continuousstochasticdifferentialequations.Inourmethod,thelocationsoftheinterfaces betweenthetwo approachesare changingaccording tothe distributionofmoleculesin aone-dimensional domain.To balancethe accuracyand efficiency, thetime stepofthe numericalmethodfor the continuousstochastic differentialequations is adaptedto the dynamicsofthemoleculesneartheadaptiveinterfaces.Thesimulationresultsforalinear systemandtwononlinearbiologicalsystemsindifferentone-dimensionaldomainsdemon- stratetheeffectivenessandadvantageofournewhybridmethodwiththeadaptivetime stepcontrol.

©^{2018ElsevierInc.Allrightsreserved.}

Introduction

Many modelsofbiological patternformationare describedby thereaction anddiffusionprocesses.The stochasticbe- haviors inthe processes often havesubstantial effect when the numbers ofthe molecules involved inthe reactions are relatively small[2,9,24,32]. Themechanismsforachievingrobustbiologicalpatternsagainstthenegativeeffectsofanoisy environmentwerediscussedindepthinsomecurrentstudies[3,20,24].Besidesthenegativeeffects,spatialstochasticper- turbations mightprovide positive effectsforachievinga robust cell polarization [21] and forminga sharper boundaryof geneexpressiondomains[33].Therefore,thedevelopmentofanumericalmethodforstudyingstochasticeffectsinbiological systemsbecomesmuchmoreimportantinourfuturestudy.

Forspatiallyinhomogeneoussystems,thereaction–diffusionstochasticsimulationalgorithm(SSA)isthepopularmethod tosimulatestochasticjumpsandreactionstoobtainthestateofthesystemaftereachoccurrenceofstochasticevents[12].

AlthoughtheSSA isconvenientintermsofitsimplementation,itscomputational costbecomeslarge whenthestochastic eventsinsomeregionsoccursignificantlymorefrequentlythantheeventsinotherregions,asinthecaseofsomeregions whichhaverelativelylargecopynumbersofmolecules.

Forimprovingtheefficiency,severalhybridmethodswereproposedforstimulatingthestochasticdynamicsofspatially inhomogeneous systems[8,10,11,16,23,26–28].Themostusual approachisaspatiallyhybridmethodwhichcombinesthe

*

Correspondingauthorat:DepartmentsofMathematics,CityUniversityofHongKong,HKSAR.

E-mailaddress:wingclo@cityu.edu.hk(W.-C. Lo).

https://doi.org/10.1016/j.jcp.2018.11.042

0021-9991/©^{2018ElsevierInc.Allrightsreserved.}

SSAandpartialdifferentialequationswhichprovideamean-fieldapproximationforthestochasticbehaviors[13,30].How- ever,arelatively smallstochasticeffectinthehighconcentrationregion mayresultinrelativelylarge perturbationinthe lowconcentrationregionbecauseofthediffusionprocessinspace.Sosomecurrentstudiesmovedtheirfocustodeveloping afullstochastichybridmethod[28].Thecontinuousapproachusingstochasticpartialdifferentialequationscanbeapplied toapproximatethestochasticsystematarelativelyhighconcentrationandprovidesanadditionalabilitytoincorporatethe stochasticityintheentirespatialdomain.Nevertheless,thecouplingofthediscreteandcontinuousapproachesandtheset- tingofthetimestepusedinthecontinuousapproachcanbeimprovedtobeadaptivetodifferentsystemsformaintaining highaccuracyandefficiency.Here,wewillfocusonthesetwoissuestodevelopamethodwhichiseasilyimplementedand adaptivetothesettingsofdifferentone-dimensionalreaction–diffusionsystems.

Inthispaper,webuildanadaptivespatiallyhybridmethodcouplingcontinuousstochasticdifferentialequationsandthe SSA. In ourhybrid method,the locationsofthe interfacesbetweenthe two numericalmethods are adapted to thecopy numbers ofmolecules in each compartment. Tobalance the accuracy andthe efficiency,the time step of the numerical methodforthecontinuousstochasticdifferentialequationsischangingaccordingtothedynamicsofthemoleculesnearthe adaptiveinterfaces.Then we applyour methodtoa linearsystemandtwononlinear biologicalsystemsin differentone- dimensionaldomainstoverifytheeffectivenessofournewapproach.Thesimulationresultsdemonstratethat,comparing withtheSSA,ourhybridmethodhasasignificantimprovementinefficiencyandtheadaptivetimestepcontrolprovidesa betterbalancebetweentheaccuracyandefficiencythanusingfixedtimestep.

Methods

Spatiallyinhomogeneousreaction–diffusionsystem

Consider a system with N molecular species {^S1,S₂,· · ·,S_N} ^{which are involved in the following M reactions} {^R1,R2,· · ·,RM}^:

Rj

:

^srj1S1

+ · · · +

^srjNSN γj

−→

^sp_j1^S1

+ · · · +

^sp_jN^SN

.

Here, s^r_ji ands^p_ji are thestoichiometric coefficients ofthereactant andproductspecies, respectively, and

γ

j is thecorre- spondingmacroscopicrateconstant.

When we are interested in the spatial distribution ofthe molecules on a one-dimensional domain, the domain with lengthLcanbepartitionedinto K compartmentswithuniformlengthh,whereh=^L/K.

Assumption1.Thediffusionprocessisfastenoughtoassumethatthesubsystemineachcompartmentisspatiallyhomoge- neous.Inotherwords,thesizeofeachcompartmentmustbesufficientlysmallthatdiffusivejumpsoccurmorerapidlythan reactionsandtheinhomogeneityinsideeachcompartmentcanbeignored[14,15,17].Moleculesindifferentcompartments aretreatedasdifferentspecies,denotedby

{

S11

,

S12

,· · · ,

S_ki

,· · · ,

SK N

},

where S_ki representstheithspeciesinthekthcompartment.The systemstate attimet isdenotedby K×^N-component vector

X

(

t

) = (

X11

(

t

),

X12

(

t

), · · · ,

Xki

(

t

), · · · ,

XK N

(

t

)),

where X_ki isthenumberofmoleculesofS_ki.

Assumption2.Weassumethat onlymoleculesinthesamecompartment canreactwitheachother.The M reactionscan beconsideredasK×^{Mreactionsinthespatialsystemanddenotedby R}kj,the jthreactioninthekthcompartment:

R_kj

:

^sr_j1^Sk1

+ · · · +

^sr_jN^SkN γkj

−−→

^sp_j1^Sk1

+ · · · +

^sp_jN^SkN

,

where

γ

kj is the reaction rateconstant of the reaction Rkj.The state of the systemtransfers fromone state to another throughreactionfiring.ThenetchangeofthestateofthesystemcausedbyoneoccurrenceofR_kj isdenotedas

ν

kj:

ν

_kj

₌ (

0

, · · · ,

0

,

s^p_j1

−

^sr_j1

, · · · ,

s^p_jN

−

^sr_jN

from ((k−1)N+1)th tokNth

,

0

, · · · ,

0

).

Assumption3. Diffusion process is treated as a reaction in which a molecule in one compartment jumps to one of its neighboringcompartments.AssumethatspeciesSi diffuseswithadiffusioncoefficient Di andtheboundaryconditionsof theone-dimensionaldomainareconsidered asreflectiveboundaryconditionsatbothends.Therefore,thediffusive jumps obeythefollowingchainreactions:

J_ikL

:

^S1i Di/h²

←−−−

Di/h²

S_2i ^Di/h

←−−−

2 Di/h²

S_3i

· · · ←−−−

^Di/h2

Di/h²

S_{K i}

,

J_ikR

:

S1i

Di/h²

−−−→

D_i/h² S2i Di/h²

−−−→

D_i/h² S3i

· · · −−−→

^Di/h2

D_i/h² SK i

.

We denotetheleft jumpofS_i fromthekthcompartmentby J_ikL andtherightjumpof S_i fromthekthcompartmentby JikR.

The probabilitythatthereaction R_kj willhappeninthenext timeinterval[^t,t+^dt)is

α

kj(X(t))dt,where

α

kj iscalled thepropensityfunctionofR_kj andisdefinedas

α

kj

(

X

(

t

)) = γ

kjX^s

r j1 k1X^s

r j2 k2

· · ·

X^s

r jN kN

;

theprobabilitiesforthejump JikL and JikRare

α

ikL(X(t))dtand

α

ikR(X(t))dt,respectively,where

α

_ikL

(

X

(

t

)) =

^Di

h²X_ki

,

for 1

<

k

^K

, α

ikR

(

X

(

t

)) =

^Di

h²X_ki

,

for 1

^k

<

K

.

This system can be simulatedthrough the stochastic simulation algorithm (SSA) [12]. At time t, X(t) is given. We first generatetwoindependentrandomnumbersr1 andr2,whichareuniformlydistributedin[⁰,1]^{andthencalculatethenext} reactionorjumptime

τ

bythefollowingformula

τ = −

¹

α

0

ln

(

r₁

),

where

α

0 isthesumofallpropensityfunctionsofthejumps J andthereactions R.At timet+

τ

,areaction Rqmoccurs whenthesmallestqandmexistforaninequality

q−1

k=¹

M j=¹

α

_kj

₊

m

j=¹

α

_qj

r2

α

0

;

⁽¹⁾

ifq andmdonotexist,aleftjump Jw1q1L mayoccurwhenthesmallestq1andw1 existforaninequality

K k=¹

M j=¹

α

kj

+

q

1−¹ k=¹

N i=¹

α

ikL

+

w₁

i=¹

α

iq1L

^r2

α

0

;

⁽²⁾

ifq,m,q1 andw1 donotexist,arightjump Jw2q2R mayoccurwhenthesmallestq2 andw2existforaninequality

K k=1

M j=¹

α

kj

+

K k=1

N i=¹

α

ikL

+

q

₂−¹ k=1

N i=¹

α

ikR

+

w2

i=¹

α

iq2R

^r2

α

0

.

(3)

Thenthestate X(t+

τ

)isupdatedaccordingtothecorrespondingstatechange.Thisprocessisrepeateduntilitreachesthe stopcriterion.

Approximationbystochasticdifferentialequations

Letube X/hwhichrepresentsthedistributionsofthemolecularconcentrations,whereuki=^Xki/hineachcomponent.

Assumethatthenumbersofthemoleculesineachcompartmentarelarge,wecanapproximatethestochasticsystembya systemofstochasticdifferentialequations(SDE)foru[18,22]:

du_ki

=

^Di h²

u_(k−1)i

−

^2uki

+

^u(k+¹)i

dt

+

M

j=¹ r_{ki j}

(

u

)

dt

+

ⁿ(k−¹)i

(

u

)

dW_{(k−1)J R}

+

ⁿ(k+¹)i

(

u

)

dW_{(k+1)J L}

−

ⁿki

(

u

)

dW_{k J R}

−

ⁿki

(

u

)

dW_{k J L}

+

M

j=¹

n_{ki j}

(

u

)

dW_kj

,

for 2

≤

^k

≤

^K

−

¹

,

(4)

whererki j(u)isthe[(k−¹)N+ⁱ]^{thcomponentof}

ν

kj

α

kj(uh)/h;nki j(u)isthe[(k−¹)N+ⁱ]^{thcomponentof}

ν

kj

α

kj(uh)/h;

n_ki(u) equals to D_i

h²u_ki/h; the variables W’s are the Wiener processes which are independent to each other. For the reflectiveboundaryconditions,whenk=^{1,thefirsttermoftherighthandsideinEq. (4) isreplacedby D}_h2ⁱ(−^u1i+^u2i)dt andthetwotermsn₍k−¹)i(u)dW₍k−¹)J R andnki(u)dWk J L areremoved;whenk=^{K,thefirsttermoftherighthandsidein} Eq. (4) isreplacedby _h^D₂ⁱ

u_(K−1)i−^uK i

dtandthetwotermsn_(k+1)i(^u)dW_{(k+1)J L}andn_ki(^u)dW_{k J R} areremoved.

Fornumericalsimulation,we simplyusethe Euler–Maruyamamethodtocalculatethe solutionforthesystemofSDE [18]:

u_ki

(

t

+

t

) −

^uki

(

t

) ≈

^Di h²

u_(k−1)i

(

t

) −

^2uki

(

t

) +

^u(k+1)i

(

t

)

t

+

M j=¹

r_{ki j}

(

u

(

t

))

t

+

ⁿ(k−¹)i

(

u

(

t

)) √

t

ζ

(k−¹)J R

+

ⁿ(k+1)i

(

u

(

t

)) √

t

ζ

_{(k+1)J L}

−

ⁿki

(

u

(

t

)) √

t

ζ

_{k J R}

−

ⁿki

(

u

(

t

)) √

t

ζ

k J L

+

M

j=¹

n_{ki j}

(

u

(

t

)) √

t

ζ

kj

,

for 2

≤

^k

≤

^K

−

¹

,

(5)

whereζ’sareindependentstandardnormalrandomvariables.Whenk=^{1,thefirsttermoftherighthandsideinEq. (5) is} replacedby ^D_h2ⁱ(−^u1i+^u2i) t,andn₍k−¹)i(u(t))√

tζ₍k−¹)J R andnki(u(t))√

tζk J Lareremoved;whenk=^{K,thefirstterm} oftherighthandsideinEq. (5) isreplacedby ^D_h2ⁱ

u_(K−1)i−^uK i

t,andn_(k+1)i(^u(t))√

tζ_{(k+1)J L}andn_ki(^u(t))√

tζ_{k J R}are removed.

Herewe applytheEuler–Maruyamamethodasitiseasy tobenumericallyimplemented.Actually,wecanapply other higherorder numericalmethods to improvethe accuracy [4,18,31]. Although thenumerical SDEapproach isan efficient methodtoapproximatethe stochasticprocesseswithhighmolecularconcentrations,theaccuracymaynot behighwhen thenumberofmoleculesbecomeslow.Inthelatersections,wewilldevelopthehybridmethodwhichcouplestheSSAand numericalSDEtobalancetheaccuracyandefficiencyinthesimulation.

AdaptiveinterfacesbetweenSSAandnumericalSDE

Todecidea methodto capturetheadvantages oftheSSA andnumericalSDE,we firstconsider awayto separatethe domain intotwo regions that satisfies:1) the methodis efficientinthe region withlarge numbersofmolecules; 2) the methodisaccurateintheregionwithsmallnumbersofmolecules.WeconsidertoapplythenumericalSDEtoapproximate thedynamicsinthekthcompartmentif

1≤i≤minN;1≤j≤M

⎧ ⎨

⎩

X_ki

^sp_ji

−

^sr_ji

⎫ ⎬

⎭ >

Nint

;

(6) inothercompartments,wewillapplytheSSAforthesimulations.

IfN_intislarge,theconditioncanreducetheprobabilitythattheapproximationprovidesanegativenumberofmolecules inthekthcompartment aftereachiteration.Aset I_C isdefinedasasetofall theindexesk satisfyingtheinequality(6);

aset ID isdefinedas{¹,2,...,K}\^ID; IB R andIB L arethesetsoftheleftboundarypointsandtherightboundarypoints, respectively,inallintervalsinIC.AsampleofthesefoursetsisillustratedinFig.1.Herewesimulatethefluxbetweenthe tworegionsbytheSSA.Therefore,thefiringtimeofthejumpbetweenIC andID arestochastic.

IntheIC region,weusethenumericalSDEtoapproximatethestochasticdynamics.Ifeachintervalin IC islargerthan onecompartment,weusetheEuler–Maruyamamethod,like(5),tobuildthefollowingiterationfork∈^IC:

u_ki

(

t

+

t

) −

^uki

(

t

) ≈

^Di h²

u_(k−1)i

(

t

) −

^2uki

(

t

) +

^u(k+1)i

(

t

)

t

+

M j=¹

r_{ki j}

(

u

(

t

))

t

+

ⁿ(k−1)i

(

u

(

t

)) √

t

ζ

_{(k−1)J R}

+

ⁿ(k+¹)i

(

^u

(

t

)) √

t

ζ

₍k+¹)J L

−

ⁿki

(

^u

(

t

)) √

t

ζ

k J R

Fig. 1.IllustrationofthedomaindecompositionbetweentheSSAandthestochasticdifferentialequations.ID(yellow)representstheregionfortheSSA;

IC(blue,pink,green)representstheregionfortheSDE;IB R(green)andIB L(blue)aretheleftboundarypointsandtherightboundarypoints,respectively, inallintervalsinIC.(Forinterpretationofthecolorsinthefigures,thereaderisreferredtothewebversionofthisarticle.)

−

ⁿki

(

u

(

t

)) √

t

ζ

_{k J L}

+

M

j=¹

n_{ki j}

(

u

(

t

)) √

t

ζ

_kj

,

fork

∈

^IC

\ (

I_{B R}

∪

^IB L

);

⁽⁷⁾

fork∈^IB L,thefirsttermoftherighthandsideinEq. (7) isreplacedby ^D_h2ⁱ

−^uki+^u(k+1)i

t,andn_(k−1)i(^u(t))√

tζ(k−1)J R and nki(u(t))√

tζk J L are removed; for k ∈ ^IB R, the first term of the right hand side in Eq. (7) is replaced by Di

h²

u_(k−1)i−^uki

t, and n_(k+1)i(^u(t))√

tζ(k+¹)J L and n_ki(^u(t))√

tζk J R are removed. If an interval in I_C has only one compartment,weapplyEq. (7) withoutthediffusionterm.

Timestepselectionfornumericaldifferentialequations

The selectionofthetime stept=t_C forthenumericalSDE(7) wasnotdiscussedinthepreviousstudiesofhybrid methods. Based on thestudyforthe efficient

τ

-selection forthe

τ

-Leapingmethod [5], we considerthat the meanand varianceoftherelativechangeofthemolecularpopulationsineachiterationisboundedbycertainthreshold :

| <

t_CX_ki

> | ≤

^max

{

X_ki

,

1

} ,

var

{

t_CX_ki

} ≤

^max

{

X_ki

,

1

} ,

(8) forallk∈^IC.Tosatisfythepreviousconditionsandtheconditionforthestabilityofcentraldifferencescheme[18]

t_C

≤

t₀

=

^min

1≤i≤N

h² 2D_i

,

weobtainthefollowingsettingfortC selection:

tC

=

min

1≤ⁱ≤^N,k∈^IC

max

{

hu_ki

,

1

}

μ

_ki

(

u

) , (

max

{

hu_ki

,

1

} )

²

σ

_ki

(

u

) ,

t0

(9)

where

μ

_ki

(

u

) =

^Di h²

u_(k−1)i

(

t

) −

^2uki

(

t

) +

^u(k+1)i

(

t

)

h

+

M j=¹

r_{ki j}

(

u

(

t

))

h

,

and

σ

_ki

(

u

) =

^Di h²

u_(k−1)i

(

t

) +

^2uki

(

t

) +

^u(k+1)i

(

t

)

h

+

M j=¹

n_{ki j}

(

u

(

t

))

h

₂

,

fork∈^IC\(IB R∪^IB L).Fork∈^IB L,thefirsttermsoftherighthandsidesof

μ

ki(^u)and

σ

ki(^u)arereplacedby D_i

h²

−

^uki

+

^u(k+¹)i

hand D_i h²

u_ki

+

^u(k+¹)i

h

,

respectively

;

fork∈^IB R,thefirsttermsoftherighthandsidesof

μ

ki(^u)and

σ

ki(^u)arereplacedby D_i

h²

u_(k−1)i

−

^uki

hand D_i h²

u_(k−1)i

+

^uki

h

,

respectively

.

If ismuchlessthan1,ourtimestepsettingcanguaranteethattherelativechangesofthenumbersofthemoleculesare smallenoughtoensurethenumericalstabilityfortheapproximation.Inthenumericaltests,wewilltake =⁰.1 whichis smallenoughtoensurethenumericalstabilityoftheEuler–Maruyamamethod.

ItisworthtoremarkthatalthoughthetimestepsizetC iscontrolledtoreducetherelativeerror,anegativevaluemay still appearintheiteration(7) withsmallprobability.Whenanegative value appears,thetime steptC canbe reduced throughdividingbytwo;ifthenegativevaluestillexistsaftert_C decreases,theprocesscanberepeateduntilthenegative valuedoesnotexistintheiteration(7).

Whenajumpofamolecule(theprocessismodeledbytheSSA)happensacrossaninterface,themolecularpopulation willchangeinthe I_C region.Tomaintaintheaccuracyoftheapproximation,weconsiderthatifajumpacrossaninterface happens,wewillresetthevalueoftC tolettheiterationrunsimultaneouslywiththejump.Thedetailswillbeexplained inthealgorithmoverview.

Algorithmoverview

Forasysteminaone-dimensionaldomainwithlengthL,givenaninitialtimet=^t0,aninitialcondition X(t₀)=^X0 and afinaltimeT,weperformthefollowingsteps:

1. Setavalue hforthespatialsizeofeach compartmentsuchthat thenumberofcompartmentsisan integer K=^L/h.

Assignanindexfrom{¹,2,...,K}^foreachcompartment.Setanerrorparameter andaninterfacethresholdN_intwhich willbeusedinEqs. (6) and(9),respectively.

2. ByEq. (6) withNint,divides{¹,2,...,K}^{intofoursetsofindexes I}D,IC,IB LandIB R.

3. If IC isnotempty,useEq. (9) tocalculatetC andsetTC=^t+tC;otherwise,settC=^TC= ∞^{andruntheSSAfor} theentirespatialdomainuntilI_C isnotempty.

4. If ID is not empty, generate two independent random numbers r1 andr2 whichare uniformly distributed in[⁰,1]^. Calculate the next reaction time tD= −_α¹₀^ln(r1) forthe SSA, where

α

0 isthe sum ofthe propensity functions of the right jumps J_ikR (for k∈^ID∪^IB R), the left jumps J_ikL (for k∈^ID∪^IB L) and the reactions R_ki (for k∈^ID) for i∈ {¹,2,...,M}^.SetTD=^t+tD.UsetheSSAmethodwiththesecond randomnumberr2 tofindthecorresponding reactionorjump.IfID isempty,settD=^TD= ∞^.

5. (a) Case1:IfT_C<T_D,runtheiteration(7) witht=t_C forallthecompartmentsin I_C.Sett=^TC.

(b) Case2:IfTC=^TD,runtheiteration(7) witht=tC forallthecompartmentsinIC.Sett=^TC.RuntheSSAfor updating X inaccordancewiththereactionorjumpfoundinStep 4.

(c) Case3:IfTC>TD andajumpacrossthe interfacesisselectedforthefiringreactionintheSSAmethod,runthe iteration (7) with t=t_C −(T_C −^TD) forall ith compartment, where i∈^IC. Runthe SSA forupdating X in accordancewiththereactionorjumpfoundinStep 4.Sett=^TD.

(d) Case4:Cases1–3arenotsatisfied,update X inaccordancewiththereactionorjumpfoundinStep 4.Sett=^TD. 6. For Cases 1, 2 and3,reset the foursets I_{B L}, I_{B R}, I_D and I_C according toEq. (6). If I_C is not empty, useEq. (9) to

calculatenewt_C andsetT_C=^t+t_C;otherwise,sett_C=^TC= ∞^{.ForCase4,ifI}C isempty,thesimilarupdateof thesetsisneeded;ifIC isnotempty,theupdateofthesetsisnotneeded.

7. GobacktoStep 4 untilt≥^T. Numericalresults

Linearsystem

Hereweapplyasimplelinearsystemtocomparetheperformanceofthehybridmethodwithdifferenttimestepsettings.

Inthelinearsystem,thereisonlyonetypeofmolecules, S₁,inaone-dimensionaldomainoflengthL=^{50.Wedividethe} domaininto100compartmentswithuniformsizeh=⁵⁰/100=⁰.5.Therearetwotypesofreactionslistedas

R_k1

:

^S1 γ1

−→ φ

fork

∈ {

¹

,

2

, ...,

100

}

^andRg2

: φ −→

^{γ2 S}1forg

∈ {

¹

,

2

, ...,

20

} .

Themolecule S1 diffuseswithacoefficientD withreflectiveboundaryconditions.Weset

γ

1=^1,

γ

2=^{500 andthediffusion} coefficient D=^{10.Theinitialconditionfor X}k1 whichrepresentsthenumberofS₁inthekthcompartmentis:

X_k1

(

0

) =

⁵⁵

−

⁰

.

5k

,

fork

=

¹

, ...

100

.

Forthislinearsystem, wecanexplicitlyobtaintheexactsolutionsofthemeanandthestandard deviation,whichwillbe usedtocalculatetheerrortoverifytheaccuracy.Also,thistype ofsimplemodelwasoftenappliedtostudythestochastic effectinbiologicalpatterning[20,24].

Ifwe usetheSSAtosimulatethesystemfromt=^{0 tot}=^6,theaveragecomputationalcostpersimulationisover60 secondsamong5,000 simulations.Whenweapply ourhybridmethodwith =⁰.1 andN_int=^10,thecomputationalcost isreducedto8secondwhichis13% ofthecostoftheSSA.

Fig. 2.Thesimulationresultsatt=^{6 forthelinearsystembythehybridmethodwithdifferent}tC settings:fixedtC=⁰.1h²/D,0.01h²/Dandan adaptivetC definedin(9).A)Themeanvaluesofthegradients.B)Thestandarddeviationsofthegradients.Foreachcase,5,000simulationsarecollected toobtainthestatisticalresults.Thedashedlinesrepresenttheexactsolutionsofthemeanandthestandarddeviation.

Fig. 3.TheaccuracyandtheefficiencyofthehybridmethodwithdifferenttCsettings.A)Thesumsoftheerrorofthemean.B)Thesumsoftherelative errorofthemean.C)Thesumsoftheerrorofthestandarddeviation.D)Thesumsoftherelativeerrorofthestandarddeviation.E)Thecomputational costsatdifferenttimet.

To show the advantage ofthe adaptive t_C, we compare the performance ofthe hybrid method with differentt_C settings.Fig.2showsthatthemeanvaluesandthestandarddeviationsofthecaseswithafixedtC=⁰.1h²/D,0.01h²/D andanadaptivetC definedin(9).Foreachcase,5,000simulationsarecollectedtoobtainthestatisticalresults.

Fig.2Ashowsthatallthreecaseshaveasimilaraccuracywhenthesolutionsarecomparedwiththeexactmeansolution;

thestandarddeviationsshowninFig.2Bdemonstratethatthecasewithafixedt_C=⁰.1h²/D doesnotperformasgood as theother two cases.Toquantify the results,we measure the sumofthe error (the absolutedifferencesbetween the approximationandtheexactsolutionineachcompartment)inallcompartmentsandtheresultsareshowninFig.3.

Fig.3Ashowsthatthesumsoftheerrordonothaveahugechangewithdifferentt_C settings.Thisresultisconsistent whenweconsiderthesumsoftherelativeerror(theabsolutedifferencesbetweentheapproximationandtheexactsolution dividedbytheexactsolutionineachcompartment)showninFig.3B.Whenwemeasuretheerrorinthestandarddeviation (Figs.3C,D),thecasewithafixedtC =⁰.1h²/D hasalarger error,alsoarelativeerror,thantheother twocaseswhich havesimilarperformance inapproximatingthestandarddeviation. However,theefficiencyofthecasewithafixedt_C = 0.1h²/D is the best among all three cases (Fig. 3E). The average computational cost for this case is around 6 seconds.

Betweentheothertwocases,theadaptivetime stepsettinghaslesscomputationalcostthanthecasewithafixedtC = 0.01h²/D (the former is 7 secondsper simulation andthe latteris over 8 seconds per simulation) although they have similar performance in the accuracy. Compared withthe SSA (60 seconds per simulation), the hybrid method with the

Fig. 4.Thesimulationresultsforthesystemofmorphogen-mediatedpatterningbytheSSAandthehybridmethodwith=⁰.1 andNint=^5.Foreach case,1,000simulationsarecollectedtoobtainthestatisticalresults.

adaptive time step setting saves over 80% of computational cost and provides an advantage on balancing between the accuracyandefficiency(Figs.3C–E).

Systemofmorphogen-mediatedpatterning

In[19], thesystemofmorphogen-mediated patterninginvolvesthree typesofmolecules, L, E and W,whichare free ligand,receptorandligand-receptorcomplex,respectively.Wedividethedomainoflength100 μm into100compartments withsizeh=^{1 μm anda}compartmentrepresentsasinglecell.Freemorphogensareproducedinalocalregion[⁰,10]^and diffuseinthedomainwitha diffusioncoefficient D1=^{10 μm2s−1;receptors and}ligand-receptorcomplexesare fixedon thecellmembrane.Ineachcompartment,thereactionsarelistedas

R_g1

: φ −→

^{γ1 L}

,

forg

∈ {

¹

, ...,

10

},

R_k2

:

^L

+

^E

−→

^{γ2 W}

,

R_k3

:

^W

−→

^{γ3 L}

+

^E

,

R_k3

:

^W

−→

^γ4

φ,

fork

∈ {

¹

, ...,

100

} .

Assumethat thenumberoftotalreceptorsislarge enough,thenthefrustrationof E+^{W isrelatively small.We simplify} the system by assuming R+^{W is a constant number E}T =^{500 ineach} compartment. The parameter values are listed asfollows:

γ

1=^{10 s−1,}

γ

2=^10−4s−1,

γ

3=

γ

4=^{10−2s−1.The initial conditionsfor X}k1 and Xk2, whichrepresentthe numbersofLandW inthekthcompartment,respectively,are:

Xk1

(

0

) =

Xk2

(

0

) =

100

−

k

,

fork

=

1

, ...

100

.

Fig.4showstheresultsofthemeansandthestandarddeviationsofthesolutionsatt=^{10 s,whichareobtainedfrom} the SSA and thehybrid method with

₌0.1 and Nint=^{5.For each case, 1,000} simulations are collected to obtain the statisticalresults.Fromthesimulationresults,wefindthatthehybridmethodhasagoodperformanceastheSSA.Onthe other hand,theaveragecomputational costofthe hybridmethodpersimulation is3.42 s butthe averagecomputational costoftheSSAis15.49 swhichis4timeslongerthanthatofthehybridmethod.Wealsoapplythehybridmethodwith twolargerthresholdvaluesNint=^{10 andN}int=^{20 andfindthattheaccuracydoesnothaveanysignificantchangebutthe} computationalcostincreasesfrom3.42 sto8.74 swhen Nint increasesfrom5to20.Moreover,whenthesmaller Nint=² isconsidered,thecomputationalcostincreasesfrom3.42 sto5.29 sasasmallertimestept_C isrequiredformaintaining theaccuracy oftheapproximation when Nint decreases.ComparedwiththeSSA, thehybrid methodwithasuitable Nint cansaveover75%ofcomputationalcostforthissimulation.

Systemofyeastpolarity

In [1], thestochastic model ofcell polarization showedthat a positive feedbackalone is sufficientto account forthe spontaneousestablishmentofasingle siteofpolarity.Weapply themodelin[1] to verifytheaccuracyofourmethod.In

Fig. 5. TheaccuracyandtheefficiencyofthehybridmethodwithdifferentnumbersofsignalingmoleculesNs.A)Asamplesimulationatt=^{20 min} throughthehybridmethodwithNs=^{4000.B)Asamplesimulationatt}=20 min throughtheSSAmethodwithNs=^{4000.C)Thepercentageofpolarized} caseswithin1,000independentsimulationsfordifferentNsvalues.D)Theaveragecomputationalcostwithin1,000independentsimulationsfordifferent Nsvalues.

the model,thereis onlyone type ofsignalingmolecules, S1,andthecomputational domainrepresents thecrosssection ofcellmembranewhichisconsideredasaone-dimensionaldomainwithlength10

π

μm (theradiusofcellis5 μm).The domain is partitionedinto 50 identical compartmentswithuniform length 0.2

π

μm. Signalingmoleculesmove between cytoplasmic states and membrane-bound states. In each compartment, there are three types of reactions: spontaneous membraneassociation(fromcytoplasmicstate tomembrane-boundstate),positive-feedbackassociation (fromcytoplasmic state tomembrane-bound state) andspontaneous membranedisassociation (from membrane-boundstate tocytoplasmic state).LetX_k1bethenumberofS₁inthekthcompartment.Ineachcompartment,wehave

Spontaneous membrane associationR_k1

:φ

^k1(Ns−

kX_k1)

−−−−−−−−−→

^S1

,

Positive-feedback associationR_k2

:φ

^{k2(Xk1/h)(Ns−}

kX_k1)

−−−−−−−−−−−−−−→

^S1

,

Spontaneous membrane disassociationR_k3

:

^S1

k3

−→ φ,

withthepropensityfunctions

α

k1

=

^k1

(

N_s

−

k

X_k1

),

α

k2

=

k2

(

Xk1

/

h

)(

Ns

−

k

Xk1

)

and

α

k3

=

k3Xk1

,

whereNsisthetotalnumberofsignalingmolecules.TheinitialconditionisXk1=^{10 forallk.Forourhybridmethod,weset}

=⁰.1 and N_int=^{5;forthebiological}parameters,weset D=¹.2 μm²/min,k₁/k₂=^10−4,k3=⁹/min andk₂/k₃=⁰.9N_s [1].TheinitialconditionforS1is

X_k1

(

0

) =

¹⁰

δ

k

,

fork

=

¹

, ...

100

,

where δk’sare independentrandomnumbersgeneratedfromthe uniformdistributionon[⁰,1]^{.Figs.5AandBshowtwo} samplesimulationsatt=^{20 min,obtainedbythehybridmethodandtheSSA,}respectively.

In[1],thestochasticmodelofcellpolarizationdemonstratedthatthefrequencyofpolarizationinverselydependsonthe numberofsignalingmoleculesN_s.Figs.5CandDshowthatthehybridmethodandtheSSAcancapturethisfeatureofthe system.Weassumethatpolarizationinsimulationsatt=^{20 min isdeterminedbywhetheranintervalof10%ofthewhole} domaincontainsmorethan50%ofthetotalnumberofsignalingmolecules(asthesamplesinFigs.5AandB).Fig.5Cshows thepercentageofpolarizedcaseswithin1,000independentsimulationsfordifferentN_svalues(redlinerepresentsthehy- bridmethod;thebluelinerepresentstheSSA).Thefrequencyofpolarizationisdecreasingfrom0.65to0whenthenumber