Banff International Research Station
for Mathematical Innovation and Discovery
Free-energy calculations. A mathematical perspective
Workshop held from July 19 to 24, 2015 at the Casa Matemática Oaxaca, Mexico
Organized by Chris Chipot (Université de Lorraine and University of Illinois), Tony Lelièvre (École des Ponts, ParisTech) and Robert Skeel (Purdue University)
List of participants
To understand fully the vast majority of physical, chemical and biological processes, it is often necessary to examine their underlying free-energy behavior. Such is the case, for instance, of molecular recognition and association (e.g. binding of a drug to a target protein; partitioning of the same drug across the cell membrane). These processes, which are of paramount importance in a variety of fields, like rational drug design, cannot be predicted accurately without the knowledge of the associated free-energy changes. Reliable determination of free-energy changes using numerical simulations based on the fundamental principles of statistical mechanics is now within reach. Developments on the methodological fronts in conjunction with the continuous increase in computational power have brought free-energy calculations to the level of robust and well-characterized modeling tools, while widening their field of applications. In particular, development of robust numerical methods for determination of free-energy differences is connected to fundamental questions in various scientific fields, ranging from rare-event simulations, coarse-graining and reduced models to the thermodynamics of out-of-equilibrium systems. Much of these methodological developments would not have been possible without the contribution of mathematics.
For the most part, the statistical-mechanical framework for calculating free-energy differences has been developed several years ago [1-4]. How to apply this framework in computer simulations in an effective fashion remains, however, an effervescent research area, in which mathematics continues to be a prominent actor. In recent years, remarkable progress has been made in this field. Noteworthy examples include the development of methods for calculating free-energy differences between two states based on non-equilibrium dynamics  and by sampling paths connecting these states , great improvements in quantum-mechanical approaches to estimating free energies, clarifications of how to perform correctly thermodynamic integration employing constrained  and unconstrained dynamics [8,9], the construction of techniques that markedly reduce non-ergodicity problems in free-energy calculations  and the development of quasi-chemical approach to calculating the free energy. Furthermore, our understanding of conceptual connections between these different methods has improved markedly, as well as our ability to estimate statistical and systematic errors in molecular numerical simulations.
(1) L. D. Landau. Statistical physics. The Clarendon Press, 1938.
(2) R. Zwanzig. J. Chem. Phys. 1954, 22, 1420-1426.
(3) G. M. Torrie ; J. P. Valleau. J. Comp. Phys. 1977, 23, 187-199.
(4) C. H. Bennett. J. Comp. Phys. 1976, 22, 245-268.
(5) C. Jarzynski. Phys. Rev. Lett., 1997, 78, 2690-2693.
(6) P. G. Bolhuis, D. Chandler, C. Dellago, P. L. Geissler. Ann. Rev. Phys. Chem. 2002, 53, 291-318.
(7) W. K. den Otter. J. Chem. Phys. 2000, 112, 7283-7292.
(8) E. Darve, A. Pohorille. J. Chem. Phys. 2001, 115, 9169-9183.
(9) J. Hénin, C. Chipot. J. Chem. Phys. 2004, 121, 2904-2914.
(10) B. J. Berne, J. E. Straub. Curr. Opin. Struct. Biol., 1997, 7, 181-189.