Molecular Dynamics (MD) simulation is a powerful approach for the study of the kinetics and thermodynamics of biochemical systems with atomic-level resolution. However, detailed investigation of processes such as protein folding and enzyme kinetics is fraught with difficulties because the time scales spanned lie beyond those accessible with current MD simulators. Detailed characterization of free energy landscapes for such systems, namely the identification of transition states, local minima, and details of the topology of the “terrain” connecting neighboring minima remains challenging. Much of the computational time in standard MD is expended in local minima with transitions between minima occurring very rarely. We have developed a coarse MD approach for energy landscape exploration using reverse integration of a ring of replicas. Reverse ring integration is directed by short bursts of normal, forward in time MD simulation providing estimates of coarse derivatives and is applied here to the study of an Alanine Dipeptide fragment dissolved in water. Exploitation of smoothness in the “correct” coarse observables is a central feature of our approach. We couple our reverse ring integration scheme (suitable for low-dimensional energy landscape exploration) with a recently developed computational technique (constructing diffusion maps on the data) that automates the discovery of good reaction coordinates by performing eigen-processing of detailed MD data from simulation bursts.

Good coarse observables have been proposed for the Alanine Dipeptide system, namely dihedral angles along the backbone of the protein molecule. We first consider reverse ring integration on the low-dimensional landscape parameterized by these coarse observables. A number of distinct "modes" of ring evolution are derived using different transformations of the independent variable in the basic ring evolution equation. The evolving positions of ring replicas during reverse integration constitute new "simulation protocols". Each integration step consists of an initialization where detailed configurations consistent with the coarse observables at the ring nodes are prepared using constrained dynamics followed by a short forward replica “burst” of MD. Data processing of these MD trajectories (using techniques such as Maximum Likelihood Estimation) provides estimates of coarse gradients along the ring; estimation of tangent vectors along the ring is also required to decompose the gradient vector into components parallel and perpendicular to the ring. Our approach allows rapid escape from local energy wells with detection of neighboring transition states (saddle points). We also present strategies for saddle point escape that facilitate transition between wells using ring integration.

Parameterization of a low-dimensional energy landscape is extremely difficult for systems where experience and intuition do not suggest a suitable set of reaction coordinates. We use a recently developed computional machinery that allows for identification of low-dimensional manifolds possibly underlying high-dimensional data. In this diffusion map approach, MD simulation data points are treated as nodes on a weighted graph with edge weights defined by a matrix of pairwise affinities between points (a "kernel"). Appropriate kernel normalization produces a Markov matrix the eigenvectors and eigenvalues of which provide meaningful information on the dataset geometry. The dimensionality reduction approach is used here to process molecular configurations generated during an MD trajectory to generate "on-the-fly" reaction coordinates. Reverse ring integration in these coordinates is illustrated and represents a promising approach to landscape exploration for systems where the reaction coordinates are unknown. This work is in collaboration with Prof. R. Coifman and M. Maggioni at Yale University.