Effect of macromolecular crowding on the rate of diffusion-limited enzymatic reaction

ManishAgrawal1,S.B.Santra2,RajatAnand1andRajaramSwaminathan1

DepartmentofBiotechnology,2DepartmentofPhysics,

IndianInstituteofTechnologyGuwahati,Guwahati-781039,Assam,India.

1

2008Thecytoplasmofalivingcelliscrowdedwithseveralmacromoleculesofdifferentshapesandsizes.Moleculardiffusioninsuchamediumbecomesanomalousduetothepresenceofmacromoleculesanddiffusivityisexpectedtodecreasewithincreaseinmacromolecularcrowding.Moreover,manycellularprocessesaredependentonmoleculardiffusioninthecellcytosol.Theenzymaticreactionratehasbeenshowntobeaffectedbythepresenceofsuchmacromolecules.Asimplenumericalmodelisproposedherebasedonpercolationanddiffusionindisorderedsystemstostudytheeffectofmacromolecularcrowdingontheenzymaticreactionrates.Themodelexplainsqualitativelysome loftheexperimentalobservations.

Ju PACSnumbers:02.50.Ey;05.40.Jc;05.60.Cd

91 INTRODUCTION

]hceTheaqueousphaseofcellcytoplasmiscrowdedwithmmacromoleculessuchassolubleproteins,nucleicacids-andmembranes[1].Theinfluenceofsuchcrowdingtaonbiochemicalreactionsinsidephysiologicalmediaaretmanifold[2].Duetocrowding,theaveragefreeen-s.ergyµofanonspecificinteractionbetweenanymoleculetainthemediumandacrowdingmoleculemaychangemconsiderablywhichmayinfluencethereactionactivity-γ=exp(µ/kBT),wherekBistheBoltzmannconstantdandTistheabsolutetemperature.Stericrepulsionisnothemostfundamentalofallinteractionsbetweenmacro-cmoleculesinsolutionatfiniteconcentrationandasan[effectofsuchrepulsionthemacromoleculesoccupyasub- 1stantialvolumefractioninthecellinterior[3].Signifi-vcantvolumefractionofmacromoleculesinthemedium8imposesaconstraintonintroducinganynewmacro-6molecule.Asaconsequenceofcrowding,macromolecular0associationreactionsbecomeincreasinglyfavorable.Be-3.causeofcrowding,themoleculardiffusioninthemedium7isexpectedtobeanomalous[4].Theeffectofmacro-0molecularcrowdingondifferentkineticstepsofenzyme80catalysissuchasformationofenzyme-substratecomplex:andenzyme-productcomplexwereanalyzedthroughdif-vferentequilibriumthermodynamicmodels[5].AnumberiXofapproacheshavebeenproposedtostudytheeffectsrofmacromolecularcrowdingonthereactionkineticratealawssuchasthelawofmassaction[5],fractallikeki-netics[6],thepowerlawapproximation[7],stochasticsimulation[8]andlatticegassimulation[9].Intheseana-lyticandnumericalmodels,theinfluenceofmacromolec-ularcrowdingonbothequilibriumthermodynamicsandreactionrateswereaddressedanditwasobservedthattheratedecaysexponentiallywithtimeasexpectedinequilibriumkinetics.Theinfluenceofmacromolecularcrowdingontheenzymaticreactionrateshasbeeninves-tigatedexperimentallyusingavarietyofcrowdingagents[10].Thesestudieshavealsoindicatedasignificantin-fluenceofcrowdingontherateparametersoftheenzy-

maticreaction.Itwasfoundthattheeffectofcrowdingontheenzymaticreactionmaybedifferentdependingonwhethertheproductformationintheenzymereac-tionislimitedbythediffusionalencounterofsubstrateandenzymeortheformationofthetransitionstatecom-plex,anassociationofenzymeandsubstrate.Moreover,moleculardiffusionisknowntobethemajordetermi-nantofmanycellularprocessesandplaysakeyroleincellmetabolismwheretheencounterofthefreesubstratewithanactivesiteoftheenzymeisoftentheratede-terminingstep.However,howthekineticsofanenzy-maticreactionisdependentonthesizeandconcentrationofthecrowdingmacromoleculesisstillnotfullyunder-stood.Themacromolecularcrowdingtilldateremainsunderappreciatedandneglectedaspectoftheintracellu-larenvironment[11].Itishenceessentialtounderstandtheexperimentalobservationsfrommicroscopicorigin.Inthispaper,anapproachbasedonnon-equilibriumdynamicsofenzymaticreactionsinthediffusionlimitedregimeisconsidered.Theaimistounderstandquali-tativelytheinfluenceofinertmacromolecularcrowdingonthediffusionlimitedenzymaticreactionsgovernedbynon-equilibriumthermodynamics.Asimplenumericalmodelintwodimensions(2d)basedonmoleculardiffu-sionindisorderedsystemscoupledwithenzymaticreac-tionisproposedhere.Thedisorderedsystemismod-eledbypercolationclusters[12].Itispredictedthattherateofadiffusion-limitedenzyme-catalyzedreactionwillexperienceamonotonicdecreasewithincreaseinthefractionalvolumeoccupancyofthecrowdingagent.Themodelexplainsqualitativelycertainexperimentalobser-vations.

THEMODEL

Inbrief,theenzymekineticreactioninthecellcyto-plasmcanbedescribedassubstratemoleculesdiffusingthroughcrowdingmacromoleculesandbindingtotheac-tivesiteofthefreelyfloatingenzymes.Subsequentlya

productisformedifthereactionisenergeticallyfavor-ableandthisproductdiffusesthroughthesamecrowdofmacromolecules.TheclassicalMichaelis-Mentenequilib-riumenzymekineticreactionisgivenas[13]

E+S⇀↽ES→E+P

(1)

whereErepresentsenzyme,Srepresentssubstrate,PrepresentsproductandESistheintermediateenzyme-substratecomplex.

Inthepresentmodel,thereactionislimitedbydiffu-siononlyandtheformationofthetransitionstatecom-plexESisnottakenintoaccount.Theconversionofsubstratetoproductisalsoassumedtobeinstantaneous.Notethatdiffusionhastheslowesttimescaleinthisprob-lem.Hence,theaboveenzymaticreactionreducestoanirreversibleoneas

E+S→E+P.

(2)

Thefinalequilibriumstatecorrespondstoconversionofallsubstratestoproducts.AMonteCarlo(MC)al-gorithmhasbeendevelopedtostudydiffusionlimitedenzymaticreactionasinEq.2inthepresenceofinertmacromolecules.Thealgorithmisdevelopedonthe2dsquarelatticeofsizeL×L.Forsimplicity,themotionofthemacromoleculesisignoredandtheseactasimmobileandinertobstacles.Theinertobstaclesdonotinteractwitheitheramongthemselvesorwiththesubstrateorproduct.Theobstacles(O),enzyme(E),substrate(S)andproduct(P)areallrepresentedaspointparticlesinthismodel.Itisalsoassumedthatthereexistsonlyoneimmobileenzymeinthewholesystem.Theenzymeisplacedatthecenterofthelattice.Afterplacingtheenzyme,theobstaclesandthesubstratesaredistributedrandomlyoverthelatticesiteswiththeirspecifiedcon-centrationsCOandCSrespectively.Arandomnumberriscalledfromauniformdistributionofrandomnum-bersbetween0and1correspondingtoeachlatticesite.Ifr≤CS,thesiteisoccupiedwithasubstrateandifCS2

topif1/4anditisatthebottomif3/4Cyclicboundaryconditionhasbeenappliedinthemo-tionofSandP.Thesimulationhasbeenperformedupto106MCtimestepsona256×256squarelattice.Thedataareaveragedover100ensembles.Thetimeevolutionofthesystemmorphologyforaf=0.1withCS=0.01isshowninFig.1atthreedifferenttime.Theblackdotsrepresentthesubstratesandthegrayboxesrepresenttheproducts.Forclarityobstaclesarenotshown.Itcanbeseenthattheinitialblackdotsareconvertedtograyboxesattheend.Thatmeans,thesubstratemoleculesarediffusing,reactingwiththeenzyme,andaregettingconvertedintoproducts.Intime,almostallthesub-stratemoleculesareconvertedtoproductsandtheprod-uctmoleculesalsodiffuseandspreadalloverthespaceuniformly.Linandcoworkers[15]simulatedsomeele-mentarykineticreactionslikeA+B→0withnoob-staclesunderreflectiveboundaryconditionandobservedZeldovichcrossover(segregationofAandB)[16].Suchsegregationisnotobservedwithperiodicboundarycon-ditioninthepresentsimulation.EffectofimpenetrableboundaryondiffusionlimitedreactionlikeA+A→0leadstodifferentbehaviordependingondifferentbound-aryconditions[17].

RESULTSANDDISCUSSION

Classicaldiffusionofatracerparticleindisorderedsys-temshasalreadybeenstudiedextensivelyandtheresultsarewellunderstood[18].Generallythediffusionismod-eledbyrandomwalkandthedisorderedsystemismod-eledbyspanningpercolationclusters[12].Forstudyingdiffusion,aquantityofinterestistherootmeansquare

3

(a)t=212(b)t=218(c)t=220

FIG.1:Thesystemmorphologyona256×256squarelatticeisshownatthreedifferenttimes(a)t=212,(b)t=218and(c)t=220forsubstrateconcentrationCs=0.01andareafractionaf=CS+CO=0.1(CO=0.09).Theblackdotsrepresentthesubstratesandthegrayboxesrepresenttheproducts.Forclarityobstaclesarenotshown.Theenzymeisrepresentedbyacrossatthecenterofthelattice.Productsareformedduetotheenzymaticreactionandinthelongtimelimitalmostallthesubstratesareconvertedintoproducts.

(rms)distancer(t)coveredbythediffusingparticleintimet.Thermsdistancer(t)in2disgivenby

r2(t)=4D×t2k

(3)

whereDisthediffusivityofthesystem.Theexponentkhasavalue1/2fordiffusiononaregularlatticeinthet→∞limit.Onthepercolationcluster,diffusionisfoundtobeanomalousandthevalueofkbecomeslessthan1/2[18].Theenzymekineticreactioninsideacellcytoplasminvolves(i)diffusionofalargenumberofsub-stratemoleculesthroughtherandomstructureofinertmacromolecules,(ii)reactionwiththeenzymetohaveproducts,and(iii)finallydiffusionofproductsfromtheenzymethroughthesamemacromolecularcrowding.Thediffusionprocessinvolvedhereisthenacollectivemo-tionofalargenumberofparticlesinpresenceofdisorderwhichisacomplicatedprocessthandiffusionofasingletracerparticleinadisorderedmedium.Self-diffusionisexpectedtoplayanontrivialrolealongwiththediffu-sionofSorPinpresenceofdisorderinthesesystems.Inordertocheckwhethertheenzymekineticreactionconsideredhereisdiffusionlimitedornot,oneneedstoanalyzethethediffusivebehaviorofeitherthesubstratesortheproducts.Tocalculatetheaveragediffusionlengthoftheproductparticles,thecoordinates{xi(t),yi(t)}ofeachproductiisrecordedwithtimet.Timeismeasuredstartingfromthebirthofaproduct.Thermsdistancer(t)traveledintimetisthencalculatedasr2(t)=

1

system.However,inordertocheckthediffusivebehav-ioroftheparticlesoneneedstoestimatetheexponentkdefinedinEq.3.Thelocalslopekt=dlogofthecurvelogt)versuslog2r(t)/dlog2tployingcentraldifference2r(method.2tisdeterminedbyem-InFig.2(b)and(c),ktisplottedagainsttimetfortwodifferentsubstratecon-centrationsCS=0.10(b)andCS=0.01(c)forthesamesetofareafractionsafasinFig.2(a).Thevalueofktsaturatesto1/2startingfromasmallervalueasttendstoalargevalue.Thus,acrossoverfromsub-diffusivetodiffusivebehaviorhasoccurredforallareafractionsinthelongtimelimit.Inthecaseoflowsubstratecon-centrationCs=0.01andhighareafractionaf=0.4,ktshowscertainanomalousbehavior.Notethat,atthispa-rameterregimethemacromolecularconcentrationis0.39whichisjustbelow1−pc≈0.41sincethepercolationthresholdfora2dsquarelatticeispc≈0.59.Theemptysitesprovidestheconnectivityforthesubstratemoleculesalloverthelattice.However,pcisdefinedonainfinitelylargesystem.Forasmallersystem,evenattheconcen-trationof0.39theconnectivityofemptysitesmaybelostforsomeoftheensemblesconsidered.Consequently,theproductmaybetrappedinalocalizedregionaroundtheenzymeandthismaybethereasonbehindtheanomalousbehaviorobservedinthisparameterregime.

Sincetheparameterregimehereislimitedbydiffu-sion,theenzymekineticreactionisthenexpectedtobediffusionlimited.Duetotheenzymekineticreaction(giveninEq.2)thesubstratesareconvertedtoprod-uctsintimewithunitprobabilityontheirencounter.Inordertocharacterizetheenzymekineticreaction,thenumberofproductsNParecountedasfunctionoftimet,theMCtimestep,fordifferentsubstrateconcentra-tionsCSandareafractionsaf=CS+CO.InFig.3,theproductnumbersNPisplottedagainsttimetfordifferentareafractionsafwithCS=0.01.Initially,NPincreaseslinearly,thenslowsdownandfinallysaturatesinthelongtimelimit.Forlowareafraction,itcanbeseenthatthereactionisalmostcompletei.e.;mostofthesubstratesgiveninitially,NS(0)=CS×L2≈655,areconvertedtoproductsexponentiallyasinclassicalequi-libriumMichaelis-Mentenkineticsthoughinthepresentmodelanon-equilibriumkineticsisconsidered.However,notethatthereisaconsiderabledecreaseintheproductyieldwithincreaseinareafractionandtheirprofilesarefoundnottofollowanexponentialincrease.Ithasal-readybeenpredictedbynumericalsimulationsthatclas-sicalMichaelis-Mentenkineticsmaynotapplytoenzy-maticreactionsincrowdedmedia[19].Ina1dmodelofreactiondiffusionwithdisorder,DoussalandMonthus[20]alsofoundlargetimedecayinthespeciesdensityviarealspacerenormalizationgroupcalculations.Themacromolecularcrowdingthencouldhaveaconsiderableandnontrivialeffectontheenzymaticreactionrate.Initialrateofenzymaticreactionsdeterminesmostofthemolecularprocessandthusisanimportantquantity

4

800

aCS=0.01

af=0.1f=0.2af=0.3pNaf=0.4

400

005

t/10

5

10

FIG.3:PlotofnumberofproductsNPversustimetfordifferentareafractionsaf=CS+COkeepingsubstratecon-centrationconstantatCS=0.01.

toestimate.Sincenon-equilibriumenzymaticreactionisconsideredhere,thereactionrateRisdefinedastheratioofthenumberofproductsNPtotimetfor10%conversionofthesubstrates.Risthensampleaveraged.AsimilaranalysishasalsobeenperformedforNPver-sustplotscorrespondingtoCS=0.1fordifferentareafractionsaf.InFig.4(a),thenormalizedreactionrateRn=R/CSisplottedagainstobstacleconcentrationCOfortwodifferentsubstrateconcentrationsCS=0.01(cir-cles)andCS=0.1(squares).Notethat,areafractionaf=CS+COisnotagoodparametertostudythere-actionratesinceafwillremainfiniteforfiniteCSevenatCO=0.Intheinset,RnisalsoplottedagainstCOinsemilogarithmicscale.Therearefewthingstonotice.First,thereactionrateisdecreasingwiththeincreaseinobstacleconcentrationCOinanonlinearfashion.Sec-ond,thereactionratesaredifferentforaparticularCOevenafternormalizingbythesubstrateconcentrationCS.Third,thereisamonotonicdecreaseofln(Rn)forsmallCOanddeviatesfromlineardecreaseforlargeCO.Thedecreaseinreactionratewithincreasingcrowdingcon-centrationisexpectedandalsoobservedinexperiments[10,22].However,thedependenceoftherateonthecrowdingconcentrationisdifferentformthepredictionmadebyMinton[5]inthetransitionstateaswellasdif-fusionlimitedenzymaticreactioninwhichahumpintheln(Rn)versusCOcurveisexpectedforanintermediateCO.Fourth,thenormalizedreactionrateisgoingtozeroasCOapproaches1−pc≈0.41.BeyondCO=0.41,theobstaclescouldblockthespanningclustersoftheemptysites.Consequentlytheenzymaticreactionwillbelocal-izedandthereactionrateisexpectedtogotozero.Theaboveobservationscanqualitativelybeunder-stoodintermsofdiffusionandpercolationphenomena.AsCOincreases,diffusivityisexpectedtodecreasebe-causeofthecrowdingduetoobstacles.Theinfluenceof

(a)

CS=0.01CS0.2

S=0.10

C/R=nR0.1−1

)nR(nl−40.0

CO

0.40.00.0

0.2

C0.4

O

0.2

(b)

CS=0.01CS=0.10

D0.1

0.00.0

0.2

C0.4

O

FIG.4:(a)PlotofnormalizedreactionrateRn=R/CSagainstCOfortwodifferentCSvalues0.01(circles)and0.1(squares).ln(Rn)isplottedagainstCOforthesameCSvaluesintheinset.ThesamesymbolsetfordifferentCSvaluesisused.(b)PlotofdiffusivityDagainstCOforCS=0.01andCS=0.1.Thesamesymbolsetof(a)isused.

macromolecularcrowdingonthediffusionofsoluteshasbeeninvestigatedinrecentexperimentsutilizingdifferentcrowdingagentsandareducedsolutediffusioncoefficientwasobservedwithincreasingsizeandconcentrationofcrowdingmacromolecules[21].AnestimateofdiffusivityD=(dr2(t)/dt)/4(asgiveninEq.3)hasbeenmadeuti-lizingthedataofdiffusionlengthr(t)fordifferentsetsofsubstrate(CS)andobstacle(CO)concentrations.InFig.4(b),DisplottedagainstCOforCS=0.01(circles)andCS=0.10(squares).Likereactionrate,diffusivityDisalsostudiedasafunctionofobstacleconcentrationCOinsteadofaf.ItcanbeseenthatDalsodecreaseswithCOinanonlinearfashion.Firstofall,itisinter-estingtonotethatthewholedependenceofRnonCOisisinaccordancewiththebehaviorofDwithCO.Theenzymaticreactionrateinthisparameterregimeisthere-foremostlygovernedbydiffusionandcanbeconsideredapurelydiffusionlimitedenzymaticreaction.Itisim-portantnowtoconsiderthelowCOvalues,especiallythecaseofCO=0.ForlowCOvalues,DisslightlylessforCS=0.1thanthatofCS=0.01forthesameCO.ThisslightdecreaseinDisduetodiffusionthroughtheselfcrowdingathigherCS.Ontheotherhand,the

5

reactionrateatzeroobstacleconcentrationisexpectedtobeproportionaltoCSandDandRcanbeobtainedasR≈CS×D.ItcanbeseenthatthenormalizedreactionrateRnobtainedhereisveryclosetothecorre-spondingvaluesofDatCO=0forboththeCSvalues.AtCO=0,theselfdiffusionofthesubstratemoleculeseventuallydeterminesthereactionratesandmightberesponsibleforaslightdecreaseinRnforCS=0.1withrespecttoCS=0.01asseeninFig.4(a).TheeffectofCSinabsenceofobstacleshasbeenverifiednumericallyforseveralhighervaluesofCSandaconsiderableeffectofself-crowdinghasbeenobservedonthereactionrateaswellasondiffusivity.Notethat,RnvaluesareslightlygreaterthanDforalmostallvaluesofCOasitcanbeseenbycomparingFig.4(a)and(b).Thismighthavehappenedfirstlyduetothefactthattheinitialyieldoc-curonlyfromthelocallyavailablesubstratemolecules.Thediffusionlengthofthesesubstratemoleculesareverylessincomparisontotheexpecteddiffusionlength.Sec-ondly,oneshouldnotethattheinitialreactionrateforagivenCOhastobecalculatedkeepingthesubstrateconcentrationCSfixed.However,inthepresentmodelthesubstrateconcentrationisdecreasingwithtimeasthesubstratesarebeingconvertedintoproducts.TheeffectwillbepredominantforlowCSandsmallsystemsize.Consequentlytheratedeterminationwillbeerroneousinthet→0limitduetolowyield.Hence,extremecarehastobetakenindeterminingtheinitialreactionrate.Theenzymaticreactionconsideredhereisthecompletelydiffusionlimitedandtheresultsobtainedareexplainableintermsofdiffusionindisorderedsystems.Itisthereforeintriguingtonotethatsuchasimplemodelofenzymaticreactionbasedondiffusionandpercolationphenomenaonly,isabletoexplainqualitativelytheexperimentalob-servations[10,22]aswellasresultsobtainedincompli-catedmodels[5,6,7,8,9].Hence,diffusionisobservedtobeplayingthecrucialroleindeterminingtheenzymaticreactionrates.

Itshouldbeemphasizedherethatenzymaticreactionsoccurin3-dimensionalspaceinlivingsystemswhereasthesimulationisperformedin2-dimensionshere.Thesimulationresultsobtainedhereagreequalitativelywiththeexperimentalobservationsanditisexpectedthatthefeaturesofthemodelwillberetainedinhigherdimen-sionsalso.Themaindifficultyin3dsimulationisinpar-allelupdatingofalargenumberofsubstrateandproductmoleculesduringtimeevolutionthroughalargenumberofMCtimesteps.Timerequiredforthefullconversionofsubstratetoproductincreasesexponentiallywiththenumberofmolecules(NS=CS×Ld)whichstronglyde-pendsonthedimensionalityofspaceforafixedsubstrateconcentration.However,forquantitativecomparisonoftheresultsobtainedinsimulationwiththatofexperi-ments,themodelmustbeextendedtothreedimensions.Thebiochemicaleventsinthedenselycrowdedmito-chondrialmatrix,thesiteforTCAcycleandfattyacid

oxidationpathwayarelargelygovernedbylargemacro-moleculesofvarioussizes.Itisthusimportanttoin-vestigatetheinfluenceofcrowdingasexertedbymacro-moleculesofdifferentsizes.Adecreaseinreactionratehasbeenobservedinexperimentsforvaryingobstaclesizeskeepingtheobstacleconcentrationconstant[22].Itseemsthatthecomplexinteractionbetweenobstaclesandthesubstrateissizedependentandmightbegov-erningtheenzymaticreactionrate.Itisexpectedthatthediffusionofsubstratesacrosslargemacromoleculesmightbeslowduetothecomplexinteractionswiththeobstacles.Inthepresentmodelofenzymaticreaction,thiscomplexinteractionbetweenobstacleandsubstratemaybeincorporatedbyintroducingaresidencetimeforthesubstratemoleculesateachencounterwiththeobsta-cle.Aslowingdowninthereactionratewithincreasingresidencetimehasbeenobservedinthesimulationinac-cordancewiththeexperimentalresults[22].Thedetailswillbereportedelsewhere.

SUMMARY

Theeffectofmacromolecularcrowdingontheenzy-maticreactionrateshasbeenmodeledbyaMCalgo-rithmbasedondiffusionandpercolationphenomena.Thesubstrates,products,obstaclesandenzymeallarerepresentedbypointparticles.Asingleimmobileen-zymeisconsideredandplacedatthecenterofthelat-tice.Theobstaclesandthesubstratesaredistributedrandomlywiththeirspecificconcentrationsfollowingauniformdistributionofrandomnumbersbetween0and1.Theobstaclesremainimmobilethroughoutthesimu-lation.Itisfoundthatthereactionissolelydiffusionlim-itedundertheseconditions.Thediffusionofsubstratesandproductsaremodeledbyacollectiverandomwalk.Theproductsformgraduallyandsubsequentlyalmostallthesubstratesareconvertedintoproductsafteralongtime.Theinitialreactionratehasbeenestimatedfordifferentsubstrateandobstacleconcentrations.Thenor-malizedreactionratehasanonlineardependenceontheobstacleconcentrationandfoundslightlydependentonthesubstrateconcentration.Thedependenceofreactionrateonthesubstrateaswellasobstacleconcentrationsisthenqualitativelyunderstoodwiththehelpofdiffusionandpercolationtheory.Theresultsqualitativelyexplains

6

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