Table Of Content

Recovery of free energy branches in single molecule experiments Ivan Junier1, Alessandro Mossa2, Maria Manosas3, and Felix Ritort2,4 ∗ 1Programme d’E´pigénomique, 523 Terrasses de l’Agora, 91034 E´vry, France 2Departament de F´ısica Fonamental, Facultat de F´ısica, Universitat de Barcelona, Diagonal 647, 08028 Barcelona, Spain 3Laboratoire de Physique Statistique, Ećole Normale Supérieure, 24 rue Lhomond, 75231 Paris, France 4 CIBER-BBN, Networking Centre on Bioengineering, Biomaterials and Nanomedicine We present a method for determining the free energy of coexisting states from irreversible work measurements. Ourapproachisbasedonafluctuationrelationthatisvalidfordissipativetransfor- mations in partially equilibrated systems. To illustrate the validity and usefulness of the approach, we use optical tweezers to determine the free energy branches of the native and unfolded states of a two-state molecule as a function of the pulling control parameter. We determine, within 0.6 k T B accuracy,thetransitionpointwherethefreeenergiesofthenativeandtheunfoldedstatesareequal. PACSnumbers: 05.70.Ln,87.80.Nj,82.37.Rs 9 0 Recentdevelopmentsinstatisticalphysics[1]havepro- Boltzmann constant and T the temperature of the envi- 0 vided new methods to extract equilibrium free energy ronment. In this relation, ˆ is the time reversal of , F F 2 differences in small systems from measurements of the whiletheaverages aretakenovertheensembleof F(R) (cid:104)•(cid:105) n mechanical work in irreversible processes (see [2, 3] for allpossibleforward(reverse)trajectories. Theparticular a reviews). In this regard, fluctuation relations [3] are case = δ(W W(Γ)) yields a relation between work J F − generic identities that establish symmetry properties for distributions along the forward and reverse processes, 5 the probability of exchanging a given amount of en- P (W) = P ( W)exp[β(W ∆G)]. This relation has F R 2 − − ergy between the system and its environment along ir- been experimentally tested and used to extract free en- reversible processes. If a system, initially in thermody- ergydifferencesinsinglemoleculeexperiments[5,6,7]. A ] t namic equilibrium, is strongly perturbed by fast varying thorough discussion on its validity domain can be found f o acontrolparameterλbetweentwovaluesλ andλ ,then in [4] 0 1 s thesystemisdrivenoutofequilibrium. Theworkexerted Fluctuation relation under partial equilibrium con- . at upon the system, averaged over the ensemble of all pos- ditions. By considering only the trajectories that go m sible trajectories, reads W = λ1 (∂ /∂λ)dλ where fromonespecificsubsetofconfigurationstoanotherone, (cid:104) (cid:105) (cid:104) λ0 H (cid:105) - is the system Hamiltonian. According to the second Maragakis et al. [8] have derived another relation use- d H (cid:82) law of thermodynamics, W is always greater than the ful to extract free energy differences between subsets of n (cid:104) (cid:105) freeenergydifferencebetweentheinitialandfinalstates, states. Inprinciple,thevalidityofEq.(1)isrestrictedto o c ∆G = G(λ1) G(λ0). The Crooks fluctuation relation initialconditionsthatareGibbsianoverthewholephase − [ [4] extends the predictive power of the Second Law by space (what we might call global thermodynamic equi- S establishing a symmetry relation for arbitrary function- librium). It is, however, possible to extend Eq. (1) to 1 v als of a trajectory Γ measured along a nonequilibrium thecasewheretheinitialstateisGibbsianbutrestricted 6 process (forward or F process) and its time reversed one overasubsetofconfigurations(whatwemightcallpartial 8 (reverseorRprocess). Intheforwardprocessthesystem thermodynamicequilibrium). Arelationmathematically 8 starts in equilibrium at λ and λ is varied from λ to λ similar to Eq. (1) can be derived, but involving nonequi- 3 0 0 1 foratimet accordingtoanarbitraryprotocolλ(t)(i.e., librium processes that are in partial (rather than global) . f 1 λ = λ(t )). In the reverse process the system starts in equilibrium. Itisusefultorephraseherethederivationin 1 f 0 equilibrium at λ and λ is varied from λ to λ following such a way to emphasize the role played by partial equi- 9 1 1 0 the time reversed scheme, given by λ(t t). In its most librium. As we will see this makes it possible to exper- 0 f − : general form the Crooks fluctuation relation reads [4] imentally determine the free energy of coexisting states v for values of λ such that the system is never globally i X exp[ β(W ∆G)] F = ˆ R, (1) equilibrated. (cid:104)F − − (cid:105) (cid:104)F(cid:105) r Let Peq ( ) denote the partially equilibrated (i.e., a where standsforanarbitraryfunctionaloftheforward Boltzmanλ,nS–(cid:48)GCibbs) distribution for a given value of λ. F trajectories the system can take through phase space, β Such distribution is restricted over a subset of con- (cid:48) is the inverse of the thermal energy k T where k is the S B B figurations contained in (i.e., (cid:48) ). C S C ∈ S ⊆ S The case = corresponds to global equilibrium: (cid:48) PEQ( ) SPeq (S). Partially equilibrated states sat- λ C ≡ λ, C isfy Peq ( ) =SPEQ( )χ ( ) / , where χ is ∗ET-moawilh:ormitocrotr@reffsnp.ounbd.eesn.ceshouldbeaddressed. the chλa,Sra(cid:48)cCteristicλfunCctioSn(cid:48) dCeZfinλe,Sd Zovλe,Sr(cid:48)the subseSt(cid:48) (cid:48) S 2 (isχSth(cid:48)(eCp)a=rtit1ioinf fCun∈ctiSo(cid:48)narnedstrziecrtoedottohetrhweisseu)b,saetnd(cid:48)Z, λi.,eS.(cid:48), Ferneeergy (a) Force N U S = exp( βE ( )), with E ( ) the energy fZuλn,cSt(cid:48)ion of Ct∈hSe(cid:48)syste−m foλrCa given λ aλndC . Given a (cid:80) C forward trajectory Γ, going from configuration 0 when N U C (b) λ =oveλr0wthoicCh1 tfhoer λsys=temλ1,isleptarSt0ia(llSy1)eqbueilitbhreatseudbsaettλof Reaction Coordinate x0 x1 0 S (λ1). Consider now the following transformation of the x1 functional in Eq. (1): (Γ) χ ( ) (Γ)χ ( ). F F → S0 C0 F S1 C1 (c) Under previous conditions the following identity can be proved (see Supp. Mat.): x0 F o (cid:104)Fexp[−β(W −∆GSS10,,λλ10)](cid:105)SF0→S1 = pSR0←S1 , (2) rwa t (cid:104)Fˆ(cid:105)SR0←S1 pSF0→S1 rd (forward) (forward) R where the average S0→S1(S0←S1) is now restricted to ev (cid:104)•(cid:105)F(R) e forward(reverse)trajectoriesthatstartinpartiallyequi- r s librated state 0 ( 1) at λ0 (λ1) and end in 1 ( 0) at e t (reverse) (reverse) λ1 (λ0). pSF0→SS1 (pSSR0←S1) stands for the probSabilSity to : Current state of the molecule (N or U) be in ( ) at the end of the forward (reverse) pro- 1 0 S S cisestshdeefifrneeedeanbeorvgey, daniffder∆enGceSS10,,bλλe10tw=eeGnS1p(aλr1t)ia−llyGSeq0(uλil0i)- FScIhGe.m1a:ticAptiwctou-rsetaotfetmheolferceueleenienrgaylpaunldlisncgapeexpienriamtewnot-.sta(ate) folder as a function of the reaction coordinate. The blue and brated states and . In the following, we will drop 0 1 S S red colors represent the subsets of configurations that define the subscript (F, R), leaving the direction of the arrow theN andU statesrespectively. (b) Dependingon thevalue to distinguish forward from reverse. Moreover, we will ofthecontrolparameterx,theshapeofthefreeenergyland- adopt the shorthand notation S1 pS0→S1/pS0←S1. If scape is tilted toward one state (either N or U). For each =1 we obtain a generalizatioPnS0of≡the Jarzynski equal- value of x, the Boltzmann–Gibbs equilibrium value of the pFitayr,tPicSSu01la(cid:104)erxcpa[s−eβ(W=−δ(W∆GSS10W,,λλ10()Γ])(cid:105))=, w1e.gWethtehreeareslafotironthe fuonrfcoeldreedst(rriecdte)dfotrocee-dacishtasntcaetebrdaenficnheess.th(ce)nDautrivineg(ablruaem)painndg F − protocol, x is changed at a constant pulling speed from x 0 PSS01PPSS00←→SS11((−WW)) =exp[β(W −∆GSS10,,λλ10)], (3) cotoardnxe1rvits(oiftrmobmeoatxshu1rsettaoGtexUs0(axisn)i(tnahdlelicfrraeetveeedernsbeeyrcgtaihessee)acraoenlodcroemtdhpecuirmtcelodelsew.cuitIlnhe respect to G (x )) at a given value of x (gray area), only which has been used in [8] in the case of global equilib- N 0 forwardandreversetrajectoriesthatareinN atx andinU 0 rium initial conditions. atxhavetobeconsidered(whichistrueonlyforthereverse Experimentaltest. HerewetestthevalidityofEq.(3) trajectory in the example shown in the picture). by performing single molecule experiments using optical tweezers. Let us consider an experiment where force is appliedtotheendsofaDNAhairpinthatunfolds/refolds at x to U ( ) at x and ii) the fraction of reverse 0 1 ≡ S in a two-state manner. The conformation of the hairpin trajectories (decreasing extension) that go from U at x can be characterized by two states, the unfolded state toN atx ,wecandetermine U. Thenbymeasuringthe 0 PN (U)andthenativeorfoldedstate(N)—seeFig.1. The corresponding work values for each of these trajectories, thermodynamic state of the molecule can be controlled we can use Eq. (3) to estimate the free energy of the by moving the position of the optical trap relative to unfolded branch G (x) as a function of x. By repeating U a pipette (Fig. 2, upper panel). The relative position of the same operation with N instead of U, the free energy thetrapalongthex-axisdefinesthecontrolparameterin of the folded branch G (x) can be measured as well. N ourexperiments,λ=x. Dependingonthevalueofxthe Note that we adopt the convention of measuring all free molecule switches between the two states according to a energies with respect to the free energy G (x ) of the N 0 rate that is a function of the instantaneous force applied native state at x . We are also able to compute the free 0 to the molecule [9]. In a ramping protocol the value of energy difference between the two branches, ∆GU(x) = N x is changed at constant pulling speed from an initial G (x) G (x). U N − value x (where the molecule is always folded) to a final We have pulled a 20 bps DNA hairpin using a minia- 0 valuex (wherethemoleculeisalwaysunfolded)andthe turized dual-beam laser optical tweezers apparatus [10]. 1 force f (measured by the optical trap) versus distance x Moleculeshavebeenpulledattwolowpullingspeeds(40 curvesrecorded. Bycomputingi)thefractionofforward and 50 nm/s) and two fast pulling speeds (300 and 400 trajectories (i.e., increasing x) that go from N ( ) nm/s), corresponding to average loading rates ranging 0 ≡ S 3 Trap Hairpin 1 Pipette 0 Handles -1 400 nm/s -2 300 nm/s Fast Unf. 50 nm/s 0.6 15 Fast Ref. 40 nm/s Slow Unf. -3 0.5 f [pN]111234 Slow Ref. 183.5 184 184.5Work 1[8k5B*T] 185.5 186 186.5 0.4 187.5 400 nm/s 11 300 nm/s 50 nm/s 0 20 40 60 80 100 187 40 nm/s 0.3 x [nm] T]186.5 0.2 kB* 186 z [185.5 0.1 185 184.5 0 176 178 180 182 184 186 182 184 186 188 190 Work [kB*T] u [kB*T] FIG. 2: (Upper panel) Experimental setup. (Lower FIG. 3: Experimental verification of the fluctuation re- pPaNn←elU)(UnWfol)diantgxan=dxr¯ef=old6i1n.g75wnormkdmisetarsiubruetdionatsPtwNo→pUu(lWlin)g, llaotgi(oPnN→EUq.(W(3)/).PN←(UU(ppWer))paasneal)fuPnclottionofofloWgP(NUin(x¯k)B+T − − speeds: 300 nm/s (red -unfolding- and green -refolding-) and units) at different pulling speeds. A least squares fitting 40 nm/s (blue -unfolding- and orange -refolding-). In the in- method of the experimental data to straight lines gives the set we show two force-distance cycles corresponding to low following slopes: 0.91(4) (400 nm/s), 1.07(8) (300 nm/s), (40nm/s)andfast(300nm/s)pullingspeeds. Thegrayarea 1.02(6) (50 nm/s), 1.07(3) (40 nm/s). (Lower panel) Ben- indicatesthemechanicalwork(=Rx0fdx)exertedalongthe nett’s acceptance ratio method. We plot the function z(u) x¯ green refolding trajectory. Statistics (number of molecules, Eq. (4) for different pulling speeds. The thick black line cor- totalnumberofunfolding/refoldingcycles): 40nm/s(2,223), respondstothefunctionz=u. Errorbarscorrespondtothe 50nm/s(2,183),300nm/s(3,337),400nm/s(1,551). More standarddeviationoftheBennettestimate,asdeterminedin detailsaboutthedataanalysisprocedurecanbefoundinthe Ref.[11]. DatastatisticsarereportedinthecaptionofFig.2. Supp. Mat. As a direct test of the validity of Eq. (3), in the upper panel of Fig. 3 we plot the quantity between2.6and26pN/s,fromx =0tox =110.26nm 0 1 log(PN U(W)/PN U( W))+log U(x¯)againstW (in (for convenience we take the initial value of the relative → ← − PN k T units). As expected, all data fall into straight lines distance trap-pipette equal to 0). A few representative B of slope close to 1. The intersections of such lines with force-distance curves are shown in Fig. 2 (inset of lower theW-axisprovideanestimateofG (x¯). Notethatboth panel). Wehavethenselectedavaluex=x¯=61.75nm, U fastandslowpullingspeedsinterceptthehorizontalaxis close to the expected coexistence value of x where both around the same value within 0.75 k T of error. With N and U states have the same free energy (see below). B this method, we estimate G (x¯) = 185.9(5) k T. Note Such value of x is chosen in order to have good statistics U B that this free energy estimate, as all the others in this for the evaluation of the unfolding and refolding work paper, refers to the whole system, comprising hairpin, distributions. The system is out of equilibrium at the handles, and trap. fourpullingspeeds. Toextractthefreeenergyoftheun- folded branch G (x¯), we have measured the work values A more accurate test of the validity of Eq. (3) and a U WN U = x¯ fdx, WN U = x0fdx along the unfold- better estimation [11] of the free energy GU(x¯) can be → x0 ← x¯ obtained through the Bennett acceptance ratio method ing and refolding trajectories, respectively, and then de- terminedth(cid:82)edistributionsPN(cid:82)U(W)andPN U( W). [12]. In Bennett’s method we define the following func- → ← − tions: In the main panel of Fig. 2 we show the work distributions obtained for a slow and fast pulling process. Note that the support of the unfolding work distributions is 1 z (u) = log , bounded by the maximum amount of work that can be UNF (cid:42)1+ nnURNEFF exp[β(W −u)](cid:43)UNF exerted on a molecule between x and x¯. This bound 0 exp( W) corresponds to the work of those unfolding trajectories z (u) = log − . that have never unfolded before reaching x¯. REF (cid:42)1+ nnURNEFF exp[−β(W +u)](cid:43)REF 4 260 go through N (or U) (ensuring that we get a reasonable 356 statistical significance). The results of the reconstruc- 354 tionofthefreeenergybranchesareshowninFig.4. The 240 352 430000 nnmm//ss two free energy branches cross each other at a coexis- 350 50 nm/s 40 nm/s tence value x at which the two states (N and U) are c 0 20 40 60 80 100 equally probable. This is defined by ∆GU(x ) = 0. We N c T]220 get x 61.28 nm, which is in good agreement with an- B* 188 c ≈ gy G [k 111888567 oexthpeerrimesetinmtaalted,atxaca=cc6o2rd.0in(7g)tnoma,soimbtpalienepdheinnotemrpenreotliongg- ner200184 ical model (see Supp. Mat.). In the insets of Fig. 4 e e 183 we zoom the crossing region. It is interesting to recall e 182 Fr 60.5 61 61.5 62 62.5 188 again the importance of the aforementioned correction 180 term( U)toEq.(3). Ifsuchtermisnotincludedinthe 187 PN 186 analysisthenthetworeconstructedbranchesnevercross 185 (bottom right inset). This result is incompatible with 184 160 183 the existence of the unfolding/refolding transition in the FUonlfdoeldde bdr abnracnhch 182 60.5 61 61.5 62 62.5 hairpin, showing that the factor PNU is key to measure free energy branches. 50 55 60 65 70 x [nm] Equation (2) is valid in the very general situation of partially equilibrated initial states which, however, are FIG.4: Free energy of the folded and unfolded branches. The arbitrarily far from global equilibrium. This makes the purplelineisGN(x),thedarkgreenlineisGU(x),bothcom- particular case Eq. (3) a very useful identity to recover puted with respect to G (x ). The black solid line and the N 0 the free energy of states that cannot be observed in con- gray area mark the expected value x = 62.0(7) nm and its c ditions of thermodynamic global equilibrium. We have standard error, while the dashed black line is the crossing shown how it is possible to apply Eq. (3) to recover free point x 61.28 nm. Both left and right insets show a mag- c nified im≈age of the coexistence region. In the right one, the energy differences of thermodynamic branches of folded dashedlinesrepresentthefreeenergiesthatwewouldobtain and unfolded states in a two-state DNA hairpin. These itfhewefuhnacdtioonvser∆loGokUNe(dx)th−e∆faGctUUo(rx)PNUan.dT∆hGeNNto(px)p−an∆elGsNUh(oxw)s, meneetrhgoiedssocfannobne-nfautritvheersteaxtteesnsduecdhtaostmheisrfoecldoevderoyroifnftreere- which,bydefinition,shouldbeindependentofxandequalto mediates states. G (x ). Again,thedashedlinesrepresenttheresultifwedo noUt in1clude the factor U. We are grateful to M. Palassini for a careful reading PN of the manuscript. We acknowledge financial support from grants FIS2007-61433, NAN2004-9348, SGR05- FromEq.(3)ithasbeenproved[11,12]thatthesolution 00688 (A.M, F.R). of the equation z(u)=u, where z(u) z (u) z (u) log U(x¯), (4) ≡ REF − UNF − PN [1] C. Jarzynski, Phys. Rev. Lett. 78, 2690 (1997). is the optimal (minimal variance) estimate of G (x¯). U [2] F. Ritort, Adv. Chem. Phys. 137, 31 (2008). In Fig. 3 (lower panel) we plot the function z(u) [3] J. Kurchan, J. Stat. Mech. (2007) P07005. for different pulling speeds. It is quite clear that the [4] G. E. Crooks, Phys. Rev. E 61, 2361 (2000). functions z(u) are approximately constant along the u [5] G. Hummer and A. Szabo, Proc. Nat. Acad. Sci (USA) axis and cross the line z = u around the same value 98, 3658 (2001). G (x¯) = 186.0(3). A distinctive aspect of Eq. (3) is the [6] D. Collin et al., Nature (London) 437, 231 (2005). U presence of the factor U. If such correction was not [7] A.Imparato,F.SbranaandM.Vassalli,Europhys.Lett. PN 82 58006 (2008). taken into account then the fluctuation relation would [8] P. Maragakis, M. Spichty, and M. Karplus, J. Phys. not be satisfied anymore. We have verified that if PU is N Chem. B 112, 6168 (2008). notincludedintheanalysis,thentheBennettacceptance [9] E. Evans and K. Ritchie, Biophys. J. 72, 1541 (1997). ratio method gives free energy estimates that depend on [10] The DNA sequence is 5’-GCGAGCCATAATCTC the pulling speed (see Supp. Mat.). ATCTGGAAACAGATGAGATTATGGCTCGC-3’. Free energy branches. After having verified that the Pulling experiments were performed at ambient tem- fluctuationrelationEq.(3)holdsandthatitcanbeused perature (24◦C) in a buffer containing Tris H-Cl pH 7.5, 1M EDTA and 1M NaCl. The DNA hairpin was to extract the free energy of the unfolded (G (x¯)) and U hybridized to two dsDNA handles 29 base pairs long folded (G (x¯), data not shown) branches, we have re- N flanking the hairpin at both sides. Experiments were peated the same procedure in a wide range of x values. done in a dual-beam miniaturized optical tweezers with Therangeofvaluesofxissuchthatatleast8trajectories fiber-coupled diode-lasers (845 nm wavelength) that 5 produce a piezo controlled movable optical trap and measure force using conservation of light momentum. Theexperimentalsetupisbasedon: C.Bustamanteand S. B. Smith, Nov. 7, 2006. U.S. Patent 7,133,132,B2. [11] M.R.Shirts,E.Bair,G.Hooker,andV.S.Pande, Phys. Rev. Lett. 91, 140601 (2003). [12] C. H. Bennett, J. Comp. Phys. 22, 245 (1976). 6 Supplementary material: Recovery of free energy branches in single molecule experiments Ivan Junier1, Alessandro Mossa2, Maria Manosas3, and Felix Ritort2,4 1Programme d’E´pigénomique, 523 Terrasses de l’Agora, 91034 E´vry, France 2Departament de F´ısica Fonamental, Facultat de F´ısica, Universitat de Barcelona, Diagonal 647, 08028 Barcelona, Spain 3Laboratoire de Physique Statistique, Ećole Normale Supérieure, 24 rue Lhomond, 75231 Paris, France 4 CIBER-BBN, Networking Centre on Bioengineering, Biomaterials and Nanomedicine I. CROOKS RELATION IN PARTIAL THERMODYNAMIC EQUILIBRIUM In this section, we provide a full derivation of the relation (2) that has been introduced in the main article: !Fexp[−β(W −∆GSS10,,λλ10)]#S0→S1 = pS0←S1 . (1) !Fˆ#S0←S1 pS0→S1 wheretheaverages S0→S1(S0←S1) aretakenoverallforward(reverse)trajectoriesthatstartinpartiallyequilibrated !•# subset of state ( ) at λ (λ ) and end in ( ) at λ (λ ). Because initial conditions of the forward and 0 1 0 1 1 0 1 0 S S S S rGeve(rλse)proGcess(eλs a)r)ebteatkweeneninsupbasretitasloefqsutialitbersiuinmf,aErqfr.o(m1)galollboawlsetqouicliobmripuumtectohnediftrieoensen(esergeymdaiiffnerteenxct)e.s (∆GSS10,,λλ00 = SI1n re1la−tionS(01),0 stands for an arbitrary functional of the forward trajectories the system can take through phase F space, ˆ is the time reversal of , W is the mechanical work exerted upon the system along the forward trajectories, F F and pS0→S1 (pS0←S1) stands for the probability to be in 1 ( 0) at the end of the forward (reverse) process. In what S S follows, ˆ always refers to the time reversal of . • • Inordertoprovethevalidityofthisrelation,weadoptaframeworksimilartotheoneusedforthederivationofthe generalized Crooks relation [1] (Eq. (1) of the main text). In this regard, we recommend [1] for a thorough discussion about the generality of the derivation. Thus, we consider a system composed of a finite set of microscopic configurations , which evolves according to S C a discrete time Markovian dynamics that verifies the following detailed balance equation: p( ) t t+1 C →C =exp β[E ( ) E ( )] (2) p( ) {− λ(t+1) Ct+1 − λ(t+1) Ct } t+1 t C →C where p stands for the probability transition between two successive configurations in time, that is and here. t t+1 C C E ( )istheenergyof whenthecontrolparameterisequaltoλandthemicroscopicconfigurationis . Thismodels λ C S C the evolution of a system that can exchange energy with a constant temperature heat bath. We define the forward ptrraojbeacbtoilriytyTfCor=t{hCe0t,rCa1j,e.c..t,oCrtyf−1}tsoootchcautrλd(0u)ri=ngλt0haenfdorλw(atfrd)=prλo1c.esPs.(TTCh)e=n byttf=u−0s1inpg(Ctthe→dCett+ai1l)edisbtahleancocrerecsopnodnitdiionng, we get: TC ! PPˆ((TTˆCCˆ)) = !ttttf=f=−−0011pp((CCˆtt →→CCˆtt++11)) = !ttttf=f=−−0011pp((CCtt+→1 →Ct+C1t)) =exp−βt$tf=−01[Eλ(t+1)(Ct+1)−Eλ(t+1)(Ct)] (3)   where we have used!ˆ = . At this po!int, it is useful to remind the definition of the thermodynamic quantities in thisMarkovianproceCsts. DCutfr−intgtheforwardprocess,foragiventrajectory onecandefinethetotalheatexchanged with the thermal bath, Q( ), and the total work exerted upon the systemT,CW( ), as [2]: TC TC tf−1 W( ) = [E ( ) E ( )] λ(t+1) t λ(t) t TC C − C t=0 $ tf−1 Q( ) = [E ( ) E ( )] (4) λ(t+1) t+1 λ(t+1) t TC C − C t=0 $ As a result, eq.(3) can be written as ( )/ˆ(ˆ ) = exp( βQ( )). Moreover by multiplying the equality (3) both ˆ atthedenominatorandatthenumeraPtoTrCbyPtheTCrespective−globaTlCequilibriumprobabilityoftheinitialstate,onegets 7 2 PEQ( ) ( )/PEQ(ˆ)ˆ(ˆ ) =exp(βW( )) / where is the partition function of . Next, a useful wλa0ytoC0rePstrTicCtthiλs1relCa0tioPnTtoCˆpartialequilibTrCiumZλ(1s,eSemZaλi0n,Stext)istZoλi,nStroducethecharacteristicfunSctiondefinedover any subset of : χ ( )=1 if and is equal to zero otherwise, so that Peq ( )=PEQ( )χ ( ) / where Peq S(!) isStheSB!oCltzmann Cwe∈igSht! of restricted to the subset of state λ,S!.CThen,λfroCm rSel!aCtioZnλs,S(3)Zaλn,Sd! (4), we gλe,St:! C C S! ∈S PPλλee10qq,,SS10((CCˆ00))PPˆ((TTˆCCˆ)) = ZZλλ10,,SS10 χχSS01((CCˆ00))eβW(TC) (5) Now, S0→S1 is an average over the forward trajectories that start in partially equilibrated states 0 at λ0 and "•# S end in at λ . Thus, for any functional of these trajectories, we have: 1 1 S F Peq ( ) ( ) χ ( ) Peq ( ) ( ) χ ( ) λ0,S0 C0 P TC • S1 Ctf λ0,S0 C0 P TC • S1 Ctf S0→S1 = !TC = !TC , (6) "•# Pλe0q,S0(C0)P(TC)χS1(Ctf) pS0→S1 !TC where the sum is over all the possible trajectories of the system. In this relation, we have defined pS0→S1 = Peq ( ) ( )χ ( ), that is the probability to be in at the end of the forward process, i.e. starting in λ0,S0 C0 P TC S1 Ctf S1 p!gTeaCtr:tialequilibriuminS0. Applyingtherelation(6)toFexp[−β(W−∆GSS10,,λλ10)]andusingrelation(5),wesuccessively Pλe0q,S0(C0)P(TC)F(TC)exp[−β(W −∆GSS10,,λλ10)]χS1(Ctf) "Fexp[−β(W −∆GSS10,,λλ10)]#S0→S1 = !TC pS0→S1 (7) = !TC Pλe1q,S1(Cˆ1)Pˆ(TˆCˆ)F(TC)exp[β∆GSS10,,λλ10)]ZZλλ10,,SS10χS0(C0) (8) pS0→S1 Peq (ˆ)ˆ(ˆ )ˆ(ˆ )χ (ˆ ) λ1,S1 C1 P TCˆ F TCˆ S0 Ctf ˆ = !TCˆ (9) pS0→S1 Peq (ˆ)ˆ(ˆ )ˆ(ˆ )χ (ˆ ) Peq (ˆ)ˆ(ˆ )χ (ˆ ) λ1,S1 C1 P TCˆ F TCˆ S0 Ctf λ1,S1 C1 P TCˆ S0 Ctf ˆ ˆ = !TCˆ !TCˆ (10) Pλe1q,S1(Cˆ1)Pˆ(TˆCˆ)χS0(Cˆtf) pS0→S1 ˆ !TCˆ = ˆ S1←S0pS1←S0 (11) "F# pS0→S1 wherefrom(7)to(8), inadditiontotherelation(5), wehaveusedχ ( )=χ (ˆ). From(8)to(9), wehaveused S1 Ctf S1 C0 tehxep[fβa∆ctGtSSh01a,,λλt10t)h]ZZeλλs10e,,SSt10of=re1v,eχrsSe0(tCr0a)je=ctoχrSie0s(Cîtsft)h,ethseamdeefiansittiohne soeftthofeftoirmwearrdevterrasjeecotfoFrie,st.hFatroims Fˆ(9()TˆCˆt)o=(10F)(,TwCe),haanvde simply multiplied by the same quantity both the denominator and the numerator. Finally, from (10) to (11) we have used the equivalent of relation (6) for the reverse process plus the definition pS1←S0 = Pλe1q,S1(Cˆ1)Pˆ(TˆCˆ)χS0(Cˆtf) . ˆ This ends the derivation of relation (1). !TCˆ 8 3 400 nm/s 300 nm/s 50 nm/s 3 40 nm/s 2 1 0 183.5 184 184.5 185 185.5 186 Work [kB*T] FIG. 1: Plot of log(PN→U(W)/PN←U( W)) as a function of W (in kBT units). Compare to the upper panel of Fig. 3 in the − main text. 185.5 400 nm/s 300 nm/s 50 nm/s 185 40 nm/s 184.5 ] T * 184 B k [ z 183.5 183 182.5 182 184 186 188 190 u [kB*T] FIG.2: Bennett’sacceptanceratiomethod,appliedtotheerroneousfreeenergy∆G˜,obtainedbyimposing U =1inEq.(12). PN This picture may be compared to the lower panel of Fig. 3 in the main text. 9 4 II. FURTHER DETAILS ABOUT THE EXPERIMENTAL DATA A. Effect of the term U on the validity of the fluctuation relation PN One of the goals of our research was to provide experimental evidence of the validity of the equation (labelled (3) in the main text of the Letter) PSS01PPSS00←→SS11((−WW)) =exp[β(W −∆GSS10,,λλ10)]. (12) A consequence of the above relation is that, since the term S1 pS0→S1/pS0←S1 is obviously influenced by the pulling speed, if we ignore it and erroneously apply the usual CPrSo0ok≡s relation PPSS00←→SS11((−WW)) =exp[β(W −∆G˜SS10,,λλ10)], (13) we are bound to find an incorrect “free energy” difference ∆G˜, the value of which depends on the pulling velocity. This dependence is illustrated in Fig. 1 and, even more clearly, in Fig. 2. In Fig. 1 the slopes of the fitting lines strongly deviate from 1 as the pulling speed changes, while Fig. 2 shows how the experimental lines intersect the line z = W at different values that are separated up to 2 k T. More important, data for fast and low pulling speeds B (orange and red lines versus blue and red lines, respectively) are clustered in two distinct regions in the figure. This shows that the free energy difference ∆G˜ systematically changes with the pulling speed. B. Note on the data analysis procedure The work probability density functions (PDF) represented in Fig. 2(b) of the Letter have been calculated by following a histogram-free recipe recently proposed by Berg and Harris [3]. The method, in a nutshell, consists of approximatingtheempiricalcumulativedistributionfunction(ECDF)withadifferentiablefunctionthatincorporates a suitable number of terms of the Fourier expansion of the data. The Kolmogorov test is used to decide when the differencebetweentheECDFanditsanalyticalapproximationmaybeascribedtochance. ThePDFisthenevaluated simply by differentiation, while error bars can be estimated using a jackknife method [4]. This procedure is especially convenient for our data, because the statistics of the forward trajectories such that the hairpin is unfolded at x = x can get so sparse that the traditional, histogram-based method becomes excessively c dependent on the number and origin of the bins. A small downside, on the other hand, is that Berg and Harris’ technique (like the Kolmogorov test it is based upon) is more sensitive in the central region of the distribution than aboutthetails. SinceweusetheanalyticvaluesofthePDFtocomputetheprobabilityratiosusedforFig.3(a)inthe Letter (or Fig. 1 in the appendices), the slight misrepresentation of the extrema of the unfolding work distributions results in the small curvature that can be noticed in those figures. III. DERIVATION OF x FROM A SIMPLIFIED MODEL c Let us consider a pulling experiment where a two-state-like molecule is driven from an initial configuration charac- terizedbythecontrolparameterx x tosomefinalx x ,wherex isthevalueofxforwhichthestatesN and 0 c 1 c c # $ U have same free energy. In this section, a simplified version of the theoretical model described in [5] is introduced, that allows the estimation of x in terms of quantities easily measured in single molecule experiments. c Let us assume that the Thermodynamic Force-Distance Curve (TFDC, i.e., the force-distance curve that we would observeifwewereperformingtheexperimentinequilibriumconditions)isshapedlikeinFig.3,thatis,itjumpsfrom the line f(x)=f +k x to the line f(x)=f +k x ∆f at some coexistence point x=x . This is not the most 0 eff 0 eff c − general case, but is a useful approximation for the experimental setting we worked with. The coordinates of the points in Fig. 3 are therefore A=(x ,f +k x f ), C =(x ,f +k x ), 0 0 eff 0 min c 0 eff c ≡ D =(x ,f +k x ∆f), F =(x ,f +k x ∆f f ). c 0 eff c 1 0 eff 1 max − − ≡ From the experimental data, it is easy to measure directly f , k , and ∆f. In the case of our experiment on the 0 eff 10 5 fmax F C f +"# f c E B f e c c r o F f !"# f c D A f min 0 0 x x !"#x x x +"#x x 0 c c c 1 Distance FIG. 3: Schematic view of the Thermodynamic Force-Distance Curve. two-state hairpin, for instance, we have f = 10.86(1) pN, 0 k = 48.3(5) pN/µm, (14) eff ∆f = 0.92(3) pN, where the uncertainty is estimated as standard error. Moreover, performing many pulling experiments and applying theCrooksrelation[1]totheresultingforwardandreverseworkdistributions,weareabletodeterminethereversible work W , that corresponds to the entire area below the TFDC from x=x to x=x . From the knowledge of the rev 0 1 three parameters (14) and of the reversible work W = 353.3(4) k T, it is possible to determine x as we explain rev B c below. First,weintroducethenotionofcoexistenceforcef ,definedastheforceatthemidpointbetweenC andD. Then, c webaptizeB andE thepointswherethecoexistenceforceisreachedonthefoldedandtheunfoldedline,respectively. In other words, B =(x ∆x/2,f ) , E =(x +∆x/2,f ), c c c c − where ∆f f =f +k x ∆f/2, ∆x= . c 0 eff c − k eff The area (red in Fig. 3) below the TFDC between B and E is what in [5] is called W . The blue area is (f + rip 0 k x )(x x ∆x), while the green area is k (x x ∆x)2/2, hence eff 0 1 0 eff 1 0 − − − − W =W (f +k x )(x x ∆x) k (x x ∆x)2/2. (15) rip rev 0 eff 0 1 0 eff 1 0 − − − − − − On the other hand, it is also true that W =f ∆x. (16) rip c Combining Eqs. (15) and (16), and solving with respect to x , we find c W (x x )(f +k x ) (x x ∆x)2 ∆x rev 1 0 0 eff 0 1 0 x = − − − − + +x , c 0 ∆f − 2∆x 2