Fluctuation theorems for discrete kinetic models of molecular motors

Fluctuation theorems for discrete kinetic models of molecular motors

Alessandra Faggionato Alessandra Faggionato. Dipartimento di Matematica, Università di Roma ‘La Sapienza’ P.le Aldo Moro 2, 00185 Roma, Italy faggiona@mat.uniroma1.it  and  Vittoria Silvestri Vittoria Silvestri. Statistical Laboratory, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK. V.Silvestri@maths.cam.ac.uk
Abstract.

Motivated by discrete kinetic models for non–cooperative molecular motors on periodic tracks, we consider random walks (also not Markov) on quasi one dimensional (1d) lattices, obtained by gluing several copies of a fundamental graph in a linear fashion. We show that, for a suitable class of quasi–1d lattices, the large deviation rate function associated to the position of the walker satisfies a Gallavotti–Cohen symmetry for any choice of the dynamical parameters defining the stochastic walk. This class includes the linear model considered in [31]. We also derive fluctuation theorems for the time–integrated cycle currents and discuss how the matrix approach of [31] can be extended to derive the above Gallavotti–Cohen symmetry for any Markov random walk on with periodic jump rates. Finally, we review in the present context some large deviation results of [17] and give some specific examples with explicit computations.

Keywords: Semi–Markov process, continuous time random walk, large deviation principle, molecular motor, Gallavotti–Cohen symmetry, time–integrated cycle current.

1. Introduction

Molecular motors are special proteins able to convert chemical energy coming from ATP–hydrolysis into mechanical work, allowing numerous physiological processes such as cargo transport inside the cell, cell division, muscle contraction [23]. They are able to produce directed transport in an environment in which the fluctuations due to thermal noise are significant, achieving nonetheless an efficiency even higher than the one of macroscopic motors. In addition synthetic molecular motors have been obtained and their improvements are under continuous investigation [19].

Molecular motors haven been extensively studied both theoretically and experimentally (cf. [24, 29, 37, 38, 40] and references therein). We focus here on the large class of molecular motors (e.g. conventional kinesin) which work non–cooperatively and move along cytoskeletal filaments [23]. Keeping in mind the polymeric structure of these filaments, two main models have been proposed. In the so called Brownian ratchet model [24, 37] the dynamics of the molecular motor is given by a one–dimensional diffusion in a spatially periodic potential randomly switching its shape (indeed, along its mechanochemical cycle the molecular motor can be strongly or weakly bound to the filament, thus leading to a change in the interaction potential). The other paradigm [20, 21, 25, 26, 27, 28, 29, 43], on which we concentrate here, is given by continuous time random walks (CTRW), along with a quasi one dimensional (quasi–1d) lattice obtained by gluing several copies of a fundamental graph in a linear fashion. CTRWs are thought in the Montroll–Weiss sense [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 with two marked vertices and (see Fig. 1, left). For simplicity we assume that has no multiple edges or self–loops. The associated quasi–1d lattice is then obtained by gluing several copies of , identifying the –vertex of one copy to the –vertex of the next copy (see Fig. 1, right). Given a vertex in and , we write for the corresponding vertex in the –th copy of in . Since , to simplify the notation we denote such a vertex by throughout. Each site corresponds to a spot in the monomer of the polymeric filament to which the molecular motor can bind. The other vertices describe intermediate conformational states that the molecular motor achieves by conformational transformations, modeled by jumps along edges in . Note the periodicity of the quasi–1d lattice .

Figure 1. The fundamental graph with marked vertices (left) the quasi–1d lattice (right).

The evolution of the molecular motor is described by a CTRW , taking values in the vertex set of the quasi–1d lattice . Once arrived at vertex , waits a random time with distribution (that we assume to have finite mean) and then jumps to a neighboring vertex in with probability . We assume that and exhibit the same periodicity of . In what follows, we call dynamical characteristics the above data , .

Warning 1.1.

In the degenerate case that is a delta measure, e.g. equals , the above CTRW reduces to the so–called discrete time random walk. We do not restrict to distributions having a probability density w.r.t. the Lebesgue measure, so that can be composed by some delta measure as well.

We remark that when is the exponential distribution with mean , then the resulting CTRW is Markov and its density distribution satisfies the Fokker–Planck equation

(1)

In what follows, we assume that the random walk starts at , i.e. .

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 vertices, we recover a CTRW on with nearest–neighbor jumps and –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).

Figure 2. Parallel chains model (left), divided–pathway model (right).

Calling the set of vertices of the fundamental graph , for we define the cell as the set of vertices in of the form with (for example, in Fig. 1 the cell is given by ). Our aim is to investigate large fluctuations and associated symmetries of the cell process , defined as if belongs to the cell, i.e. if for some . Trivially, the cell process determines the 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 (i.e. almost surely) and has Gaussian fluctuations. A large deviation principle is proved in [17] (see Section 5 for more details). We call the associated large deviation function:

(2)

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 dynamical 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 , or similar, being a constant and 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. Fluctuations 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 generic Markov CTRW on with nearest–neighbor jumps and dynamical characteristics of periodicity two, which corresponds to (1) with of the following form: if is even and if is odd, for generic constants . This GC symmetry for the velocity reads

(3)

being the rate function of the cell process modulo rescaling by the length of monomers in the polymeric filaments. For the above 2-periodic Markov CTRW it holds .

Since the above CTRW with period is a simplified model for the motion of real 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 in fact (3) is only universal in the subclass of 1d lattices whose fundamental graph is –minimal in the following sense: there exists a unique self–avoiding path in from to . An example of –minimal graphs is given in Fig. 3. Note that the graphs associated to the quasi–1d lattices in Fig. 2 are not –minimal.

Figure 3. Example of a –minimal graph with .

We can now recall the characterization provided in [17]:

Theorem 1 ([17]).

Suppose that is a Markov CTRW on the quasi–1d lattice , in particular it has exponentially distributed waiting times and transition rates as in (1). Then the following holds:

  • If is –minimal, then the cell process satisfies the GC symmetry

    (4)

    where

    (5)

    and , with and , is the unique self–avoiding path from to in .

  • Vice versa, if is not –minimal, then the set of transition rates for which the GC symmetry (4) holds for some (which can depend on ) 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 is not –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 is analyzed for a larger class of random processes, having a suitable regenerative structure. Moreover, it has been proved (cf. Theorems 4 and 8 in [16]) that the GC symmetry (3) holds if and only if and are independent, where the random time is defined as . For the Markov random walk on a linear chain this 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) for non –minimal fundamental graphs (see Section 2.1), (b) derive fluctuation theorems for time–integrated cycle currents in the case of generic CTRWs and –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.

2.1. Extension of Theorem 1–(i) to generic CTRWs

We consider generic CTRWs on , i.e. also non Markov. As a first result we give a sufficient condition assuring that the 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 –minimal graphs , thus leading to the extension of Theorem 1–(i) to non Markov CTRWs.

Theorem 2.

Consider a generic CTRW on the quasi–1d lattice with dynamical characteristics and . Then the cell process satisfies the GC symmetry (4) for some constant if

(6)

for any self–avoiding path from to in the fundamental graph (, ).

As a consequence, if is –minimal, then the cell process satisfies the GC symmetry (4) where now

(7)

and , with and , is the unique self–avoiding path from to in .

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 which are not –minimal and such that the GC symmetry (3) holds for any choice of the jump probabilities . We refer to Section 7 for an example.

2.2. GC symmetries for cycle currents

As next result we show that, for –minimal fundamental graphs, the GC symmetry (3) is indeed a special case of a fluctuation theorem for cycle currents (see e.g. [2, 4, 14, 15]). As a consequence we give, among others, an alternative derivation of (3) for –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 to be –minimal and we denote by the new finite graph obtained from by gluing together and in a single vertex called (see Fig. 4).

Figure 4. The fundamental graph and the associated graph obtained by gluing and .

We denote by the cycle in corresponding to the unique self–avoiding path from to in , and we call the other cycles in which form, together with , a cycle basis according to Schnackenberg’s construction. We also define the affinity of a cycle as

(8)

Due to the periodicity of the dynamical characteristics, the CTRW naturally induces a CTRW on . We then consider the path in given by the vertices visited by up to time and complete it to get a cycle in , e.g. by adding an extra path of minimal length ending at the initial point. Finally we decompose the random cycle in the above cycle basis: . The random coefficients ’s are also called time–integrated cycle currents, and for them we derive in Appendix B the following fluctuation theorems:

Theorem 3.

Suppose that is –minimal and let be a generic CTRW on the associated quasi–1d lattice . Then the random vector satisfies a LDP with speed and good111”Good” means that the level sets of are compact rate function . Calling the associated rate function, roughly we have

(9)

Moreover the following GC symmetries hold:

(10)
(11)
(12)

As a consequence, the LD rate function of the cell process introduced in (2) fulfills the GC symmetry

(13)

Let us also remark that for Markov CTRWs the symmetry (13) reduces to (4), since .

For Markov CTRWs [15], but also for a larger class of CTRWs, one can show that the function

is well posed and it holds

(14)

Moreover, via Legendre transform, the above identities (10), (11) and (12) correspond respectively to the following (15), (16) and (17):

(15)
(16)
(17)

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 the LD rate function of the cell process can be expressed as the Legendre transform 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 be the linear chain graph of Fig. 5, i.e. with , and , . If denotes the associated quasi–1d lattice, then can be identified with with periodic jump rates. We therefore take to be the vertex set of , and denote by , , the rate associated to the edge . Finally, set . Then the Markov CTRW waits at an exponentially distributed time of mean , and then jumps to either or with probability and respectively. Note that and for any and that the constant in (5) is now given by .

Figure 5. The linear chain graph (up), and the associated quasi–1d lattice (down).

Let us first consider the case . Given , we introduce the matrix , defined as follows for :

(18)

For example, for we have

Following the approach of [31] for the –periodic linear model, we introduce the function

where, we recall, is the cell number of . By the Markov property of we have . Using that and , we conclude that

(19)

and therefore . When , (19) remains valid with defined as

Since on the other hand , the Perron–Frobenius theorem gives222 denotes the real part of the complex number .

(20)

By Gärtner–Ellis theorem, the cell process satisfies a LD principle with rate function given by

(21)

Having (21) the GC symmetry (4) follows from the equality

(22)

with defined according to (5). This is in turn a consequence of the following result:

Proposition 2.2.

Let . Then there exists an invertible matrix such that

(23)

being the transpose of . In particular, for the linear chain graph identity (22) is satisfied as well as the GC symmetry (4).

It is known that any square matrix is similar to its transpose [42], i.e. an invertible matrix such that . Hence, once proved (23), one immediately gets that and have the same spectrum and therefore the conclusion of the proposition becomes trivial by the above discussion.

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 mathematically involved. On the other hand, in Section 6 we consider a parallel chains model (whose fundamental graph is not –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 which is not –minimal and such that the GC symmetry (3) holds for any choice of the jump probabilities . Finally, in Sections 8 and 9 we consider spatially homogeneous CTRWs on 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 stochastic processes exhibiting such a regenerative structure, not necessarily CTRWs). 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 aim we introduce a coarse–grained version of , called skeleton process with values in . More precisely, we set if is the last vertex of the form visited by (see the example in Fig. 6). In the applications to molecular motors, 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.

Figure 6. Example of a trajectory and the associated trajectory referred to the quasi–1d lattice of Fig. 1.

Note that , and therefore the skeleton process and the cell process have 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 of jump times for the skeleton process , set , call the inter–arrival times and the jumps of the skeleton process (see Fig. 7). By our assumptions on , we get that the sequence is given by independent and identically distributed random vectors and it fully characterizes the skeleton process itself.

Figure 7. Jump times and inter–arrival times for the trajectory of the skeleton process of Fig. 6. Note that and .

4. Time integrated cycle currents and affinity

In this section we restrict to –minimal fundamental graphs 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 the unique self–avoiding path from to in , hence with , . We assume that without loss of generality, since the cases 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 denote a new finite graph obtained from by gluing together and in a single vertex called (see Fig. 4). We denote by the natural graph projection (see Fig. 8) and introduce the projected process having values in . As explained in formula (26) below, one can recover the asymptotic behavior of the skeleton process (and therefore of the cell process ) by analyzing the currents of the projected process .

Figure 8. The natural projection .

Let us briefly recall some concepts from cycle theory (see e.g. [2, 8, 15, 39]). A cycle in is described by a path along edges of such that . Given a cycle and two neighboring vertices in , we define as the number of appearances of the string in minus the number of appearances of the string in (i.e. the number of jumps from to minus the number of jumps from to performed by the cycle ). We can make the cycle space into a real vector space by considering formal linear combinations of cycles and using the identification

(24)

whenever for any neighboring vertices .

To the path we associate the cycle in the graph . Let us now fix a cycle basis in with . This can be done according to Schnackenberg’s construction as follows. We take a spanning tree (i.e. a subgraph of without loops which contains all vertices of ) containing the linear chain . Given an edge in not belonging to the spanning tree, there exists a unique self–avoiding cycle (apart from orientation and starting point) in the graph obtained by adding the edge to the spanning tree. Just take one , fixing arbitrarly orientation and starting point. The collection of cycles obtained by varying the edge in this procedure forms a cycle basis. Note that this basis contains the cycle , which is indeed associated to the edge (see Fig. 9). Note also that, since is –minimal, the cycle has no edge in common with .

Figure 9. Schnackenberg’s construction of the cycle basis . In the left picture, the spanning tree is drawn in bold lines.

We can finally define the affinity of a cycle . To this aim recall that the CTRW is defined in terms of the dynamical characteristics and and that we are considering also non Markov CTRWs. Note that the jump probabilities defined on can be projected on the graph without any ambiguity since we are assuming (see Fig. 10 for an example).

Figure 10. The quasi–1d lattice with the jump rates (left), and the projected graph with the induced jump rates (right).

Finally, recall the definition of cycle affinity (cf. (8)).

Let us now go back to the dynamics. Since we have (recall that ). We now associate to each trajectory a cycle in as follows. Consider the projected path . If , then is given by the string of vertices visited by , taken in chronological order. If , then we complete the above string by adding a path in from to (this additional path depends only on : the same final point , the same additional path). Finally, we take the decomposition of the random cycle in our fixed basis, i.e.

(25)

The fundamental link between the above construction and the original skeleton process is given by the following formula:

(26)

This is obtained observing that, since the graph is –minimal, it holds (and therefore by (25)), and that differs from by at most .

Having clarified the content of Theorem 3, we refer to Appendix B for its proof.

5. Previous results on the asymptotic velocity and large deviations

In this section we review some results of [16, 17]. We point out that a key ingredient in their derivation has been the regenerative structure discussed in Section 3. In Sections 8 and 9 we will discuss specific random walks for which the LD rate functions entering in Theorem 4 below can be computed. On the other hand, Theorem 5 below will be very useful in the rest of the paper.

Recall that and that (cf. Section 3) denotes the first jump time for the skeleton process , i.e.

(27)

Recall also that we have assumed that all the waiting times of have finite mean, i.e. has finite mean for all vertices in . It is then simple to show that . As derived in [16], since , almost surely the skeleton process and therefore also the cell process admit an asymptotic velocity:

(28)

We refer the interested reader to [16] for what concerns the Gaussian fluctuations of . In the rest of this section we concentrate on large deviations.

Theorem 4 ([17]).

Call the first time the skeleton process hits , i.e.

(29)

Then the following holds:

  • As the random variables satisfy a LDP with speed and convex rate function

    (30)

    where

    (31)
  • As , the random variables and satisfy a LDP with speed and good333The rate function is good in the sense that is compact, for any and convex rate function given by

    (32)

    and .

Roughly, we have

for large, respectively.

It is useful for applications to reduce the computation of to the one of simpler functions. The following characterization of is provided in [17], Proposition 4.3. Recall the definition of given in (27), and let be defined by

(33)

Then the functions in (31) satisfy

(34)

for , where is the unique value of such that , while for .

In addition to Theorem 1 the following characterization of the GC symmetry for the cell process is provided in [17, Theorems 4,8]:

Theorem 5 ([17]).

The following facts are equivalent:

  • For some the GC symmetry holds for all ;

  • The random variables and are independent.

  • The functions and are proportional, i.e.  such that for all .

Moreover, when (i),(ii) hold it must be .

Remark 5.1.

As a byproduct of Theorem 5 with Theorem 2 (or, equivalently, with (13) in Theorem 3) the independence stated in Theorem 5–(ii) holds for any –minimal fundamental graph.

Remark 5.2.

Note that, by (34), Item (iii) is equivalent to the proportionality of and , which is often easier to check. Indeed, for , if and only if .

Remark 5.3.

Consider a CTRW on with –periodic rates. Since the fundamental graph is given by a finite linear graph and is therefore –minimal, we know that the GC symmetry of Theorem 5–(i) is satisfied (cf. Theorems 2 and 3). As byproduct with Theorem 5 we get in particular that the time needed by to hit the set does not depend on which site is visited when first arriving in . This property was already derived in [10] for CTRWs on with –periodic rates.

6. Example: Violation of GC symmetry with a non –minimal fundamental graph

Considering a Markov CTRW, the violation of the GC symmetry for almost any choice of jump rates in the case of non –minimal fundamental graphs has a non trivial derivation, based on complex analysis [17]. We discuss here an example, given by a parallel–chains model [25], confirming the thesis.

Let us consider the fundamental graph in Fig. 11 (left), in which to each pair of neighbouring vertices we have assigned a positive rate in . The associated quasi–1d lattice is represented in Fig. 11 (right).

Figure 11. The graph together with the rates (left) and the associated quasi–1d lattice (right).

Let denote the Markov CTRW on with periodic jump rates induced by . Finally, let and denote the cell process and skeleton process associated to .

By Theorem 4, as , the random variables and satisfy a LDP with speed and rate function , defined in (32). Since the fundamental graph is not –minimal, we aim to show that satisfies the GC symmetry (3) only for a set of transition rates of zero Lebesgue measure in . According to Theorem 5, to this end it suffices to show that and are not proportional for almost any choice of the jump rates.

The computation of the ratio can be reduced to a single cell analysis as follows. Let be the first time that the process , starting at , reaches after performing at least one jump, and set