Statistics of Infima and Stopping Times of Entropy Production and Applications to Active Molecular Processes

We study the statistics of infima, stopping times, and passage probabilities of entropy production in nonequilibrium steady states, and we show that they are universal. We consider two examples of stopping times: first-passage times of entropy production and waiting times of stochastic processes, which are the times when a system reaches a given state for the first time. Our main results are as follows: (i) The distribution of the global infimum of entropy production is exponential with mean equal to minus Boltzmann’s constant; (ii) we find exact expressions for the passage probabilities of entropy production; (iii) we derive a fluctuation theorem for stopping-time distributions of entropy production. These results have interesting implications for stochastic processes that can be discussed in simple colloidal systems and in active molecular processes. In particular, we show that the timing and statistics of discrete chemical transitions of molecular processes, such as the steps of molecular motors, are governed by the statistics of entropy production. We also show that the extreme-value statistics of active molecular processes are governed by entropy production; for example, we derive a relation between the maximal excursion of a molecular motor against the direction of an external force and the infimum of the corresponding entropy-production fluctuations. Using this relation, we make predictions for the distribution of the maximum backtrack depth of RNA polymerases, which follow from our universal results for entropy-production infima.

Popular Summary

A glass falling to the ground will smash into many pieces, but the shattered shards will never spontaneously come back together. Entropy production quantifies how unlikely it is to observe a sequence happening in reverse with respect to the natural order of events. A breaking glass, for example, produces entropy whereas entropy would decrease if the glass were to re-form. In macroscopic systems, entropy always increases with time. There is a small chance, however, that entropy can decrease in mesoscopic systems, those with an intermediate size such as proteins, though little is known about such events. We find that there are universal laws for entropy production in mesoscopic systems and show how these laws can describe behavior in some of the molecular machinery found in living cells.

An interesting quantity that characterizes entropy-decreasing events is the infimum, or negative record, of entropy. Our work shows that the mean of this infimum value is greater than or equal to the negative of the Boltzmann constant, a physical constant that relates energy to temperature. We also show that the statistics for the times of events that produce negative records are identical to the statistics for the times of positive-record events with the same magnitude. We call the former the infimum law and the latter the stopping-time fluctuation theorem.

Our results hold, regardless of the physical properties of the system, and can be a powerful tool for understanding extreme-value and timing statistics of mesoscopic biological processes. For example, using the infimum law, we make predictions for the distribution of the maximum net number of cycles an enzyme performs against the direction of a chemical bias.

Figure 1

Illustration of three key results of the paper. (a) Schematic representation of the infimum law for entropy production. Several stochastic trajectories of entropy production are shown (solid lines), and their infima are indicated (filled circles). The infimum law implies that the average infimum of the entropy production (green solid line) is larger than or equal to $-k_{\mathrm{B}}$ (orange line). (b) First-passage-time fluctuation theorem for entropy production with two absorbing boundaries. We show examples of trajectories of stochastic entropy production as a function of time, which first reach the positive threshold $s_{\mathrm{tot}}$ (horizontal thick blue line) and which first reach the negative threshold $-s_{\mathrm{tot}}$ (horizontal thick red line). The probability distribution $p_{T_{+}}(t;s_{\mathrm{tot}})$ to first reach the positive threshold at time $t$ and the probability distribution $p_{T_{-}}(t;-s_{\mathrm{tot}})$ to first reach the negative threshold at time $t$ are related by Eq. (8). (c) Waiting-time fluctuations: The statistics of the waiting times between two states I and II are the same for forward and backward trajectories that absorb or dissipate a certain amount of heat $Q$ in isothermal conditions.

Figure 2

Entropy-production passage-probability ratio $\ln [{\mathsf{P}}_{+}^{(2)}/{\mathsf{P}}_{-}^{(2)}]$ as a function of the thresholds $s_{\mathrm{tot}}^{-}$ and $s_{\mathrm{tot}}^{+}$, obtained from Eqs. (45) and (46). The solid white line is the curve for which ${\mathsf{P}}_{+}^{(2)}={\mathsf{P}}_{-}^{(2)}=1/2$, and it follows from Eq. (55). This curve converges asymptotically to the value $s_{\mathrm{tot}}^{-}=k_{\mathrm{B}}\ln 2$ for $s_{\mathrm{tot}}^{+}\to +\infty$ (white dashed line).

Figure 3

Illustration of the first-passage times for entropy production. (a) First-passage time $T_{+}^{(1)}$ for entropy production with one positive absorbing boundary $s_{\mathrm{tot}}$ (top) and first-passage time $T_{-}^{(1)}$ for entropy production with one negative absorbing boundary $-s_{\mathrm{tot}}$ (bottom). (b) First-passage times $T_{+}^{(2)}$ and $T_{-}^{(2)}$ for entropy production with two absorbing boundaries at $\pm s_{\mathrm{tot}}$. At the time $T_{+}^{(2)}$, entropy production passes the threshold $s_{\mathrm{tot}}$ for the first time without having reached $-s_{\mathrm{tot}}$ before. At the time $T_{-}^{(2)}$, entropy production passes the threshold $-s_{\mathrm{tot}}$ for the first time without having reached $s_{\mathrm{tot}}$ before.

Figure 4

Illustration of a Smoluchowski-Feynman ratchet. A Brownian particle (gray sphere) immersed in a thermal bath of temperature ${\mathsf{T}}_{\mathrm{env}}$ moves in a periodic potential $V(\varphi )$ (black shaded curve) with friction coefficient $\gamma$. The coordinate $\varphi$ is the azimuthal angle of the particle. When applying an external force $F=\gamma v$ in the azimuthal direction, the particle reaches a nonequilibrium steady state. In this example, $V(\varphi )=k_{\mathrm{B}}{\mathsf{T}}_{\mathrm{env}}\ln [\cos (\varphi )+2]$, $\alpha =R\cos (\varphi )$ and $\beta =R\sin (\varphi )$, with $R=2$.

Figure 5

Cumulative distributions of the infimum of entropy production for a Smoluchowski-Feynman ratchet in a steady state for different values of the mean entropy change $\overline{s}(t)=⟨S_{\mathrm{tot}}(t)⟩/k_{\mathrm{B}}$. Simulation results for the Smoluchowski-Feynman ratchet with the potential (79) (solid lines) are compared with analytical results for the drift-diffusion process given by Eq. (81) (dashed lines), and with the universal bound given by $1-e^{-s}$ (rightmost yellow curve). Simulation parameters: $10^4$ simulations with ${\mathsf{T}}_{\mathrm{env}}=300\mathrm{K}$, $\ell =2\pi \mathrm{nm}$, $\gamma =8.4\mathrm{pN}\text{s}/\mathrm{nm}$, $V_0=k_{\mathrm{B}}{\mathsf{T}}_{\mathrm{env}}$, simulation time step $\mathrm{\Delta }t=0.00127\mathrm{s}$, and total simulation time $t=0.4\mathrm{s}$. The external force $F$ and the drift velocity $v$ are determined by the average entropy production $\overline{s}(t)$.

Figure 6

Mean of the finite-time entropy-production infimum $⟨S_{\inf }(t)⟩$ as a function of the mean entropy change $\overline{s}(t)=⟨S_{\mathrm{tot}}(t)⟩/k_{\mathrm{B}}$ for a Smoluchowski-Feynman ratchet in a steady state. Simulation results for the Smoluchowski-Feynman ratchet with the potential (79) (symbols) are compared with analytical results for the drift-diffusion process given by Eq. (82) (dashed lines), and with the universal infimum law given by $-k_{\mathrm{B}}$ (yellow thick bar). Different symbols are obtained for different simulation time steps $\mathrm{\Delta }t$. Simulation parameters: $10^4$ simulations with ${\mathsf{T}}_{\mathrm{env}}=300\mathrm{K}$, $\ell =2\pi \mathrm{nm}$, $\gamma =8.4\mathrm{pN}\text{s}/\mathrm{nm}$, $V_0=k_{\mathrm{B}}{\mathsf{T}}_{\mathrm{env}}$, and total simulation time $t=0.4\mathrm{s}$.

Figure 7

Empirical two-boundary first-passage-time distributions of entropy production to first reach a positive threshold $p_{T_{+}^{(2)}}(t;s_{\mathrm{tot}})$ (blue squares) and the rescaled distribution for the negative threshold $p_{T_{-}^{(2)}}(t;-s_{\mathrm{tot}})e^{s_{\mathrm{tot}}/k_{\mathrm{B}}}$ (red circles) for the Smoluchowski-Feynman ratchet with the potential (79) in a steady state. The distributions are obtained from $10^4$ numerical simulations, and the threshold values are set to $\pm s_{\mathrm{tot}}=\pm 2.2k_{\mathrm{B}}$. The simulations are done using the Euler numerical scheme with the following parameters: $F=0.64\mathrm{pN}$, ${\mathsf{T}}_{\mathrm{env}}=300\mathrm{K}$, $\ell =2\pi \mathrm{nm}$, $\gamma =8.4\mathrm{pN}\text{s}/\mathrm{nm}$, $V_0=k_{\mathrm{B}}{\mathsf{T}}_{\mathrm{env}}$, and simulation time step $\mathrm{\Delta }t=0.0127\mathrm{s}$. Top inset: Empirical passage probabilities of entropy production to first reach the positive threshold (${\mathsf{P}}_{+}^{(2)}$, blue squares) and to first reach the negative threshold (${\mathsf{P}}_{-}^{(2)}$, red circles) as a function of the threshold value $s_{\mathrm{tot}}$. The analytical expressions for ${\mathsf{P}}_{+}^{(2)}$ (blue solid line), given by Eq. (47), and for ${\mathsf{P}}_{-}^{(2)}$ (red dashed line), given by Eq. (48), are also shown. Bottom inset: Logarithm of the ratio between the empirical passage probabilities ${\mathsf{P}}_{+}^{(2)}$ and ${\mathsf{P}}_{-}^{(2)}$ as a function of the threshold value $s_{\mathrm{tot}}$ (magenta diamonds). The solid line is a straight line of slope 1.

Figure 8

Empirical one-boundary first-passage-time distributions of entropy production to first reach a positive threshold $p_{T_{+}^{(1)}}(t;s_{\mathrm{tot}})$ (blue squares), and the rescaled distribution for the negative threshold $p_{T_{-}^{(1)}}(t;-s_{\mathrm{tot}})e^{s_{\mathrm{tot}}/k_{\mathrm{B}}}$ (red circles), for the Smoluchowski-Feynman ratchet with the potential (79) in a steady state. The estimate of $p_{T_{+}^{(1)}}(t;s_{\mathrm{tot}})$ [$p_{T_{-}^{(1)}}(t;-s_{\mathrm{tot}})$] is obtained by measuring the time when entropy production first reaches the single absorbing boundary $s_{\mathrm{tot}}=2.2k_{\mathrm{B}}$ ($-s_{\mathrm{tot}}=-2.2k_{\mathrm{B}}$) in $10^4$ simulations. The simulations are done with the same parameters as in Fig. 7, and the empirical probabilities are calculated over a total simulation time of ${\tau }_{\max }=20\mathrm{s}$. Top inset: Empirical passage probabilities of entropy production with a positive-threshold (${\mathsf{P}}_{+}^{(1)}$, blue squares) and with a negative-threshold (${\mathsf{P}}_{-}^{(1)}$, red circles) as a function of the values of the thresholds. For comparison, the expression for ${\mathsf{P}}_{+}^{(1)}$ given by Eq. (53) (blue solid line) and the expression for ${\mathsf{P}}_{-}^{(1)}$ given by Eq. (54) (red dashed line) are also shown. Bottom inset: Logarithm of the ratio between ${\mathsf{P}}_{+}^{(1)}$ and ${\mathsf{P}}_{-}^{(1)}$ as a function of the threshold value (magenta diamonds). The solid line is a straight line of slope 1.

Figure 9

First-passage statistics of entropy production with two absorbing and asymmetric boundaries at values $s_{\mathrm{tot}}^{+}=2s_{\mathrm{tot}}^{-}$ for the Smoluchowski-Feynman ratchet with potential (79) in a steady state. (a) The first-passage-time distribution $p_{T_{+}^{(2)}}(t;s_{\mathrm{tot}}^{+})$ (blue open squares) for entropy production to first reach $s_{\mathrm{tot}}^{+}=2k_{\mathrm{B}}$ given that it has not reached $-s_{\mathrm{tot}}^{-}=-k_{\mathrm{B}}$ before, and the first-passage-time distribution $p_{T_{-}^{(2)}}(t;-s_{\mathrm{tot}}^{-})$ (red open circles) to first reach $-s_{\mathrm{tot}}^{-}$ given that it has not reached $s_{\mathrm{tot}}^{+}$ before. The data are from $10^4$ simulations of the Smoluchowski-Feynman ratchet using the same simulation parameters as in Fig. 7. (b) Passage probabilities for entropy production ${\mathsf{P}}_{+}^{(2)}$ (blue filled squares) to first reach the positive threshold $s_{\mathrm{tot}}^{+}$ given that it has not reached $-s_{\mathrm{tot}}^{-}$ before, and the passage probability for entropy production ${\mathsf{P}}_{-}^{(2)}$ (red filled circles) to first reach the positive threshold $-s_{\mathrm{tot}}^{-}$ given that it has not reached $s_{\mathrm{tot}}^{+}$ before. We show the passage probabilities as a function of the negative threshold value $s_{\mathrm{tot}}^{-}$; the positive threshold is $s_{\mathrm{tot}}^{+}=2s_{\mathrm{tot}}^{-}$. The curves represent the analytical expressions for the passage probabilities given by Eqs. (5) and (6) (${\mathsf{P}}_{+}^{(2)}$, blue solid curve; ${\mathsf{P}}_{-}^{(2)}$, red dashed curve).

Figure 10

Waiting-time statistics between the states $X_{\mathrm{I}}$ and $X_{\mathrm{II}}$ as illustrated in panel (a) for the Smoluchowski-Feynman ratchet with the potential (79). In panel (b), simulation results are shown for the normalized waiting-time distributions of transitions $\mathrm{I}\to \mathrm{II}$ (blue squares) and of reversed transitions $\mathrm{II}\to \mathrm{I}$ (red circles). The simulations are done with the same parameters as in Fig. 7, and the distributions are obtained from $10^4$ trajectories. Inset: Logarithm of the ratio between ${\mathsf{P}}_{+}^{\mathrm{I}\to \mathrm{II}}$ and ${\mathsf{P}}_{-}^{\mathrm{II}\to \mathrm{I}}$ as a function of the environment-entropy change in the transition $\mathrm{I}\to \mathrm{II}$ (magenta diamonds). The transition probability ${\mathsf{P}}_{+}^{\mathrm{I}\to \mathrm{II}}({\mathsf{P}}_{-}^{\mathrm{II}\to \mathrm{I}})$ is the fraction of the trajectories starting from $X_{\mathrm{I}}$ $(X_{\mathrm{II}})$ at $t=0$ that reach $X_{\mathrm{II}}$ $(X_{\mathrm{I}})$ at a later time $t>0$ without returning to $X_{\mathrm{I}}$ $(X_{\mathrm{II}})$ before. The solid line is a straight line of slope 1. The data in the inset are obtained from $10^5$ simulations starting from state I and $10^5$ simulations starting from state II.

Figure 11

Coarse-graining procedure from a continuous Langevin process (top) to a discrete Markov process (bottom). The horizonal axis denotes either a chemical coordinate, which quantifies the progress of a chemical reaction, or a position coordinate, which quantifies the position of a molecular motor. In the Langevin description, this coordinate is described by the position of a Brownian particle (gray circle) moving in a periodic potential $V(X)$ of period $\ell$ (blue curve) and driven by an external bias $F=\mathrm{\Delta }G/\ell -F_{\mathrm{mech}}$. In the discrete process, this coordinate is described by the state of a Markov jump process with stepping rates $k_{+}$ and $k_{-}$, which are related to stopping times and passage probabilities of entropy production through $k_{+}=P_{+}^{(2)}⟨1/T_{+}^{(2)}⟩$ and $k_{-}=P_{-}^{(2)}⟨1/T_{-}^{(2)}⟩$. Our coarse-graining procedure corresponds to a partitioning of the coordinate axis into intervals (green shaded areas) centered on the minima of the potential (dashed lines).

Figure 12

Stochastic model for transcription. (a) Sketch of an elongating RNA polymerase. The polymerase can either incorporate a new nucleotide in its active site (red shaded square) or enter a backtracked state. (b) Sketch of a backtracking RNA polymerase. The polymerase diffuses along the DNA template until its active site (red shaded area) aligns with the 3’ end of the transcribed RNA. (c) Network representation of a Markov jump process describing the dynamics of RNA polymerases. The $X$ coordinate gives the position of the polymerase along the DNA template, and the $Y$ coordinate gives the number of nucleotides transcribed. Green nodes represent elongating states, and brown nodes represent backtracked states. The inset shows all possible transitions from or to an elongating state. The green solid arrow denotes active translocation of the RNA polymerase, and the dashed green arrow denotes its time-reversed transition. Because the heat dissipated during the forward translocation is much larger than $k_{\mathrm{B}}{\mathsf{T}}_{\mathrm{env}}$, the rate of the backward transition is very small. The blue arrow corresponds to the entry in a backtracked state, and the brown arrow is the exit from a backtracked state into the elongating state. In the backtracked state, motion is biased towards the elongating state.

Figure 13

The dynamics of RNA polymerase during backtracking. The motion of the polymerase is represented as a diffusion process in a tilted periodic potential with an absorbing boundary on the right (which corresponds to the transition from the backtracked state into the elongating state). The period of the potential is equal to the length of one nucleotide (nt). The potential is tilted towards the absorbing boundary due to an energetic barrier $\mathrm{\Delta }{V}_{\mathrm{RNA}}$, and an external opposing mechanical force $F_{\mathrm{mech}}$ pushes the polymerase away from the absorbing boundary. This process can be coarse grained into a discrete one-dimensional hopping process with waiting times $T_{+}^{(2)}$ and $T_{-}^{(2)}$.