A Reversibility Parameter for a Markovian Stepper

Recent experimental studies on the stepwize motion of biological molecular motors have revealed that the “characteristic distance” of a step is usually less than the actual step size. This observation implies that the detailed-balance condition for kinetic rates of steps is violated in these motors. In this letter, in order to clarify the significance of the characteristic distance, we study a Langevin model of a molecular motor with a hidden degree of freedom. We find that the ratio of the characteristic distance to the step size is equal to unity if the dominant paths in the state space are one dimensional, while it deviates from unity if the dominant paths are branched. Therefore, this parameter can be utilized to determine the reversibility of a motor even under a restricted observation. Single-molecule measurement techniques have expanded the possibilities for studying biological macromolecules from physical points of view. For instance, molecular motors, which move along protein filaments in a stepwise manner fueled by the hydrolysis of adenosine tri-phosphate (ATP), have been investigated extensively. Through many studies, several of the important properties of molecular motors, including their step sizes and the kinetic rates of their forward and backward steps, have been experimentally determined. In this letter, we study the modulation of the kinetic rates by an external force. Such phenomena have been examined in recent experiments [1–5]. Interestingly, the results of these experiments suggest that the detailedbalance (DB) condition for the kinetic rates is violated in the stepwise motion of a motor. Here, we start from the DB condition for a system that has several discrete states [6]. Let m1 and m2 denote two of these states, and let k21 and k12 represent the transition rates from m1 to m2 and vice versa, respectively. Then, the DB condition is given as k21 k12 = exp(β∆G12), (1) where β ≡ (kBT ) −1 is the inverse temperature of the heat bath (kB being the Boltzmann constant), and ∆G12 ≡ G1 −G2 represents the difference in the (free) energy between the states m1 and m2. Note that the DB condition given in eq. (1) is derived from a more microscopic condition termed the local detailed-balance (LDB) condition with several assumptions. Let r1 and r2 denote microstates in the states m1 and m2, respectively, and let [r] represent a microscopic path connecting r1 and r2 in an interval t. Let P ([r]|r1) represent the transition probability of a path [r] with given r1. Then, the LDB condition is expressed as P ([r]|r1) P ([r̃]|r2) = exp [β∆G(r1, r2)] , (2) where [r̃] is the time-reversed path of [r], and ∆G(r1, r2) ≡ G(r1) − G(r2) represents the difference in the (free) energy between the microstates r1 and r2. Eq. (1) is derived from eq. (2) provided that the thermal energy is sufficiently less than the hight of a barrier separating m1 and m2, and the curvature of G(r) around these states are the same. It has been known that the LDB condition is an essential condition in order to make a stochastic model compatible with thermodynamics, and to derive several nonequilibrium equalities, including the fluctuation theorems and the Jarzynski equality [7,8]. Furthermore, it has been argued that eq. (2) is satisfied in a wide class of models, including Markov chains, Langevin equations, and deterministic systems connected to heat baths [7–10]. When one wants to test the validity of eq. (1) for a given system, one has to know ∆G12. From experimental point of view, this is sometimes difficult. Instead, eq. (1) can be indirectly verified by changing the magnitude of the external force f applied to the system. If the magnitude of the external force is changed by a small amount δf , ∆G12 would be modified as ∆G12 → ∆G12+δf ·l, where l is a vector connecting m1 and m2. Then, according to eq. (1), the ratio of the kinetic rates would be modified as k21(f + δf ) k12(f + δf ) = k21(f) k12(f) exp (βδf · l) . (3) Note that this condition can be verified without determining ∆G12.