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## Free energy calculations: From theory to applications

#### Ecole des Ponts, Champs-sur-Marne, France.

#### Organisers

To understand fully the vast majority of chemical processes, it is often necessary to examine their underlying free-energy behavior. This is the case, for instance, in protein-ligand binding and drug partitioning across the cell membrane. These processes, which are of paramount importance in the field of computer-aided, rational drug design, cannot be predicted reliably without the knowledge of the associated free-energy changes. The 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 contributed to bringing free-energy calculations to the level of robust and well-characterized modeling tools, while widening their field of applications. In particular, the development of robust numerical methods for free energy computations is strongly connected to fundamental questions in various scientific fields: rare event simulations, coarse-graining and reduced models, or the thermodynamic of out-of-equilibrium systems, for example.

Much of the statistical-mechanical framework for calculating free-energy differences has been developed some time ago [1-4]. How to apply this framework in computer simulations in an effective fashion remains, however, a very active area of research. 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 [5] and by sampling paths connecting these states [6], great improvements in quantum mechanical approaches to estimating free energies, clarifications of how to perform correctly thermodynamic integration employing constrained [7] and unconstrained dynamics [8,9], the construction of techniques that markedly reduce non-ergodicity problems in free-energy calculations [10] 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.

These advances justify the organization of a new CECAM workshop at the EPFL. From a historical perspective, the proposed workshop follows a long heritage of scientific events focused on free-energy calculations, made possible by the CECAM and initiated over 25 years ago in Amersfoort, the Netherlands by Herman Berendsen. Yet, rapid progress witnessed in the past decade in the arena of free-energy calculations was made possible by novel developments on both mathematics and statistical-mechanics

fronts. The proposed workshop builds on previous events organized by Chris Chipot and David Pearlman in Lyon in 2000, by Chris Chipot and Andrew Pohorille in Lyon in 2004, and by Éric Darve and Chris Chipot in Banff in June 2008, at a time where the CECAM was undergoing managerial reorganization. With four years separating the last event, the planned 2012 workshop, interdisciplinary by design, is expected to attract all current experts in the field to survey progress achieved recently, focusing not only on methodological and algorithmic issues, but also on relevant applications that contribute

to push the field forward.

Their contributions to the area of free-energy calculations [11,12] qualify the authors of the present proposal to provide an authoritative assessment of the current state of the art and organize a successful workshop.

(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.

(11) C. Chipot, A. Pohorille. Free energy calculations. Theory and applications in chemistry and biology. Springer, 2007.

(12) T. Lelièvre, M. Rousset, G. Stoltz. Free energy computations. A mathematical perspective. Imperial College Press, 2010.

To understand fully the vast majority of chemical processes, it is often necessary to examine their underlying free-energy behavior. This is the case, for instance, in protein-ligand binding and drug partitioning across the cell membrane. These processes, which are of paramount importance in the field of computer-aided, rational drug design, cannot be predicted reliably without the knowledge of the associated free-energy changes. The 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 contributed to bringing free-energy calculations to the level of robust and well-characterized modeling tools, while widening their field of applications. In particular, the development of robust numerical methods for free energy computations is strongly connected to fundamental questions in various scientific fields: rare event simulations, coarse-graining and reduced models, or the thermodynamic of out-of-equilibrium systems, for example.

Much of the statistical-mechanical framework for calculating free-energy differences has been developed some time ago [1-4]. How to apply this framework in computer simulations in an effective fashion remains, however, a very active area of research. 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 [5] and by sampling paths connecting these states [6], great improvements in quantum mechanical approaches to estimating free energies, clarifications of how to perform correctly thermodynamic integration employing constrained [7] and unconstrained dynamics [8,9], the construction of techniques that markedly reduce non-ergodicity problems in free-energy calculations [10] 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.

These advances justify the organization of a new CECAM workshop at the EPFL. From a historical perspective, the proposed workshop follows a long heritage of scientific events focused on free-energy calculations, made possible by the CECAM and initiated over 25 years ago in Amersfoort, the Netherlands by Herman Berendsen. Yet, rapid progress witnessed in the past decade in the arena of free-energy calculations was made possible by novel developments on both mathematics and statistical-mechanicsfronts. The proposed workshop builds on previous events organized by Chris Chipot and David Pearlman in Lyon in 2000, by Chris Chipot and Andrew Pohorille in Lyon in 2004, and by Éric Darve and Chris Chipot in Banff in June 2008, at a time where the CECAM was undergoing managerial reorganization. With four years separating the last event, the planned 2012 workshop, interdisciplinary by design, is expected to attract all current experts in the field to survey progress achieved recently, focusing not only on methodological and algorithmic issues, but also on relevant applications that contributeto push the field forward.

Their contributions to the area of free-energy calculations [11,12] qualify the authors of the present proposal to provide an authoritative assessment of the current state of the art and organize a successful workshop.

The proposed workshop will be decomposed into five different thrusts, one for each day of the event.

Thrust 1: Numerical analysis of free energy calculation methods.

In recent years, much progress has been made in the determination of free-energy differences in chemical and biological systems, which, to a large extent, has been facilitated by the emergence of a large variety of methods that have improved significantly both the efficiency and the reliability of the measured quantities. This progress has, however, come at a price. It is increasingly difficult for the practitioner to find its way through the maze of available computational schemes. Moreover, the reliability and efficiency of these techniques have not been examined in depth for all of them. A fundamental understanding of the numerical behavior of a free-energy method is, however, pivotal, not only from a practical perspective, but also for the development of new approaches. This theme sets out to address the following questions: What is the statistical error and the systematic, finite-length bias of a free-energy method? How can the efficiency and the reliability of free-energy calculations be improved? How is it possible to assess the quality of a computation when the exact answer is not known? How are the so many methods conceptually related? Which method is the best for a specific problem? We propose to address these questions by reaching a consensus on the best possible current practices for free-energy calculations.

Thrust 2: Non-equilibrium methods for equilibrium free-energy calculations.

Chris Jarzynski recently proposed a new equality to calculate the equilibrium free-energy difference between two states based on non-equilibrium simulations, wherein the reference state is switched at constant velocity towards the target one. 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, the Jarzynski equality holds for arbitrary switching velocities. Detailed mathematical analyses have, however, revealed that in its current implementation this approach often suffers from large statistical errors. Several questions remain, therefore, open, such as: What is the optimal choice of parameters likely to minimize the error? Are there cases, wherein this approach can be shown to be superior to equilibrium approaches? How can forward and backward trajectories be used routinely to improve its efficiency?

Thrust 3: Coarse-graining and reduced models.

Free energy is a pivotal quantity to build reduced models in statistical mechanics. It is indeed, in some sense, the effective energy associated to the considered reaction coordinate. A very important ans related question is: How to use the free energy to build dynamically consistent reduced models ? Techniques of coarse-graining are required both for computational reasons (to work with smaller systems) and for theoretical purposes (to get a better understanding on complex systems). Many techniques have been proposed in the literature, based on generalized Langevin equations, proper orthogonal decompositions, or the Mori-Zwanzig projection formalism, for example. This is still a very active field of research, and it would be interested to have an overview of the current techniques, with applications to chemical processes.

Thrust 4: Methods for ergodic sampling.

One of the most important shortcomings faced in free-energy calculations is the existence of a broad range of free-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, time-averages are close to thermodynamic-ensemble averages. In many cases, however, as a result of finite sampling, this assumption is not satisfied, regions of conformational space becoming disconnected and the system being trapped in metastable regions. It is, therefore, of paramount importance to design sampling schemes, which increase the rate of conformational sampling in such situations. Techniques based on adaptive biasing, for instance, have proven to be very useful and have recently benefited from a good mathematical understanding of the underlying assumptions ensuring their efficiency, which paves the road to propose improvements of classical approaches. A new research area concerns the development of algorithms to sample stationary measures for non-reversible dynamics (out-of-equilibrium systems).

Thrust 5: Transition path sampling.

Many important physical, chemical or biological processes span time scales that exceed significantly those amenable to molecular-dynamics simulations. A possible route to address this shortcoming is consists in selecting a putative reaction coordinate, from whence the free energy and other related quantities can be determined. In contrast, transition path sampling is a reaction-coordinate-free method, in which the ensemble of transition pathways is sampled using a Monte Carlo procedure. A set of dynamical pathways is obtained, 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 non-equilibrium methods. In particular, biased path sampling techniques can overcome some of the limitations of the Jarzynski identity. More generally, numerical techniques aimed at generating equilibrium trajectories from a metastable state to another one are needed for free-energy calculations, and to understand reactive paths.

## References

**France**

Tony Lelievre (INRIA and Ecole des Ponts ParisTech) - Organiser

**United States**

Chris Chipot (University of Illinois at Urbana-Champaign, USA) - Organiser

Andrew Pohorille (NASA) - Organiser