Dissipative Heat Decomposition in Stochastic Energetics: Implication of the Instantaneous Diffusion Coefficient in Nonequilibrium Steady States

We give a decomposition expression for dissipative heat using the instantaneous diffusion coefficient in a nonequilibrium steady state. The dissipative heat can be expressed using three diffusion coefficients: instantaneous, equilibrium, and drift. An experimental application of the decomposition expression permits us to evaluate the heat dissipation rate from single-trajectory data only. We also numerically demonstrate this method.

analyzing the tracked trajectory data, {x 0 , x 1 , · · · , x n , · · · , x N−1 }, for which the time resolution is ∆t and x n ≡ x(t n ) for t n ≡ n ∆t, as much information as possible is expected to be extracted from the data. For diffusion in living cells, the most common and conventional analytical method is the mean-squared displacement (MSD) analysis. Not only the diffusion coefficient but also the drift velocity and a characteristic diffusion length within a limited area can be estimated. 1 However, as the MSD analysis is based on stationary stochastic processes, we cannot examine its relationship to nonequilibrium biological activity in living cells.
Meanwhile, the development of stochastic energetics has enabled us to energetically describe and estimate phenomena in the fluctuating world. 2,3 Violations of the fluctuationdissipation relation in nonequilibrium steady states (NESSs) can be expressed in terms of the heat dissipation rate into the thermal environment without a prior knowledge about energy potential, 4 and this theoretical expression has been applied to an evaluation of the singlemolecule energetics of a rotary molecular motor, F 1 -ATPase. 5 However, in general, measurement of the mechanical responses of molecules is a difficult task. In order to overcome these difficulties, Toyabe et al. developed an analytical method to evaluate the potential profile at each chemical state from a single-molecule trajectory only, without the need for response measurement, and applied it to F 1 -ATPase. 6 In addition, we have also suggested an analytical method for a single-diffusion trajectory based on stochastic energetics, 7 which can be used to estimate the enhanced motion due to active diffusion in living cells. 8 Specifically, the in- is an energetically well-defined physical quantity for active diffusion trajectories. However, when the response takes the form of contributions to a potential, the implication of the IDC remains unclear.
Here, we study a system described by an overdamped Langevin equation where γ is the friction coefficient, U(x) is an arbitrary potential including NESS cases, T is the wherex n ≡ (x n + x n+1 )/2, {ξ n } is a sequence of the Gaussian random variables satisfying ξ n = 0 and ξ n ξ m = δ nm , and o(∆t) denotes terms of order greater than ∆t.
Next, let us consider the dissipative heat into the thermal environment, ∆Q n , during the time interval, [t n , t n+1 ). According to stochastic energetics, 3 the heat at time t is defined as . 10 Therefore, for discrete data, ∆Q n corresponds to −U ′ (x n )∆x n . Since Eq. (2) can be transformed into the identical relation we can obtain the following expression This equation leads to an expression for the dissipative heat where Since the average of Eq.
(2) at a given initial position z becomes ∆x n z = −U ′ (z)∆t/γ + o(∆t), 11 we can obtain for z =x n . As ξ 2 n converges to (dB t ) 2 /dt = 1 in the limit ∆t → dt, 3,9 the dissipative heat at t and position,x ≡ x(t) + dx(t)/2, can be formally expressed as where the three diffusion coefficients are defined as We refer to these three terms as the instantaneous, equilibrium, and drift diffusion coefficients, respectively. Equation (7) clearly shows that the IDC is intimately related to the dissipative heat, according to the difference, D eq − D drift (x), in NESSs. Note that Eq. (7) is derived in the limit ∆t → dt. Unfortunately, in experiments with a finite time resolution, ∆t, we cannot reach this limit. Instead, Eq. (6) is practical. However, the influence of noise with finite order, we can remove the finite effect of noise. This is because ξ 2 n = lim N→∞ 1 N∆t N−1 n=0 ξ 2 n = 1, and J out converges to the same limit, 1 N∆t N−1 n=0 ∆Q n . This means that we can evaluate the heat dissipation rate from single-trajectory data only without measuring response. The key to this evaluation is to calculate ∆x n x n with a high space resolution by using as much data as possible. 13 Meanwhile, the friction coefficient, γ, can be easily calculated from trajectories. 6 Finally, we give a numerical demonstration to evaluate the heat dissipation rate. As an example, let us consider a tilted periodic potential, U(x) = 20k B T cos(2πx/l) − F x/l, with force, F = 120k B T/l N, and periodic length, l = 2 µm. Figure 1(a) shows the graph of this potential, where the unit of energy is k B T . Unless the numerical calculation of the integral of the Langevin equation (Eq. (1)) is in the form of a Stratonovich-type SDE, excess energy is generated in contradiction to the law of energy conservation. 3 Here, we computed the integral using the following semi-implicit algorithm: 9 A numerical trajectory for the parameters, γ = 1.0 × 10 −8 kg/s, T = 300 K, and ∆t = 0.001 s, is shown in Fig. 1(b). Here, ∆x n z was calculated within each bin, where the space resolution was 10 nm, for a single trajectory with N = 10 6 . Figure 1(c) indicates that the calculated ∆x n z (points) show good agreement with the theoretical curve (solid curve). In order to determine the accuracy of our method of evaluating the heat dissipation rate, we considered the energy rate balance between the output, J out , and the input, which is defined as J in ≡ For each trajectory, we calculated the relative error, δ ≡ (J out − J in )/J in . The probability densities of δ for N = 1 × 10 5 , 2 × 10 5 , 5 × 10 5 , 1 × 10 6 , 2 × 10 6 are shown in Fig. 1(d). Each ensemble consists of 10 5 trajectories, and the ensemble average of N, δ N , is shown in Fig. 1(e). In our simulations, the average didn't converge to just zero for not only the semi-implicit algorithm but also the Heun algorithm. 3 The accurate convergence might strongly depend on the algorithm for solving the SDE.
The development of stochastic energetics by Sekimoto has changed researchers' perspective on the fluctuating world. For example, diffusion phenomena in molecular motors or living cells can be analyzed in terms of the stochastic energetics based on single-trajectory data only, 6,7 and the dissipative heat can be related to some form of nonequilibrium biological activity. 14 The method presented in this note has potential application in the analysis of singlemolecule trajectories in living cells, which will facilitate the discovery of the relationship between molecular function and cell environment.

Acknowledgment
This work was inspired from communications with Shoichi Toyabe, who sent comments following the publication of our previous paper, 7 for which I am sincerely grateful. I also gratefully acknowledge Akinori Awazu and Yuichi Togashi for helpful discussions. This work was supported by MEXT, Japan (Platform for Dynamic Approaches to Living System; KAK-ENHI 23115007).