The_Chemical_Thermodynamics_for_Growing_Systems

《The_Chemical_Thermodynamics_for_Growing_Systems》由會員上傳分享，免費在線閱讀，更多相關內容在學術論文-天天文庫。

TheChemicalThermodynamicsforGrowingSystemsYukiSughiyama,?AtsushiKamimura,?DimitriLoutchko,andTetsuyaJ.Kobayashi?InstituteofIndustrialScience,TheUniversityofTokyo,4-6-1,Komaba,Meguro-ku,Tokyo153-8505Japan(Dated:January25,2022)Weconsidergrowingopenchemicalreactionsystems(CRSs),inwhichautocatalyticchemicalreactionsareencapsulatedina?nitevolumeanditssizecanchangeinconjunctionwiththereac-tions.ThethermodynamicsofgrowingCRSsisindispensableforunderstandingbiologicalcellsanddesigningprotocellsbyclarifyingthephysicalconditionsandcostsfortheirgrowingstates.Inthiswork,weestablishathermodynamictheoryofgrowingCRSsbyextendingtheHessiangeometricstructureofnon-growingCRSs.ThetheoryprovidestheenvironmentalconditionstodeterminethefateofthegrowingCRSs;growth,shrinkingorequilibration.Wealsoidentifythermodynamicconstraints;onetorestrictthepossiblestatesofthegrowingCRSsandtheothertofurtherlimittheregionwhereanonequilibriumsteadygrowingstatecanexist.Moreover,weevaluatetheentropyproductionrateinthesteadygrowingstate.Thegrowingnonequilibriumstatehasitsoriginintheextensivityofthermodynamics,whichisdi?erentfromtheconventionalnonequilibriumstateswithconstantvolume.Theseresultsarederivedfromgeneralthermodynamicconsiderationswithoutassuminganyspeci?cthermodynamicpotentialsorreactionkinetics;i.e.,theyareobtainedbasedsolelyonthesecondlawofthermodynamics.I.INTRODUCTIONandgrowthshouldaccompanythethermodynamiccost.However,welackatheoreticalbasistoaddressthesefun-Self-replicationisahallmarkoflivingsystemsbydamentalproblemsofgrowingsystems.whichtheyaredi?erentiatedfromnonlivingones.SinceInthiswork,weestablishthethermodynamicsforvonNeumann’sformulationofself-reproducingautomatagrowingsystems.Thedi?cultyindevelopingitliesin[1,2],thephysicalandchemicalbasisofself-replicationthatthechangeinthevolumea?ectsallreactionsinit.hasbeenpursuedtheoreticallyandexperimentallyinor-Intheconventionaltheoryofchemicalreactions,reactiondertounderstandandsynthesizelivingsystems[3–24].?uxesaredescribedasfunctionsofdensitiesofchemicalsOfthevariouscomponentsnecessaryforself-replication,(concentrations)[43–50],whichpresumesaconstantvol-autocatalyticreactioncycles,thoughtofasthedrivingume.However,ifthevolumechanges,thedensitiescanengine,formacentralpart[25–31].However,thepres-changeeventhoughthenumbersofchemicalsremainun-enceofcyclesisnotsu?cientforself-replication.Becausechanged.Hence,itisnecessarytoreturntoathermo-thecyclesshouldbecon?nedinanencapsulatingvolumedynamicformulationinwhichthenumbersofchemicalswhichde?nesthereplicationunit,thesizeofthevolumeandthevolumearetreatedseparately.Inotherwords,shouldalsogrowinaccordancewiththeproductionofwehavetoexplicitlyaccountoftheextensivityofther-chemicalsbythecycles.modynamicfunctions,whichisscaledoutwhentheden-Inspiteoftheactiveinvestigationofautocatalyticre-sitiesaloneareconsidered.Nevertheless,weshouldalsoactioncyclesinthelastdecades[25–31],thegrowthofretainthedensityrepresentationanditsdualrepresenta-volumeanditscouplingwiththeautocatalyticcyclestionbythechemicalpotentialstoappropriatelycharac-havenotbeenthoroughlyinvestigatedsofar.Althoughterizesteadygrowingstatesandtheconditionsimposedtherecentrediscoveryofgrowthlawsofbacteria[32]ini-bytheintensivevariablesoftheenvironment.tiatedasurgeofnewcoarse-grainedautocatalyticmodelsWeclarifythisentangledrelationamongthetriadof[10–12,33–38],thevolumegrowthinthesemodelsiscon-chemicalnumbers,densitiesandpotentialsbyclarifyingsideredonlyheuristically[5,39–42],e.g.,byrepresentingthegeometricstructuretheyform.ThisstructureisbuiltarXiv:2201.09417v1[cond-mat.stat-mech]24Jan2022itwithalinearfunctionofchemicalsinit.ontherecentlydiscoveredHessiangeometricstructureInthelightofchemicalthermodynamics,thechangeinbetweenchemicaldensitiesandpotentialsinaconstantvolumeandthein?uxandout?uxofchemicalsdrivenbyvolume[51,52]byadditionallyintroducingthespacethecyclesaremutuallydependentandshouldbether-ofthenumbersofchemicals.Basedonthesecondlawmodynamicallyconsistent.Thisinterdependenceofre-ofthermodynamics,ourtheoryclassi?esthethermody-actionsandvolumeinevitablyconstraintheirpossiblenamicconditionsunderwhichthesystemgrows,shrinksstatesanddynamics.Inaddition,thecyclesthemselvesorequilibrates.Italsorevealstheregioninwhichthemaynotalwaysproceedintheforwarddirection,whichchemicaldensityisconstrainedtoasteadygrowth.Fur-canresultinshrinking,dependingontheenvironmentalthermore,itenablesustoevaluatetheentropyproduc-conditions.Itisnontrivialunderwhatthermodynamictionrate,i.e.,thephysicalcostofthesteadygrowth.Ourconditionsacoherentforwardcycledynamicsandvol-nonequilibriumsystemwithvolumegrowthhasitsoriginumegrowthcanbeachieved.Moreover,asteadycyclingintheextensivityofthermodynamics,whichisdi?erent

12fromtheconventionalnonequilibriumsystemswithcon-stantvolume[43–50].Weemphasizethatourderivationisperformedbasedonapurelythermodynamicargument[51–54].Asare-sult,itdoesnotdependonanyparticularformofthermo-dynamicpotentialsorreactionkinetics.Thus,ourtheoryiswidelyapplicableandcontributestounderstandingtheoriginsoflifeandconstructingprotocells[13–24]aswellasseekingtheuniversallawsofbiologicalcells[10–12,32–38].Thispaperisorganizedasfollows.WedevoteSec.IItooutlineourmainresultswithoutshowingthedetailsoftheirderivation.FromSec.IIIonward,westartwiththederivationofourmainresults.InSec.III,weanalyzethebehaviorofthetotalentropyfunctionwithrespecttotimeforchemicalreactiondynamics.WedevoteSec.IVtothepreparationforthegeometricstructureofgrowingsystems.InSec.V,weclassifytheenvironmentalcon-ditionstodeterminethefateofthesystembasedontheformofthetotalentropyfunction.InSec.VI,wecon-siderthesteadygrowingstateandevaluatetheentropyproductionrateinthisstate.WeillustrateourtheoryinSec.VIIontheidealgasasaspeci?cexampleofthermodynamicpotentials.InSec.VIII,wenumericallyFIG.1.DiagrammaticrepresentationofopenCRSs.Theverifyourtheorybyconsideringaspeci?cexampleofachemicalreactionsoccurwiththereaction?uxesJ(t)=chemicalreactionsystemcomposedoftheidealgasandr{J(t)},therthreactionofwhichisrepresentedasthechem-obeyingmassactionkinetics.Finally,wesummarizeouricalequationatthebottom.Here,A={Ai}arethelabelsworkwithfurtherdiscussionsinSec.IX.ofthecon?nedchemicals,andB={Bm}aretheonesoftheopenchemicalswhichcanmoveacrossthemembranemwiththedi?usion?uxesJD(t)={JD(t)}.ThenumbersII.OUTLINEOFTHEMAINRESULTSofthecon?nedandopenchemicalsinthesystemarede-imnotedbyX={X}andN={N},respectively.Also,imA.Thermodynamicsetup(S+)rand(O+)rdenotestoichiometriccoe?cientsofthere-imactantsinrthreaction,whereas(S?)rand(O?)raretheonesoftheproducts.ThestoichiometricmatricesaregivenLetusstartwiththepresentationofthesettingoftheiiimmmasSr=(S?)r?(S+)randOr=(O?)r?(O+)r.Fortheo-system(FIG.1).Consideragrowingopenchemicalreac-reticalsimplicity,weignorethetensionofthemembraneandtionsystem(CRS)surroundedbyareservoir.Weassumeassumethatitneverbursts.thatthesystemisalwaysinawell-mixedstate(alocalequilibriumstate),andthereforewecancompletelyde-scribeitbyextensivevariables(E,?,N,X).Here,EforthereservoirasΣ??[E,???,N?],andthereforetheto-T,Π?,μ?and?representtheinternalenergyandthevolume;talentropycanbeexpressedasN={Nm}denotesthenumberofchemicalsthatcanmoveacrossthemembranebetweenthesystemandthehiΣtot=Σ[E,?,N,X]+Σ??E,???,N?,(1)reservoircalledopenchemicals;meanwhile,X=XiT,Π?,μ?isthenumberofchemicalscon?nedwithinthesystem;theindicesmandirespectivelyrunfromm=1toNNwhereweusetheadditivityoftheentropy.Furthermore,andfromi=1toNX,whereNNandNXarethenum-duetothehomogeneityoftheentropyfunctionforthebersofspeciesoftheopenandcon?nedchemicals.Thesystem,withoutlossofgenerality,wecanwriteitasreservoirischaracterizedbyintensivevariables(T,?Π?,μ?),whereT?andΠarethetemperatureandthepressure;?Σ[E,?,N,X]=?σ[?,n,x],(2)μ?={μ?m}isthechemicalpotentialcorrespondingtotheopenchemicals.Also,wedenotethecorrespondingex-tensivevariablesby(E,???,N?).whereσ[?,n,x]istheentropydensityand(?,n,x):=Inthermodynamics,theentropyfunctionisde?ned(E/?,N/?,X/?).Sincethisworkonlytreatsasituationon(E,?,N,X)asaconcave,smoothandhomogeneouswithoutphasetransitions,weassumethatσ[?,n,x]isfunctionΣ[E,?,N,X].Wewritetheentropyfunctionstrictlyconcave.

23Next,wede?nethedynamicsforthesystemasdEd?=JE(t),=J?(t),dtdtdNmdXi=OmJr(t)+Jm(t),=SiJr(t),(3)rDrdtdtwhereJ(t),J(t),J(t)={Jm(t)}andJ(t)=E?DD{Jr(t)}representtheenergy,thevolume,thechemicaldi?usionandthechemicalreaction?uxes,respectively;S=SiandO={Om}denotestoichiometricmatri-rrcesforthecon?nedandtheopenchemicals(seeFIG.1).Theindexrrunsfromr=1toNR,whereNRisthenumberofreactions.Also,inEq.(3),weemployedEin-stein’ssummationconventionfornotationalsimplicity.ThedynamicsofthereservoirisgivenasFIG.2.Diagrammaticrepresentationof(a)isochoricand(b)isobaricsituations.(a)Intheisochoriccase,theexternaldE?d??dN?mpressureΠvariestokeepthevolume?constant.Theinternal?=?J(t),=?J(t),=?Jm(t).(4)?E?Dpressure?(y),whichalwaysbalanceswithΠ,canconvergeto?dtdtdt?EQEQthechemicalequilibriumpressure?(y)=Π?.(b)IntheInthiswork,weassumethatthetimescaleoftheisobariccase,thevolume?variestokeeptheinternalpres-?reactionsismuchslowerthanthatoftheothers(thatsure?(y)alwaysequaltotheconstantexternalpressureΠ.??is,JE(t),J?(t),JD(t)?J(t)).Therefore,ourdynam-Consequently,theinternalpressure?(y)=Πmaynotbal-??EQancewiththechemicalequilibriumpressure?(y),whichicsise?ectivelygovernedonlybythereaction?uxJ(t)isspeci?edbythechemicalpotentials?μinthereservoir.This(seeSec.IIIfordetails).Inaddition,weconsiderimbalancedrivesgrowingorshrinkingofthevolume.minimalmotifsofautocatalyticcycles,whichwereiden-ti?edin[26]andcharacterizedbytheregularityofthestoichiometricmatrixSforthecon?nedchemicals,i.e.,x.Thisdualisticrepresentationiscentraltoourtheory.NX=NR=Rank[S](seeAppendixAfordetails).Inaddition,??(y)canbeinterpretedasthepressureofthesystematthestateywhosecorrespondingdensityisx=???(y).B.Thermodynamicpotentials,duality,andtotal?Ifthevolumeis?xed,theinternalpressure?(y)al-entropycharacterizingthegrowingsystemswaysbalanceswiththeexternalpressureΠincurredby?theboundarytokeepthevolume?constant(seeFIG.?Withtheabovesetup,weobtainaconjugatepairof2(a)).Furthermore,theinternalpressure??(y)=Πcon-??thermodynamicpotentials,?(x)and?(y),whichplayvergestothepressure??(yEQ)=Π?EQatthechemicalpivotalrolesinourtheory.ThepartialgrandpotentialequilibriumstateyEQ.ThestateyEQisdeterminedbydensity?(x)=?[T,?μ?;x]isde?nedasthechemicalpotentials?μinthereservoirashino