FLUCTUATION THEOREMS FOR DISCRETE KINETIC MODELS OF MOLECULAR MOTORS ALESSANDRA FAGGIONATO AND VITTORIA SILVESTRI 7 Abstract. Motivatedbydiscretekineticmodelsfornon–cooperativemolecularmotors 1 onperiodictracks,weconsiderrandomwalks(alsonotMarkov)onquasionedimensional 0 (1d)lattices,obtainedbygluingseveralcopiesofafundamentalgraphinalinearfashion. 2 We show that, for a suitable class of quasi–1d lattices, the large deviation rate function n associated to the position of the walker satisfies a Gallavotti–Cohen symmetry for any a choice of the dynamical parameters defining the stochastic walk. This class includes J the linear model considered in [31]. We also derive fluctuation theorems for the time– 4 integrated cycle currents and discuss how the matrix approach of [31] can be extended 1 toderivetheaboveGallavotti–CohensymmetryforanyMarkovrandomwalkonZwith periodic jump rates. Finally, we review in the present context some large deviation ] h results of [17] and give some specific examples with explicit computations. c Keywords: Semi–Markov process, continuous time random walk, large deviation princi- e m ple, molecular motor, Gallavotti–Cohen symmetry, time–integrated cycle current. - t a t s 1. Introduction . t a Molecular motors are special proteins able to convert chemical energy coming from m ATP–hydrolysis into mechanical work, allowing numerous physiological processes such - d as cargo transport inside the cell, cell division, muscle contraction [23]. They are able n to produce directed transport in an environment in which the fluctuations due to ther- o mal noise are significant, achieving nonetheless an efficiency even higher than the one of c [ macroscopic motors. In addition synthetic molecular motors have been obtained and their improvements are under continuous investigation [19]. 2 v Molecular motors haven been extensively studied both theoretically and experimentally 1 (cf. [24, 29, 37, 38, 40] and references therein). We focus here on the large class of molec- 2 ular motors (e.g. conventional kinesin) which work non–cooperatively and move along 7 1 cytoskeletal filaments [23]. Keeping in mind the polymeric structure of these filaments, 0 two main models have been proposed. In the so called Brownian ratchet model [24, 37] . 1 the dynamics of the molecular motor is given by a one–dimensional diffusion in a spatially 0 periodic potential randomly switching its shape (indeed, along its mechanochemical cycle 7 the molecular motor can be strongly or weakly bound to the filament, thus leading to a 1 : change in the interaction potential). The other paradigm [20, 21, 25, 26, 27, 28, 29, 43], v on which we concentrate here, is given by continuous time random walks (CTRW), along i X with a quasi one dimensional (quasi–1d) lattice obtained by gluing several copies of a r fundamental graph in a linear fashion. CTRWs are thought in the Montroll–Weiss sense a [35], and are also known as semi–Markov processes satisfying the condition of direction– time independence in the physical literature [44], as well Markov renewal processes in the mathematical one [5]. The above fundamental graph used to build a quasi–1d lattice is a finite connected graph G with two marked vertices v and v (see Fig. 1, left). For simplicity we assume that G has no multiple edges or self–loops. The associated quasi–1d lattice is then obtained G 1 2 A.FAGGIONATOANDV.SILVESTRI by gluing several copies of G, identifying the v–vertex of one copy to the v–vertex of the next copy (see Fig. 1, right). Given a vertex v in G and n Z, we write v(n) for the ∈ corresponding vertex in the n–th copy of G in . Since v(n 1) = v(n), to simplify the − G notation we denote such a vertex by v(n) throughout. Each site v(n) corresponds to a spot inthenth monomerofthepolymericfilamenttowhichthemolecularmotorcanbind. The other vertices v(n) describe intermediate conformational states that the molecular motor achieves by conformational transformations, modeled by jumps along edges in . Note G the periodicity of the quasi–1d lattice . G G a b a( 1) b( 1) a(0) b(0) a(1) b(1) − − G v v v(−1) v(0) v(1) v(2) c c( 1) c(0) c(1) − Figure 1. The fundamental graph G with marked vertices v,v (left) the quasi–1d lattice (right). G The evolution of the molecular motor is described by a CTRW (X ) , taking values in t t 0 ≥ the vertex set of the quasi–1d lattice . Once arrived at vertex x, X waits a random time t G withdistributionψ (thatweassumetohavefinitemean)andthenjumpstoaneighboring x vertex y in with probability p(x,y) > 0. We assume that ψ and p(x,y) exhibit the x G same periodicity of . In what follows, we call dynamical characteristics the above data G ψ , p(x,y). x Warning 1.1. In the degenerate case that ψ is a delta measure, e.g. ψ equals δ , the x x 1 above CTRW reduces to the so–called discrete time random walk. We do not restrict to distributions ψ having a probability density w.r.t. the Lebesgue measure, so that ψ can x x be composed by some delta measure as well. We remark that when ψ is the exponential distribution with mean 1/λ(x), then the x resulting CTRW is Markov and its density distribution P (t) := P(X = x) satisfies the x t Fokker–Planck equation d (cid:88) (cid:88) P (t) = r(y,x)P (t) r(x,y)P (t), r(a,b) := λ(a)p(a,b). (1) x y x dt − y y In what follows, we assume that the random walk starts at v(0), i.e. X = v(0). 0 As observed in [43], the above formalism allows us to treat at once several specific examples analyzed in the literature. For example, when the fundamental graph is given by a finite linear chain with N vertices, we recover a CTRW on Z with nearest–neighbor jumps and N–periodic dynamical characteristics [12, 20, 21]. Supported by experimental results, CTRWs on more complex quasi–1d lattices have been considered in the biophysics literature [11, 25] (see Fig. 2 for two examples). v(−1) v(0) v(1) v(2) v(−1) v(0) v(1) v(2) Figure 2. Parallel chains model (left), divided–pathway model (right). FLUCTUATION THEOREMS FOR DISCRETE KINETIC MODELS OF MOLECULAR MOTORS 3 Calling V the set of vertices of the fundamental graph G, for n Z we define the nth ∈ cell as the set of vertices in of the form v(n) with v V v (for example, in Fig. 1 the G ∈ \{ } 0th cell is given by a(0),b(0),c(0),v(0) ). Our aim is to investigate large fluctuations and { } associated symmetries of the cell process (N ) , defined as N = n if X belongs to the t t 0 t t nth cell, i.e. if X = v(n) for some v V v .≥Trivially, the cell process determines the t ∈ \{ } position of the molecular motor along the filament apart from an error of the same order of the monomer size, which is negligible when analyizing velocity, Gaussian fluctuations and large deviations. As shown in [16], the cell process admits a limit velocity v (i.e. N /t v almost lim t lim → surely) and has Gaussian fluctuations. A large deviation principle is proved in [17] (see Section 5 for more details). We call I : R [0,+ ] the associated large deviation → ∞ function: P(N ϑt) e I(ϑ)t, t 1. (2) t − ≈ ∼ (cid:29) In the last decades some general principles, called fluctuation theorems and common to out–of–equilibrium systems, have been formulated and intensively studied first for dy- namical systems and then also for stochastic processes (see for example [2, 4, 8, 9, 13, 18, 30, 36, 41]). For stochastic systems, they often correspond to relations of the form J(ϑ) = J( ϑ) cϑ, or similar, c being a constant and J being the rate function of − − an observable changing sign under time inversion. These last relations are also called Gallavotti–Cohen type symmetries, shortly GC symmetries in what follows. Fluctua- tions theorems have also been investigated for small systems such as molecular motors [1, 15, 17, 31, 32, 33, 34, 40], and GC symmetries (in particular, for the velocity) have been obtained for some special models. In particular, in [31, 32, 34], the authors derive a GC symmetry for the rate function of the velocity of a molecular motor described by a genericMarkovCTRWonZwithnearest–neighborjumpsanddynamicalcharacteristicsof periodicitytwo,whichcorrespondsto(1)withr(a,b)ofthefollowingform: r(a,a 1) = ξ ± ± if a is even and r(a,a 1) = ζ if a is odd, for generic constants ξ ,ζ > 0. This GC ± ± ± ± symmetry for the velocity reads I(ϑ) = I( ϑ) cϑ, ϑ R, (3) − − ∈ I beingtheratefunctionofthecellprocessmodulorescalingbythelengthofmonomersin the polymeric filaments. For the above 2-periodic Markov CTRW it holds c = 1 ln ξ+ζ+. 2 ξ ζ Since the above CTRW with period 2 is a simplified model for the motion of−r−eal molecular motors, a natural question concerns the validity of (3) for a larger class of CTRWs, or even for all possible CTRWs on quasi–1d lattices. For Markov CTRWs we have shown in[17] that(3) is not universal, and infact (3) isonly universalin thesubclass of 1d lattices whose fundamental graph G is (v,v)–minimal in the following sense: there existsauniqueself–avoidingpathγ inGfromv tov. Anexampleof(v,v)–minimalgraphs G is given in Fig. 3. Note that the graphs G associated to the quasi–1d lattices in Fig. 2 are not (v,v)–minimal. We can now recall the characterization provided in [17]: Theorem 1 ([17]). Suppose that X is a Markov CTRW on the quasi–1d lattice , in t G particular it has exponentially distributed waiting times and transition rates r( , ) as in · · (1). Then the following holds: (i) If G is (v,v)–minimal, then the cell process N satisfies the GC symmetry t I(ϑ) = I( ϑ) ∆ϑ, ϑ R, (4) − − ∀ ∈ 4 A.FAGGIONATOANDV.SILVESTRI z1 z2 z3 z4 z5 v v Figure 3. Example of a (v,v)–minimal graph G with γ = (v,z ,...,z ,v). 1 5 where r(z ,z )r(z ,z ) r(z ,z ) 0 1 1 2 n 1 n ∆ = ln ··· − (5) r(z ,z )r(z ,z ) r(z ,z ) 1 0 2 1 n n 1 ··· − and (z ,z ,z ,...,z ,z ), with z = v and z = v, is the unique self–avoiding 0 1 2 n 1 n 0 n path from v to v in G−. (ii) Vice versa, if G is not (v,v)–minimal, then the set of transition rates r( , ) for · · which the GC symmetry (4) holds for some ∆ (which can depend on r( , )) has · · zero Lebesgue measure. It is simple to verify (see Section 6) that the GC symmetry (3) can be satisfied for very special choices of the jump rates when G is not (v,v)–minimal. In this case, due to the above theorem, a small perturbation of these rates typically breaks the GC symmetry. We point out that in [16] the GC symmetry for the LD rate function of the cell process isanalyzedforalargerclassofrandomprocesses, havingasuitableregenerativestructure. Moreover, it has been proved (cf. Theorems 4 and 8 in [16]) that the GC symmetry (3) holds if and only if X and S are independent, where the random time S is defined as (cid:8) S1 1 (cid:9) 1 S := inf t 0 : X v( 1),v(1) . For the Markov random walk on a linear chain this 1 t − ≥ ∈ { independence had been pointed out already in [10] (see Remark 5.3). We also point out the above Theorem 1 is related to the theorem on page 584 of [3] (see also the discussion on cycle currents in Section 4). On the other hand, in the derivation of the equivalence stated in that theorem, some additional arguments are necessary to get the difficult implication. The aim of the present work is the following: (a) extend Theorem 1–(i) to generic CTRWs (i.e. non Markov) and give some sufficient condition assuring the GC symmetry (3)fornon(v,v)–minimalfundamentalgraphs(seeSection2.1),(b)derivefluctuationthe- orems for time–integrated cycle currents in the case of generic CTRWs and (v,v)–minimal fundamental graphs and, as a consequence, recover the GC symmetry (3) independently from [17] (see Section 2.2), (c) extend the matrix approach outlined in [31] to Markov CRTWs on general linear chain models, getting also the GC symmetry (4) (see Section 2.3), (d) give a short presentation of some results of [17] in a less sophisticated language (see Section 5), (e) give specific examples with explicit computations (see Sections 6, 7, 8 and 9). 2. Main results In this section we present our main results, postponing their derivation to the next sections and to the appendixes. FLUCTUATION THEOREMS FOR DISCRETE KINETIC MODELS OF MOLECULAR MOTORS 5 2.1. Extension of Theorem 1–(i) to generic CTRWs. We consider generic CTRWs on , i.e. alsononMarkov. Asafirstresultwegiveasufficientconditionassuringthatthe G GC symmetry (3) holds for some constant ∆ (for a sufficient and necessary condition see Criterion 1 in Appendix A). This condition is trivially satisfied in (v,v)–minimal graphs , thus leading to the extension of Theorem 1–(i) to non Markov CTRWs. G Theorem 2. Consider a generic CTRW (X ) on the quasi–1d lattice with dynamical t t 0 characteristics p(x,y) and ψ . Then the cell pr≥ocess N satisfies the GC sGymmetry (4) for x t some constant ∆ if m 1 m 1 (cid:89)− (cid:89)− p(x ,x ) = e∆ p(x ,x ) (6) i i+1 i+1 i i=0 i=0 for any self–avoiding path (x ,x ,...,x ) from v to v in the fundamental graph G (x = v, 0 1 m 0 x = v). m As a consequence, if G is (v,v)–minimal, then the cell process N satisfies the GC t symmetry (4) where now p(z ,z )p(z ,z ) p(z ,z ) 0 1 1 2 n 1 n ∆ = ln ··· − (7) p(z ,z )p(z ,z ) p(z ,z ) 1 0 2 1 n n 1 ··· − and (z ,z ,z ,...,z ,z ), with z = v and z = v, is the unique self–avoiding path from 0 1 2 n 1 n 0 n v to v in G. − Note that for Markov CTRWs expressions (5) and (7) indeed coincide. The theorem is a immediate consequence of Criterion 1 discussed in Appendix A. Remark 2.1. When considering discrete time RWs (recall Warning 1.1) it is possible to exhibit examples of fundamental graphs G which are not (v,v)–minimal and such that the GC symmetry (3) holds for any choice of the jump probabilities p(x,y). We refer to Section 7 for an example. 2.2. GC symmetries for cycle currents. As next result we show that, for (v,v)– minimalfundamentalgraphs,theGCsymmetry(3)isindeedaspecialcaseofafluctuation theoremforcyclecurrents(seee.g.[2,4,14,15]). Asaconsequencewegive, amongothers, an alternative derivation of (3) for (v,v)–minimal fundamental graphs, which is based on cycle theory and does not use preliminary facts from [17] as the above cited Criterion 1. We present here our result giving more details and precise definitions in Section 4. To this aim, we assume G to be (v,v)–minimal and we denote by G˜ the new finite graph obtained from G by gluing together v and v in a single vertex called v (see Fig. 4). ∗ G G˜ v z1 z2 v v z1 z2 ∗ Figure 4. ThefundamentalgraphGandtheassociatedgraphG˜ obtained by gluing v and v. 6 A.FAGGIONATOANDV.SILVESTRI Wedenoteby thecycleinG˜ correspondingtotheuniqueself–avoidingpath(z ,z ,...,z ) 1 0 1 n C from v to v in G, and we call ,..., the other cycles in G˜ which form, together with 2 m C C , a cycle basis according to Schnackenberg’s construction. We also define the affinity 1 C ( ) of a cycle as A C C k 1 (cid:89)− p(xi,xi+1) ( ) := ln if = (x ,x ,...,x ), x = x . (8) 0 1 k 0 k A C p(x ,x ) C i+1 i i=0 Duetotheperiodicityofthedynamicalcharacteristics, theCTRWX naturallyinduces t a CTRW Y on G˜. We then consider the path in G˜ given by the vertices visited by Y up to t timetandcompleteittogetacycle inG˜,e.g. byaddinganextrapathofminimallength t C ending at the initial point. Finally we decompose the random cycle in the above cycle t basis: = (cid:80)m a (t) . The random coefficients a (t)’s are also caClled time–integrated Ct i=1 i Ci i cycle currents, and for them we derive in Appendix B the following fluctuation theorems: Theorem 3. Suppose that G is (v,v)–minimal and let (X ) be a generic CTRW on the t t 0 associated quasi–1d lattice . Then the random vector 1(a (t≥),a (t),...,a (t)) satisfies a G t 1 2 m LDP with speed t and good1 rate function . Calling (ϑ ,ϑ ,...,ϑ ) the associated rate 1 2 m I I function, roughly we have (cid:20) (cid:21) 1 P (a (t),...,a (t)) (ϑ ,...,ϑ ) e t (ϑ1,ϑ2,...,ϑm). (9) 1 m 1 m − I t ≈ ∼ Moreover the following GC symmetries hold: m (cid:88) (ϑ ,ϑ ,...,ϑ ) = ( ϑ , ϑ ,..., ϑ ) ϑ ( ), (10) 1 2 m 1 2 m i i I I − − − − A C i=1 m (cid:88) (ϑ ,ϑ ...,ϑ ) = (ϑ , ϑ ,..., ϑ ) ϑ ( ), (11) 1 2 m 1 2 m i i I I − − − A C i=2 (ϑ ,ϑ ,...,ϑ ) = ( ϑ ,ϑ ,...,ϑ ) ϑ ( ). (12) 1 2 m 1 2 m 1 1 I I − − A C As a consequence, the LD rate function I(ϑ) of the cell process introduced in (2) fulfills the GC symmetry I(ϑ) = I( ϑ) ( )ϑ, ϑ R. (13) 1 − −A C ∀ ∈ Let us also remark that for Markov CTRWs the symmetry (13) reduces to (4), since ( ) = ∆. 1 A C For Markov CTRWs [15], but also for a larger class of CTRWs, one can show that the function Q(λ1,λ2,...,λm) := lim 1 lnE(cid:104)e−(cid:80)mi=1λiai(t)(cid:105) , (λ1,λ2,...,λm) Rm t −t ∈ →∞ is well posed and it holds m (cid:8) (cid:88) (cid:9) (ϑ ,ϑ ,...,ϑ ) := sup ϑ λ +Q(λ ,λ ,...,λ ) , (14) 1 2 m i i 1 2 m I (λ1,λ2,...,λm)∈Rm −i=1 1”Good” means that the level sets of I are compact FLUCTUATION THEOREMS FOR DISCRETE KINETIC MODELS OF MOLECULAR MOTORS 7 Moreover, via Legendre transform, the above identities (10), (11) and (12) correspond respectively to the following (15), (16) and (17): Q(λ ,λ ,...,λ ) = Q( ( ) λ , ( ) λ ,..., ( ) λ ), (15) 1 2 m 1 1 2 2 m m A C − A C − A C − Q(λ ,λ ,...,λ ) = Q(λ , ( ) λ ,..., ( ) λ ), (16) 1 2 m 1 2 2 m m A C − A C − Q(λ ,λ ,...,λ ) = Q( ( ) λ ,λ ,...,λ ). (17) 1 2 m 1 1 2 m A C − 2.3. Derivation of the GC symmetry (4) for Markov CTRWs on the linear chain by the matrix approach. When the CTRW on the quasi–1d lattice is Markov, then G the LD rate function I of the cell process N can be expressed as the Legendre transform t of the maximal eigenvalue of a suitable matrix depending by a scalar parameter. In [16, Theorem 3] a general formula is derived by generalizing the matrix approach used in [31]. We restrict here to Markov CTRWs on a linear chain and show how one can derive the GC symmetry (4) by the matrix approach. To make the discussion self–contained we briefly recall how to express the LD rate function in terms of the above maximal eigenvalue. To this aim let G be the linear chain graph of Fig. 5, i.e. G = (V,E) with V = 0,1,...,N , E = (x,x+1),x = 0,...,N 1 and v = 0, v = N. If denotes the assoc{iated quasi–}1d latti{ce, then can be identifi−ed}with Z with periodic juGmp rates. We therefore take Z to be the vertexGset of , and denote by ξ , x Z, the rate associated to G x± ∈ the edge (x,x 1). Finally, set r(x) = ξ +ξ+. Then the Markov CTRW X waits at x an ± x− x t exponentiallydistributedtimeofmean1/r(x), andthenjumpstoeitherx+1orx 1with − probability ξ+/r(x) and ξ /r(x) respectively. Note that ξ = ξ and r(x) = r(x+N) x x− x± x±+N for any x Z and that the constant ∆ in (5) is now given by ∆ = ln ξ0+ξ1+···ξN+−1. ∈ ξ0−ξ1−···ξN−−1 G ξx− ξx+ 0 1 x N ξ+ ξ+ ξ+ ξ+ ξ+ G N−1 0 1 0 1 1 0 1 N 1 N N+1 ξN−−1 ξ0− ξ1− − ξ0− ξ1− − Figure 5. The linear chain graph G (up), and the associated quasi–1d lattice (down). G Let us first consider the case N 3. Given λ R, we introduce the N N matrix ≥ ∈ × (λ), defined as follows for 0 i,j N 1: A ≤ ≤ −  r(i) if i = j,  ξ−+ if 0 < i N 1, j = i 1,  j ≤ − −  ξ if 0 i < N 1, j = i+1, (λ) = j− ≤ − (18) A i,j ξ0−e−λ if i = N −1,j = 0, ξ+ eλ if i = 0,j = N 1, 0N−1 otherwise. − 8 A.FAGGIONATOANDV.SILVESTRI For example, for N = 3 we have  r(0) ξ ξ+eλ − 1− 2 A(λ) =  ξ0+ −r(1) ξ2−  . ξ e λ ξ+ r(2) 0− − 1 − Followingtheapproachof[31]forthe2–periodiclinearmodel,weintroducethefunction F(x,λ,t) := (cid:88)eλkP(X = x+kN) = E(cid:2)eλNt1(X = x+N N)(cid:3), t t t k Z ∈ where, we recall, N is the cell number of X . By the Markov property of X we have t t t ∂ F(x,λ,t) = ξ+ F(x 1,λ,t)+ξ F(x+1,λ,t) r(x)F(x,λ,t). UsingthatF( 1,λ,t) = t x 1 − x−+1 − − eλF(N 1,λ,t)−and F(N,λ,t) = e λF(0,λ,t), we conclude that − − (cid:88) ∂ F(x,λ,t) = (λ) F(y,λ,t), 0 x N 1, (19) t x,y A ≤ ≤ − 0 y N 1 ≤ ≤ − andthereforeF(x,λ,t) = (cid:80) [et (λ)] F(y,λ,0). WhenN = 2, (19)remainsvalid 0 y N 1 A x,y with (λ) defined as ≤ ≤ − A (cid:18) r(0) ξ+eλ+ξ (cid:19) (λ) = − 1 1− A ξ0++ξ0−e−λ −r(1) Since on the other hand E(eλNt) = (cid:80) F(x,λ,t), the Perron–Frobenius theorem 0 x N 1 gives2 ≤ ≤ − E(eλNt) etΛ(λ), Λ(λ) := max (γ) : γ eigenvalue of (λ) . (20) ≈ {(cid:60) A } By G¨artner–Ellis theorem, the cell process satisfies a LD principle with rate function I given by I(ϑ) = sup ϑλ Λ(λ) , ϑ R. (21) λ R{ − } ∈ ∈ Having (21) the GC symmetry (4) follows from the equality Λ(λ) = Λ( ∆ λ), λ R, (22) − − ∈ with ∆ defined according to (5). This is in turn a consequence of the following result: ξ+ξ+ ξ+ Proposition 2.2. Let ∆ = ln 0 1··· N−1. Then there exists an invertible matrix U such that ξ0−ξ1−···ξN−−1 U 1 (λ)U = T( ∆ λ) λ C, (23) − A A − − ∀ ∈ T( ∆ λ) being the transpose of ( ∆ λ). In particular, for the linear chain graph A − − A − − identity (22) is satisfied as well as the GC symmetry (4). It is known that any square matrix A is similar to its transpose AT [42], i.e. an ∃ invertible matrix U such that U 1AU = AT. Hence, once proved (23), one immediately − gets that (λ) and ( ∆ λ) have the same spectrum and therefore the conclusion of A A − − the proposition becomes trivial by the above discussion. 2(cid:60)(x) denotes the real part of the complex number x. FLUCTUATION THEOREMS FOR DISCRETE KINETIC MODELS OF MOLECULAR MOTORS 9 2.4. Further results. Four specific examples are discussed in Sections 6, 7, 8 and 9. We briefly comment on them. The derivation of Theorem 1–(ii), given in [17], is mathemati- cally involved. On the other hand, in Section 6 we consider a parallel chains model (whose fundamental graph is not (v,v)–minimal) and show by direct computations that usually the GC symmetry (3) is not satisfied. In particular, we recover in a specific example the content of Theorem 1–(ii). In Section 7, by considering discrete time RWs (recall Warning 1.1), we exhibit an example of fundamental graph G which is not (v,v)–minimal and such that the GC symmetry (3) holds for any choice of the jump probabilities p(x,y). Finally, in Sections 8 and 9 we consider spatially homogeneous CTRWs on Z with waiting times having respectively exponential and gamma distribution, and compute explicitly several quantities related to large deviations introduced in Section 5 (in particular, the LD rate function for the hitting times and the LD rate function for the cell process). 2.5. Outline of the paper. As already pointed out, a crucial feature of the CTRWs on quasi–1d lattices is a regenerative structure (several results of [17] are indeed valid for stochasticprocessesexhibitingsucharegenerativestructure,notnecessarilyCTRWs). We explain this regenerative structure in Section 3. In Section 5 we recall the main results of [17] applied to the present context, while in Section 4 we recall some basic facts on cycle currents and discuss in detail the objects involved in the cycle fluctuation theorems stated in Theorem 3. Some of these results will be used in our proofs. In Sections 6, 7, 8 and 9 we discuss the above mentioned example. Appendixes A, B and C will be devoted to the derivation of Theorem 2, Theorem 3 and Proposition 2.2 respectively. Finally, Appendix D contains some minor technical facts. 3. Regenerative structure and skeleton process In this section we explain the regenerative structure behind the CTRWs on . To this G aim we introduce a coarse–grained version of X , called skeleton process (X ) with t t∗ t 0 values in Z. More precisely, we set X = n if v(n) is the last vertex of the fo≥rm v(k) t∗ visited by (X ) (see the example in Fig. 6). In the applications to molecular motors, s 0 s t ≤ ≤ the skeleton process contains all the relevant information, since it allows to determine the position of the molecular motor up to an error of the same order of the monomer size. X v(0) a(0) b(0) v(1) a(1) v(1) c(0) v(0) c(−1) t 0 1 0 X t∗ Figure 6. Example of a trajectory (X ) and the associated trajectory t t 0 ≥ (X ) referred to the quasi–1d lattice of Fig. 1. t∗ t≥0 G Note that N X 1, and therefore the skeleton process and the cell process have | t − t∗| ≤ the same asymptotic behaviour and large deviations. The technical advantage of dealing with the skeleton process instead of the cell process comes from the following regenerative structure. Consider the sequence S < S < ... of 1 2 jump times for the skeleton process X , set S := 0, call τ := S S the inter–arrival t∗ 0 i i− i−1 times and w := X X 1,+1 the jumps of the skeleton process (see Fig. 7). i S∗i − S∗i 1 ∈ {− } By our assumptions on X ,−we get that the sequence (w ,τ ) is given by independent t i i i 1 ≥ 10 A.FAGGIONATOANDV.SILVESTRI and identically distributed random vectors and it fully characterizes the skeleton process itself. 0 1 0 X t∗ S S S 0 τ 1 τ 2 τ 1 2 3 Figure 7. Jump times S and inter–arrival times τ for the trajectory of i i the skeleton process of Fig. 6. Note that w = +1 and w = 1. 1 2 − 4. Time integrated cycle currents and affinity In this section we restrict to (v,v)–minimal fundamental graphs G and apply the cycle theory (see e.g. [2, 4, 14, 15]) to formulate fluctuation theorems for cycle currents also for non–Markovian CTRW (cf. Theorem 3). We denote by γ = (z ,z ,...,z ) the unique self–avoiding path from v to v in G, hence 0 1 n with z = v, z = v. We assume that n 3 without loss of generality, since the cases 0 n ≥ n = 1,2 can be reduced to the one above by doubling or tripling the fundamental cell, as explained in Appendix D (see also Fig. 16 therein). Let G˜ denote a new finite graph obtained from G by gluing together v and v in a single vertex called v (see Fig. 4). We denote by π : G˜ the natural graph projection (see Fig. 8) and intr∗oduce the projected process Y :=Gπ→(X ) having values in G˜. As explained t t in formula (26) below, one can recover the asymptotic behavior of the skeleton process X t∗ (and therefore of the cell process N ) by analyzing the currents of the projected process t Y . t G v(0) v(1) v(2) v(3) π G˜ v z z 1 2 ∗ Figure 8. The natural projection π : G˜. G → Letusbrieflyrecallsomeconceptsfromcycletheory(seee.g. [2,8,15,39]). Acycle in C G˜ isdescribedbyapath(x ,x ,...,x )alongedgesofG˜ suchthatx = x . Givenacycle 0 1 k 0 k and two neighboring vertices x,y in G˜, we define N ( ) as the number of appearances x,y C C of the string (x,y) in minus the number of appearances of the string (y,x) in (i.e. C C the number of jumps from x to y minus the number of jumps from y to x performed by the cycle ). We can make the cycle space into a real vector space by considering formal C