High-barrier isomerization beyond the free-energy surface: AIMMD's symbolic committor reveals a stepwise S-shaped path in retinal and p-coumaric acid

The reaction coordinate of a high-barrier chemical reaction can now be learned from unbiased molecular dynamics trajectories and written as a readable analytical expression, not left inside a neural network. Previously committors were learned almost exclusively for diffusive biomolecular self-organization, where barriers are broad and momenta relax quickly; for high, narrow chemical barriers the committor is effectively a step function, and fitting only the pB=0.5 isocommittor surface would look accurate on most configurations while missing mechanistic information. In the retinal model, 9.1% of 32,727 generated paths were transition paths; against 438 starting structures from an independent AIMMD run with 200 two-way shooting paths each, neural-network predictions agreed closely with estimates over qB in [−4,4], corresponding to pB in [0.02,0.98].

The mechanism appears as a stepwise S-shaped path: near reactant and product the system moves mainly along the heavy-atom dihedral θ1, and near the transition state the hydrogen-bearing θ2 takes over. The minimum free-energy path in (θ1,θ2) space is a straight line connecting cis and trans, suggesting a concerted torsion; the transition-path ensemble instead gives an S-shaped path. In the retinal transition-path ensemble, θ1 reaches maximum velocity on either side of the transition state but is relatively slow at it, while θ2 has overall higher velocities and peaks in the immediate vicinity of the transition state; transition paths last 0.128 ± 0.054 ps on average, too short to thermalize with surrounding degrees of freedom.

Symbolic regression distills the neural-network committor into compact expressions containing only cosines of four proper dihedrals, and only the four-variable nonlinear coupling reproduces the S-shaped contours. A one-variable cos θ4 model and a two-variable sum of cos θ1 and cos θ2 align with the network at pB=0.5 but smooth the path into a straight line, losing the stepwise feature. The four-variable model's loss is about 8% lower than the two-variable model's, and in the visual comparison of predicted and sampled logit committor it is the only analytical model reaching accuracy comparable to the reference neural network; symbolic regression ran 10 times with the same loss function to observe repeating patterns.

Mass asymmetry was used to predict a second molecule: the p-coumaric acid anion has a larger effective-mass difference between its heavy-atom and hydrogen-bearing dihedrals, so its S-shape should be more pronounced. In retinal θ1 is defined by four heavy atoms and θ2 contains the hydrogen at C14; p-coumaric acid has a 4-hydroxyphenyl group on one side and a hydrogen on each alkene carbon, a larger mass difference. For the p-coumaric acid anion, AIMMD with a transfer-learned ANI-2x potential and 50,000 two-way shooting simulations yielded 7,479 reactive paths, and its (θ1,θ4) transition-path ensemble shows a pronounced S-shape, with velocity profiles again separating a heavy coordinate accelerating first from a light coordinate fastest at the barrier top.

The demonstrated scope is two cis-trans isomerizations in vacuo: N-retinylidene-lysine with an AMBER-type force field and the p-coumaric acid anion with a transfer-learned ANI-2x potential; the conclusions are limited to the configurational committor and these two systems.

Accuracy of the configurational committor does not imply inertia is negligible, and velocities near the transition state may still carry information; for p-coumaric acid the symbolic loss remains above the neural-network loss at low and intermediate complexity, and whether compact expressions are general awaits more chemically diverse reactions; extrapolation of the analytical forms lacks sampling to check it.