Pathway-resolved observable memory reveals hidden mediation from sparse conformational measurements

Biomolecular conformational transitions are often characterized by measuring a small number of observables, leading to memory functions due to the fact that the underlying dynamics involve many coupled and hidden degrees of freedom. Here, we discuss the extraction of effective memory from molecular simulation data with a transfer tensor approach. We then use a hierarchy of models to examine what mechanistic information can be gleaned from such extracted memory. An analytically solvable model that involves coupled harmonic oscillators shows how memory modes encode local and shared hidden-bath couplings in an ideal limit. A nonlinear double-well potential coupled bilinearly to a harmonic bath provides an analytical Zwanzig memory kernel, which we compare with transfer tensors extracted from trajectory data to distinguish physical bath friction from effective observable memory. A two-dimensional double-well pathway model then introduces a missing structural coordinate and shows that comparing observable sets can flag whether an omitted degree of freedom is structural or bath-like. With a three-coordinate conformational model, we then show that measuring multiple observables converts scalar memory analysis into a pathway-resolved description of delayed influence among measured coordinates. Variations in barrier height, bath timescales, observable set, and system-bath coupling patterns indicate that pathway power is more robust than assigning global memory periods to individual hidden modes. A hidden-mediator variant further highlights that pathway-resolved memory can reveal delayed communication produced by unobserved mediators, especially with the analysis of how pathway-resolved memory responds to perturbations. These results motivate observable-memory analysis as a framework for extracting mechanistic information from sparse experimental or simulation measurements, particularly in allosteric systems where hidden mediation and delayed communication are central to function.