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Original citation:
Sayyid, Faiz and Kalvala, Sara. (2016) On the importance of modelling the internal spatial
dynamics of biological cells. Biosystems, 145 . pp. 53-66.
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ContentslistsavailableatScienceDirect
BioSystems
j ou rn a l h o m epa g e :w w w . e l s e v i e r . c o m / l o c a t e / b i o s y s t e m s
On
the
importance
of
modelling
the
internal
spatial
dynamics
of
biological
cells
Faiz
Sayyid,
Sara
Kalvala
∗DepartmentofComputerScience,UniversityofWarwick,Coventry,WestMidlands,UnitedKingdom
a
r
t
i
c
l
e
i
n
f
o
Articlehistory: Received15July2015
Receivedinrevisedform25May2016 Accepted31May2016
Availableonline2June2016
Keywords:
Whole-cellsimulation Reaction–diffusion GPGPU
Single-celldynamics
a
b
s
t
r
a
c
t
Spatialeffectssuchascellshapehaveveryoftenbeenconsiderednegligibleinmodelsofcellular path-ways,andmanyexistingsimulationinfrastructuresdonottakesucheffectsintoconsideration.Recent experimentalresultsarereversingthisjudgementbyshowingthatverysmallspatialvariationscanmake abigdifferenceinthefateofacell.Thisisparticularlythecasewhenconsideringeukaryoticcells,which haveacomplexphysicalstructureandmanysubtlecontrolmechanisms,butbacteriaarealsointeresting forthehugevariationinshapebothbetweenspeciesandindifferentphasesoftheirlifecycle.
Inthisworkweperformsimulationsthatmeasuretheeffectofthreecommonbacterialshapeson thebehaviourofmodelcellularpathways.ToperformtheseexperimentswedevelopReDi-Cell,ahighly scalableGPGPUcellsimulationinfrastructureforthemodellingofcellularpathwaysinspatiallydetailed environments.ReDi-Cellisvalidatedagainstknown-goodsimulations,priortoitsuseinnewwork.We thenuseReDi-Celltoconductnovelexperimentsthatdemonstratetheeffectthatthreecommonbacterial shapes(Cocci,BacilliandSpirilli)haveonthebehaviourofmodelcellularpathways.Pathwaywavefront shape,pathwayconcentrationgradients,andchemicalspeciesdistributionaremeasuredinthethree differentshapes.Wealsoquantifytheimpactofinternalcellularclutteronthesamepathways.Through thisworkweshowthatvariationsintheshapeorconfigurationofthesecommoncellshapesaltermodel cellbehaviour.
©2016TheAuthors.PublishedbyElsevierIrelandLtd.ThisisanopenaccessarticleundertheCCBY license(http://creativecommons.org/licenses/by/4.0/).
1. Introduction
Theintricate networks of complementaryphysiological pro-cessesthatcomposecellularbehaviourhavebeenmodelledina numberofstudies(LoewandSchaff,2001;Karretal.,2012).These processesarebroadlyknownascellularpathways.Someofthis modellingisreaction-only(Ortonetal.,2005;Gongetal.,2010; denBreemsetal.,2014)andthusisnotabletosimulatethe spa-tiallyheterogeneousnatureofthecell.Whilstreaction-onlymodels ofcellularpathwayshave provided greatinsight there is scope toimprove therepresentation of the cellby capturing thecell environment’sspatialdetails,suchascellgeometryandorganelle placement.Furtherworkisrequiredtounderstandtheinfluence ofspatialorganisationoncellularprocesses(Kholodenko,2009). Reaction–diffusionsystemscanextendthesereactiononlymodels
Abbreviations:ReDi-Cell,reaction–diffusioncell;GPGPU,generalpurpose com-putingongraphicalprocessingunits.
∗Correspondingauthor.
E-mailaddresses:[email protected](F.Sayyid),[email protected]
(S.Kalvala).
byallowingthechemicalspeciesandorganellestobeaccurately positionedinthecytoplasmandbypermittingthevirtualcellto takeitsnaturalshape.Insilicostudiesallowadegreeof measure-mentandcontrolthatisimpossibleinvitro.
In this work we simulate the effect of cell shape and organelleplacementonabstractedcellularpathwaysmodelledby reaction–diffusionsystems(Sections3and4).Whilethe simula-tionsperformedinthisworkarecomputationallyexpensive,the numericalmethodschosenareinherentlysuitedtomassively par-allelhardwaresuchasaGPU.
We developReDi-Cell (Section5),a performantand scalable GPGPUcellsimulationenvironmentsuitableforthiswork. ReDi-Cell is validated against VCell, a known-good cell simulation (Sections6.1–6.3).While VCellwould allowus to perform the sameexperimentsasthoseundertakeninthisworktheywould takemuchlongertoperformduetoVCell’sinabilitytorunona GPGPU.Inthevalidationexperimentsperformedinthisworkwe findthatReDi-Cellisapproximately24×fasterthanVCell.VCell isalsoclosedsource,makingitimpossibletoportittoanyother architecture.
Wethenperformnovelexperimentsthatexaminetheeffectthat threecommoncellshapes(Cocci,BacilliandSpirilli)haveonmodel
http://dx.doi.org/10.1016/j.biosystems.2016.05.012
cellularpathways.Pathwaywavefrontshape,pathway concentra-tiongradients, and chemical species distributionare measured inthree differentshapesin ordertoquantify spatialsensitivity (Sections6.5and6.6).Finally,thedampeningpotentialofclutter onintracellularsignallingpathwaysisexplored(Section6.7).
Byperformingthesesimulationsweseektoquantifytheeffect ofthreecommoncellshapes,andinternalclutteronthe dynam-icsofvirtualcellularpathways.Numericalsimulationallowsusto answerthesequestionswhenitwouldbechallengingtofindclosed formanalyticalsolutions.
2. Background
Inthissectionwereviewtherelatedbiologicalbackgroundfor thiswork.Cellulardynamicsandtheevidence(bothexperimental andtheoretical)fortheirspatiallysensitivenaturearediscussedin detail.
2.1. Cellulardynamics
Cellsreceiveandprocesssignalsrelatedtointernaland exter-nalchangesintheenvironment,suchasinfection,stress,injury, ornutrientneed.Thisbehaviourisachievedthroughmechanisms knownasbiologicalpathways,aseriesofinternalmolecular inter-actionsthat leadtoa specificchangein thecell.The effects of theseinteractionsvarywidely,butcanincludetheproductionofa newchemicalspecies,suchasafatoraprotein,orgeneratingcell movement.
Most cellular pathways fall into one of two categories: metabolic,orsignaltransduction.Metabolicpathwaysinvolvethe transformationofonespeciesintoanother,eithertobeused imme-diately,tobestored,ortoinitiateothermetabolicpathways.Signal transductionpathwaysamplify extracellular signals receivedat surfacereceptorsandpropagatethem throughthecell.Cellular behaviourisalsocontrolledbygeneregulatorynetworks,which areresponsibleforturninggenesonandoff.Thisswitchinggoverns theproductionofproteins.
Ithasbecomeclearthatthesereactionpathwaysarehighly spa-tiallyorganizedwithinthecell,withmanyreactionsoccurringonly inspecificregions(Srere,2000).Forexample,withinsignal trans-ductionpathways,spatialgradientsandmicrodomainsofsignalling occurduetolocalisedchemicalspecies,suchasphosphatases.As intracellulardistancesincrease,activesignallingmessengersare morelikelytobecomedeactivatedontheirjourneytothecell inte-rior(Meyersetal.,2006).Thedoseresponsecurvesoftheyeast MitogenActivatedProteinKinase(MAPK)cascadediffer depend-inguponboththegeometryofthecellandthesubtlechangein feedbackparameters(Zhaoetal.,2011).
2.2. Examplesofspatiallysensitivesystems
Spatiallydependentbehaviours,includingpathwaydynamics, havebeendocumentedinbothcomputationalandexperimental studies.Flourescenceresonanceenergytransfer-based technolo-gieshave been used to find concentration gradients and have givenexperimentalproofofspatiallysensitiveprocesses(Meyers, 2012).Concentrationgradientshavebeenfoundtoexistina num-berofsystemsincludingphosphorylatedstathmin/oncoprotein18 (Niethammeretal.,2004),MAPKFus3(Maederetal.,2007),and AuroraBkinase(Fulleretal.,2008).
Somecellularprocessescannottakeplacewithouttheexistence ofconcentrationgradients.Thepropagationofanactionpotential alongtheaxonisanextremeexampleofasystemdependenton concentrationgradients.Asthepotentialmovesalong theaxon, spatialgradientsinmembranepotentialandcurrentsareformed.
Theinteractionbetweenthegradientsandtheionchannels con-tinuetoforcethewavealongtheaxon.Thebehaviourofthissystem isintrinsicallyspatiallydependant,andnonspatialmodelsmight failtocapturethepropagation.
Calciumwaveandsparkmodelsareanotherexampleofa sys-temthatis spatially sensitive.Calcium wavesare necessaryfor thecontinuationofdevelopmentinsomeeggsafterfertilization. Recentexperimentalandtheoreticalworksuggeststhatcalcium wavepropagationmaybeaffectedbythepositioningof intracel-lularcalciumreleasechannels(Chenetal.,2014).InsilicoMyocyte behaviourhasalsobeendemonstratedtodependonspatial orga-nisation(Wengetal.,1999).Spatialmodelshavegivenusthefirst mechanisticinsights,showingmicro-domaindynamicstobea sys-temlevelpropertyarisingfromthecomplexinteractingbehaviour ofthecell(NevesandIyengar,2009).
The translocation of Nuclear FactorB (orNF-B) in tumor cellsisalsospatiallysensitive.Ithasbeendemonstrated experi-mentallythatcellshapeandmicro-environmentalfactorsregulate thetranslocationofNF-B,whichmayinturnplayakeyrolein thedevelopmentandthereforepotentiallythetreatmentofsome formsofcancer(Dolcetetal.,2005).
2.3. Mechanismsforspatialsensitivity
It is established in Section 2.2 that many cellular processes arespatiallysensitive.Inordertomodelthesespatiallysensitive systemsinanappropriatewaywereviewwhichmajorphysical propertiesofthecellenvironmentcausethissensitivity.
2.3.1. Concentrationgradients
Some models of cell biology implicitly assume that chemi-calspeciesconcentrationsarehomogeneouslydistributedacross space(Orton etal., 2005;Gong etal., 2010;den Breems etal., 2014).Howeverwhenthereactionspacebecomesrelativelylarge andchemicallyconnectedtoothersystems,concentrationsmay becomeheterogeneous,givingrisetoconcentrationgradients.It hasbecomeclearthatthesespatialgradientsexistinawiderange ofcellularprocesses(Howard,2006).
Inadditiontoreactantlocality,reactionlocality,suchas activa-tionatthemembraneanddeactivationinthecytoplasm,givesrise toconcentrationgradients.Ifonereactiondependsonchemicals producedattwodifferentlocationsthenthosetwochemicalsmust diffusetothereactionsiteforthereactiontooccur.Asthe concen-trationofthediffusingchemicalislessatthereactionsitethanat theproductionsite,agradientisformed.Insomecases concentra-tiongradientscanbeattributedtothedifferentdiffusivitiesofthe sameproteininactiveandinactiveforms(Kholodenko,2009). Con-centrationgradientschangereactionsystemdynamics.Nonspatial modelsmakethesimplifyingassumptionthatconcentrationis uni-formthroughthecellandthereforemightpredictincorrectvalues ofthediffusingchemical.
2.3.2. Cellshape
2.3.3. Cellularsub-structure
Thegeometryandpositionofsubstructureswithinthecellcan playanimportantroleinthespatialrestrictionofchemical reac-tionsandhencephysiologicalprocesses.Somechemicalprocesses areonlyabletotakeplacewithinspecificorganelles.Forexample, organellessuchasendosomesandtheGolgiapparatusplaya crit-icalroleinregulatingsignaltransmissiontothenucleus(Hwang etal.,2014).Precisepositioningoftheseorganelleswithinthecell isofcentralimportancetoeffectivecellregulation(Hwangetal., 2014).It followsthataccuraterepresentationof organelle posi-tionandgeometryisnecessaryforthecorrectnessofaninsilico
experiment.
3. Spatialmathematicalmodelsofcellularsystems
Having establishedtheimportanceof spatialorganisationin cellularsystemswenowreviewmathematicalmodelscapableof representingtheheterogeneousnatureofthecellenvironment.
3.1. Reaction–diffusionmodels
Reactionshavebeenmodelledbyvarioustechniques includ-ingordinarydifferentialequations(ODE)andstochasticprocesses. However diffusion, an intrinsic physical phenomenon some-times remains neglected. The aforementioned spatial factors (Section 2)would notbe importantifchemicals didnotmove. Reaction–diffusion models extendthe basic reaction model by addingadiffusivetransportsystemthatrepresentsthemovement ofchemicals.
Somecells haveevolvedtotakeadvantageofdiffusion’slow energyrequirementforchemicaltransport(Sohetal.,2010).Itis importanttonotethatwhenthediffusionprocessismuchfaster thanthereactionprocesstheeffectofdiffusiononthedynamicsof thesystemisdecreased.Inthesecasesreaction-onlysystemsmay besufficient.Wethereforelimitdiscussionsofreaction–diffusion systemstothoseinwhichbothreactionanddiffusionhavea signif-icantimpactonthedynamicsofthesystem.Duetohighchemical concentrations,metabolicpathwayssuchasglycolysisrelyon dif-fusionasameansoftransportacrossthecytoplasm.Inaddition, anyprocessinvolvingcytoplasmic/nuclearshuttlingmayalsobe affectedbydiffusion.
Reaction–diffusionsystemscanmodelcomplexchemical con-figurationswhich are impossible withreaction-onlysystems. A systemofreactinganddiffusingchemicalscanthusevolvefroman initial,relativelyuniformstatetoacomplexstate(Turing,1990). Forexample,Gray–Scottsystemshavebeenshowntobecapable ofmimickingbiologicalpatternformation(Mainietal.,1997).
Suchstudiesdemonstratethatreaction–diffusionsystemsare intrinsicallyspatialandsotheyarewellsuitedtomodellingthe arbitrarilyshapedboundariesofinsilicorepresentationsofcells. Theabilitytovarythediffusioncoefficientsinreaction–diffusion modelscanbeusedtocomputationallyidentifyparametersthat are likely to reproduce experimentally observed microdomain behaviour.
Therearemanydifferentwaystoconstructareaction–diffusion system.Areaction–diffusionmodelcanbeformulatedbyadding adiffusive componenttoanalready testedreactionmodel.The threemostcommonreactionsystemsaretheStochastic Simula-tionAlgorithm(SSA),theChemicalLangevineEquation(CLE)and theReactionRateEquation(RRE).TheSSAcanbemadespatialby discretisingthereactingspaceandtreatingadiffusiveevent,from onespatialquantatoanother,asareaction(Erbanetal.,2016).The CLEcanbemodifiedbyaddingatermforwhiteGaussiannoiseto simulatediffusion(Andrewsetal.,2009).TheRREissubjectto per-hapsthemostrecognisablemodification;byincorporatingFick’s
lawofdiffusionRREsystemsbecomePartialDifferential Equation-OrdinaryDifferentialEquation(PDE-ODE)hybridsystems.
Finally, there exist particlebased reaction–diffusion models. Thesesystemsrepresentparticlesexplicitly,withphysicallaws, suchas MolecularDynamics (MD) orBrownian Dynamics (BD) (PlimptonandSlepoy,2005)(orapproximationsthereof)dictating theevolutionofthesystem.
3.2. PDEbasedreaction–diffusionmodels
PDE-ODEbasedreaction–diffusionsystemsareasuitable tech-niquetosimulatereaction–diffusionmodels insidecells (Neves andIyengar,2009).PDE-ODEreaction–diffusionmodelsallowthe formation of spatial gradientsby permittingthe localisationof chemicalspeciesandthedescriptionofcellsubstructure morphol-ogywithineasilyexpressibleboundaryconditions.
AsthePDE-ODEmodelextendstheODEmodel,comparisons areoftenmadebetweenthetwo;ODEsdescribethechangein con-centrationofasetofspeciesovertime,whereasPDE-ODEmodels describethechangeinconcentrationofasetofspeciesovertime andspace(EungdamrongandIyengar,2004).PDE-ODEmodelstake accountofthediffusiveprocessesthatoccurwithinasystemas wellaschangesinconcentrationduetoreaction.Mathematically, wedescribeaPDE-ODEreaction–diffusionsystemas:
∂
u∂
t =D∇
2
u+R(u) (1)
oralternatively:
∂
u∂
t =D∂
2u∂
x2 +∂
2u∂
y2 +∂
2u∂
z2+R(u) (2)
whereuistheconcentration,DisthediffusioncoefficientandRis asystemofreactionsdescribedbyasetofODEs.ThePDEs(
∇
u)and ODEs(R(u))requiredifferentsolvingtechniques.Inrealsystemstherecanbemanydifferentchemicalspecies. Asthenumberofspeciesandthenumberofreactionsincreases thecomputationalcostforsolvingthePDE-ODEreaction–diffusion equation also increases. This is due to the increase in the numberoftermsinR(u)andthenumberoftimesthePDE equa-tionissolved.PDEreaction–diffusionsimulations(likeallother reaction–diffusion simulationtechniques) are not withouttheir drawbacks, requiring diffusion coefficients and initial reactant concentrations,andlocations,whicharesometimesunknown. The-oretical diffusion coefficients can becalculated from molecular weightsand,withtheadventoffluorescencerecoveryafter pho-tobleaching experiments, it is possible to directly measure the diffusioncoefficientsofcellularcomponents.
3.3. SolvingPDE-ODEreaction–diffusionmodels
Eq.(1)hasbothanODEcomponentandaPDEcomponent.Inthis sectionwedescribemethodsforcomputationallysolvingthesetwo typesofdifferentialequation.HyperbolicPDEssuchasthediffusion equationcanbesolvedinmanydifferentways.Twoofthemost commonaretheexplicitfinitedifferencemethod(FDM)andthe implicitCrank–Nicholsonmethod.FDMisboundbythecondition thatt/x2<0.5(Smith,1985);thatis,thesizeofthetimestep (t)dividedbythesquareofthesmallestmeasurabledistance(x) mustbelessthan0.5.
Thealternativetousinganexplicitschemeisanimplicitscheme, wherethereaction–diffusionsystemissolvedasasetof simulta-neousequationsbyalinearsolver.Implicitmethodsdonothaveto observethesametime/spaceratiorequirement,howeveritismore challengingtoimplementboundaryconditions(suchascomplex cellandorganelleshapes)inanimplicitregime.
ODE systems, such as the RRE component of the reaction–diffusionequation,haveenjoyedutilityacrossarangeof disciplinesandassuchmanyrobustsolvingtechniquesexist.Most numerical methods for solving ODEs approximate the solution by numerical integration. Reaction–diffusion systems generally taketheformoffirst-orderinitialvalueproblems(IVP).Thesecan readilybesolvedusingRunge–Kutta(RK)methods.
3.4. Summary
We have seen in the literature that a wide variety of cel-lular processes are affected by cell shape, organelle placement andinitialconcentrationofchemicalspecies.Itisalsoseenthat reaction–diffusionsystemscanaccuratelymodeltheseprocesses’ spatialdynamics.
4. Arepresentativeabstractionforcellularpathways
Cellularpathwayscaninvolvelargenumbersofchemicalspecies takingpartincomplexnetworksofchemicalreactions.Often, ini-tialconditionsandparametersinthesenetworksareunknown.In thisworkwearenotinterestedinaspecificreactionsystem,only theeffectof thecellenvironment ona generalreactionsystem. Fundamentallyallcellprocesses arechemical,thereforewecan modelanyphysiologicalprocess(atanabstractlevel)witha gen-eralreactionsystemcoupledwithdiffusion.Thereforewesimulate
abstractedcellularpathwaysthatobviatetheneedforunknown parameters,yetarerepresentativeofasimplecellularpathway.We selecttheLotka–Volterra(Lotka,1925;Volterra,1927)(LV)reaction modelastheabstraction.
The LV model was independently proposed to explain two different behaviours; the increase in the number of predatory fishintheAdriaticand atheoretical,oscillating,chemical reac-tion. The model is the simplest example of a predator-prey system(McLaughlin and Roughgarden, 1991)and its properties have been widely studied in the literature. In a predator-prey system,agents competewith each other and theenvironment tosurvive.Ingeneral,predator-preysystemsdescribecompetitive interactions.Thenatureoftheagentsvarybyapplicationbut com-monlytaketheformofcarnivoresandherbivores,orasinthiscase chemicalspecies.
TheLVsystemisdescribedas:
dx
dt =x(˛−ˇ∗y) (3)
dy
dt =−y(−ı∗x) (4)
wherexisthenumberofprey,yisthenumberofpredators,dx/dt
anddy/dtaretheratesofgrowthofthepopulations.˛,ˇ,,ıare parametersdescribingtheinteractionsofthetwospecies.
LVsystemshavepreviouslybeencoupledwithdiffusion(LVD) (Hastings,1978).Suchinvestigationsarenormallyconcernedwith themathematicalpropertiesofthesystem,suchasexistenceof solutionsorstabilityanalysis,ratherthansimulationorapplication. LVDsimulationshaveoccasionallybeenusedtomodelecological migration;puttingtheLVsystembackinitstraditionalsetting.An exampleofsuchastudycanbeseenin(Modelica,2014).
Ifitisfoundthatthespatialorganisationofthevirtualcellalters thedynamicsofasimple modelcellularpathway(LVD),then it
Fig.1. Commonbacterialshapesreproducedfrom(BacterialShapes,2014).
ishighlylikelythatmorecomplexpathwayswillhavemore pro-nouncedeffects.
5. ReDi-Cell
InSection2wesawthatthecellsdisplayheterogeneous inter-nalandexternalgeometries,andthataccuraterepresentationof cellstructureiscrucialtosimulation.Section3demonstratesthat whilstcomputationallyexpensive,reaction–diffusionsystemsare highlysuitedtomodellingcellularpathwaysincomplex geome-tries.FinallySection4demonstratesthatifweareinterestedin theeffectofcellgeometryonageneralreactionsystemthat com-plexpathwayscanbereplacedwithabstractions,suchastheLVD system.
These conclusions suggest that cellular pathways simu-lations should be capable of representing abstractions of reaction–diffusionsystemsinsidecomplexcellgeometryina per-formant manner. In this section we present ReDi-Cell, a novel simulationpackagethatfulfilstheserequirements.Thesoftware isavailableathttp://github.com/FaizSayyid/ReDi-Cell.
5.1. Description
ReDi-Cellisa3DhighperformanceGPGPUreaction–diffusion simulation of arbitrary cellular geometry.It is the first GPGPU PDE-ODEcellsimulation infrastructure specificallydesigned for investigating the effect of realistic cell shapes and internal components. ReDi-Cell is designed to take cellular morphol-ogy information, store it as voxels, and numerically solve a reaction–diffusionsystemrepresentingaphysiologicalprocesses. Each voxel represents a quanta on a regularly spaced, three-dimensionalgrid.
Systems incorporating PDEsare more challenging to imple-ment,withfewlargescaleprojectsavailableforuse(Nevesand Iyengar,2009).VCell(LoewandSchaff,2001)isonesuch simu-lationpackage.VCellincludesavarietyofsimulationtechniques notlimitedtoPDEsimulationandhasbeenusedinavarietyof publishedmodels.
ReDi-Cell,asahighperformanceGPGPUsolvertailormadefor PDE-ODEsystems,scalestofarmoredetailedproblemsovermore timestepsthanarefeasiblewiththeavailableVCellinterface.The usermaydedicateasmanyGPUsastheyhaveavailabletothetask; spatialdecompositionoccursoverallavailableGPUs.
5.2. Realisticcellshapes
Fig.2. ApproximationstocellshapesinFig.1usingtheViSiTsoftwaretoolkit.
Fig.3.CutawayofaReDi-CellmodelCoccusshowingthemembrane,cytoplasm andnucleusproducedusingVisIt.Thedifferentcoloursrepresentdifferent biolog-icalmaterialtypeswhichhavedifferentsimulationproperties.Thesesimulation propertiesallowfordifferingratesofreactionanddiffusioninthoseareasofthe cell.
5.3. Cellularsub-components
ReDi-Cellallowsinternalcellcomponents,suchasorganelles,to berepresented.Eachcomponent’spermeabilitycanbealteredby changingthediffusionparameterinthatlocationinthecell.This allowsReDi-Cellcomponentstomimictheirbiologicalanalogues inthecell,withdifferentreagentshavingdifferentresponsesto dif-ferentcellmaterials.Forinstance,somereagentsareconfinedtothe plasmamembrane,whereasothersmightbeabletopassthrough. InFig.5weseeachemicaldiffusinginacrowdedcellular envi-ronmentwithcellularsub-components.Thechemical,incapableof diffusingthroughtheimpermeablemembraneofthecellnucleus isforcedtodiffusearoundtheclutterthatrepresentsanorganelle. Fig.3showsan“onion-peel”decompositionofanexample simu-lationenvironment.ThisillustratesthewayReDi-Cellcomponents arecapableofcapturingtheinternalstructureofthecell.Fig.4 showsanexampleofa2DslicethroughthesameReDi-Cell envi-ronment.Fig.4includesgridlinestoshowthewayinwhichspace hasbeendiscretised.Thelevelofdiscretisationisconfigurable.
Itisunlikelythatallsub-cellularstructureswillhave homoge-neousmorphology;organellesareofmanydifferentshapesand sizes.ReDi-Cellallowsinternalcellcomponentshapestobe repre-sentedasaccuratelyastheexternalcellshape.Weseeanexample ofthiscapabilityinFig.6,whichshowssphericalandcylindrical shapedorganellesinsideaSpirillum.
5.4. Simulation
Simulationexecution inReDi-Cell is dividedintotime steps. A single time step consists of a diffusion process and a
Fig.4. 2DslicethroughthecentreoftheReDi-CellmodelofaCoccusdepictedin
Fig.3.ThisfigureillustratesthewayinwhichVisItcanchangeperspectivetoshow specificregionsofinterest.2Dsliceresultsareusedextensivelyinthiswork.
reaction process.First, theODEs representing thereactionsare solved.Thisprocessoccursovereachpartofthecellinwhich dif-fusionisallowedtotakeplace.Thenthe3DPDEdiffusionequation issolvedfor eachreagent,againintheappropriateparts ofthe cell.
TherearemanydifferentwaysofsolvingPDEsandODEs,as dis-cussedinSection3.3.WechoosetosolvePDEsusinganexplicit FDMalgorithm (Smith,1985)andODEs using aRunge–Kutta 4 (RK4)algorithm(Pressetal.,2002).TheFDMalgorithmisa com-monmethodofsimulatingreaction–diffusionandisusedinmany modernstudiesofsuchsystems(Blagodatskietal.,2015).RK4isan equallypopularmethodofsolvingODEsanddespiteenjoyingless successonstiffsystemsissuitableforallofthesystemsthatwe useinthiswork.Weshowthistobetruebyverifyingallsimulated reactionsystemsagainstMATLAB’sexplicitode45solver.Matlab documentationsuggeststhatifode45solvesthesystempromptly thenthesystemisnotstiff(Matlab,2016).Inadditiontothiswe verifyourresultsagainstMathematica’sWolframAlphaODEsolver anditisonceagaindemonstratedthatanon-stiffsolverissuitable forthesystemsusedinthiswork.
ReDi-CellusesVisualiseIt(VisIt)forvisualisation,apopularopen sourcetoolforvisualizingandanalyzingdata(Childsetal.,2012). VisItiscapableofdisplaying2Dslicesof3Dshapes.InFigs.4and6 weseeexamplesofsuchcutawaycells.ReDi-Cellhastwodifferent implementationsofthereaction–diffusionprocess,aparallel algo-rithmexecutedontheGPUandaserialalgorithmexecutedonthe CPUforwhenacompatibleGPUisnotavailable.
5.5. Implementation
Fig.5.2Dslice,producedusingVisIt,throughthecentreofaReDi-Cellmodelof aBacillus,withamodelchemicaldiffusingaroundacentralorganelle.Thisimage showshowdifferentmaterialscanhavedifferentsimulationpropertiesandhow VisItcanrendermaterialsindifferentways.Insteadofthesolidmaterial represen-tationsfoundinFig.4andFig.3onlytheboundariesofmaterialsaredrawnallowing bothmaterialtypeandchemicalconcentrationtoberenderedinthesameimage. Theboundariesofthedifferentbiologicalmaterialsarerepresentedbythedifferent colouredlines.Redrepresentstheboundaryofacentralorganelle,lightgreenline theinternalboundaryofthecellandpurpletheexternalboundaryofthecell.The boundariesintheimageareoverlayedontoaheatmapofchemicalconcentration, thisisthereasonthatmostoftheimageisdarkblue(representing0 concentra-tion).Themulti-colouredcircletotheleftofthecentralorganelleshowstheonly non-zeroconcentrationportionoftheheatmap;areactinganddiffusingcircleof chemicalconcentrationthathasnotreachedtheinternalboundary.Att=0this chemicalwasasinglepointofconcentration.Thedifferentmaterialshavedifferent simulationpropertiesasdiscussedinFig.3.Inthiscasethematerialenclosedby theredboundarydoesnotpermitreactionordiffusiontooccurandthechemicalis haltedatthemembrane.Themembranepermitsbothreactionanddiffusionbutthe rateofdiffusionishalved.Beyondthepurplelinerepresentingtheexternal bound-aryofthemembraneonlydiffusionprocessesarepermitted.(Forinterpretationof referencetocolorinthisfigurelegend,thereaderisreferredtothewebversionof thisarticle.)
Fig.6.2DslicethroughthecentreofaReDi-CellmodelSpirillumproducedusing VisIt.TheboundariesinthisimagearesolidasinFigs.3and4andsono concen-trationsarevisible.Thisfiguredemonstratesdifferentshapedorganellesinsidethe modelcell.Greenrepresentsthecytoplasminwhichchemicalsarepermittedto reactanddiffuse.Redshapesrepresentorganellesthroughwhichchemicalcannot diffuseorreactinside.
theExplicitFDM-RK4reaction–diffusionmethod.IntheGPU algo-rithmmanyreactionordiffusioneventsarerunatthesametime, insteadofoneaftertheother,asisthecaseintheCPUalgorithm. Inasingletimestepallreactioneventsarerunafteralldiffusion eventshavetakenplace.Duringthesimulation,dataiswrittento thediskforofflineanalysis.TransferringdatatoandfromtheGPUis
Table1
Descriptionofvalidationexperiments.Theparametersusedintheexperimentsare showninthecaptionsofthefiguresdepictingexperimentalresults.
System Reactionequation Initialconditions
Diffusion A:10
A+B→C dA
dt =−k∗A∗B A:10
dB
dt=−k∗A∗B B:10
dC
dt =k∗A∗B C:10
expensiveandthuswritingtodiskeverytimestepcandegrade per-formance.Thereforewecompromise,transferringdataandwriting tothediskatconfigurablysizedintervals.
Theexecutionoftimestepsis“batched”togetherinconfigurably sizedintervals.Executionispausedbetweenbatcheswhilstresults arewrittentothedisk.Thelargerthebatchsizethefasterthe sim-ulationwillrun.Timestepbatchsizesof100stepsarecommonly used,whichprovidesagoodtrade-offbetweenspeedandoutput granularity.
TheGPUalgorithmhastwostages:Thekernellauncher,andthe kernelitself.ThekernellauncherrunsontheCPU,itdividesthe simulationintobatchesandwritesoutputtothedisk.Thekernel runsontheGPUandisresponsibleforexecutingthereaction pro-cessinparallelandthenthediffusionprocessesinparallelacross alloftheGPUcores.
WealsoincludeaCPUalgorithm.Asingletimestepconsistsofa serialimplementationoftheexplicitFDM-RK4reaction–diffusion method.Aftereachtimestepthestateofthesystemiswrittento disk.
Interms ofspeed, wefindthatReDi-Cell canperformfaster thanVCell.Usingverificationexperiments1(Figs.7and10(a)),2 (Figs.8and10(b))and3(Figs.9and10(c))asbenchmarkswefind thatReDi-Cellis approximately24×fasterthanVCell.ReDi-Cell benchmarksimulationswereperformedonanNVIDIATeslaK20.
6. Experiments
In this sectionwe detail three sets of experiments that we runwithReDi-Cell.ThefirstisfocusedonvalidatingReDi-Cell’s implementationagainstthestateoftheart,VCell.Thevalidation experiments use two different prototypes: diffusion only and
A+B→C.AsummaryofthereactionsystemsisshowninTable1.In thesecondsetweinvestigatethebehaviourofabstractedcellular pathwaysindifferentcellshapes.Inthethirdsetweinvestigate theimpactofclutteronthesamepathways.Foradiscussionofthe choiceofdiffusioncoefficientsseeAppendixC.
Experimentstakeplaceinavarietyofclutteredanduncluttered environments.Theresultstaketheformofconcentrationheatmaps andconcentrationagainsttimegraphs.
Graphsof concentration against time areused when study-ingthedynamicsofareactioninasinglevolume.Heatmapsare usedwhenmeasuringtheconcentrationinmanyvolumesatonce, at a singlepoint in time. We exploit this property tomeasure the effect of space onthe distribution of concentration. Many heatmapsrecordedatsequentialtimescanbestitchedtogether intoananimationtoshowthesameinformationasa concentra-tiongraph,butformultiplevolumesatonce.Bothconcentration against time graphsand heatmapsare commonin cell simula-tion,withVCell(againstwhichwevalidateourresults)usingthese measurements.
6.1. Uncluttereddiffusion
Fig.7.VerificationExperiment1–uncluttereddiffusion.HeatmapsofspeciesAafter3000timestepsofdiffusion.TheexperimentisruninbothReDi-Cell(Fig.7(a))and VCell(Fig.7(b)).ItcanbeseenthattheheatmapsareidenticalvalidatingtheaccuracyofReDi-Cell’sdiffusioncomputation.k=1.412,t=10−14s,D=1.0,x=10−6m.
Fig.8.VerificationExperiment2–cluttereddiffusion.HeatmapsofspeciesAafter3000timestepsofdiffusion.TheexperimentisruninbothReDi-Cell(Fig.8(a))andVCell (Fig.8(b)).ItcanbeseenthattheheatmapsareidenticalvalidatingtheaccuracyofReDi-Cell’scluttereddiffusioncomputation.k=1.412,t=10−14s,D=1.0,x=10−6m.
uncluttered setting in ReDi-Cell. Results from this experiment are shown in Figs. 7(a) and 10(a). Thissame systemwas then run again in an identical uncluttered setting in VCell. Results fromthis experiment are shown in Figs. 7(b) and 10(a).These figures showthat theresultsof theseexperiments arein good agreement.
6.2. Cluttereddiffusion
Thissetofexperiments examineshowReDi-Cell modelsthe interactionof chemicalswiththeimpermeable boundariesthat modeltheinternalcellwallandorganelles.Thediffusionsystem fromTable1wasruninaclutteredenvironment,consistingofa
Fig.10.Graphsofspecies(A)concentrationinReDi-CellandVCellmeasuredina sin-glesub-volumeover3000timesteps.Concentrationismeasuredinthesub-volume
Table2
Descriptionofnewworkexperiments.Theparametersusedintheexperimentsvary andsoareshowninthecaptionsofthefiguresdepictingexperimentalresults.
System Reactionequation Initialconditions
LVD dA
dt =A(˛−ˇ∗B) A:10
dB
dt =−B(−ı∗A) B:10
singleimpermeablecube.Thespeciesareinitiallylocatednextto thecube.TheexperimentisfirstruninReDi-Cell.Resultsfromthis experimentareshowninFigs.8(a)and10(b).Thesamesystemwas runinVCell.VCellresultsareshowninFigs.8(b)and10(b).Once again,theresultsfromReDi-CellandVCellareconsistentwitheach other.
6.3. ClutteredA+B→C
InthissetofexperimentsweinvestigateReDi-Cell’sbehaviour whensimulatingreaction–diffusion nexttoimpermeable mem-branes. TheA+B→C reaction–diffusionsystemfrom Table1 is runinaclutteredenvironment.Thiscluttertakestheformoffour impermeablecubes,separatedbynarrowchannels.Thespeciesare initiallylocatedatthecentreofthechannels.Theexperimentis firstruninReDi-Cell,theresultsofthisexperimentareshownin Figs.9(a)and10(c).Thisexperimentisrunagaininanidentical environmentinVCell.Theresultsofthisexperimentareshownin Figs.9(b)and10(c).ReDi-Cell’sresultsareshowntobeconsistent withVCell’s.
HavingvalidatedReDi-CellagainstVCellwenowuseReDi-Cell toperformnovelexperimentsinSections6.5–6.7.
In all three sets of experiments run so far we find a good agreementbetweenReDi-CellresultsandVCellresults.The con-centrationtrajectoriesforVCellandReDi-Cellarenearlyidentical inallthreeexperiments.Verylittlevariationisshownbetweenthe twoalgorithms.Heatmapresultsalsoshowgoodagreement.The shapesoftheReDi-CellandVCellconcentrationprofilesinallthree experimentsshowgoodagreementsbetweeneachother.The val-uesdisplayedintheheatmapalsoshowaverygoodagreement.In summary,itcanbeseenthatReDi-Cellisanaccuratesimulationof reaction–diffusionsystems.
6.4. Theimportanceofspatialsimulation
Having discussed importance of spatial simulation we show the impact that removing spatial component has on a reaction–diffusionsystem.Thesystemusedinthisexperimentis theLVDsystemdetailedinTable2.Todemonstratethiswerun two experiments. In the firstexperiment we turnoff diffusion allowingonlyreaction,thusmakingtheexperimentnonspatial.In thesecondexperimentweallowforbothreactionanddiffusionin aCoccus.TheresultsofthisexperimentareshowninFig.11.
6.5. LVDwavefrontsinnaturalcellshapes
Thissetof experimentsmeasurestheeffectofcellshape on wavefrontproperties,specificallywavefrontshapeandwavefront concentration.Wedefinethewavefrontasthelocusofpointsthat areattheinterfacebetweenzeroandnonzeroconcentrations.The systemusedinthissetofexperimentsistheLVDsystemdetailed inTable2.
thatthechemicalwasinitiallyconfinedto.Thelinesthatrepresentthe concen-trationinReDi-CellandVCellareontopofeachothervalidatingtheaccuracyof ReDi-Cell’sclutteredreaction–diffusioncomputation.k=1.412,t=10−14s,D=1.0,
Fig.11.AcomparisonofspatialandnonspatialsystemsinReDi-Cell.Concentration ismeasuredinthesub-volumethatthechemicalwasinitiallyconfinedto.Withboth reactionsystemsitcanbeseenthatthereisalargedifferenceinresultsbetween spatialandnon-spatialsystems.Boththefinalvalueoftheconcentrationandthe tra-jectoryofthereactionovertimevarysubstantially.Fixedparameters:t=10−14s,
x=10−6m,D=0.02.
Table3
LVDwavefrontsinnaturalcellshapes.Concentrationatthewavefrontafter1000 timesteps.˛=1.5,ˇ=1.0,=3.0,ı=1.0,t=10−14s,x=1.0*10−6m,reaction
rate=1012,D=1.0.
Shape Concentrationatthewavefront
Coccus 1.06
Bacillus 1.38
Spirillum 2.61
TheresultsofthissetofexperimentsareshowninFig.12and Table3.TheReDi-Cellreactionvesselsareapproximationsofreal cellshapes.Threeexperimentswererun,oneineachcellshape. Thevolume,initialconcentrationsandinitialpositionsofchemical speciesarekeptanalogous.
Fig.12.LVDwavefrontsinnaturalcellshapesafter1000timesteps,simulated usingReDi-CellandrenderedinVisIt.Inthisfigureweseetheeffectofrunningan LVDsysteminmodelnaturalcellshapes.˛=1.5,ˇ=1.0,=3.0,ı=1.0,t=10−14s,
Table4
Peakconcentrationafter2000timesteps.˛=1.5,ˇ=1.0,=3.0,ı=1.0,t=10−14s,
x=1.0*10−6m,reactionrate=1012,D=1.0.
Shape Peakconcentrationof speciesAafter2000 timesteps
Coccus 2.99
Bacillus 5.49
Spirillum 7.26
First,wavefrontshapewasinvestigated.Theshapeofthe wave-frontsaredifferentinallthreeexamplesin Fig.12.TheCoccus’ wavefrontisextremelycircular,whilsttheBacillus’wavefrontis almostflat.TheSprillum’swavefront,curvesmoresteeplytoward thebottomofthecell.
Next,weexaminetheeffectthatcellshapehasonaverage wave-frontconcentration.TheresultsarerecordedinTable3.Theaverage concentrationatthewavefrontinallthreebacterialshapesis dif-ferent.Theconcentrationgradientsbehindthewavefrontarealso different,forexamplewhilsttheconcentrationbehindthe wave-frontintheBacillusandintheCoccusarehomogeneous,theirvalues aredifferent.
Differentsurfaceareatovolumeratiosmayberesponsiblefor theuniquepatternsofreflectioninsidevirtualcellsandsocause thedifference in concentration gradients and wavefrontshape betweencellshapesseeninthissection.Ingeometrieswithlinesof symmetry,suchastheCoccusandBacillus,concentrationwavesare reflectedinauniform,symmetricfashion.TheSpirillumis nonuni-forminshapeandthusreflectionsarenotnecessarilysymmetric aboutthelongestaxisofthecell.Thismayproducethe asymmet-ricwavefrontsseenintheresults.Changesinreflectiveproperties mayaltermicrodynamicsinequallydistinctwaysandsochange theglobalresponseofthesystem.Inaddition,incellsofequivalent volumesignalsinthesimulatedCoccustravelagreaterfractionof thecell’slength.
6.6. LVDspeciesdistribution
Thissetofexperimentsexaminestheeffectcellshapehason finalspeciesdistribution.Speciesdistributionismeasuredas fol-lows.The3Dcellisdividedinto2Dslicesalongthecell’slongest axis,witheach slicehavingathicknessequaltoonesimulation unitlength,x.Thetotalamountofagivenspeciesineachslice isfoundbyiteratingovereachsub-volumeinthesliceand sum-mingthecontributions. Finallythetotal amountofa particular speciesineachsliceisplottedagainstthepositionofthatslicealong thecell.
TheLVDsystemdetailedinTable2isruninthethreedifferent cellshapes.Thespeciesdistributionprofileisrecordedafter2500 timesteps.ThespeciesdistributionsareshowninFig.13andthe peakconcentrationareshowninTable4.
We see several differences between the graphs in Fig. 13. Different cellular geometries result in different concentration profiles and peak concentrations. The shapes of the concen-tration profiles are different in each example. The Coccus has a more concentrated profile withthe greatest peak concentra-tion, the Spirillum has a mostly uniform profile with a lower peak, but the Bacillus shows the most difference. Discounting thetapering edge,weachieve aflatprofile,witha spikeatthe wavefront.
6.7. LVDwithcellularclutter
Inthissetofexperimentswemeasuretheeffectofintra-cellular clutter on the dynamics of the LVD system. Two experiments wererun,theinitialconditionsforwhichareshowninFig.14(a)
Fig.13.LVDconcentrationdistribution after 1500timesteps,simulatedusing ReDi-Cell and rendered in VisIt. ˛=1.5, ˇ=1.0, =3.0, ı=1.0, t=10−14s,
Fig.14. LVDwithcellularclutter.Thisfigureshowstheclutteredandunclutteredmodelcellinitialconditions,renderedinVisIt.Measurementsiteisthesmallredsquare atthetop.
Fig.15.LVDwithcellularclutter.SpeciesAconcentration,inclutteredandunclutteredsettings,measuredover4000timestepsatthemeasurementsiteinthemodelcellsas detailedinFig.14.ThissimulationwasperformedusingReDi-Cell.Fixedexperimentalparameters:A=1,B=10,˛=0.5,ˇ=1.0,=10.0,ı=1.0,t=10−14s,x=1.0*10−6m,
reactionrate=1012.
(unclutteredsystem)and14(b)(cluttered).Theresultsareshown inFig.15(a).Theonlyvariablechangedistheinteriorvolume,by addingthreespherestorepresentclutter.Thevolumeofthecell,the initialconcentrationsandinitialpositionsofspeciesAandspecies
Bareallkeptconstantineachexperiment.Concentrationis mea-suredatthetopofthevirtualcell,thisislocationismarkedbythe redoutlinedsquareatthetopofFig.14(a).
Fig.15(a)shows thedifference incluttered and uncluttered environmentsonthearrivalofspeciesAatalocationimmediately nexttothemembraneandoppositetothesource.
Clutterwithinvirtualcellscanchangethewayinwhichinsilico
biochemicalprocessesprogressbychangingtheflowofachemical speciesthroughthesystem.Specifically,clutterhastwoimpacts onthissystem:firstly,thetimeofarrivalofspeciesAisdifferent, andsecondly,theshapeoftheconcentrationprofilesaredifferent. Therangeofconcentrationvaluestakenismuchsmaller.Thesignal hasbeendampedbytheclutterinthesystem.Allofthesefactors contributetothedifferencesin concentrationgradient between dampedandundampedsystems.
7. Conclusionsandfurtherwork
Wehaveperformedsimulationsthatshowhowthebehaviour ofmodelcellularpathwayschangesbetweenthreecommoncell shapes,andhowtheconfigurationofinternalcellularcluttercan alterthewayinwhichsignalspropagatethroughavirtualcell.In doingsoweshowthatmodelcellularpathwaysaresensitivetothe shapeandtheinternalgeometryofthecell.Wealsoshowedwhich physicalpropertiesofthemodel pathwaychangewhenthecell shapechanges;specifically,wavefrontshape,peakconcentration andconcentrationdistributionareallaffectedbytheshapeofthe cell.Finally,wehavedevelopedanovelGPGPUhighperformance cellsimulationtoolkitsuitableforperformingtheseexperiments.
contrastReDi-Cell is opensource allowingusers tochangethe sourcecodetotakeadvantageofanyfuturedevelopmentsin com-putingarchitecture.In additiontheclient/servernatureofVCell meansthatjobsarescheduledbeforeexecution,possiblyincurring anadditionaldelay.Finally,ReDi-Cellusesthededicated“VisIt” visualisationtoolkit which enablesresultstobepresented in a varietyofwaysnotpossiblewiththeVCellviewer.
One issue in trying to model cellular phenomena in detail hasbeenthelackofexperimentaldatatoinformthesearchfor parametersaswellastovalidatetheresultingsimulations.Recent developments,suchasphotobleachingandsuperhighresolution imaging(Balletal.,2012)arestartingtoprovidethisdata.Detailed informationonfeaturessuchas3Dcellgeometry,reactant con-centrations,andreactantlocationsareprovidingabasisfor this parametersearch.
Theconversionofexperimentaldataintosimulation param-eters canprovequite challengingin terms of setting upthein silicoinitialconditions:itisdifficulttospecifythree-dimensional shapesanddistributionsinanintuitiveway.Oneapproachistouse imageprocessingalgorithmstoextractqualitativefeaturesfrom microscopyimagesandvideosatthesametimeasnoise-reduction algorithmsareapplied.Asanexample,PhenoPlot(Sailemetal., 2015)enablestheautomatic,accuratetranslationofhighresolution microscopyimagesofcellstoconvenientglyphbased representa-tionsofthecellintermsoffeaturessuchasnucleartextureandsize, degreeofroundedness,etc.Thesefeaturescanthenbeusedtotailor thecharacteristicsofthevirtualcelltobesimulatedusing ReDi-Cell.Thus,withthesecombinedtechniquesthesimulationofmore realisticsystemswithinanintegratedexperimental/computational cyclebecomesfeasible.
Oneavenueoffurtherworkwouldinvestigatehowtheratioand distributionofclutterwithinthecytosolchangesthedynamicsof cellularpathways.
Anotherinterestingandchallengingdirectionforfuturework istheunderstandingtheeffectofchangingcellmorphologyonthe dynamicsofabiologicalprocess.Manyrealcellsexhibit plastic-ityduring theirlife cycle,changing fromone shape toanother inorder to bestsuit theirenvironment (Yinet al., 2014).Such changes can be caused by a variety of environmental factors (Young,2006)withtime-scalesanddegreesofplasticityof vary-ingmagnitudes. For example,smallshape changes canbeseen inmostcellsdue toinstabilitiesin cytoskeletalstructure (Kueh and Mitchison, 2009)with larger changes occurring in scenar-iossuchasstemcelldifferentiation.Nutritionalstressisanother exampleofashapechangingfactor,causingcellstoundergo fila-mentation(Young,2007).AsseeninSection6.6,ourexperiments indicatethatfilamentationaltersthetimebetweenexternal stim-ulusandinternalstatechange,asthesignalhasfurthertotravel. Macroscopicbehaviour,suchasmotility,maybealteredin inter-estingwaysby thistiming change.Forexample, weknow that
Escherichiacolimotordirectionswitchesbetweentwostates, for-wardand tumble (Darnton et al., 2007), due toenvironmental cues.WearecurrentlyperformingsimulationsusingReDi-Cellto answerthequestionofwhetherfilamentationofE.colialtersthe timebetweenexternalstimulusandmotilitystatechange,andif thischangemayaccountforawideningofthebacteria’ssearch area.
WebelievethatReDi-Cellcanplayaroleintheadvanced under-standingofthesecomplexcellularphenomena.ReDi-Cellcanbe usedtorapidlyvalidatemodelsofhow behaviouris modulated bysubtlespatialproperties suchas theshapeand sizeof cells. Weexpectthatfurtheradvancesinexperimentaltechniqueswill providearichandextensivesourceofdata,andsimulation tech-niquessuchasourswillprovidethenecessarytoolstomakesense ofthisexplosionofdataintrulyunderstandingthecomplexyet elegantfunctioningofcells.
Acknowledgements
We are grateful for financial support from EPSRC, under grantsEP/J500586/1andEP/I03157X/1(ROADBLOCK:Towards Pro-grammableDefensiveBacterialCoatingsandSkins).
AppendixA. TheeffectofgeometryontheTuringsystem
Inthissectionweshowtheimpactofintroducingclutterona Turingsystem.Herethesinglepieceofcluttercauses discontinu-itiesinthegeneratedpattern.
Fig.A.16
AppendixB. Normalisedscaleheatmaps
InthissectionwereproducetheheatmapsfoundinFig.12with normalisedheatmapscales.
Fig.B.17
Fig.A.16.ConcentrationmapofspeciesVinaturingsystemafter10000timesteps. TheconcentrationofspeciesVisproportionaltothebrightnessoftheblue.au=0.08,
bu=−0.08,cu=0.04,du=0.03,reactionrateu=1012,Du=0.02,av=0.1,bv=0.0,cv=
Fig.B.17.LVDwavefrontsinnaturalcellshapesafter1000timesteps.˛=1.5,ˇ=1.0,
=3.0,ı=1.0,t=10−14s,x=1.0*10−6m,reactionrate=1012,D=1.0
AppendixC. Choiceofdiffusioncoefficients
Thevalueofthediffusioncoefficientforachemicalspeciescan dramaticallyalterthebehaviourofthereaction–diffusionsystem
TableC.5
Tableofdiffusioncoefficientsrepresentativeofcellularenvironments.Reproduced from(MiloandPhillips,2015).
Molecule Medium Diffusion
coefficient (m2/s)
H+ Water 7000
H2O NucleusofChicken
Erythrocyte
200
Protein(≈30kDaGFP) eukaryoticcell(CHO) cytoplasm
30
Protein(≈30kDa) E.colicytoplasm 7–8 Morphogen(bicod-GFP) D.melanogasterembryo
cytoplasm
7
Protein(≈40kDa) E.colicytoplasm 2–4 Protein(≈70–250kDa) E.colicytoplasm 0.4–0.2 Protein(≈140kDaTar-YFP) E.colicytoplasm 0.2 Protein(≈70kDaLacY-YFP) E.colicytoplasm 0.03 mRNA variouslocations 0.005–0.001
associated withit. Despite thedifficultyin measuringdiffusion coefficientssomevaluesofbiologicallyrelevantentitieshavebeen recordedintheliterature;acompilationofcommoncellular chem-icals withtheirdiffusioncoefficientscanbefoundin(Miloand Phillips,2015).Cellulardiffusioncoefficientsareproportionalto boththemassofthespeciesand tothedensityof themedium through whichtheytravel, resultingin a wide rangeof values. For example in thedata in (Milo and Phillips, 2015) diffusion coefficientsrangefrom7000m2/sforH+ionsto0.001m2/sfor mRNA.Diffusioncoefficientsforproteintranslocationwithinthe cytoplasmhavebeenmeasuredinE.colitobebetween30m2/s and0.03m2/s.Inthisworkwesimulateabstractedsignallingand metabolicpathwaysbothofwhichcaninvolveproteinsofvarying size;withthisinmindwechoosediffusioncoefficientsof1and 0.1representingvaluesin-betweentheminimumandmaximum ordersofmagnitudeforproteindiffusion.Thesevaluesrepresent biologicallyrealisticdiffusioncoefficientsforthetypesofsystems that weare interestedin exploring.We reproduceaportionof thediffusioncoefficientsreportedin(MiloandPhillips,2015)in TableC.5.
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