Stochastic Thermodynamics of Cooperative Biomolecular Machines: Fluctuation Relations and Hidden Detailed Balance Breaking

TL;DR

This work addresses how cooperative biomolecular machines with hidden conformational dynamics conform to stochastic thermodynamics. By analyzing a generic kinetic model, it shows that the first-passage time fluctuation theorem for an observable process is recovered only when hidden flux through the initial state manifold vanishes (hidden detailed balance); violations of the theorem then signal hidden detailed balance breaking. An enzymatic model reveals that such violations arise from breakdowns of local detailed balance in steps between coarse-grained states, with mesoscopic local detailed balance restoring when hidden current is absent. The findings connect thermodynamic quantities to kinetic measurements, provide experimentally accessible signatures (e.g., first-passage-time PDFs and kinetic branching ratios), and illuminate how coarse-graining influences the thermodynamic description of driven cooperative networks.

Abstract

We examine a biomolecular machine involving a driven, observable process coupled to a hidden process in a kinetically cooperative manner. A stochastic thermodynamics framework is employed to analyze a fluctuation theorem for the first-passage time of the observable process under nonequilibrium steady-state conditions. Based on a generic kinetic model, we demonstrate that, along first-passage trajectories, entropy production remains constant when the changes in stochastic entropy and free energy of the machine are balanced, which corresponds to zero net hidden flux through the initial state manifold. Under this condition, which we define quite generally, this first-passage time fluctuation theorem can be established, with its violation serving as an experimentally detectable signature of hidden detailed balance breaking (which we subsequently characterize). In addition, using an enzymatic model, we show that the violation of our first-passage time fluctuation theorem can be thought of as a consequence of the breakdown of local detailed balance in the steps linking coarse-grained states that correspond to the initial and intermediate state manifolds. In the absence of hidden current, the fluctuation theorem is restored, and a mesoscopic local detailed balance condition can be established, which has implications for the thermodynamic analysis of driven, coarse-grained systems. This work sheds significant light on the unique connections between stochastic thermodynamic quantities and kinetic measurements in complex cooperative networks.

Figures (3)

• Figure 1: (a) Generic model for a biomolecular machine with kinetic cooperativity under NESS conditions. The machine undergoes a driven, observable, cyclic process cooperatively coupled to a hidden process with dynamics occurring within the state manifolds, $\left\{ B_{m}\right\}$, which have largely arbitrary internal topologies and connectivities (see text for details). Transitions between manifolds are designated here as $\left\{ \mathbf{K}_{\pm m}\right\}$. (b)--(c) Examples of underlying schemes corresponding to the generic kinetic model in (a). The rates of the transitions between the discrete states are represented by $\left\{ k_{\pm m}^{k,l}\right\}$ and $\left\{ \gamma_{k,l}^{\left(m\right)}\right\}$.
• Figure 2: (a) Adaptation of the generic kinetic model in Figure \ref{['fig:gen-model']}a to single-enzyme turnover with conformational interconversion. The enzyme reversibly catalyzes the conversion of a substrate to a product (see text for details). The reaction is cooperatively coupled to a hidden process, with the enzyme undergoing slow conformational changes within $B_{1}$ (state manifold for the free enzyme) and $B_{2}$ (state manifold for the substrate-bound enzymatic complex). The right-hand side is a representation of the scheme wherein the two steps in each reaction pathway are folded onto each other, resulting in a conformational loop with a corresponding population current $J$. (b) Depiction of two first-passage trajectories for the model in (a), each producing a different amount of entropy, with the blue one starting and ending in the same underlying state, and the red one doing so in different states.
• Figure 3: Plots of $p_{+}/p_{-}$ against $J$ (a) and $\gamma_{1}^{\left(1\right)}$ (b) and (c) for the model in Figure \ref{['fig:enz-model']}a (see text for details). In (a) and (b), the second-order estimation and the exact solution are shown; (c) shows the full range of the exact solution in (b). In all panels, since $w>0$, it is seen that the bound $p_{+}/p_{-}\leq\exp\left[\beta w\right]$ is obeyed.