Quasi_species_Error_Threshold_and_Hypercycles

Quasi-species,ErrorThresholdandHypercyclesAModelforMolecularEvolutionMorDolevJanIhmelsIntroductionDarwinianevolution:diversityofspeciesStartingpoint:cellCell:molecularmechanisms,complexmachinery,highlyconservedOriginoflife:processesthatleadtoemergenceofthecellEigen+Schuster:molecularevolutionDoweneedlivingthingsforevolution?ConditionsforNaturalSelectionMetabolism:Spontaneous,independentformationanddegradationfarfromequilibriumSelf-reproduction:Abilitytoinstructownsynthesis(autocatalysis),conservationofinformationMutability:Limitedfidelityofself-reproductiveprocessRibonucleicAcidSatisfiesrequirementsfornaturalselectionEigen&Schuster’scandidatefortheoriginoflifeDynamicsofSelectionSimplestmodel:iikkikiiiixwxQA)1(iikkikiiiiixwxDQAdtdx)(kkconstxkkkkkkktxExDA)(Reproducibleglobalconditions:…DynamicsofSelectionSystematconstantoverallconcentration:ikkikiiiixwxtEWdtdx)(kkkkkiiiiixxEtEDQAW)(valueselectivewhereQuasi-speciesSinglespeciesarecoupled.Competition,butco-operationbetweenrelatedspecies.Reorganize:NspeciesNquasi-species.NkkNkkyx11iiiytEdtdy)(kkkkkyytE)(Transformedequation:…Quasi-speciesSteadystate:Naturalselection:onequasi-speciessurvivesTargetofselection:quasi-species(distribution),notsinglemolecularspeciesCharacterizedbymastersequencemax)(tEExample:RNAClonesWildtype:averagesequenceSpeciesisadistributionModelSummaryNaturalselectionofinanimateobjectsOpenquestions:•Darwiniandiversityvs.prebioticunity•Canwehaveintegrationofseveralfunctions?•ErrorthresholdTheerrorthresholdmxjx-therelativeconcentrationofthemastersequence.-therelativeconcentrationofalltheothers.Wecansimplifyourequations:)()1()(1jjmmjmmjjjjjmmmmmmjmxAxAxxQAxAdtdxxAxAxQxAdtdxxx…TheerrorthresholdWewanttohaveasteadystate,atwhich)0(dtdxdtdxjm.0mxSolvingtheequationsweget:,jmjmmAAAQAxDemanding,weget:0mx…TheerrorthresholdsAAQmj1Wheresisthesuperiorityofthemastersequence.Letqbethereplicationfidelityperdigit.Then:)1(1log)1log(logqsNeeqQqNqNNThemaximallengthofamoleculeislimitedbytheerrorrate,1-q.Themaximallengthofagenomeisinverselyproportionaltotheerrorrate:Theerrorthreshold:Isthereacorrelationbetweenerrorrateandgenomesize?TheerrorcatastropheInRNAreplicationwithoutareplicasetheerrorrateisaround0.01.Thisallowsamaximallengthofnotmorethan100nucleotides.Toincreasethefidelityofreplication,anenzymeisneeded.Eigen’sparadox:noenzymewithoutalargegenome,andnolargegenomewithoutenzymes.Howcanweincreasetheamountofinformationmaintainedinthesystem,withoutanincreaseintheerrorrate?Asinglelongmoleculevs.twoshortones:Ifwecouldhaveastablecoexistenceofmorethanonequasi-speciesinthesystem,thissystemwouldkeepmoreinformation.Theproblem:competitionbetweenquasi-speciespreventsastablecoexistence.TheHypercycleOrganizedarrangementofcatalyticmoleculesPSECatalyst:Catalyticcycle:Autocatalyst…HypercycleNexthigherorder:CatalytichypercycleCycliclinkageofSelf-replicativeunitsIntermediatesInareCatalyticcyclesPropertiesAllowscollectivegrowthofcooperativesystem(Eigen’sparadox).Competitionbetweenmemberspeciesisswitchedoff.Threebasictypesofgrowthrates:Constant:coexistence.Linear:exponentialgrowth.Higherthanlinear:‘onceforalltime’-selection.…PropertiesEntireensemblecapableofexpansionandoptimization.Replacementisdifficult.Geneticcode,mechanisms,chirality.1iiiixbxdtdxHypercyclesexhibithyperbolicgrowth:Anexampleforahypercycle:RNA-phageinfection:HypercyclesareallaroundusProblemsWithHypercyclesMutantsinhypercyclesA’BABAA”Selfish:BetterTargetAltruistic:BetterReplicaseLarge(4)cyclesfluctuategreatly:instabilityShortcutscandestroycyclesParasitescandestroyhypercycleCompartmentsEigensuggestscompartmentsFunctionaladvantagescanbeselectedforFeedbackfromphenotypetogenotypeCompartmentsImprovedreplicasemutationscanbeselectedforDowestillneedhypercycles?SummaryQuasispecies:SimplemolecularevolutionErrorCatastrophe:Eigen’sparadoxHypercyclesasasolution:Cooperationbetweenquasi-speciesCanovercomeerrorthreshold‘Onceandforall’-selectionProblemsofthehypercycle:noselectionfavoring“altruistic”mutantsneedforcompartmentsReferencesM.Eigen,P.Schuster,TheHypercycle,Springer-Verlag,1979M.Eigen,“Stepstowardslife:APerspectiveonEvolution”,OxfordUnivPress,1996J.Hofbauer,K.Sigmund,“Thetheoryofevolutionanddynamicalsystem”,CambrigeUnivPress,1988J.M.Smith,Hypercyclesandtheoriginoflife,Nature20:445-6,1979J.M.Smith,E.Szathmary“Themajortransitionsinevolution”,W.H.Freeman,1995