08w5074 Mathematical and Numerical Methods for Free Energy Calculations in Molecular Systems

Arriving Sunday, June 15 and departing Friday, June 20, 2008

Organizers: Christophe Chipot (Universite Henri Poincare, CNRS), Eric Darve (Stanford University).

Confirmed Participants

Press Release: Mathematical and Numerical Methods for Free Energy Calculations in Molecular Systems

Information for Participants

Schedule and Abstracts (PDF file)

Final Report (PDF file)

Objectives

We are proposing 5 thrusts, one for each day of the workshop.

Thrust 1: symplectic integrators for improved molecular dynamics simulations. It is now well established that symplectic integrators are among the best integrators for molecular dynamics as they accurately preserve the energy of the system and produce accurate statistics over long time scales. E. Hairer and others (notably R. Skeel and J. Izaguirre) have shown that this could be proved mathematically by considering modified or shadow Hamiltonians of order N, H^N_h, such that the discrete numerical solution with time step h coincides nearly exactly with the exact trajectory with H_h^N. However many issues remain open such as: how can high-order modified Hamiltonians be numerically computed? Can they be used to study the stability of numerical methods for non-linear differential equations?

Thrust 2: non-equilibrium methods for equilibrium free energy calculations. C. Jarzynski recently proposed a new equality to calculate the equilibrium free energy difference between system A and B based on non-equilibrium simulations where the system is switched at constant speed from A to B. Previous methods such as slow growth only converge when the switching speed is very small, which is computationally very expensive in some cases. In contrast, Jarzynski's equality holds for an arbitrary switching speed. However, detailed mathematical analyses have revealed that in its current implementation this approach often suffers from a large statistical error. Consequently, several research questions remain open, such as: what is the optimal choice of parameters which will minimize this error? Are there cases in which this approach can be shown to be superior to equilibrium approaches? How can forward and backward switching trajectories be used to improve its efficiency?

Thrust 3: Error analysis, estimation of accuracy and modeling of the density of states. Despite the fact that methods to compute free energy have been around for decades, the reliability and efficiency of the approaches have not been considered in depth. A fundamental understanding of a free energy method's behavior is important not only for simulation practice but also for the development of new methodologies and comparison among methodologies. This thrust will attempt to address the following questions: what are the statistical error and/or systematic bias of a free energy method? How can the efficiency and reliability be improved? How is it possible to assess the quality of a computation when the true answer is unknown?

Thrust 4: Methods for enhanced ergodic sampling. One of the most important problems faced by free energy calculations is the existence of a broad range of energy barriers at multiple scales, both lower and higher than thermal energy. All calculations rely on the assumption that during the short time span of the simulation, the time average is close to the thermodynamic ensemble average. In many cases, as a result of the finite sampling, this assumption is broken and various regions of the conformational space become disconnected and the system gets trapped in metastable regions. It is therefore imperative to design methods which increase the rate of conformational sampling in such situations.

Thrust 5: Transition path sampling, ordering parameters v. reaction coordinates. Many important physical, chemical or biological processes occur on time scales that exceed those accessible by direct simulation. One approach to deal with this issue is to select a putative reaction coordinate from which free energy and related quantities can be obtained. In contrast, the transition path sampling method is a reaction coordinate-free method in which the ensemble of transition pathways is sampled using a Monte Carlo procedure. One obtains a set of dynamical pathways which can then be further analyzed to obtain information about the reaction mechanism. An interesting avenue of research is the application of transition pathway techniques to the non-equilibrium method of C. Jarzynski (see above). In particular, biased path sampling techniques can overcome some of the shortcomings of Jarzynski's method. Whether such an approach can yield a technique competitive with other free energy methods is an open question of current research.

The focus of the proposed workshop is clearly mathematical and numerical in nature. Contributions are expected to be presented from a theoretical perspective, rather than a mere application of an already well-established and characterized approach. The goal of the workshop is to provide a much clearer view on the range of applicability of free energy methods and on their inherent limitations. It is expected that the workshop will lead to new mathematical insights and methods, making free energy calculation an important tool for scientific discovery and allowing computer simulations to keep pace with rapidly developing experimental methods.