Harvesting chemical power from cyclic environments

TL;DR

The paper develops a thermodynamic framework to harvest chemical power from a cyclic environment by coupling a non-dilute reacting liquid to reservoirs that periodically swap solvent or reactant. It combines a continuum model with a reduced phase-equilibrium description, and maps the dynamics to two driven harmonic oscillators to reveal a resonance at $Ω_*$ that maximizes chemical power. The authors show how power and efficiency depend on cycle amplitude, reservoir and reaction rates, and whether the mixture undergoes phase separation, with wet-dry cycles resembling Carnot engines and reactant cycles resembling stochastic heat engines. The results suggest that cyclic environments could have powered early metabolic machinery, delivering up to about $10^1$–$10^2$ W per mole under favorable conditions, and offer design insights for molecular energy-harvesting systems and selection of reaction pathways in prebiotic settings.

Abstract

Life relies on a sophisticated metabolic molecular machinery that turns over high-energy molecules to evolve complex macromolecules and assemblies. At the molecular origin of life, such machinery was absent, implying the need for simple yet robust physical mechanisms to harvest energy from the environment and perform chemical work or produce chemical power. However, the mechanisms involved in harvesting energy from a macroscopic cyclic environment to drive chemical processes on the molecular scale remain elusive. In this work, we propose a theory that describes the kinetics of chemical reactions in a system subject to a cyclic reservoir with varying properties. We compare cycles of solvent (wet-dry cycles), with cycles of a component participating in a chemical reaction (reactant cycle). We find that for both wet-dry and reactant cycles, resonance frequencies exist at which the chemical power is maximal. We identify which cycle type is more beneficial in harvesting chemical power for different molecular interactions. Our findings of harvest efficiencies around ten percent suggest that the cyclic environment could have played a key role in catalyzing the metabolic molecular machinery at the molecular origin of life.

Figures (11)

• Figure 1: Harvesting chemical power from a cyclic reservoir. (a) A mixture with a chemical reaction is coupled to a cyclic reservoir. The reservoir periodically exchanges solvent (wet-dry cycles) or reactant (reactant cycles). Such components are either added (ii) or removed (i) oscillating between a state rich or poor in the component exchanged with the reservoir ("reservoir-rich/poor"). In certain conditions, the system can reach a non-dilute regime, where interactions among the components lead to liquid-liquid phase separation. (b) The cyclic property of the reservoir is governed by an oscillating chemical potential of the exchanging component. (c) We calculate the chemical power and find that there is a resonance frequency $\Omega_*$ at which the chemical power is maximal. (d) The value of the resonance frequency and maximal chemical power vary depending on the cycle type (wet-dry cycle or reactant cycle).
• Figure 2: Illustration of chemical work. To cycle the chemical potential $\mu^r(t)$ of the reservoir, work $\mathcal{W}^r$ is performed on the reservoir. As a result, molecules associated with the chemical potential $\mu_i^r$ are continuously exchanged with the mixture (blue shaded domain), with $\dot{N}^r$ as the exchange rate of such molecules. This exchange of molecules drives the mixture away from equilibrium. Thus, the chemical reaction (here between red and green molecules) is altered, leading to chemical work $\mathcal{W}^c$ performed by the chemical reactions. The harvest of energy or chemical power from the reservoir on the molecular scales is characterized by the efficiency $\eta=\mathcal{W}^c/\mathcal{W}^r$.
• Figure 3: Reactant and wet-dry cycles can induce oscillations of the mixture's composition with intermittent periods of phase separation. (a,d) Periodically exchanging molecular components with the reservoir can lead to intermittent periods of phase separation, highlighted by yellow domains. While the chemical potential of the mixture is maintained close to the oscillating value of the reservoir (top), the volume fractions of the components $A$ and $B$, exhibit oscillations containing multiple frequencies. (b,e) At any given moment along the trajectory, the mixture's composition experiences two effective driving fluxes: one from the external reservoir $\boldsymbol{h}_r\,\equiv\,(\bar{h}_A,\bar{h}_B)$ (black arrow) and another from chemical reactions $\boldsymbol{r}_c\,\equiv\,(r_A,r_B) = (-r,r)$ (orange arrow). Each of the force components points towards its respective equilibrium composition, which is shown by the black dot (reservoir equilibrium) or orange dot (chemical equilibrium), respectively. As a result, the system evolves in a net direction, i.e, $\boldsymbol{\dot\phi}_\text{tot}=\bm{h}_r\,+\,\bm{r}_c$ (Eq. \ref{['eq:phitotdot']}). (c,f) After multiple oscillations, the system's trajectory forgets its initial condition and settles into a stable and closed loop in the thermodynamic phase diagram, shortly referred to as "orbit".
• Figure 4: The oscillating chemical and reservoir equilibria act like compositional harmonic springs in the thermodynamic phase diagrams. (a) Decreasing the amplitude of reservoir cycles $\hat{\mu}_A$, decreases the orbit size. (b) Each point along the orbit (red line), system (open white symbol) experiences a net flux $\bm{\dot{\phi}}_\text{tot}$ which is tangent to the orbit. $\bm{r}_c$ is the flux due to chemical reaction. $\bm{h}_r$ is the flux from the reservoir mean chemical potential. Inset: Comparison between orbit obtained from numerical solution (red) and analytic calculation using the mapping on a two-spring model (black) agrees well with each other. (c) Same system and time point as in (b) but illustrated using a spring model with elongations $\delta \bar{\phi}_A$ and $\delta \bar{\phi}_B$. The harmonic springs are characterized by the spring matrices $\underline{\underline{\bm{\mathbf{K}}}}_c$ and $\underline{\underline{\bm{\mathbf{K}}}}_r$ related to chemical reactions and the reservoir. They exert effective forces directed toward the reservoir and the chemical equilibrium point at each. (d,e) The larger the amplitude of the respective oscillating reservoir $\hat{\mu}_A$, the larger the average chemical power $\mathcal{P}^c/(k_c k_\text{B}T)$, while the resonance frequency of the maximal power remains approximately unchanged. The data points are results from numerical simulations, while the solid lines represent analytic predictions using the two-spring model. (f) The maximum chemical power increases linearly with increasing oscillation amplitude.
• Figure 5: Larger exchange rates $k_r$ between reservoir and mixture enhance harvested chemical power. (a,b) There exists a resonance frequency that maximizes the chemical power. This resonance frequency is strongly affected by changing the reservoir kinetic rates. (c) Maximal chemical power increases and saturates for both types of reservoir cycles. Interestingly, there is a crossover below which reactant cycles lead to more chemical power, while above wet-dry cycles dominate.