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Mechanics of collective unfolding M.Caruela,b,J.-M.Allainc,L.Truskinovskyc aInriaSaclayIle-de-France,M3DISIMteam,Palaiseau,France bUniversite´ParisEst,LaboratoireMode´lisationetSimulationMultiEchelle,MSME-UMR-CNRS-8208,Biomeća-61AvenueduGeńe´ralde Gaulle,94010Cre´teil,France cLMS,CNRS-UMR7649,EcolePolytechnique,RoutedeSaclay,91128Palaiseau,France 5 Abstract 1 0 Mechanicallyinducedunfoldingofpassivecrosslinkersisafundamentalbiologicalphenomenonencounteredacross 2 thescalesfromindividualmacro-moleculestocytoskeletalactinnetworks. Inthispaperwestudyaconceptualmodel n ofathermalload-inducedunfoldinganduseaminimalisticsettingallowingonetoemphasizetheroleoflong-range a interactions while maintaining full analytical transparency. Our model can be viewed as a description of a parallel J bundleof N bistableunitsconfinedbetweentwosharedrigidbackbonesthatareloadedthroughaseriesspring. We 7 showthatthegroundstatesinthismodelcorrespondtosynchronized,singlephaseconfigurationswhereallindividual unitsareeitherfoldedorunfolded. Wethenstudythefinestructureofthewigglyenergylandscapealongthereaction ] h coordinate linking the two coherent states and describing the optimal mechanism of cooperative unfolding. Quite p remarkably, our study shows the fundamental difference in the size and structure of the folding-unfolding energy - barriersinthehard(fixeddisplacements)andsoft(fixedforces)loadingdeviceswhichpersistsinthecontinuumlimit. o i Wearguethatboth,thesynchronizationandthenon-equivalenceofthemechanicalresponsesinhardandsoftdevices, b havetheirorigininthedominanceoflong-rangeinteractions. Wethenapplyourminimalmodeltoskeletalmuscles . s wherethepower-strokeinacto-myosincrossbridgescanbeinterpretedaspassivefolding. Aquantitativeanalysisof c i themusclemodelshowsthattherelativerigidityofmyosinbackboneprovidesthelong-rangeinteractionmechanism s allowing the system to effectively synchronize the power-stroke in individual crossbridges even in the presence of y h thermalfluctuations. Inviewoftheprototypicalnatureoftheproposedmodel, ourgeneralconclusionspertaintoa p varietyofotherbiologicalsystemswhereelasticinteractionsaremediatedbyeffectivebackbones. [ 1 v 9 1. Introduction 5 4 Incontrasttoinertmatter, distributedbiologicalsystemsarecharacterizedbyhierarchicalnetworkarchitectures 1 with domineering long-range interactions. Even in the absence of metabolic fuel this leads to a highly nontrivial 0 cooperativemechanicalbehaviorinbothstaticsanddynamics. Passivecollectiveeffectsareusuallyrevealedthrough . 1 synchronizedconformationalchangesinterpretedhereasgenericfolding-unfoldingtransitions.Theexperimentshows 0 that such systems exhibit coherent macroscopic hopping between folded and unfolded configurations resisting the 5 destabilizingeffectoffinitetemperatures(DietzandRief,2008;ThomasandImafuku,2012;Erdmannetal.,2013). 1 : Aminimalmechanicalmodelshowingthecooperativebehaviorisaparallelbundleofbistableunitslinkedbytwo v sharedbackbonesanditsmostnaturalbiologicalprototypeisamusclehalf-sarcomereundergoingthepower-stroke i X (Carueletal.,2013). Anotherexampleistheunbindingoffocaladhesionswithindividualadhesiveelementscoupled r throughacommonelasticbackground(ErdmannandSchwarz,2007). Similarbehaviorhasbeenalsoassociatedwith a mechanical denaturation of RNA and DNA hairpins where the long-range interactions are due to the prevalence of stem-loopstructures(Liphardtetal.,2001;Bosaeusetal.,2012).Othermolecularsystemswithcooperativeunfolding includeproteinβ-hairpins(Munõzetal.,1998)andcoiledcoils(Bornschlo¨glandRief,2006). Thebackbonedomi- natedinternalarchitectureinallthesesystemsleadstoamean-fieldtypemechanicalfeedbackwhichisalsoubiquitous inbi-stablesocialsystems(KometaniandShimizu,1975;DesaiandZwanzig,1978). Emailaddress:[email protected](M.Caruel) PreprintsubmittedtoJournaloftheMechanicsandPhysicsofSolids January8,2015 Inthispaperwesystematicallystudythemechanicsoftheminimalmodelinthesettingwhichcanbedirectlyasso- ciatedwithskeletalmuscles. Werecallthatthemechanicallyinducedconformationalchange(power-stroke)inskele- talmusclestakesplaceinmyosinheads(cross-bridges)thatareboundinparalleltoactinfilaments(Smithetal.,2008; Linarietal.,2010;Guerinetal.,2011;ErdmannandSchwarz,2012;Piazzesietal.,2014). The(thermo)mechanical behavior of this system was first analyzed by Huxley and Simmons (Huxley and Simmons, 1971) who interpreted thepre-andpost-power-strokeconformationsofthemyosinheadsasdiscretechemicalstates(spinmodel). Similar ideashavebeenindependentlyadvancedinthestudiesofbistableadhesionclusters(Bell,1978)andinotherapplica- tionsrangingfromJahn-Tellereffectandripplesingraphenesheets(Bonillaetal.,2012)tounzippingofbiological macromolecules(Guptaetal.,2011;Pradosetal.,2012). Since in (Huxley and Simmons, 1971) the behavior of the spin model was studied only in a hard device (prescribed displacements), the cooperative effects were not found. To understand this surprising result we study the zero-temperatureanalogoftheHuxleyandSimmonsmodel. Weshowthatforthissystemthestructuresoftheenergy landscape in hard and soft (prescribed forces) loading conditions are rather different. In particular, we explain why inthismodelthecollectivebehavioratfinitetemperaturecanbeexpectedinthesoftbutnotintheharddevice. To capture the coherent hopping in the hard device case, we regularize the spin model in two ways. First, following (MarcucciandTruskinovsky,2010)wereplacethediscretechemicalstates(hardspins)byacontinuousdouble-well potential with a finite energy barrier (snap-spring model). Second, to take into account the elastic interactions between individual crosslinkers, we introduce a series spring mimicking the backbone elasticity (Wakabayashi et al., 1994;Huxleyetal.,1994)andbringinginmean-fieldinteractions. Weshowthatinthesnap-springmodeltheenergy barriersseparatingthesynchronizedstatesarestillmarkedlyhigherinasoftdevicethaninaharddevicewhichpro- videsanexplanationfortheobservedretardedrelaxationinisotonicexperimentsonskeletalmuscles(Reconditietal., 2004;Piazzesietal.,2002;Decostreetal.,2005). Aninterestingpeculiarityofthesnap-springmodelisthattherelaxedpotential,representingtheglobalminimum of the energy, is always convex in a soft device but is only convex-concave in a hard device. This means that the macroscopicstiffness,whichisalwayspositiveinasoftdevice,canbecomenegativeinaharddevice. Thenegative stiffness(metamaterial)response(NicolaouandMotter,2012),whichpersistsinthecontinuumlimit,clearlycontra- dictstheintuitiondevelopedinthestudiesofsystemswithshort-rangeinteractions. Thenon-convexityoftheground stateenergyinaharddevicemeansthatthesystemisnon-additiveandcannotberelaxedthroughthemixingoffolded andunfoldedunits. Toillustrateourgeneralresults,weconsiderinsomedetailthespecialcaseofskeletalmuscleswherewecanmake quantitative estimates by using realistic parameters. Our analysis shows that the height of the microscopic energy barriers for the power-stroke in individual cross-bridges is of the order of the energy of thermal fluctuations. This impliesthatthecross-bridgescanundergoconformationalchangesinanon-cooperativestochasticmanner. However, the presence of long-range interactions creates a bias in the individual folding-unfolding equilibria in the form of a macroscopic barrier on top of which the microscopic barriers are superimposed. We call this barrier macroscopic becauseitsheightisproportionaltothenumberofelementsinthesystem. Duetothepresenceofthemacroscopic barrier, the folding transitions in individual cross-bridges become energetically preferable only after the top of this barrier has been reached which means that individual cross-bridges have been synchronized. These observations suggestthattheelementarycontractileunitofskeletalmuscleshasevolvedtocontrolthestateofalargeassemblyof folding elements through a mean-field type mechanical feedback. Due to such passive synchronization, the power- stroketakesplacecollectivelywhichobviouslyamplifiesthemechanicaleffect. Whilewefocusinthispaperontheathermalbehavior,ouranalysisrevealstheoriginoftheanomalousthermo- dynamicsandkineticsofskeletalmusclesandsimilarsystemsobservedatfinitetemperatures(Carueletal.,2013). A detailedstudyofthetemperatureeffectsonthecollectiveunfoldingwillbepresentedelsewhere. Thepaperisorganizedasfollows. InSection2,westudytheequilibriummechanicalbehaviorofthespinmodel andshowthatalreadyinthisminimalsettingthebehaviorofthesysteminsoftandharddevicesisdifferent.Thesnap- springmodelisintroducedinSection3wherewedemonstratethatitremovesthedegeneraciesofthespinmodeland effectivelyinterpolatesbetweenthesoftandharddevicebehaviors.Inthesamesection,wealsostudythefinestructure oftheenergylandscapeseparatingthecoherentstatesofthesystemandintroduceareactioncoordinatetodescribe the successive individual folding-unfolding transitions constituting the collective unfolding. The adaptation of the snap-springmodelforskeletalmusclesispresentedinSection4. FinallyinSection5wepresentourconclusions. 2 2. Thehardspin(HS)model Consider the behavior of an elementary cluster of N bistable units connecting two rigid backbones. In the spin model (Huxley and Simmons, 1971), each crosslinker is represented by a bistable potential connected to a series spring. The potential, representing two folding configurations, is assumed to have infinitely narrow energy wells representingtwochemicalstates,seeFig.1. Thepotentialdescribingindividualspinunitscanbewrittenintheform  uHS(x)=v00 iiffxx==0,a. (1) − Herethespinvariablextakestwovalues,0and a,describingtheunfoldedandthefoldedconformations,respectively. − Byawedenotedthe“reference”sizeoftheconformationalchangeinterpretedasthedistancebetweenthetwoenergy wells. Withtheunfoldedstateweassociateanenergylevelv whilethefoldedconfigurationisconsideredasazero 0 energystate. Thepotential(1)isshownschematicallyinFig.1(a). a u (x) b HS u k rigid HS backbone T T N v 0 x a 0 z − Figure1:Hardspinmodelofaparallelbundleofbistablecrosslinkers.(a)Energylandscapeofanindividualcrosslinker;(b)Ncrosslinkersloaded inasoftdevice. In addition to a spin unit with energy (1) each cross-bridge contains a linear shear spring with stiffness k; see Fig.1(b). Theenergyoftheelasticspringisu (x)=kx2/2andtheenergyofthewholecrosslinkeris E u=u (x)+u (z x). (2) HS E − Without loss of generality, we can assume that the reference length of the linear spring is already incorporated into thedefinitionoftheelongationz. Noticethatthemechanicalsystemwithenergy(2)hasamulti-stableresponse,see Fig.2. To non-dimensionalize the resulting model, which one can associate with the names of Huxley and Simmons even though they never considered such parallel connection explicitly (for this representation, see Marcucci and Truskinovsky (2010)), we choose a as a characteristic length, associate the characteristic energy scale with ka2 and normalizeforcesbyka. Theonlyremainingdimensionlessparametersofthemodelare N andv andwecanwrite 0 thedimensionlessenergyofthesystem(percrosslinker)intheform 1 (cid:88)N (cid:34) 1 (cid:35) v(x;z)= (1+x)v + (z x)2 . (3) N i 0 2 − i i=1 Here,forconvenience,wepreservedthesamenotationsfornon-dimensionalquantities. In a hard device each crosslinker is exposed to the same total elongation z and thus the individual units are independent. Inthesoftdevicecase,wherethecontrolparameteristhetotaltensionT,theenergypercrosslinkeris 1 (cid:88)N (cid:34) 1 (cid:35) w(x,z;t)=v(x,z) tz= (1+x)v + (z x)2 tz , (4) − N i 0 2 − i − i=1 3 a b 0.5 1 B1 0 0.5 A B2 A xˆ 0.5 tˆ 0 − 1 0.5 − B2 − B1 1.5 1 − 1.5 1 0.5 0 0.5 − 1.5 1 0.5 0 0.5 − − − − − − z z Figure2: Behaviorofasinglecrosslinkerinthespinmodel.(a)Equilibriumstatesforvariousz;(b)Correspondingtensionlevels.Dashedlines, metastablestates;boldline,globalminimum. Arrowsshowtheresponsetosuddenshorteningincludingafrozenelasticphase(A → B1)anda subsequentphaseequilibration(B1→B2).Hereweusedv0=0. wheret=T/Nistheforcepercrosslinker. Nowforeachcrosslinkerbothx andzareinternaldegreesoffreedomand i the individual units can no longer be considered as independent. Indeed, if we minimize out the global continuous variablezbysolving ∂w/∂z =0,weobtain |t,{xi} 1 (cid:88)N z=t+ x, (5) i N i=1 (cid:80) whichafterinsertinginEq.4showsthatthepartiallyminimizedenergydependsquadraticallyon N x, i=1 i w˜(x;t)= N1 (cid:88)x2i2 −txi+(1+xi)v0− t22− 21N1 (cid:88)xi2. i i One can see that the transition from hard to soft device introduces a mean-field interaction among the crosslinkers which,asweshowlater,isultimatelyresponsibleforthecooperativebehavior. 2.1. Mechanicalequilibriuminaharddevice TodescribetheequilibriumresponseoftheHSmodelinaharddevice,weneedtocomputethelocalminimaof themechanicalenergy(3)atfixedz. Sinceeachoftheinternaldegreesoffreedom x, for1 i N cantakeonly i ≤ ≤ twodiscretevalues, x = xˆ = 0and x = xˆ = 1, agivenequilibriumstateischaracterizedbythedistributionof i 0 i 1 − the N crosslinkers between the two spin configurations. Due to the permutational invariance of the problem, each equilibrium state is fully characterized by a discrete order parameter representing the fraction of crosslinkers in the foldedstate, 1 (cid:88)N p= α, (6) i N i=1 whereα =1ifx = 1andα =0ifx =0. i i i i − InAppendix Aweshowthatallequilibriumconfigurationsofthistypecorrespondtolocalminimaoftheenergy (3). Atagivenvalueoftheorderparameter p,theenergiesofallsuchmetastablestatesareequalto 1 (cid:0)1 (cid:1) vˆ(p;z)= p (z+1)2+(1 p) z2+v . (7) 2 − 2 0 Thisisalinearcombinationoftheenergiesoftwolimitingconfigurations,onefullyfoldedwith p=1andtheenergy 1(z+1)2 andtheotheronefullyunfoldedwith p = 0andtheenergy 1z2+v . Theabsenceofamixingenergyisa 2 2 0 manifestationofthefactthatthetwocoexistingpopulationsofcrosslinkersdonotinteract. 4 a c 3 N=5 A 1 2 N B1 vˆ p=0 vˆ z 0.5 1 B2 p=1 0 0 1 0.5 0 0.5 1 1 0.5 0 − − − − b d 2 1 B2 A p=1 t/t 1 t/t ∗ p=0 ∗ 0 B1 0 1 0.5 0 0.5 1 1 0.5 0 − − − − δz=z z δz=z z − ∗ − ∗ Figure3: MechanicalresponseoftheHSmodelinaharddevicewithN = 5. (a)Energylevelsofthemetastablestates(p = 0,1,...,1)for 5 differentappliedelongations.(b)correspondingtension-elongationrelations.(c)and(d)aredetailsof(a)and(b)withanillustratedresponsepath toafastloadingexperimentincludingafrozenelasticphase(A→B1)followedbyasubsequentphaseequilibration(B1→B2).Thicklines,global minimumcorrespondingtop=0(resp. p=1)forz>z∗(resp.z<z∗).t∗=v0andz∗=v0−1/2.Parametersarev0=1andN=5. Thetension-elongationrelationsalongmetastablebranchesparameterizedby pcanbewrittenas ∂ tˆ(p;z)= vˆ(p;z)=z+p, (8) ∂z soforgiven pweobtainequidistantparallellines,seeFig.3. ApeculiarfeatureoftheHSmodelisthatthedomain ofhystereticbehaviorextendsindefinitelybecausethespinsystemdoesnothaveanystressthresholds. Tofindtheglobalminimumoftheenergyweneedtoperformateachvalueofzanadditionalminimizationover thediscretevariable p. Ifwecomputethederivative ∂ 1 vˆ(p;z)=z+ v . ∂p 2 − 0 whichdoesnotdependon p,weobtainthatforz > z ,wherez = v 1/2,theglobalminimizeristhefoldedstate ∗ ∗ 0− with p=1andforz<z itistheunfoldedstatewith p=0,seeFig.3. Theglobalminimumenergyprofileexhibitsa ∗ kinknearthecrossing(folding)point. Similarkinksassociatedwithunfoldingofhairpinsandotherfoldingpatterns have been observed in the energy profiles reconstructed from single molecule force spectroscopy measurements of proteinsandnucleicacids(Guptaetal.,2011). Observethatthegroundstateenergyinthismodelisnonconvexindependentlyofthenumberofunits. Thismeans thattheenergyisnotconvexifiedthroughtheformationofmixturesasinsystemswithshort-rangeinteractions(see, forinstance,PuglisiandTruskinovsky(2000)). Thereasonisthatthismean-fieldsystemisstronglynon-additiveand allmixedstatesareenergeticallyunfavorableduetohighcostofmixing. Somewhatsimilarsituationtakesplacein theory of elastic phase transitions where the relaxed energy is also nonconvex in the general case (it is only quasi- convex)whichisagaintheconsequenceoflong-rangeelasticinteractionsexemplifiedbythegradientconstraint(see, forinstance,Ball(2002)). A consequence of the energy nonconvexity is the non-monotonicity of the force-elongation relation shown in Fig.3(b,d). Morespecifically,thesystemexhibitsanegativestiffnessatthepointwhereallcrosslinkerscollectively 5 flip from the folded to the unfolded state. Similar mechanical behavior has been recently artificially engineered in metamaterialsbydrawingontheBraessparadoxfordecentralizedgloballyconnectednetworks(CohenandHorowitz, 1991; Nicolaou and Motter, 2012). As in our case, the mean-field type coupling in metamaterials is achieved via parallelconnectionswithmultiplesharedlinks. Biologicalexamplesofsystemswithnegativestiffnessareprovided byunzippingRNAandDNAhairpins(Woodsideetal.,2008;Bosaeusetal.,2012). 2.2. Mechanicalequilibriuminasoftdevice ConsidernowtheHSmodelloadedinasoftdevicewhentheequilibriumstatescorrespondtolocalminimaofthe mechanicalenergy(4). Anequilibriumstateisagainfullycharacterizedbythefractionofunitsinthefoldedstate p, definedbyEq.6. Wecanthenwritethe(marginal)energyofthepartiallyequilibratedsystemas w˜(p,z;t)=vˆ(p,z) tz, − wherevîstheenergyofthemetastablebranchparametrizedby pintheharddevicecase,seeEq.7. ThenweeliminatezbyusingEq.5. Eachvalueoftheorderparameter pdefinesabranchoflocalminimizersof theenergy(4)parameterizedbyt,seeAppendix B.Atagivenvalueof p,theenergyofametastablestatereads 1 1 wˆ(p;t)= t2+pt+ p(1 p)+(1 p)v . (9) −2 2 − − 0 a c 1 N=5 0.8 0.5 C1 A wˆ p=1 wˆ 0.6 0 T T N p=0 0.4 C2 0.5 1 1.5 0.4 0.6 0.8 1 b d 1.2 A 2 1 t/t 1 p=1 t/t 0.8 ∗ p=0 ∗ C2 0.6 C1 0 0.4 1 0.5 0 0.5 1 1 0.5 0 − − − − t/t δzˆ=zˆ z ∗ − ∗ Figure4: MechanicalresponseoftheHSmodelinasoftdevice. (a)Energylevelsofthemetastablestates(p=0,1,...,1)fordifferentapplied 5 forces.(b)Correspondingtension-elongationrelations.In(c)and(d)wezoomintodomainneart=t =v0andshowschematicallytheresponse ofthesystemtoasuddenapplicationofaloadincrementwithanelasticphase(A→C1)followedbya∗foldingphaseC1→C2.Thicklines,global minimumcorrespondingtop=0(p=1)fort>t (t<t ).Parametersare,v0=1andN=5. ∗ ∗ Incontrasttothecaseofaharddevice,herethereisanontrivialcouplingterm p(1 p)describingtheenergyof − mixing. Thepresenceofthistermisasignatureofamean-fieldinteractionamongindividualcrosslinkers. Indeed,if oneelementchangesconfiguration,itscontributiontothecommontensionchangesaccordinglyandtheotherelements mustadjusttomaintaintheforcebalance. Thetension-elongationrelationassociatedwithasetofmetastablestates sharingthesamevalueoftheparameter pcanbewrittenintheform ∂ zˆ(p;t)= wˆ(p;t)=t p. −∂t − 6 Atagivenvalueof pthisrelationisidenticalwithitscounterpartinthecaseofaharddevice,seeEq.8. Theglobally stablestatescanbefoundbyminimizing(9)over p. Since∂2wˆ/∂p2 = 1,thisfunctionisconcavein p. Therefore, − theglobalminimumisagainattainedeitherat p = 1or p = 0. Theenergiesofthetwocoherentconfigurationswith p=1and p=0coincidewhent=t =v . 0 ∗ Our Fig. 4 illustrates the structure of the energy-tension and the tension-elongation relations corresponding to different values of p for the system with N = 5. Notice that in contrast to the case of a hard device, the force- elongationrelationcharacterizingtheglobalminimuminasoftdeviceexhibitsaplateauindicatingadiscontinuityin elongationasthecrosslinkersswitchcollectivelyatt=t fromunfoldedtofoldedconformation. ∗ Notice that the plateau replaces the region of negative stiffness detected in the hard device case and the force- elongationrelationbecomesmonotone. Thisshowsthateveninthecontinuumlimitthestable“material”responses of our system in hard and soft devices remain different. Such behavior would be completely unexpected from the perspective of classical statistical mechanics, however, it is characteristic of systems with domineering long-range interactions(Dauxoisetal.,2003). 2.3. Energylandscape As we have seen in the previous sections, the globally stable state of the HS system correspond to one of the twocoherentconfigurationscharacterizedby p = 1andby p = 0. Wecannowposethequestionaboutthesizeof theenergybarrierseparatingthesetwoconfigurations. Toaccesstheenergybarriersandtofindthetransitionstates (saddlepointsoftheenergy)westudytheenergydependenceon p. Forgeneralvaluesoftheloadingparametersthis dependencewasfoundtobelinearinaharddevice,indicatingthatthereisnoconventionalbarrier,andconcaveina softdevicewhichmeansthatthereisapotentialenergybarrier. Thisobservationshowsthataswitchfromunfolded tofoldedconfigurationinasoftdevicecarriesanenergeticcostwhileinaharddevicethetransitioniscost-free. a b C+C B+B A+A t 1 0.1 wˆ(p;t ) ∗ B z z − ∗ 1 C A 1 0.05 − D vˆ(p;z ) 1 0 ∗ − 0 0.2 0.4 0.6 0.8 1 p A+C D+D B+D Figure5:(a)EnergylandscapeattheglobalminimumtransitionfortheHSmodel.Solidlines,harddeviceatz=z ;Dashedlines,softdeviceat t =t . DotsrepresenttheenergyofthedifferentconfigurationsforasystemwithN =5;linescorrespondtothel∗imitN . Parametersare vo0ft=he∗1galonbdaNlm=in5im.(ubm)Rteenpsrieosne-netlaotniognatoiofnthceubrveeha(tvhiiocrkolfinaes)yosctceumrrwinigthintwaostcrreossss-flirnekeecrosnwfiigthurva0tio=n0inimapsoofstindgevti∗ce=.0Daanshdezd∗l→i=ne−s∞1,m/2e,ttahsetatbrlaenssittaitoens p=0andp=1.Theintermediatestress-freeconfigurationisobtainedeitherbymixingthetwogeometricallycompatiblestatesBandDinahard devicewhichresultsinaB+DstructurewithoutadditionalinternalstressorbymixingthetwogeometricallyincompatiblestatesAandCinasoft devicewhichresultsinaA+Cstructurewithinternalresidualstress. The(collective)transitiontakesplaceatz=z inaharddeviceandt=t inasoftdevice. Thedifferencebetween ∗ ∗ thecorrespondingenergylandscapesatthethresholdvaluesofparametersisillustratedinFig.5(a). Fortheeaseof comparison the energy minima are shifted to zero in both loading conditions. Each black dot represents the energy levelofaparticularmetastablestateandthedottedlinerepresentsthesetofsuchmetastablestatesinthecontinuum limit N when p becomes a continuous variable. The barriers separating individual metastable states are not → ∞ defined in the spin model. A simple way to recover microscopic barriers by switching from hard to soft spins is discussedinSec.3;foranotherregularizationapproachsee(BenichouandGivli,2013). 7 Tounderstandtheoriginofthispeculiarbehavioroftheenergylandscapeitisinstructivetoconsidertheminimal HSsystemwith N = 2; seeFig.5(b). Hereforsimplicityweassumedthatv = 0implyingt = 0andz = 1/2. 0 ∗ ∗ − Thetwo“pure”configurationsarelabeledasA(p=0)andC (p=1)att =t =0andasD(p=0)andB(p=1)at ∗ z=z = 1/2.Inaharddevice,wherethetwoelementsdonotinteract,thetransitionfromstateDtostateBatagiven ∗ − z = z goesthroughtheconfiguration B+Dwhichhasthesameenergyasconfigurations Dand B: thecrosslinkers ∗ infoldedandunfoldedstatesaregeometricallyperfectlycompatibleandtheirmixingrequiresnoadditionalenergy. Instead,inasoftdevice,whereindividualelementsinteract,atransitionfromstateAtostateCtakingplaceatagiven t = 0requirespassingthroughthetransitionstateA+C whichhasanonzeroresidualstress. Individualcrosslinkers inthismixturestatehavedifferentvaluesofzandthereforetheenergyofthestressed“mixed”configurationA+Cis largerthantheenergiesofthe“pure”unstressedstates AandC. Weconcludethatthemacroscopicbarrierinasoft deviceishigherthaninaharddevicebecauseinasoftdeviceatransitionisagenuinelycooperativeeffectrequiring essentialinteractionofindividualelementswhileinaharddevicetheconformationalchangeindifferentunitstakes placeindependently. ·10−2 1 ·10−2 A F e vic 40 0.5 C 40 de vˆ 0 D d 20 20 ar 0.5 H 0 B − 0 E 0 0.5 1 0 0.5 1 0 0.5 1 p p p Softdevice 30 ·10−2 P 20 wˆ F 10 0 Q cbarrieroftdevice t+ B Q D P 10 ·10−2 copinas t∗ S E wˆ 5 Macrosdomaini t− C R 300 ·E10−2 B A 20 R wˆ 10 0 S z 0 0.5 1 p ∗ Figure6: SummaryofthebehavioroftheHSmodel. Themainframeshowsthetension-elongationrelationscorrespondingtop=0,0.1,...,1 forasystemwithN =10(graylines)andthetension-elongationrelationintheglobalminimum(thicklines). Thehorizontaldottedlinesshow thelimitofthedomainwhereamacroscopicenergybarrierispresentinasoftdevice.Thesatelliteframes(above,harddevice;right,softdevice) showtheenergybarrierscorrespondingtovarioustransitions(A B,C D,...)shownbydashedlinesinthemainframe. → → Forgeneralvaluesoftheloadingparameters,weconsidertheN limitwheretheenergylandscapebecomes →∞ smoothandthebarrierislocatedatthesaddlepoint p = p ,where p = t v +1/2isthesolutionof∂wˆ/∂p = 0. ∗ ∗ − 0 Since0 p 1,weobtainthatthemacroscopicbarrierandtheensuingcooperativeeffectsexistfort t t with + ≤ ∗ ≤ − ≤ ≤ t =t 1/2andt =t +1/2,inthesoftdevicesetting. Noticethattheboundariesofthisintervalcorrespondtothe + th−resh∗o−ldtˆ(0;z )andtˆ(1∗;z )delimitingtheregionwithnegativestiffnessintheharddevicecase,seeFig.3. ∗ ∗ We summarize the results obtained within the HS model in Fig. 6. In the main frame, we show the tension- elongationrelationscorrespondingto p = 0,0.1,...,1,forasystemwith N = 10(graylines). Thicklinesshowthe tension-elongationrelationsintheglobalminimumcharacterizedbyatransitionlocatedatz inaharddeviceandt ∗ ∗ inasoftdevice. Thehorizontaldottedlinesshowthelimitsofthedomainwhereamacroscopicbarrierispresentina 8 softdevice. Onthesatelliteframes(top-foraharddeviceandright-forasoftdevice)weshowtheenergybarriers betweenthehomogenousconfigurationsinthecaseN (solidlines)andfordiscretevaluesofp(dots)atdifferent →∞ valuesoftheloading. To conclude, we have shown that the HS model exhibits different mechanical responses in a hard and a soft devices even though in both cases the global minimum of the energy is achieved for homogenous configurations This behavior originates from the presence of long-range interactions between the crosslinkers. These interactions introduceanelasticfeedbackinthesoftdevicecasewhichcreatesamacroscopicenergybarrierforthecooperative folding-unfoldingprocess. 3. Thesnap-springmodel Theminimaldescriptionofunfoldingtransitionprovidedbythespinmodelhasitslimitations. Forinstance,the descriptionofthebarriersintheHSmodelisincompletebecauseitdoesnottakeintoaccountthemicroscopicenergy barriersbetweenthestateswithdifferentvaluesoftheorderparameter p. Anotherproblemisthatintheharddevice settingthespinmodelisdegeneratebecauseconfigurationswithdifferentvaluesoftheorderparameterareequivalent. a b uSS(x) uSS k elastic backbone k0 kb T T N k1 v 0 y a l 0 x − z Figure7:Snap-springmodelofacluster.(a)Dimensionalenergylandscapeofabistablecrosslinker.(b)Structureofaparallelbundlecontaining Ncrosslinkersinasoftdevice. To deal with these problems we regularize the spin model by introducing two additional physical mechanisms. First,following(MarcucciandTruskinovsky,2010)wereplacehardspinsbysnap-springsalsoknownassoftspins, sothat xbecomesacontinuousvariable. Forsimplicityweassumethatthecorrespondingdouble-wellpotentialcan berepresentedasaminimumoftwoparabolas,seeFig.7(a). Byusingnon-dimensionalvariableswecanthenwrite  uSS(x)=121kk0((xx)+2+1)v20 iiffxx>ll 2 1 ≤ Herelisthedimensionlesspositionoftheenergybarrier,v isthedimensionlessenergybiasoftheunfoldedstateand 0 k andk ,aredimensionlesselasticmoduliofthefoldedandunfoldedstates,respectively. Interestingly,acomparison 1 0 withthereconstructedpotentialsforunfoldingbiologicalmacro-moleculesshowsthatthisapproximationmaybein factverygood(Guptaetal.,2011). Aspinodalregioncanbeobviouslyaddedtothepotentialofthebi-stableunit, however,inthiscasewelosetransparencywithoutgainingessentialeffects. The important observation is that while in the snap-spring model the bottoms of the energy wells remain the sameasinthespinmodel, thebarrierseparatingthetwoconformationalstatesisnowwelldefined, seeFig.7. The mechanicalresponseofasinglecrosslinkerwithanattachedseriesspringisgovernedbythedimensionlessenergy 1 u=u (x)+ (y x)2, SS 2 − whereyisthetotalelongation.Thesolutionsoftheequilibriumequationu (x)=y xareshowninFig 8.Noticethe (cid:48)SS − multi-valuednessoftherelationlinkingthevariablesxandyandtheappearanceofthespinodalbranchx connecting ∗ thetwostablebranchesx (y)andx (y). 1 0 9 a b 1 0 x0 B1 A 0.5 t1 x B2 x 0.5 ∗ t 0 t − ∗ A x1 t0 0.5 1 B2 − B1 − 1 1.5 1 0.5 0 0.5 − 1.5 1 0.5 0 0.5 − − − − − − y y Figure8: Behaviorofasinglecrosslinkerinthesnap-springmodel. (a)Equilibriumpositionsforvariousy;(b)Correspondingtensionlevels. Solidlines,metastablestates;dashedlines,unstablestate;boldline,globalminimum.Arrowsindicatetheresponsetoasuddenshorteningwithan elasticunloadingintheunfoldedstate(A→B1)followedbytheconformationalchangetothefoldedstate(B1→B2).Parametersare,λ1=0.4, λ0=0.7,l=−0.3 The mechanical independence of the crosslinkers in a hard device disappears if we take into account the finite stiffnessofbackbonewhich,inthecaseofskeletalmuscle,correspondstothecombinedstiffnessofactinandmyosin filaments(Wakabayashietal.,1994;Huxleyetal.,1994;Fordetal.,1981;Mijailovichetal.,1996;deGennes,2001). Following(Ju¨licherandProst,1995),weusealumpdescriptionofbackboneelasticitybyintroducinganadditional elastic spring with stiffness λ = k /(Nk) and the energy u (x) = Nλ x2/2. If we attach this spring in series to our b b b b parallelbundleofcrosslinkers,seeFig.7(b),wecanwritethetotalenergyofthesystempercrosslinkerintheform 1 (cid:88)N (cid:34) 1 λ (cid:35) v(x,y;z)= u (x)+ (y x)2+ b(z y)2 . (10) N SS i 2 − i 2 − i=1 In the hard device case z is the control parameter, x are the continuous microscopic internal variables generalizing i thespinvariablesintheHSmodelandyisanewcontinuousmesoscopicinternalvariable. Noticethatnowevenin aharddevicetheindividualcrosslinkersarenotindependent;theimplicitmean-fieldinteractionbecomesobviousif (cid:12) (cid:12) thevariableyisadiabaticallyeliminated(minimizedout)bysolving ∂∂vy(cid:12)(cid:12)z,x =0. Weobtain   y(x;z)= 1+1λ λbz+ N1 (cid:88)N xi, (11) b i=1 whichafterinsertingintoEq.10givesthepartiallyequilibratedenergy v˜(x;z)= N1 (cid:88)uSS(xi)+ x2i2 − (1λ+bzλ )xi+ 2(1λb+z2λ )− 2(1+1 λ )N1 (cid:88)xi2. i b b b i (cid:80) Observe that the quadratic term in x vanishes when the elasticity of the backbone becomes infinite (λ ) i b → ∞ showingthatinthislimitthelong-rangeinteractionsdisappear. Theresultingsnap-springmodelcanbeviewedasaregularizationoftheHSmodel. TorecovertheHSmodelin aharddevicecasefromEq.10,weneedtoperformthedoublelimit: k andλ . Thefirstoftheselimits 1,0 →∞ b →∞ ensuresthatxbecomesaspinvariablewhilethesecondguaranteesthaty=z. ToobtaintheHSmodelinasoftdevice weneedtoconsiderthetripleasymptotics: k ,λ 0andz wherethelasttwolimitsmustbelinkedin 1,0 →∞ b → →∞ thesensethatλ z twhichensuresthattheforcepercrosslinkertremainsfinite. b → Inasoftdevice,thetotalenergypercrosslinkerinthesnap-springmodelcanbewrittenas 1 (cid:88)N (cid:34) 1 λ (cid:35) w(x,y,z;t)=v(x,y,z) tz= u (x)+ (y x)2+ b(z y)2 tz , − N SS i 2 − i 2 − − i=1 wheret=T/N isagaintheappliedforcepercrosslinker. 10