[Frontiers in Bioscience S5, 478-495, January 1, 2013]

A statistical mechanics handbook for protein-ligand binding simulation

Walter Rocchia1, Sara Bonella2

1Drug Discovery and Development, Istituto Italiano di Tecnologia, via Morego 30, 16163 Genova, Italy, 2Department of Physics, University of Rome "La Sapienza" and CNISM Unit 1, Ple A. Moro 2, 00185 Rome, Italy


1. Abstract
2. Introduction
3. Basics
3.1. Fixed energy systems
3.2. Microscopic versus thermodynamic description
3.3. Fixed temperature systems
3.3.1. Consequences of the Boltzmann distribution
3.3.2. Time evolution in the canonical ensemble
3.4. From microscopic quantities to macroscopic observables
3.5. Complementary material to section 3.
3.5.1. Derivation of Boltzmann distribution for the canonical ensemble
3.5.2. The Fokker-Plank equation
4. Tools and concepts for the description of the binding process
4.1. Role of the free energy and of the internal constraint
4.1.1. Obstacles to absolute free energy calculation
4.1.2. Free energy differences calculation
4.2. Free energy profiles and reaction paths
4.3. The definition of bound and unbound states and the reaction coordinate
4.4. Potential versus free energy surface
4.5. A didactic example
4.6. Volumetric effect on the unbound state
4.7. Complementary material to section 4.
4.7.1. The zero temperature limit for free energy
5. Conclusions
6. Acknowledgements
7. References


In this work, the fundamental elements of statistical mechanics underlying the simulation of the protein-ligand binding process, such as statistical ensembles and the concept of microscopic estimators of macroscopic observables and free energy, are summarized in a self consistent fashion. Particular attention is then devoted to the introduction of some mathematical tools that are used in atomistic simulations aimed at estimating binding affinities and free energy profiles, and to the illustration of the origins of the difficulties encountered in this endeavor.