# Methodological Development

## Long Timestep Integrators

Long timestep integrators for stiff oscillatory ODEs are found in classical mechanics. For the past decade, my group has produced theory that explains characteristics that good multiscale integrators for long time molecular dynamics (MD) simulations should have, such as preserving the geometric structure of the underlying Hamiltonian equations, as well as the reason why molecular dynamics works even in the presence of chaotic solutions that are overwhelmed by numerical error. We have also explained linear and nonlinear resonances that severely limit the time step possible for multiscale numerical integrators.

## Coarse Graining of Dynamical Systems

Our understanding of the solution of MD equations of motion has allowed us to pursue more aggressive techniques for lengthening the time step and allowing the study of time scales in the millisecond range. In particular, we have postulated the existence of an invariant density of normal modes for proteins, which provides a space that can be used to project the fine grained equations of motion. We have used the Mori-Zwanzig projection formalism to transform the ODE of Newton's equations of motion to a stochastic differential equation (SDE) that works in low frequency motion space. Our resulting methodology results in a speedup of 2 to 3 orders of magnitude over conventional MD, while preserving long time scales PDF and time correlations (the computational objectives as explained above). Short fine grained simulations that solve a time-dependent formulation of the Fokker-Planck PDE allow us to derive the kinetic parameters of our SDE. We have applied our dimensionality reduction technique to the construction of most probable paths between molecular states as well as to dramatically improve the convergence of Monte Carlo Markov Chain (MCMC) methods to sample the PDFs of molecular simulation.

## Construction and Analysis of Kinetic Networks

As previously mentioned, our main goal is to compute a network of states that interconvert in the dynamics of a protein, their populations, and the timescales of these transitions. A convenient framework for constructing such a network is to create Markov State Models (MSM) out of many simulations. MSM provide an intuitive framework for understanding the results of simulations, as well as providing many opportunities for analysis. We are studying methods to adaptively construct MSM that preserve detailed balance, applying graph theory to extract most probable paths and other kinetic information, and comparing to Nuclear Magnetic Resonance (NMR) and other experiments for validation.

## Numerical Analysis of Stochastic Differential Equations

Just as we have shown that numerical methods for Hamiltonian systems that can be interpreted in a backward-error way to be solving a perturbed, modified Hamiltonian have much better behavior at long times, we are studying the properties that methods to solve SDE should have. For example, statistical mechanics requires that our coarse-grained equations of motion satisfy fluctuation-dissipation in order to have an equilibrium distribution. We are studying the effect of time step and other parameters on kinetics and sampling, with preliminary results suggesting different requirements on the method depending on the application. This has led us to design superior integrators for SDEs and to make progress in the construction of computational error estimators.