Caliber Corrected Markov Modeling (C2M2): Correcting Equilibrium Markov models

TL;DR

C2M2 is a maximum entropy-based method for updating Markov models to correct observable rate predictions, by incorporating trajectory-based constraints and relating them to diffusion coefficients in continuous processes.

Contribution

It introduces C2M2, a novel approach that corrects Markov models using maximum entropy principles and connects corrections to diffusion coefficients in continuous stochastic processes.

Findings

C2M2 effectively corrects rate predictions in Markov models.

The method relates model corrections to position-dependent diffusion coefficients.

Illustrations include 2D diffusion and harmonic oscillator examples.

Abstract

Rate processes are often modeled using Markov-State Models (MSM). Suppose you know a prior MSM, and then learn that your prediction of some particular observable rate is wrong. What is the best way to correct the whole MSM? For example, molecular dynamics simulations of protein folding may sample many microstates, possibly giving correct pathways through them, while also giving the wrong overall folding rate, when compared to experiment. Here, we describe Caliber Corrected Markov Modeling (C2M2): an approach based on the principle of maximum entropy for updating a Markov model by imposing state- and trajectory- based constraints. We show that such corrections are equivalent to asserting position-dependent diffusion coefficients in continuous-time continuous-space Markov processes modeled by a Smoluchowski equation. We derive the functional form of the diffusion coefficient explicitly in terms of the trajectory-based constraints. We illustrate with examples of 2D particle diffusion and an overdamped harmonic oscillator.