A continuous-space analytical framework for committor functions from molecular dynamics

Many biomolecular processes are governed by rare transitions between metastable states on complex free-energy landscapes. The committor function, the probability that a trajectory initiated from a given configuration reaches one metastable state before another, is regarded as the optimal reaction coordinate for describing such transitions. Committor functions are often computed using Transition Path Theory (TPT) combined with Markov State Models (MSMs), which discretize configuration space into states and, therefore, limit spatial resolution. Here, we introduce an analytical framework for computing continuous-space committor functions directly from molecular dynamics trajectories using a Liouville propagator formulated in a basis-function representation. The key insight is that the slow eigenmodes of the propagator can be represented as linear combinations of basis functions, yet the resulting committor equations do not depend explicitly on the unknown expansion coefficients. Consequently, the Liouville propagator and committor functions can be constructed directly from an arbitrary set of basis functions without explicitly solving for eigenfunctions. This formulation enables continuous committor functions, iso-committor surfaces, and other kinetic quantities such as mean first passage times to be computed without discretizing configuration space. Applications to a two-dimensional model potential, alanine dipeptide, and the FIP35 WW domain demonstrate that the resulting committor functions agree with MSM-TPT results while providing substantially higher spatial resolution, enabling precise identification of transition states. Because our theoretical framework accommodates general basis representations, including polynomial or Fourier expansions of physical coordinates, tensor-based collective variables (CVs), and machine-learning-derived CVs, it provides a versatile foundation for analyzing dynamics in biomolecular systems and other complex dynamical processes.