Estimating Entropy Production from Waiting Time Distributions

Living systems operate far from thermal equilibrium by converting the chemical potential of ATP into mechanical work to achieve growth, replication, or locomotion. Given time series observations of intra-, inter-, or multicellular processes, a key challenge is to detect nonequilibrium behavior and quantify the rate of free energy consumption. Obtaining reliable bounds on energy consumption and entropy production directly from experimental data remains difficult in practice, as many degrees of freedom typically are hidden to the observer, so that the accessible coarse-grained dynamics may not obviously violate detailed balance. Here, we introduce a novel method for bounding the entropy production of physical and living systems which uses only the waiting time statistics of hidden Markov processes and, hence, can be directly applied to experimental data. By determining a universal limiting curve, we infer entropy production bounds from experimental data for gene regulatory networks, mammalian behavioral dynamics, and numerous other biological processes. Further considering the asymptotic limit of increasingly precise biological timers, we estimate the necessary entropic cost of heartbeat regulation in humans, dogs, and mice.

Figure 1

A Markovian process on discrete states (black circles), observed at a coarse-grained level, with only the metastates $A$ and $B$ visible to the observer (left). From observing this system, we can deduce the distribution of time spent in metastates $A$ and $B$ (right). A nonmonotonic waiting time distribution signals [41] out-of-equilibrium dynamics.

Figure 2

Bounding the EPR from a single limiting curve. (a) All realizable systems lie above the curve $\mathrm{\Lambda }$ (black) which represents the optimal variance-entropy production trade-off. Minimizing over increasing numbers of internal states leads to rapid convergence, with results shown for up to eight internal states (green curves). Given measured values of the waiting time variance, a lower bound on entropy production can be read off, as demonstrated here for experimental data in (b) (dashed line) and (c) (dotted line). (b) Genes stochastically switch from being active to inactive, mRNA is produced when the gene is active, proteins are produced when mRNA is present, and both stochastically degrade (left). The distribution of time the gene spends in the inactive state as measured in recent experiments, for the genes glutaminase and Bmal1 (right; see also Fig. 2C in Ref. [16]). (c) Distribution of time cows spent lying before standing across three experiments from Ref. [27].

Figure 3

Bounding the EPR of a model active sensor. (a) Cells sense through receptors on their surface, which detect concentration of some molecular species (black squares) by switching between unbound (top left) and bound (bottom left) configurations. We consider a simple model of a sensor, with four internal states for the bound receptor configuration [9]. Clockwise transitions occur at a rate $W_{+}$, and counterclockwise with rate $W_{-}$. (b) Distribution of time spent bound for $\sigma =0$, 1, 2, 3, 4 with $k_B=1=W_{+}$, $W_{-}\le 1$. (c) Estimates of the EPR as a function of the exact EPR $\sigma$, as calculated from 200 trajectories of length $T=2000$ for each value of $\sigma$. The ${\sigma }_T$ and TUR estimators are shown with 95% range of predictions.

Figure 4

EPR in the small variance limit. (a) Numerical optimization (green curves) is feasible only for small $\sigma$, although extrapolating to $N=\infty$ extends this range (purple line). Alternatively, the asymptotic limit of a continuous Langevin system extends beyond the range of finite networks (dashed line). Using the asymptotic relation, we can examine the entropic cost of building an oscillator as precise as a heartbeat across different species (gray dotted lines). (b) Typical electrocardiogram measurements of heartbeats for humans, dogs, and mice [33, 59]. (c) Distribution of time between beats, scaled so the mean time is 1, from a single experiment from each species.