Optimal Dimensionality Reduction of Multistate Kinetic and Markov-State Models

We develop a systematic procedure for obtaining rate and transition matrices that optimally describe the dynamics of aggregated superstates formed by combining (clustering or lumping) microstates. These reduced dynamical models are constructed by matching the time-dependent occupancy-number correlation functions of the superstates in the full and aggregated systems. Identical results are obtained by using a projection operator formalism. The reduced dynamic models are exact for all times in their full non-Markovian formulation. In the approximate Markovian limit, we derive simple analytic expressions for the reduced rate or Markov transition matrices that lead to exact auto- and cross-relaxation times. These reduced Markovian models strike an optimal balance between matching the dynamics at short and long times. We also discuss how this approach can be used in a hierarchical procedure of constructing optimal superstates through aggregation of microstates. The results of the general reduced-matrix theory are illustrated with applications to simple model systems and a more complex master-equation model of peptide folding derived previously from atomistic molecular dynamics simulations. We find that the reduced models faithfully capture the dynamics of the full systems, producing substantial improvements over the common local-equilibrium approximation.