520028 VU Advanced molecular simulation: free energies, enhanced sampling and rare events (2024S)

A full quantitative understanding of physical, chemical and biological processes often requires knowledge of the underlying free energetics. For instance, phase diagrams of materials can be analyzed in terms of the Gibbs free energies of the different phases and the dissociation of protein/ligand complexes is governed by the binding free energy. While statistical mechanics provides a rigorous framework for the calculation of free energies, their computation with the help of molecular simulations such as molecular dynamics or Monte Carlo is often difficult as it involves the sampling of rare but important regions of configuration space. Analogous problems arise in the investigation of the kinetics and mechanics of rare transitions between long-lived states such as first order phase transitions or activated chemical reactions. Also in this case, rarely occurring microscopic state play a crucial role, determining the possibly very low rate at which the process occurs.

In this lecture course, we will discuss simulation methods developed for the calculation of free energies and the simulation of rare events. A tentative outline of the topics to be treated in the course is given below, but may be adapted according to the interests of the participants. The course essentially consists of three parts, each on building on the previous ones. The first part consists of a brief review of the statistical mechanics basics needed for the rest of the course. The second part will focus on methods for calculating free energies and in the third part various approaches for the study of the mechanism and kinetics of rare events will be discussed. The lectures will be accompanied by practical exercises in which the concepts discussed in the lectures will be applied to solve specific problems. Prerequisites: basic knowledge of statistical physics, some experience with molecular simulations (MD and/or MC), command of a higher programming language (e.g., C, C++, Python).

Table of contents:

1. Statistical mechanics in a nutshell: ensembles, free energies, collective variables and all that
1.1 Elements of probabiity
1.2 Statistical mechanical ensemble and free energies
1.3 Time evolution: molecular dynamics and Monte Carlo simulation

2. Methods for the calculation of free energies
2.1 Why is it difficulat to calculate free energies
2.2 Thermodynamic integration
2.3 Free energy perturbation
2.4 Umbrella sampling
2.5 Weighted histogram analysis methods (WHAM)
2.6 Bennett acceptance ration methods (BAR)
2.7 Replica exchange

3 Methods for studying rare events
3.1 Rare but important events
3.2 Rate equations
3.3 The reaction coordinate
3.4 Microscopic expression of the rate constant
3.5 Reactive flux
3.6 Transition state theory (TST)
3.7 TST in the harmonic approximation
3.8 RRKM theory
3.9 Dynamical corrections: reactive flux formalism

4 Transition path sampling
4.1 Introduction
4.2 Transition path ensemble
4.3 Sampling the transition path ensemble
4.4 The shooting algorithm
4.5 Analyzing transition pathways: reaction coordinate and committor
4.6 Kinetics from the transition path ensemble