(581a) Effective Free Energy Landscape Exploration Using Coarse Terrain Methods

Detailed characterization of free energy landscapes for complex systems described by atomistic/stochastic simulators is a crucial issue in many contexts. ?Mapping? such landscapes typically involves identifying features such as transition states, local minima and their basins of attraction, and minimum energy paths between neighboring minima. Two significant challenges in this endeavor are (a) finding a suitable collection of ?correct? coarse observables (coordinates) in which to conduct this search, and (b) efficiently ?navigating? such a landscape.

Exploitation of smoothness in a ?correct? set of coarse observables is also an important feature of our approach. We discuss a computational technique (constructing diffusion maps on the simulation data) that automates the discovery of good observables (reaction coordinates) by performing eigen-processing of high-dimensional data from bursts of simulation. This approach allows for parameterization of a low-dimensional energy landscape for systems where experience and intuition do not suggest a suitable set of reaction coordinates and allows for coarse terrain exploration for systems where the reaction coordinates are unknown.

We extend our coarse reverse ring integration approach (useful for traveling uphill starting from the base of a local minimum) to incorporate a terrain search that facilitates intra-well exploration. Reverse ring integration uses short replica bursts of a (standard) forward-in-time simulator to provide estimates of coarse derivatives allowing a large backward-in-time (energy) ring step to be taken. Although this approach allows rapid escape from local energy wells and detects neighboring saddle points it is unsuitable for mapping transitions between minima since ring evolution stagnates at stationary points. Terrain methods allow reverse integration to proceed when such stagnation occurs by following valleys and ridges on the landscape, which connect stationary points. These landscape features are mathematically characterized by the stationary points of a suitably defined set of objective functions and provide candidate search directions to use in pursuit of neighboring minima.

We first examine reverse ring integration on a low-dimensional landscape. The evolving positions of ring replicas during reverse integration constitute "simulation protocols" for the forward-in-time (inner) simulator. Each integration step requires initialization of a detailed configuration consistent with the coarse observables at the ring nodes followed by a short forward replica ?burst?. Data processing of these simulation trajectories (using techniques such as Maximum Likelihood Estimation) provides estimates of coarse gradients along the ring, as well as the local coarse quantities required for terrain-method type effective computations. Estimation of tangent vectors along the ring is also required to decompose the gradient vector into components parallel and perpendicular to the ring. Reverse integration proceeds in a single well until a stationary point is detected. A terrain search, either uphill or downhill, is then invoked from the stationary point using eigen-information from either a primary (related to the gradient norm) or secondary objective function. A ?catalogue? of connections is constructed for each stationary point as the search proceeds to ensure ?revisiting? of previously explored minima does not occur. Upon location of a new, neighboring, stationary point, a new round of reverse integration can be initialized. We present results using such an approach for a stochastic (Gillespie) simulator and a system of stochastic differential equations.