Coarse Graining Photo-Isomerization Reactions: Thermodynamic Consistency and Implications for Molecular Ratchets

TL;DR

The work develops a thermodynamically consistent framework for coarse-graining photo-driven molecular systems, ensuring local detailed balance across both elementary and effective transitions. It introduces sequential coarse-graining steps—first collapsing vibrational states to electronic states, then collapsing excited electronic states to ground-state species—while preserving exact dissipation and LDB via a cycle-based construction and a modified matrix-tree theorem. The resulting effective photo-isomerization reactions obey LDB and enable a unified, information-ratchet description of both photo-driven and chemically driven molecular ratchets; directionality arises from flux differences of the free-energy-harnessing cycles rather than energetic biases. The theory connects experimentally accessible quantities (absorption spectra, quantum yields, photon chemical potentials) to both the dynamics and energetics of these systems, offering design principles that prioritize flux differentiation for achieving directed motion. Overall, the framework bridges photo-driven and chemically driven nonequilibrium thermodynamics, providing a general tool for analyzing efficiency and transduction in light-activated molecular machines.

Abstract

We formulate thermodynamically consistent coarse-graining procedures for molecular systems undergoing thermally and photo-induced transitions: starting from elementary vibronic transitions, we derive effective photo-isomerization reactions interconverting ground-state species. Crucially, the local detailed balance condition, that constrains reaction kinetics to thermodynamics, remains satisfied throughout the coarse-graining procedures. It applies to the effective photo-isomerization reactions just as it does to the elementary vibronic transitions. We then demonstrate that autonomous photo-driven molecular ratchets operate via the same fundamental mechanism as chemically driven ones. Because the local detailed balance remains satisfied, autonomous photo-driven molecular ratchets, like chemically driven ones, operate exclusively through an information ratchet mechanism. This reveals that their design and optimization should prioritize molecular properties governing the information ratchet mechanism, rather than those influencing energetic bias.

Figures (8)

• Figure 1: (a) Pictorial illustration of an ideal dilute solution maintained at constant temperature by the solvent (represented by a thermometer) and immersed in incoherent light generated by a light source (represented by a bulb). Two different molecular species (represented by disks and squares) can exist in different electronic (represented by different colors) and vibrational (represented by the different position of the thickest line) states. (b) Some transitions (arrows) between different vibrational and electronic states of two species. The first (resp. second) species exists in two electronic states $\alpha$ and $\beta$ (resp. $\alpha'$ and $\beta'$). Transitions interconverting states of the same species (vertical arrows) can be thermally induced (arrows close to a thermometer) and/or photo-induced (dashed arrows close to a bulb). Transitions interconverting states of different species (oblique arrows) can only be thermally induced (arrows close to a thermometer). Each transition has always a backward counterpart, even if it is not represented.
• Figure 2: Thermally induced (arrows close to a thermometer) and photo-induced (dashed arrows close to a bulb) transitions between two electronic states of the same species $\alpha$ and $\beta$. (a) Thermally and photo-induced transitions between (some of) the vibrational states (represented by horizontal lines) of the electronic states $\alpha$ and $\beta$. (b) All thermally induced (resp. photo-induced) transitions are combined together to form an effective thermally induced (resp. photo-induced) transition between the electronic states of the same species $\alpha$ and $\beta$. Each transition has always a backward counterpart, even if it is not represented.
• Figure 3: A species in the electronic ground state $\ch{Z}$ can be interconverted into its electronic excited state $\ch{Z^{*}}$ via a thermally induced (vertical arrow close to a thermometer) and a photo-induced (vertical dashed arrow close to a bulb) transition . The electronic excited state $\ch{Z^{*}}$ can also be interconverted via a thermally induced (oblique arrow close to a thermometer) transition into the the electronic ground state of a different species $\ch{E}$. Chemically speaking, the species $\ch{Z}$ and $\ch{E}$ are different isomers. Recall that no photo-induced transition can interconvert the ground state of species $\ch{E}$ into the excited state of a different species $\ch{Z^{*}}$ as a result of the Franck-Condon principle Balzani2014. Note that each transition has always a backward counterpart, even if it is not represented.
• Figure 4: Cycles (represented by sequences of arrows) for the case in Fig. \ref{['fig:HDM']}. Each cycle can be run in two directions: one direction is illustrated by the arrows while the other is the opposite direction. By following the sequences of transitions of each cycle, the excited-state species $\ch{Z^{*}}$ is never interconverted as it is always produced by one transition ad consumed by another. Note that cycle (a) can been obtained by running cycle (c) in the opposite direction with respect to the one determined by the arrows followed by cycle (b) (namely, cycle (a) is linearly dependent on cycle (b) and (c)). Note also that cycle (d) results from a sequence of two photo-induced transitions involving photons with different frequency (represented by different bulbs of different colors).
• Figure 5: Reactions between ground-state species corresponding to the cycles in Fig. \ref{['fig:HDM_cycles']}. Each reaction has always its backward counterpart even if it is not represented. Thermometers and bulbs specify whether the underlying transitions are thermally or photo-induced or a combination of both. (a) Isomerization reaction of $\ch{Z}$ into $\ch{E}$ via a sequence of only thermally induced transitions . (b) Isomerization reaction of $\ch{Z}$ into $\ch{E}$ via a sequence of both photo- and thermally induced transitions . (c) Futile reaction of $\ch{Z}$ into itself via a sequence of both photo- and thermally induced transitions . Its net effect is the interconversion of photons into heat. (d) Futile reaction of $\ch{Z}$ into itself via a sequence of only photo-induced transitions . Its net effect is the interconversion of photons with one frequency into photons with a different frequency. Recall that each reaction is defined by the sequence of transitions in Fig. \ref{['fig:HDM_cycles']} that involves the excited-state species $\ch{Z^{*}}$.