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A dimensional Geometry for Biological Time

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A 2-dimensional Geometry for Biological Time Francis Bailly?, Giuseppe Longo†, Mael Montevil‡ April 5, 2012 Abstract This paper1 proposes an abstract mathematical frame for describing some features of biological time. The key point is that usual physical (linear) representation of time is insufficient, in our view, for the understanding key phenomena of life, such as rhythms, both physical (circadian, seasonal . . . ) and properly biological (heart beating, respiration, metabolic . . . ). In particular, the role of biological rhythms do not seem to have any counterpart in mathematical formalization of physical clocks, which are based on frequencies along the usual (possibly thermodynamical, thus oriented) time. We then suggest a functional representation of biological time by a 2-dimensional manifold as a mathematical frame for accommodating autonomous biological rhythms. The “visual” representation of rhythms so obtained, in particular heart beatings, will provide, by a few examples, hints towards possible applications of our approach to the understanding of interspecific differences or intraspecific pathologies. The 3-dimensional embedding space, needed for purely mathematical reasons, allows to introduce a suitable extra-dimension for “representation time”, with a cognitive significance. Keywords: biological rhythms, allometry, circadian rhythms, heartbeats, rate variability. Contents 1 Introduction 2 1.1 Methodological remarks . . .

  • rhythms

  • major invariants

  • rhythms associated

  • biological rhythms

  • physical temporal

  • dimensional time

  • physical time

  • distinction between

  • between replicative


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A2-dimensionalGeometryforBiologicalTimeFrancisBailly,GiuseppeLongo,MaelMontevilApril5,2012AbstractThispaper1proposesanabstractmathematicalframefordescribingsomefeaturesofbiologicaltime.Thekeypointisthatusualphysical(linear)representationoftimeisinsufficient,inourview,fortheunderstandingkeyphenomenaoflife,suchasrhythms,bothphysical(circadian,seasonal...)andproperlybiological(heartbeating,respiration,metabolic...).Inparticular,theroleofbiologicalrhythmsdonotseemtohaveanycounterpartinmathematicalformalizationofphysicalclocks,whicharebasedonfrequenciesalongtheusual(possiblythermodynamical,thusoriented)time.Wethensuggestafunctionalrepresentationofbiologicaltimebya2-dimensionalmanifoldasamathematicalframeforaccommodatingautonomousbiologicalrhythms.The“visual”representationofrhythmssoobtained,inparticularheartbeatings,willprovide,byafewexamples,hintstowardspossibleapplicationsofourapproachtotheunderstandingofinterspecificdifferencesorintraspecificpathologies.The3-dimensionalembeddingspace,neededforpurelymathematicalreasons,allowstointroduceasuitableextra-dimensionfor“representationtime”,withacognitivesignificance.Keywords:biologicalrhythms,allometry,circadianrhythms,heartbeats,ratevariability.Contents1Introduction21.1Methodologicalremarks...................................32Anabstractschemaforbiologicaltemporality.32.1Premise:Rhythms......................................32.2Externalandinternalrhythms................................43Mathematicaldescription63.1Qualitativedrawingsofourschemata............................63.2Quantitativeschemeofbiologicaltime...........................84Analysisofthemodel94.1Physicalperiodicityofcompaciedtime..........................94.2Biologicalirreversibility...................................104.3Allometryandphysicalrhythms...............................114.4RateVariability........................................114.4.1Renormalization...................................124.4.2RateVariability....................................14Physics,CNRS,MeudonInformatique,CNRS–ENSandCREA,Paris,longo@di.ens.fr,http://www.di.ens.fr/users/longoInformatique,ENSandEDFrontièresduvivant,ParisV,Paris1PublishedinProgressinBiophysicsandMolecularBiology,106(3):474484,2011.doi:10.1016/j.pbiomolbio.2011.02.001.1
5Morediscussiononthegeneralschema1.155.1Theevolutionaryaxis(τ),itsangleswiththehorizontalϕ(t)anditsgradientstan(ϕ(t))155.2ThehelicoidalcylinderofrevolutionCe:itsthreadpe,itsradiusRi.........165.3ThecircularhelixCionthecylinderanditsthreadpi..................165.4Ontheinterpretationoftheordinatet0...........................171IntroductionLivingphenomenadisplaysrathercharacteristicandspecifictraits;amongthese,manifestationsoftemporalityandofitsroleareparticularlyremarkable:development,variegatedbiologicalrhythms,metabolicevolution,aging,....Thisiswhywebelievethatanyattemptatconceptualizinglifephenomena—beitonlypartially—cannotavoidaddressingsuchtemporalaspectsthatarespecifictoit.Inthatwhichfollows,wewillexaminethisquestionfromdifferentanglesinviewofprovidingafirstattemptatsynthesis.Theintuitive“geometryoftime”inphysicswas(andoftenstillis)based,first,ontheabsoluteNewtonianstraighttimeline.ThiswaslaterenrichedbytheorderstructureofCantortyperealnumbers,anorderedsetofpoints,topologicallycomplete(denseandwithoutgaps).Thermodynamicsandthetheoriesofirreversibledynamics(phasetransitions,bifurcations,passingintochaos,...)haveimposedan“arrow”uponclassicaltime,byaddinganorientationtothetopologicalandmetricstructure.Butitiswithrelativityandquantumphysicsthatthetheorizationoftimehasledtoratheraudaciousreflections.Inthefirstcase,togiveonlyoneexamplefromaveryrichdebatewhichgoessofarastointroduceacirculartime(proposedbyGödelasapossiblesolutiontoEinstein’sequations)toMinkowskispace:bymeansofitsfamouscausalitycone,thisspaceexplains,withintheframeworkofaunifiedgeometryofspace-time,thestructureofanypossiblecorrelationbetweenphysicalobjects,inspecialrelativity.Inquantumphysicsthesituationismaybeevenmorecomplexor,inanyevent,lessstable.Wegofromessentiallyclassicalframeworkstoasometimestwo-dimensionaltime(inaccordancewiththestructureofthefieldofcomplexnumberswithregardtowhichHilbertspacesaredefined,thetheoreticallociofquantumdescription),uptotheaudacityofFeynman’stemporal“zigzags”[FG67].Thislatterapproachisaveryinterestingexampleofintelligibilitybymeansofa“geometric”restructuringoftime:thecreationofantimatterwouldcausewithintheCPTsymmetry(charge,parity,time)asymmetrybreakingintermsofcharge,whileleavingparityunchanged.Globalsymmetryisthenachievedbylocallyinvertingthearrowoftime.Anotherapproach,withsimilarmotivations,isthatofthefractalgeometryofspace-time,specifictothe“scalerelativity”proposedby[Not93]:init,timeisreorganizedupona“broken”line(afractal),whichiscontinuousbutnon-derivable.Furtherinterestingreflections,alongsimilarlines,maybefoundin[LMNN98].Physicshoweverwillremainbutamethodologicalreferenceforourwork,becausetheanalysisofthephysicalsingularityoflivingphenomena[BL06,BL11]requiresasignificantenrichmentoftheconceptualandmathematicalspacesbywhichwemakeinertmatterintelligible.Oneofthenewfeatureswhichweintroduceconsistsintheusagethatwewillmakeofthe“compactification”ofatemporalstraightline:inshort,wewilltrytomathematicallyunderstandrhythmsandbiologicalcyclesbymeansoftheadditionof“fibers”(aprecisemathematicalnotion,introducedsummarilybelow)whichareorthogonaltoaphysicaltimethatremainsaone-dimensionalstraightline.Fromourstandpoint,alivingbeingisatrue“organizer”oftime;byitsautonomyandaction,itconfersitamorecomplexstructurethanthealgebraicorderofrealnumbers,butalsomorethananyorganizationonecouldproposeforthetimeofinertmatter.Inshort,thetimeofalivingorganism,byitsspecificrhythms,intimatelyarticulatesitselfwiththatofphysicsallthewhilepreservingitsautonomy.Wewouldthereforeliketocontributetomakingthemorphologicalcomplexityofbiologicaltimeintelligible,bypresentingapossiblegeometryofitsstructure,asatwodimensionalmanifold.2