A generalized work theorem for stopped stochastic chemical reaction networks

TL;DR

This work generalizes non-equilibrium work relations to stochastic chemical reaction networks (CRNs) by introducing a compensated Poisson description and a backward process $\psi(\mathbf{n},t)$ that defines a generalized entropy $\Sigma(t)$. A key result is that, for single-molecule CRNs, the exponentiated negative generalized entropy $e^{-\theta(t)}$ forms a martingale, yielding a generalized work theorem $\mathbb{E}[e^{-\theta(\tau)}|\mathbf{n}_0]=1$ for any bounded stopping time $\tau$. For multi-molecule CRNs, a multiplicity-corrected free energy $\tilde{E}(\mathbf{n})$ restores the martingale property with $e^{-\tilde{\theta}(t)}$, giving a generalized work relation in this broader setting. Numerical studies on kinetic proofreading networks illustrate how the generalized theorem accommodates stopping conditions and initial singularities, offering rigorous thermodynamic constraints for biological circuits. These results bridge stochastic thermodynamics and information theory in discrete, mesoscopic biochemical systems, with potential implications for energy-accuracy tradeoffs in signaling and regulation.

Abstract

We establish a generalized work theorem for stochastic chemical reaction networks (CRNs). By using a compensated Poisson jump process, we identify a martingale structure in a generalized entropy defined relative to an auxiliary backward process and extend nonequilibrium work relations to processes stopped at bounded arbitrary times. Our results apply to discrete, mesoscopic chemical reaction networks and remain valid for singular initial conditions and state-dependent termination events. We show how martingale properties emerge directly from the structure of reaction propensities without assuming detailed balance. Stochastic simulations of a simple chemical kinetic proofreading network are used to explore the dependence of the exponentiated entropy production on initial conditions and model parameters, validating our new work theorem relationships. Our results provide new quantitative tools for analyzing biological circuits ranging from metabolic to gene regulation pathways.

Figures (4)

• Figure 1: Schematic of a two-step kinetic proofreading network. The enzyme (E) can bind "correct" substrate S or "incorrect" S$^\prime$ with equal rates $k_{\rm on}$. Correct and incorrect enzyme-substrate complexes (e.g., ES and ES$'$) dissociate with rates $k_{\rm off}$ and $k_{\rm off}'$, respectively. Bound complexes can undergo two successive ATP-dependent activation (e.g., phosphorylation) steps, each with rate $k_+^*$. Reverse steps (e.g., dephosphorylation) occur at rate $k_-^*$. Free enzyme (including following release of activated substrate) can also undergo phosphorylation/dephosphorylation with rates $k_+$ and $k_-$, respectively. Activation (phosphorylation) levels of all species are indicated by the superscript.
• Figure 2: Numerical simulation of the single-enzyme two-step KPR model to evaluate $e^{-\sigma(T)}$ and $e^{-\theta(T)}$ up to a fixed duration $T=1$. (a) Mean and standard error of $e^{-\sigma(T)}$ (the standard Jarzynski form, red squares) and $e^{-\theta(T)}$ (from our chemical reaction work theorem, blue circles) versus $\log(1/\varepsilon)$. The initial distribution is $p_0(x) \propto \varepsilon + \delta_{x,\text{E}}$, approaching a singular $\mathrm{E}^{(0)}$ initial condition as $\varepsilon \to 0$ (i.e., $\log(1/\varepsilon) \to \infty$). Statistics are taken over $N=10^6$ samples. For the generalized theorem, $\psi(x,t)$ was set to the NESS distribution $p^*(x)$. (b) The same expected values and standard errors as a function of sample size $N$, for the strictly singular initial condition $p_0(x) = \delta_{x,\text{E}}$ (corresponding to $\varepsilon=0$). These results show the deviation from the Jarzynski equality (Eq. \ref{['eq:jarzynski_crn']}) and validate our generalized work theorem (Eq. \ref{['eq:generalized_jarzynski_crn']}). The parameter values used are $\beta=A_0=1$, $\mu_{\text{ATP}}=3.0$, $\mu_{\text{ADP}}=\mu_{\text{Pi}}=\mu_{\text{S}}=\mu_{\text{S}'}=0$, $E_{\text{E}}=0$, $E_{\text{ES}}=1.5$, $E_{\text{ES}'}=1.9$, and $\Delta E=1$ is the energy increase associated with each phosphorylation, e.g., $E_{\mathrm{E}^{(k)}\mathrm{S}} - E_{\mathrm{E}^{(k-1)}\mathrm{S}} = \Delta E$. These energy values correspond to $k_{+}/k_{-} = e^{-1}<1$ and $k_{+}^{*}/k_{-}^{*}=e^{2} >1$.
• Figure 3: Starting from a singular initial condition at $\mathrm{E}^{(0)}$, we simulated the single-enzyme KPR process up to a stopping time $\tau$ defined as the first passage time to any one of the fully phosphorylated states ($\mathrm{E}^{(2)}$, $\mathrm{E}^{(2)}\mathrm{S}$, or $\mathrm{E}^{(2)}\mathrm{S}'$), bounded by a maximum time $T=4$. Different values of the ATP chemical potential $\mu_{\rm ATP}$ are used. Evaluating $e^{-\sigma(\tau)}$ and $e^{-\theta(\tau)}$ over $N=10^6$ independent samples, we plot the mean and standard errors of $e^{-\sigma(\tau)}$ and $e^{-\theta(\tau)}$ to show the deviation from the Jarzynski equality (Eq. \ref{['eq:jarzynski_crn']}) and again validate the generalized work theorem (Eq. \ref{['eq:generalized_jarzynski_crn']}). Parameters values are set to those used in Fig. \ref{['fig:crn_fig1']} except that $\mu_{\rm ATP}$ is varied.
• Figure 4: Numerical validation of the KPR system shown in Fig. \ref{['fig:two_step_detailed_KPR']} when multiple enzyme molecules participate. In this example we set $\mu_{\rm ATP}=3$, use a uniform initial distribution, and evaluate the entropy produced up to fixed time $T=1$ over $10^6$ simulated realizations. This analysis was performed for systems with 1,2,3,4, and 5 enzyme molecules. The results validate the generalized work theorem in Eq. \ref{['eq:generalized_jarzynski_crn_v2']} based on the multiplicity-corrected free energy $\tilde{E}$.