Optimization of nonequilibrium free energy harvesting illustrated on bacteriorhodopsin

Harvesting free energy from the environment is essential for the operation of many biological and artificial systems. We use techniques from stochastic thermodynamics to investigate the maximum rate of harvesting achievable by optimizing a set of reactions in a Markovian system, possibly under various kinds of topological, kinetic, and thermodynamic constraints. This question is relevant for the optimal design of new harvesting devices as well as for quantifying the efficiency of existing systems. We first demonstrate that the maximum harvesting rate can be expressed as a constrained convex optimization problem. We illustrate it on bacteriorhodopsin, a light-driven proton pump from Archaea, which we find is close to optimal under realistic conditions. In our second result, we solve the optimization problem in closed-form in three physically meaningful limiting regimes. These closed-form solutions are illustrated on two idealized models of unicyclic harvesting systems.

Figure 1

Left: Bacteriorhodopsin is a biomolecular free energy harvesting machine [18], illustrated in its trimer configuration by Goodsell (CC-BY-4.0) [19]. Right: during each turn of the bacteriorhodopsin photocycle, the molecule absorbs a photon and pumps a proton against the cell's membrane potential.

Figure 2

(a) Comparison of the actual harvesting rate ${\dot{\mathcal{G}}}^{\mathrm{tot}}$ at different electrical potentials $\mathrm{\Delta }\psi$, versus maximum rate $\mathcal{G}$ achieved by optimizing five intermediate transitions (color scheme from Fig. 1, right). (b) Efficiency ${\dot{\mathcal{G}}}^{\mathrm{tot}}/\mathcal{G}$ computed while separately optimizing each transition, with colors as in (a). (c) The actual steady state $\mathit{\pi }$ versus the optimal distribution ${\mathit{p}}^{*}$ when optimizing the $N\leftrightarrow O$ transition (at $\mathrm{\Delta }\psi =120$ mV).

Figure 3

(a) Comparison of the actual harvesting rate ${\dot{\mathcal{G}}}^{\mathrm{tot}}$ at different electrical potentials $\mathrm{\Delta }\psi$, versus maximum rate $\mathcal{G}$ achieved by fixing the bacteriorhodopsin cycle as baseline and allowing any additional transitions as control. (b) Efficiency of the actual bacteriorhodopsin cycle with respect to the optimized cycle. (c) The actual steady state $\mathit{\pi }$ and optimal distribution ${\mathit{p}}^{*}$ (at $\mathrm{\Delta }\psi =120$ mV).

Figure 4

(a) Unicyclic system where free energy $\mathrm{\Theta }$ is harvested by a single transition. (b) Unicyclic system where free energy per unit time $\theta$ is harvested when the system is in a particular mesostate.

Figure 5

(a) Maximum harvesting rate $\mathcal{G}$ for the unicyclic system from Fig. 4, as a function of supplied free energy $\mathrm{\Theta }$. Exact value is found numerically, LR, D, and ND are calculated using approximations described in the text. Exact and approximate optimal distributions ${\mathit{p}}^{*}$ in ND (b) and LR (c) regimes are shown, with the optimal state $i^{*}=1$ located in the middle of the histograms. (d) $\mathcal{G}$ and its LR approximation for different $\mathrm{\Theta }$ and system sizes $n$.

Figure 6

(a) Maximum harvesting rate $\mathcal{G}$ for the unicyclic system from Fig. 4, as a function of supplied free energy rate $\theta$. Exact value is found numerically, LR, D, and ND are calculated using approximations described in the text. Exact and approximate optimal distributions ${\mathit{p}}^{*}$ in ND (b) and LR (c) regimes are shown, with the optimal state $i^{*}=1$ located in the middle of the histograms. (d) $\mathcal{G}$ and its LR approximation for different $\theta$ and system sizes $n$.