生物功能的随机热力学综述 Stochastic thermodynamics for biological functions

Received:29February2024

Accepted:2July2024

DOI:

10.1002/qub2.75

REVIEWARTICLE

QuantitativeBiology

openAccess

Stochasticthermodynamicsforbiologicalfunctions

YuanshengCao

1

|

ShilingLiang

2

1DepartmentofPhysics,TsinghuaUniversity,Beijing,China

2InstituteofPhysics,SchoolofBasicSciences,ÉcolePolytechniqueFédéraledeLausanne

(EPFL),Lausanne,Switzerland

Correspondence

YuanshengCaoandShilingLiang.Email:

yscao@

and

shiling.liang@epfl.ch

Fundinginformation

NationalNaturalScienceFoundationofChina,Grant/AwardNumber:12374213;

SchweizerischerNationalfondszurFörderungderWissenschaftlichenForschung,Grant/

AwardNumber:200020_178763

Abstract

Livingsystemsoperatewithinphysicalconstraintsimposedbynonequilib-riumthermodynamics.Thisreviewexploresrecentadvancementsinapplyingtheseprinciplestounderstandthefundamentallimitsofbiologicalfunctions.Weintroducetheframeworkofstochasticthermodynamicsanditsrecentdevelopments,followedbyitsapplicationtovariousbiologicalsystems.Weemphasizetheinterconnectednessofkineticsandenergeticswithinthisframework,focusingonhownetworktopology,kinetics,andenergeticsinfluencefunctionsinthermodynamicallyconsistentmodels.Wediscussexamplesintheareasofmolecularmachine,errorcorrection,biologicalsensing,andcollectivebehaviors.Thisreviewaimstobridgephysicsandbiologybyfosteringaquantitativeunderstandingofbiologicalfunctions.

KEYWORDS

biologicalfunctions,physicalconstraints,stochasticthermodynamics

1|INTRODUCTION

Physicsandbiology,whilesharingacommongoalofunderstandingthenaturalworld,employdistinctpara-digmsandterminologiestoachievethisobjective.Physicsstrivestoidentifyfundamental,universallawsgoverningthebehaviorofmatterandenergy,oftenexpressedusingconceptssuchasenergy,entropy,forces,andrates.Incontrast,biologydelvesintotheintricatecomplexitiesoflife,focusingonthestructures,functions,andinformationflowwithinlivingsystems,oftenutilizingtermssuchassignals,codes,transcrip-tion,andtranslation.

Modernbiologicalphysics,orthephysicsoflivingsystems,bridgesthedisciplinarygapbyapplyingtheprinciplesandmethodologiesofphysicstoelucidatebiologicalphenomena.Acentralfocusofthisinterdis-ciplinaryfieldistheinvestigationofbiologicalfunctions.Fromthemolecularmachinerywithinacelltothebiomechanicsoforganismalmovement,biological

physicsseekstounraveltheunderlyingphysicalprin-ciplesgoverningtheseprocesses.

Thefunctionalityoflivingsystemsisconstrainedbythefundamentallawsofphysics,chemistry,andbiology.Withinsignalingnetworks,forexample,re-ceptorspecificityensuresthattheyonlyrespondtospecificligands.Thisbiochemicalspecificityarisesfromthemolecularstructuresshapedbyevolution.However,aswillbedetailedlater,thefundamentallimitationsgoverningmanybiologicalfunctionsstemnotonlyfrombiologicalandchemicalfactorsbutalsofromthebasicphysicalprinciplesgoverningcellularprocessesandtheirenvironment.

Amongtherelevantsubfieldsofphysics,thermo-dynamicsprovidessomeofthemostuniversalcon-straintsonbiologicalsystems.Forinstance,theconservationofenergy(thefirstlawofthermody-namics)andmateriallimitstheyieldofcellularmeta-bolism[

1,2

].Thesecondlawofthermodynamics,whichstatesthatisolatedsystemstendtoward

Thisisanopenaccessarticleunderthetermsofthe

CreativeCommonsAttribution

License,whichpermitsuse,distributionandreproductioninanymedium,providedtheoriginalworkisproperlycited.

©2024TheAuthor(s).QuantitativeBiologypublishedbyJohnWiley&SonsAustralia,LtdonbehalfofHigherEducationPress.

QuantitativeBiology.2025;e75.

/10.1002/qub2.75

-

/journal/qb

1of19

20954697,

2of19

■

CAOandLIANG

maximumdisorder(andthereforeincompatiblewithlife),haslongchallengedphysicistsseekingtounder-standhowlivingsystemsmaintaintheirorderedstate.ErwinSchrödinger,inhisinfluentialbookWhatisLife?,assertedthatlivingcellsmustabsorb“negativeen-tropy”or“freeenergy”asitistermedinstandardther-modynamics,tocounterbalancetheentropyproducedwithinbiologicalprocessesandmaintaintheirlow-entropy(highlyordered)states.Thermodynamically,akeydistinctionbetweenacellandaboxofgasisthatacellrepresentsanopensystem,continuouslyexchangingenergywithitssurroundingstoremaininastateofdynamicorder.

Traditionalequilibriumthermodynamicsoffersawell-suitedframeworkfordescribingaboxofgasmol-ecules.However,duetotheircontinuousexchangeoffreeenergywiththeenvironment,cellsareinherentlynonequilibriumsystems.Notably,acentralthemeofthisreviewisthatthecostoffreeenergyimposesfundamentallimitsontheperformanceofbiologicalfunctions.Furthermore,cellularsystemsoftenoperateatthemicroscopicormesoscopicscale,wherethenumberofparticipatingmolecules,typicallyrangingfrom10to105,fallswellbelowthethermodynamiclimitof1023particles.Thischaracteristicclassifiesatypicalcellasamicroscopicormesoscopicnonequilibriumsystem.Consequently,theperformanceofbiologicalfunctionsisalsoinherentlylimitedbytheinevitablefluctuationsarisingfromthesmallnumberofmoleculesinvolved.

Recentadvancesinstochasticthermodynamicsprovideapowerfultooltoaddressthetwofeatureslistedabove.Stochasticthermodynamicsinvestigatesthebehaviorofmesoscopicsystemsgovernedbythermalfluctuations,bothinandoutofequilibrium,establishingafundamentalrelationshipbetweenther-modynamicprinciplesandstochastickinetics.ThisfieldtracesitsrootstoEinstein’sseminalworkonBrownianmotion,inwhichheestablishedthefluctuation–dissipationtheorem(FDT),revealingaprofoundconnectionbetweenfluctuationanddissipationatthermodynamicequilibrium.Forsystemsoutofequi-librium(suchasaBrownianparticlecoupledtomultiplethermalreservoirsorsubjectedtotime-dependentenvironmentalchanges),abroaderthermodynamicframeworkisnecessary,particularlyforunderstandinginherentlynonequilibriumbiologicalprocessesatthesubcellularlevel.Fromthelatterhalfofthe20thcen-tury,significanteffortsweremadetoconnecttheirre-versibledynamicsofstochasticprocesseswithentropyproduction[

3–5

],leadingtoacomprehensiveframe-workwithprecisedefinitionsofthermodynamicquanti-tieslikeworkandheatatthemesoscopiclevel[

6

].Buildinguponthisframework,researchershaveappliedadvancedmathematicalandphysicaltoolstoinvesti-gatefundamentalthermodynamicconstraints,resultinginkeyfindingssuchasthefluctuationtheoremwhich

2025,1,Downloadedfrom

/doi/10.1002/qub2.75byCochraneChina

,WileyOnlineLibraryon[24/01/2025].SeetheTermsandConditions(

/terms-and-conditions

)onWileyOnlineLibraryforrulesofuse;OAarticlesaregovernedbytheapplicableCreativeCommonsLicense

quantifiestheirreversibilityofmesoscopicsystems[

7–

9

],thethermodynamicuncertaintyrelation(TUR)whichidentifiesthethermodynamiccostofsuppressingcur-rentfluctuations[

10

],andnonequilibriumresponseboundsdemonstratinguniversalconstraintsonthesensitivityofnonequilibriumresponse[

11

].Thesead-vancesinstochasticthermodynamicsilluminatethethermodynamicprinciplesgoverningmesoscopicsto-chasticsystemsandsetphysicalconstraintsonmesoscopicbiologicalprocesses.

Thisreviewdelvesintorecentadvancementsinunderstandingthephysicallimitations,particularlyfromanonequilibriumthermodynamicsperspective,ontheperformanceofvariousbiologicalfunctions.Webeginbyintroducingthefundamentalframeworkandrecentdevelopmentsinstochasticthermodynamics.Subse-quently,weshowcaseadvancementsinapplyingtheseprinciplestodiversebiologicalsystems.Recognizingthedeepconnectionbetweenkineticsandenergeticsinstochasticthermodynamics(e.g.,localdetailedbalance[LDB]),thereviewspecificallyfocusesonstudiesthatelucidatehowkinetics,energetics,andnetworktopol-ogydeterminebiologicalfunctionwithinthermody-namicallyconsistentmodels.Threekeyquestionswillbeaddressedanddiscussed:(1)howtodefineaspe-cificbiologicalfunctionwithinthecontextofbiologicalnetworksandthermodynamics,(2)howstochasticthermodynamics,particularlyfreeenergydissipation(orentropyproduction),imposesconstraintsonbiologicalfunctionality,and(3)howtoapproachtheselimitationsandoptimizeperformance.Thesekeyaspectswillbeexaminedwithspecificexamplesinvariousareas,includingtheprecisionofmolecularmachines,errorcorrectionmechanisms,biologicalsensing,andtheemergenceofcollectivebehaviors.Thisreviewaimstobridgethegapbetweenphysicsandbiology,fosteringaquantitativeunderstandingofbiologicalfunctions.

2|INTRODUCTIONTOSTOCHASTICTHERMODYNAMICS

2.1|Thermodynamicsofstochasticprocesses

Manybiologicalprocesses,suchasmolecularmotormotion,transcription,andproteinfolding,canbemodeledasMarkoviandynamicswherethermalnoisetriggerstransitionsbetweenmesoscopicstates[

12–

15

].Stochasticthermodynamicsoffersacomprehen-siveframeworkfordescribingthethermodynamicsofstochasticprocessesbyestablishingconstraintsonki-netics.Consequently,itenablesprecisedefinitionsofthermodynamicquantities,suchasentropyproduction,atbothtrajectoryandensemblelevels[

16–18

].Thisframeworkfurtherallowsforthederivationofthermo-dynamicconstraintsonvariousphysicalobservables,

20954697,2025,1,Downloadedfrom

/doi/10.1002/qub2.75byCochraneChina

,WileyOnlineLibraryon[24/01/2025].SeetheTermsandConditions(

/terms-and-conditions

)onWileyOnlineLibraryforrulesofuse;OAarticlesaregovernedbytheapplicableCreativeCommonsLicense

■

3of19

STOCHASTICTHERMODYNAMICSFORBIOLOGICALFUNCTIONS

illustratinghowphysicallawslimitbiologicalfunctions.Here,wewillintroducethebasicframeworkofsto-chasticthermodynamicsandhighlightseveralrecentadvancesinthisfield.

2.1.1|Masterequation

Adiscrete‐statestochasticprocessisgovernedbyamasterequationthatdescribesthetimeevolutionoftheoccupationprobabilitiesofstates:

p(t)=Wp(t);(1)

wherep(t)=[p1(t),p2(t),…,pn(t)]representsthevectorofprobabilitydistributionacrossallstatesinthesystemattimet.Here,Wisthetransitionratematrix,withitselementWijindicatingthetransitionratefromstatejtostatei.Thetimeevolutionoftheprobabilityforstateicanbederivedfromtheequationasfollows:

dX

dtpi=j≠i(Wijpj−Wjipi);(2)

whereJij=Wijpjrepresentstheprobabilitycurrentfromstatejtostatei,contributedbyalltransitioneventsalongtheedgeeijfromjtoi.ThenetcurrentfromstatejtostateiisJij=Jij−Jji.Inthelongtimelimit,thesystemreachesastationarystate,denotedbypst,inwhichall

netcurrentsarebalanced,suchthatΣj≠iJt=0forany

iorinmatrixproductformWpst=0.Ifallnetcurrents

arezero,thatis,Jt=0forallpairsi,j,thesteadystate

isanequilibriumstate.

Thesteady‐stateprobabilitydistributioncanbederivedusingagraph‐theoreticapproach,expressedasfollows:

(3)

p=;

whereTiisadirectedspanningtreerootedatstatei,andw(Ti)istheweightofthedirectedtreeTi,calcu-latedastheproductofthetransitionratesalongtheedgesofthetree.Figure

1

illustratesthespanningtreerepresentationofthesteady‐stateprobabilitydistribu-tionfora3‐statesystem.

Thisgraph‐theoreticsolutionhasbeendiscoveredseveraltimesthroughouthistory[

19–21

].Itcircumventsthedirectcomputationofmatrixinversionsandfacili-tatestheoreticalanalysisbasedonthegraph‐theoreticpropertiesofthenetwork.Thisanalysisservesasafoundationalelementforexploringthethermodynamicsofstochasticprocesses.Suchexplorationshaveledtonumerousresults,includingcycle‐decompositionofentropyproduction[

20

],boundsonnonequilibrium

responses[

11,22

],constraintsonsymmetrybreakinginbiochemicalsystem[

23

],andthedevelopmentofthespanning‐treerepresentationforfirstpassagetimesstatisticsinMarkovchains[

21,24

].

2.1.2|Localdetailedbalance

Forsmallsystems,theprincipleofmicroscopicreversibilitymandatesthatforanytransition,anasso-ciatedbackwardtransitionmustexist.Inasystemgovernedbythemasterequation,thermodynamicsisintroducedoneverypairoftransitionratesthroughtheLDBcondition

[18,25

],

=eΔSnv/kB;(4)

wherekBistheBoltzmannconstantandΔSnvrepre-

sentstheentropyproductionintotheenvironmentforthetransitionj→i.Theentropyproductionintotheenvironmentisdeterminedbytheenergyexchangebetweenthesystemanditsenvironment,incorporatingtwocontributions:theenergydifferencebetweenthetwostates,∈j−∈i,andthedrivingforceFij,

ΔSnv=(∈j−∈i)/T+Fij;(5)

whereTisthetemperatureoftheenvironment.Theentropyproductionquantifiestheirreversibilityofatransition—iftheentropyproductionispositive(i.e.thetransitionfromstatejtostateiincreasestheentropyoftheenvironment),thenWij>Wji,indicatingapreferencefortheforwardtransitionoverthebackwardone.Thelocalthermodynamicdefinitionoftransitionratesallowsustoassesswhetherasystemisinoroutofequilibriumonagloballevel.Onecancalculatethenonequilibriumdrivingforcealongacyclec=[m0,m1,m2,…,mn,m0]inthenetwork,

,、

Fc=ln;(6)

referredtoascycleaffinity,orcyclicdrivingforce.WhenFc≠0,thetime‐reversalsymmetryisbroken,indicatingthesystemisoutofequilibrium—astraversingacycleresultsinnonzeroentropyproduction.Conversely,ifFc=0forallcyclesinthenetwork,knownasKolmo-gorov’scriterion[

26

],thesystemisanequilibriumsystemandpreservestime‐reversalsymmetry.Theequilibriumnatureofasystemallowstheconstructionofanenergylandscapeinwhichtheentropyproductionofatransitionissolelydeterminedbytheenergydif-ferencebetweentheinitialandfinalstates,thatis,

ΔSnv=(∈j−∈i)/T.Consequently,theLDBcondition

reducestothedetailedbalancecondition

20954697,2025,1,Downloadedfrom

/doi/10.1002/qub2.75byCochraneChina

,WileyOnlineLibraryon[24/01/2025].SeetheTermsandConditions(

/terms-and-conditions

)onWileyOnlineLibraryforrulesofuse;OAarticlesaregovernedbytheapplicableCreativeCommonsLicense

4of19

■

CAOandLIANG

FIGURE1Graph‐theoreticsolutionforMarkovchain.(A)Athree‐stateMarkovchain.(B)Thegraph‐theoreticsolutionofstationarystate

probability.

==e−β(∈i−∈j);(7)

foranypairofstates,whereβ=1/(kBT)istheinversetemperature.DetailedbalanceensuresthesystemcanrelaxtoanequilibriumBoltzmanndistribution,

pieq=;(8)

wherethesummationinthedenominatorisoverallstates.Withdetailedbalance,thegraph‐theoreticalsolution,Equation(

3

),canbereducedtotheBoltz-manndistribution.

2.1.3|Entropyproductionrate

TheLDBconditionlinksthekineticsofaMarkovpro-cesswithitsthermodynamicproperties.Beyondtheentropyproductionofindividualtransitions,itispossibletodefinetheentropyproductionrate(EPR)attheensemblelevel,whichquantifiestheaveragerateofentropyproductionacrosstheentiresystem.DenotingtheEPRasΣ.,itcanbeexpressedasfollows:

Σ.tot=kB←JijlnJij

i;j>iJji

=kB←JijlnWij+kB←Jijlnpj.(9)

i;j>iWjiij>ipi

、 、尺----------√、-----;----、尺---------√

environmentEPR;Σ.envsystemEPR;Σ.sys

ThetotalEPR,Σ.tot,canbedecomposedintotheenvironmentEPR,Σ.env,andthesystemEPR,Σ.sys.Notably,thesystemEPRequalsthetimederivativeofthesystem’sShannonentropy,Σ.sys=−kBΣipilnpi).

Atanonequilibriumstationarystate,theprobabilitydistributionremainsunchangedovertime,resultinginzerosystemEPR.ThesignofJij=Jij−Jjialwaysalignswiththatofln(Jij/Jji),ensuringthenon‐negativetotalEPR.Thispropertyisconsistentwiththesecondlawofthermodynamics,whichstatesthatthetotalentropyofanisolatedsystem(herethesystemandenvironmenttogetherconstituteanisolatedsystem)isanon‐decreasingfunction.

2.2|Stochastictrajectoriesand

fluctuationtheorems

Themasterequationprovidesadeterministicdescrip-tionofthetimeevolutionofprobabilitydistributionsinaMarkovchain,obtainedbyaveragingoverthemanypossiblestochastictrajectorieswhicharegeneratedbythetransitionratematrixW.Ontheotherhand,theMarkovchainitselfmodelstherandomtransitionsbe-tweenstates,capturingtheinherentfluctuationsatthelevelofindividualtrajectories.AsdepictedinFigure

2B

,astochastictrajectory(andthetime‐reversedtrajectoryinFigure

2C

)withinathree‐statenetworkprovidesavisualrepresentationoftheseconcepts.Understandingthethermodynamicpropertiesofstochastictrajectoriesandextractinginformationfromthemarecentralproblemsinstochasticthermodynamics.Toaddressthesechallenges,wefirstintroducetheprobabilityofastochastictrajectoryforageneraldiscrete‐statesto-chasticprocess.Inthemostgeneralcase,asystemcanbesubjectedtoexternalcontrol,λt,leadingtoatime‐dependenttransitionmatrix,Wλ(t).Astochastictrajectoryisasequenceofstates[γ0,γ1,…,γn]alongwiththecorrespondingtimesoftransitionevents[t0,t1,t2,…,tn,tn+1],wheret0=0andtn+1=τdenotetheinitialandfinaltimesofthetrajectory,respectively,andeachtirepresentsthetimeforthetransitionγi−1→γi.Theprobabilityofatrajectorycanbewrittenasfollows:

20954697,2025,1,Downloadedfrom

/doi/10.1002/qub2.75byCochraneChina

,WileyOnlineLibraryon[24/01/2025].SeetheTermsandConditions(

/terms-and-conditions

)onWileyOnlineLibraryforrulesofuse;OAarticlesaregovernedbytheapplicableCreativeCommonsLicense

■

5of19

STOCHASTICTHERMODYNAMICSFORBIOLOGICALFUNCTIONS

FIGURE2StochastictrajectoriesforaMarkovchain.(A)Athree‐stateMarkovchain.(B)Astochastictrajectoryonthethree‐state

network.Theprobabilityofatrajectorycontainsthecontributionfromtransitioneventsandthesurvivalprobabilityamongthestatesalongthetrajectory.(C)Thetime‐reversedtrajectory.

Wiγi(t)dt

n

Pγ=p(γ0;0)∏i=1

(10)

;

i=0

Wiγi−1(ti)e∫+1

wherep(γ0,0)istheprobabilityatstateγ0withthe

initialdistribution,Wiγi−1(ti)istheprobabilityoftransition

Wγiγi(λ(t))dt

isthesurvival

γi−1→γiattimeti,ande∫+1

probabilityonstateγibetweentheinandouttransitions.Denotingtheprobabilityofthereversedtra-jectoryγundertime‐reversedprotocolλ(t)=λ(tn+1−t)asPγ,onecanfindthattheratioofthesetwoprobabilitiesisdeterminedbythetotalentropyproductionalongthetrajectory[

27

].

Pγ;

Pγ=eΔSot/kB(11)

where,

ΔSot=kBln+kBln.(12)

Equation(

11

)quantifiestheirreversibilityontrajec-torylevel,andcanalsobeunderstoodasadirectconsequenceofLDBcondition[

25

].Byintegratingoveralltrajectorieswithequalentropyproduction,wecanfindthedetailedfluctuationtheorem[

28

].

,、

PΔStotΔStot

P−ΔStot

,、=ekB;(13)

whichstatesthattheprobabilityofobservationofen-tropyproductionofanamountΔStotiseΔStot/kbmorelikelythanobservingthesameamountofnegativeentropyproductionunderatime‐reversalcontrolpro-tocol.Thisisoneofthemostfundamentalrelationsin

stochasticthermodynamics.Byaveraginge−ΔSot/kB

overallpossibletrajectories,onecanobtaintheinte-gratedfluctuationtheorem,

e−kB=Pγe−kBdγ=Pγdγ=1.(14)

,ΔSot、ZΔSotZ

γγγ

Thefluctuationtheoremhasbeenfoundseveraltimesattheendofthelastcentury

[7,8,28

].ItsapplicabilityextendsbeyondMarkovianprocesses,encompassingdeterministicHamiltoniansystems[

29

]andquantumsystems[

30

].Forexample,Jarzynskiequalityasoneoftheveryfirstintegratedfluctuationtheorems[

8

]revealsarelationbetweenworkandfreeenergychangeinanonequilibriumprocessinwhichasystemisdrivenfromaninitialequilibriumdistributionpinittoafinalequilibriumdistributionpfin.Theworkdonetothesystemfordifferentrealizationsvariesduetofluctuations,thusanexactrelationbetweenworkandfreeenergychangecannotbeestablished.However,anequalityexistsfortheaveragevaluebasedonthefluctuationtheorem.Forsuchaprocess,thetotalen-tropyproductionisΔStot=(W−ΔF)/T,whereWistheworkdoneonthesystemandΔF=Ffin−Finitisthefreeenergychangeofthesystem.Therefore,thedetailedfluctuationtheorem,Equation(

13

),leadstoCrooksrelationforwork[

28

].

=eβ(W−ΔF);(15)

whereP(W)istheprobabilityofapplyinganamountofworkWduringaforwardprocess,andP(−W)istheprobabilityofapplyinganamountofwork−Wduringatime‐reversedprocess.Similarly,theintegratedfluctua-tiontheorem,Equation(

14

),leadstotheJarzynskiequality[

8

],

20954697,

6of19

■

CAOandLIANG

he−βW〉=e−βΔF.(16)

ByapplyingtheJensen’sinequality,e−βhW〉≤he−βW〉,thesecondlawofthermodynamicsisrecoveredasfollows:

ΔF≤hW〉;(17)

whichmeansthattheaverageworkdoneonthesys-tem,takenoverallrealizationsofstochastictrajec-tories,providesanupperboundforthechangeinfreeenergy.Theequalsignistakeninthequasi‐staticlimit.Thisinequalitycanbeseenasastatistical‐levelmani-festationofthesecondlawofthermodynamics.

Onthebasisofthefluctuationtheoremandincor-poratingtheconceptofinformationbyfeedbackcontrol,SagawaandUedaintroducedageneralizedversionofJarzynskiequality[

31

],

he−β(W−ΔF)−I〉γ=1;(18)

whereIisthemutualinformationintroducedbyfeed-backcontrol.

2.3|Thermodynamicuncertaintyrelation

Atthemesoscopicscale,physicalobservablesareal-wayssubjecttofluctuationduetothermalnoise.Therelationbetweenfluctuationsanddissipationisacorefocusofstochasticthermodynamics,andthecostofsuppressingfluctuationisthecentralproblemofthestudyofthethermodynamicsofbiochemicalsystems.In2015,BaratoandSeifertproposedauniversalthermo-dynamicboundonthefluctuationofstochasticcurrents[

10

].Intheirwork,theystudiedabiasedrandomwalk(arandomwalkwheretheprobabilitiesofmovingindifferentdirectionsarenotequal)inonedimensiontointroducetheuncertaintyrelationofcurrents.Insuchasystem,theforwardandbackwardtransitionratesarek+andk−,respectively,whichgeneratestochastictrajec-toriesasshowninFigure

3

.Startingfromtheorigin,themeanandvarianceofthepositionoftherandomwalkerattimeτareVar[Xτ]=2Dτ=(k++k−)τandhXτ〉=vτ=(k+−k−)τ,respectively.Thebiasednatureofrandomwalkisassociatedwithacostofentropypro-ductionaccordingtotheLDBconditionEquation(

4

),asΔS=kBln(k+/k−)perstep.ThetotalentropyproductionaftertimeτisΣτ=hXτ〉ΔS.Combiningtheseexpressionsonecanfindarelationbetweendissipationandprecisionasfollows:

=≥.(19)

Itcanalsobeformulatedintermsofvelocity,diffu-sioncoefficient,andEPRasfollows:

2025,1,Downloadedfrom

/doi/10.1002/qub2.75byCochraneChina

,WileyOnlineLibraryon[24/01/2025].SeetheTermsandConditions(

/terms-and-conditions

)onWileyOnlineLibraryforrulesofuse;OAarticlesaregovernedbytheapplicableCreativeCommonsLicense

FIGURE3Stochastictrajectoriesofabiasedrandomwalk.

≥;(20)

whereΣ.=vΔSistheEPR.

AlthoughTURwasoriginallyobtainedfromspecificmodels,itwasconjecturedtoholdformoregeneralstochasticcurrentsinstochasticprocess,andhadlaterbeenprovenusingthelargedeviationtheoryinMarkovjumpprocesses[

32

],martingaletheoryincontinuousstochasticprocess[

33

],Cramér–Raoboundformulti‐dimensionalcurrents[

34,35

]andmanyotherap-proaches[

36–38

].Thegeneralformreadsasfollows:

≥;(21)

whichsetsatrade‐offrelationbetweentheuncertaintyofcurrentsandentropyproduction.Thismeansthatachievinghigherprecisioninacurrentgenerallyre-quiresagreaterdissipation.However,itisworthnotingthatTURcanbebrokeninunderdampedsystems,asillustratedwithanexampleofanunderdampedclock

[39

].Thus,thevalidityofTURisestablishedwithintheoverdampedregime,contingentuponanumberofadditionalassumptions[

40

].TheextensionofTURbeyondtheoverdampedlimitneedstoincorporateadditionaltre