Variational cross-validation of slow dynamical modes in molecular kinetics

Markov state models (MSMs) are a widely used method for approximating the eigenspectrum of the molecular dynamics propagator, yielding insight into the long-timescale statistical kinetics and slow dynamical modes of biomolecular systems. However, the lack of a unified theoretical framework for choosing between alternative models has hampered progress, especially for non-experts applying these methods to novel biological systems. Here, we consider cross-validation with a new objective function for estimators of these slow dynamical modes, a generalized matrix Rayleigh quotient (GMRQ), which measures the ability of a rank-$m$ projection operator to capture the slow subspace of the system. It is shown that a variational theorem bounds the GMRQ from above by the sum of the first $m$ eigenvalues of the system's propagator, but that this bound can be violated when the requisite matrix elements are estimated subject to statistical uncertainty. This overfitting can be detected and avoided through cross-validation. These result make it possible to construct Markov state models for protein dynamics in a way that appropriately captures the tradeoff between systematic and statistical errors

Met-enkephalin MD Trajectories

<p>Ten ~50 ns molecular dynamics (MD) simulation trajectories of the 5 residue Met-enkaphalin peptide. The aggregate sampling is 499.58 ns. Simulations were performed starting from the 1st model in the 1PLX PDB file, solvated with 832 TIP3P water molecules using OpenMM 6.0.</p> <p>The coordinates (protein only -- the water was stripped) are saved every 5 picoseconds. Each of the ten trajectories is roughly 50 ns long and contains about 10,000 snapshots.  </p> <p>Forcefield: amber99sb-ildn; water: tip3p; nonbonded method: PME; cutoffs: 1nm; bonds to hydrogen were constrained; integrator: langevin dynamics; temperature: 300K; friction coefficient: 1.0/ps; pressure control: Monte Carlo barostat (interval of 25 steps); timestep 2 fs.</p> <p> </p