Interrelation between precisions on integrated currents and on recurrence times in Markov jump processes

TL;DR

The paper addresses how to relate the precision of the steady-state integrated current ${\cal N}_{\alpha\beta}(t)$ in a finite Markov jump process to the intrinsic timing precision of recurrence events ${\cal P}_{\alpha\beta}^\tau$ and ${\cal P}_{\beta\alpha}^\tau$. Using a moment-generating-function approach, the authors derive an explicit, numerically tractable expression for ${\cal P}_{\alpha\beta}^{\cal N}(t)$ that depends on the rectification efficiency $\epsilon$, the steady current $J_{\alpha\beta}$, and a time-dependent function $\gamma(t)$ built from conditional occupancies $\chi_{is_0}(t)$. They then establish a finite-time interrelation (Eq. corr) between ${\cal P}_{\alpha\beta}^{\cal N}(t)$ and the recurrence-time precisions ${\cal P}_{\alpha\beta}^\tau$, ${\cal P}_{\beta\alpha}^\tau$ for bidirectional channels and provide the one-directional limit, along with kinetic and thermodynamic bounds (including TUR-based inequalities). A four-site example illustrates the behavior and validates the bounds, highlighting the practical relevance for biochemical networks where finite-time and reversible dynamics matter. The results offer a framework for accurate numerical evaluation of precision in dynamical outputs and point to future work connecting the full distributions of currents and recurrence times.

Abstract

For Markov jump processes on irreducible networks with finite number of sites, we derive a general and explicit expression of the squared coefficient of variation for the net number of transitions from one site to a connected site in a given time window of observation (i.e., an `integrated current' as dynamical output). Such expression, which in itself is particularly useful for numerical calculations, is then elaborated to obtain the interrelation with the precision on the intrinsic timing of the recurrences of the forward and backward transitions. In biochemical ambits, such as enzyme catalysis and molecular motors, the precision on the timing is quantified by the so-called randomness parameter and the above connection is established in the long time limit of monitoring and for an irreversible site-site transition; the present extension to finite time and reversibility adds a new dimension. Some kinetic and thermodynamic inequalities are also derived.

Figures (2)

• Figure 1: a) Pictorial representation of the $\alpha \leftrightarrow \beta$ jumps; $k_{\alpha \to \beta}$ and $k_{\beta \to \alpha}$ are the jump rate constants for the specific channel under consideration. b) The precision coefficients concerning the $\alpha \leftrightarrow \beta$ channel. The circular dashed arrows stand for the repetition of the transitions $\alpha \to \beta$ or $\beta \to \alpha$; note that before a transition is repeated, the backward one (if feasible) could occur several times.
• Figure 2: a) The network chosen for the illustrative calculations. All site-site jumps are assumed to occur via single transition channels; the numbers close to the arrows are the values of the corresponding rate constants. b) Temporal profile of $t \, {\cal P}_{\alpha\beta}^{\cal N}(t)$ for $k_{1 \to 3} = 1$. c) Profiles of $(\epsilon J_{\alpha\beta})^{-1}$ and ${\cal T}_{\infty}$ (respectively, the short-time and long-time limits of $t \, {\cal P}_{\alpha\beta}^{\cal N}(t)$) varying $k_{1 \to 3}$. d) Illustration of the bound Eq. \ref{['eq_kinbound']} for $10^4$ randomly generated instances of the network (see the text for details); the dashed line has unit slope. In all cases, the maximum value on the ordinate axis was found to be $(\epsilon J_{\alpha\beta})^{-1}$ or ${\cal T}_{\infty}$. e) Illustration of the bounds Eq. \ref{['eq_b3']} (the dashed line has unit slope) and Eq. \ref{['eq_b4']} (spread on the abscissa, here truncated at the value 3).