A Machine-Learned Symbolic Committor for a Chemical Reaction: Isomerization in Retinal and p -Coumaric Acid

The thermal cis-trans isomerization of C═C double bonds in chromophores is a prototypical high-barrier reaction whose mechanism hinges on subtle out-of-plane bending motions. Combining machine-learning-guided path sampling (AIMMD) with symbolic regression, we learn an interpretable committor for two chemically distinct chromophores, N-retinylidene-lysine and p-coumaric acid anion, in vacuo from unbiased molecular dynamics trajectories generated by two-way shooting. Parametrizing the committor with a neural network and sampling it uniformly across the full range allows the model to resolve the reaction coordinate across the full transition region, not only at the isocommittor surface pB(x) = 0.5. A feature-attribution analysis identifies four proper dihedrals around the reactive bond as the informative coordinates, while the improper dihedrals at the sp2-hybridized carbon atoms prove unsuitable because reactant, transition, and product states share the same values. Symbolic regression then distills the committor into compact analytical expressions─to our knowledge the first committor both learned and distilled into closed form for a chemical reaction─and shows that a coupling of all four dihedrals is required to reproduce the S-shaped, stepwise pathway seen in the transition-path ensemble. This S-shape is absent from the minimum free-energy path: it arises from the dynamics of the short transition events combined with the mass asymmetry between heavy-atom and hydrogen-bearing dihedrals, and the larger asymmetry of p-coumaric acid yields the more pronounced S-shape this picture predicts. An interpretable, machine-learned committor, though a function of configuration alone, thus exposes dynamical features of the mechanism to which the free-energy surface is blind. The workflow requires no prior assumptions about the reaction coordinate and extends naturally to other high-barrier isomerizations and, combined with a reactive potential, opens a path toward bond-breaking chemical reactions.