Fig. 3: Peaked dwell-time densities are indicative of entropy production.

a Graph of the random process model for internal ON (or OFF) periods. Microstates are represented by circles. A blue rectangle indicates a mesostate due to coarse-graining. Labels on transition arrows are rate constants. b Steady-state entropy production per clockwise cycle, \({\Delta }_{c(+)}s\), and clockwise probability current, \({J}^{+}\), as a function of \(\alpha\), ranging from \(\alpha=0\), maximally irreversible process, to \({\alpha }_{{eq}}={\left({{w}_{32}w}_{21}{w}_{13}/{w}_{12}\right)}^{1/2}\), equilibrium process (which satisfies the cycle condition for detailed balance, \({{w}_{32}w}_{21}{w}_{13}={\alpha }^{2}{w}_{12}\)). The rate of entropy production (increase in total entropy per unit time) is given by the Schnakenberg equation⁵⁶, which for a cyclic steady-state process reduces to \(\sigma={J}^{+}{\Delta }_{c(+)}s\)¹¹ with \({\Delta }_{c(+)}s=2{k}_{{{{\rm{B}}}}}{{\mathrm{ln}}}\left({\alpha }_{{eq}}/\alpha \right)\) and \({J}^{+}={w}_{13}{p}_{3}-{w}_{31}{p}_{1}\), where \({p}_{i}\) is the steady state probability of state \(i={{\mathrm{1,3}}}\). Colored dots mark \(\alpha\)-values for which dwell-time densities in mesostate \(A\) are plotted in the panel below: c Steady-state dwell-time densities in mesostate \(A\) for selected \(\alpha\)-values. d Transition graph including states for both internal OFF periods (short-lived) and external OFF periods (long-lived). Labels on directed edges indicate the dependence of transitions on activator (Pho4) and/or remodelers (Isw2 and Chd1). The red arrow indicates the orientation of the net probability current in both the upper and lower cycles of the graph.