AMC-SS: Transition Pathways and Free Energy Calculations in Complex Systems

New York University, New York NY

Investigators

Abstract

Ren DMS-0806401 In this project, the investigator develops numerical methods for the study of complex energy landscapes and barrier-crossing events (rare events) as well as free energy calculations in multiple-dimensional spaces. In the past few years, he and his collaborators developed a theoretical framework and numerical methods (the string method) for dealing with barrier-crossing events in complex systems. In the first part of the project, he applies the string method to problems arising from material sciences. In particular, the dislocation motions and the energy landscapes of silicon will be studied at the atomistic scale. The minimum energy paths at low temperature and the minimum free energy paths at finite temperature are investigated. In the second part of the project, the investigator studies the problem of mapping free energy landscapes in multiple-dimensional spaces. For problems arising from theoretical chemistry and structural biology, very often the interesting dynamics can be characterized by a number of collective variables. For example, conformation changes of bio-molecules are usually characterized by several torsion angles or distances between different chemical groups. The computation of the free energy associated with these collective variables is of great practical interest. Unfortunately such a calculation has been limited to situations when there are only a few collective variables (typically no more than 3), mainly due to the enormous number of grid points or bins involved in the calculation based on a regular grid. In this project, the investigator uses a sparse representation for the free energy function based on sparse grids. This to a certain extent overcomes the difficulty and allows the mapping of the free energy landscape in spaces up to about 30 dimensions. Many problems from structural biology, theoretical chemistry, and material sciences can be abstractly formulated as a system navigating over a complex energy landscape. The system spends most of time in metastable states (the local minima of the energy landscape), with thermally activated transitions from one metastable state to another. The questions of interest are the mechanism of such transition and the transition rates. These transitions happen on a time scale specified by Arrhenius' law, which is much longer than the intrinsic time scale of the dynamical system. For example, the individual atoms in a bio-molecule vibrate on the scale of femtoseconds (1e-15 s); in contrast, conformation changes of the molecule (which is the interesting event) typically happen on the scale of milliseconds (1e-3 s -- a trillion times longer than a femtosecond) or even longer. The disparity in the time scale makes traditional approaches (e.g. the direct simulation of molecular dynamics or Langevin dynamics) prohibitively expensive. In this project, the investigator develops efficient numerical tools (the string method) for the study of such transition events and the associated energetics. The numerical method is applied to investigate the dislocation motions in crystalline solids at the atomistic scale. The mapping of free energy landscapes of biological systems in multiple dimensions using sparse representations is studied as well.