Asymmetric Spatiotemporal Evolution of Prebiotic Homochirality

The role of asymmetry on the evolution of prebiotic homochirality is investigated in the context of autocatalytic polymerization reaction networks. A model featuring enantiometric cross-inhibition and chiral bias is used to study the diffusion equations co

AsymmetricSpatiotemporalEvolutionofPrebioticHomochirality

MarceloGleiser

DepartmentofPhysicsandAstronomy,DartmouthCollegeHanover,NH03755,USA

Theroleofasymmetryontheevolutionofprebiotichomochiralityisinvestigatedinthecontext

ofautocatalyticpolymerizationreactionnetworks.Amodelfeaturingenantiometriccross-inhibition

andchiralbiasisusedtostudythedi usionequationscontrollingthespatiotemporaldevelopment

ofleftandright-handeddomains.Boundsonthechiralbiasareobtainedbasedonpresent-day

constraintsontheemergenceoflifeonearlyEarth.Theviabilityofbiasingmechanismssuchas

weakneutralcurrentsandcircularlypolarizedUVlightisdiscussed.Theresultscanbeappliedto

anyhypotheticalplanetaryplatform.

arXiv:astro-ph/0606593v2 13 Nov 2006I.INTRODUCTIONTheemergenceofbiomolecularhomochiralityinprebioticEarthisacrucialstepintheearlyhistoryoflife[1,5].Itiswell-knownthatchiralselectivityplaysakeyroleinthebiochemistryoflivingsystems:aminoacidsinproteinsareleft-handedwhilesugarsareright-handed.However,laboratorysynthesesproduceracemicresults.Thisissomewhatsurprising,giventhatstatistical uctuationsofreactantswillinvariablybiasoneenantiometerovertheother[4]:eventhougheverysynthesisisabinitioasymmetric[7],theenantiometricexcessisneverthelesserasedasthereactionsunfold.AnimportantexceptionisthereactionbySoaiandcoworkers,whereasmallinitialenantiometricexcessise ectivelyampli edintheautocatalyticalkylationofpyrimidylaldehydeswithdialkylzincs[31].AsstressedbyBlackmond[4],Soai’sreactionsucceedsbecauseitfeaturestheneededautocatalyticbehaviorproposedoriginallybyFrank[9]withenantiometriccross-inhibitioncatalysedbydimers.Itisunlikelythatthespeci cchemistryoftheSoaireactionoccurredinearly-Earth.However,itdisplaystherelevantsignaturesofarealistichomochirality-inducingreactionnetwork:autocatalysis,enantiometriccross-inhibition,andenzymaticenhancementperformedbydimersorbylargerchirally-purechains.Inthepresentwork,wewillinvestigatethespatiotemporaldynamicsofareactionnetworkrecentlyproposedbySandarswhichsharesthesefeatures[30].Toit,wewilladdanexplicitchiralbias,inordertoinvestigatethee cacyofintrinsicandextrinsicbiasingmechanismsproposedintheliterature.A rststepinthisdirectioncanbefoundintheworkbyBrandenburgetal.,whostudiedanextensionofSandar’smodelincludingbiasbutnospatialdependence[24].Byintrinsicwemeaneitherbiasinge ectsrelatedtofundamentalphysics,suchasparity-violatingweakneutralcurrents(WNC)[16,35]or–giventhatweknowlittleofearly-Earth’sprebioticchemistryandevenlessofotherpossiblelife-bearingplanetaryplatforms[25]–tosomeasyetunknownchemicalprocess.Byextrinsicwemeanpossibleenvironmentalin uences,suchascircularly-polarizedUVlight(CPL)from,forexample,activestar-formationregions[23]ordirectseedingofchiralcompoundsbymeteoriticbombardment[8,26].Thespatiotemporaldynamicsofthereactionnetworkwillbeshowntobeequivalenttoatwo-phasesystemundergoingasymmetry-breakingphasetransitioncharacterizedbytheformationofcompetingdomainsofoppositechirality.TheevolutionofthedomainnetworkissensitivetoEarth’searlyenvironmentalhistoryandtothemagnitudeofthechiralbias.Usingthetime-scaleassociatedwiththe

emergenceoflifeonEarthitispossibletoobtainalowerboundonthebias.Inparticular,itwillbeshownthattheverysmallbiasfromWNCisine cienttogeneratehomochiralconditions.ForCPLthesituationislessclearduetouncertaintyinthenatureanddurationofsources,butstillhighlyunlikely.Theformalismissetuptobeapplicabletoanyplanetaryplatform.

Thepaperisorganizedasfollows:Insection2weintroducethebiasedpolymerizationmodelanditspertinentrateequations.Insection3wedescribethereduced(n=2)modelandhowthenetchiralitycanbeinterpretedasacontinuousorderparametersatisfyingane ectivepotential.Thisallowsustointroduceexplicitlyspatialdependenceinthestudyofbiasedpolymerization.Insection4wedescribethedynamicsofhomochiralityusingtechniquesfromthetheoryofphasetransitionswithmean- eldGinzburg-Landaumodelsinthelimitofnobias.Insection5wegeneralizeourresultstoincludeasmallbias,describingindetailthewalldynamicsinthiscaseandthetime-scalesassociatedwiththedevelopmentofhomochiralityinearly-Earth.Wealsoinvestigateiftheonsetofprebiotichomochiralityonearly-Earthcouldhavebeentheresultofanucleationevent.Weconcludeinsection6withasummaryofourresultsandanoutlooktofuturework.