Reviews in Computational Chemistry, Volume 28

Abby L. Parrill, Kenny B. Lipkowitz, Abby L. Parrill, Kenny B. Lipkowitz

• Publisher: Wiley
• Year: 2015
• Print ISBN:
  9781118407776
• eBook ISBN:
  9781118889930
• Edition: 1
• Topic: Physical Sciences
• Subtopic: Physical & Theoretical Chemistry

The Reviews in Computational Chemistry series brings together leading authorities in the field to teach the newcomer and update the expert on topics centered around molecular modeling, such as computer-assisted molecular design (CAMD), quantum chemistry, molecular mechanics and dynamics, and quantitative structure-activity relationships (QSAR). This volume, like those prior to it, features chapters by experts in various fields of computational chemistry. Topics in Volume 28 include:

• Free-energy Calculations with Metadynamics
• Polarizable Force Fields for Biomolecular Modeling
• Modeling Protein Folding Pathways
• Assessing Structural Predictions of Protein-Protein Recognition
• Kinetic Monte Carlo Simulation of Electrochemical Systems
• Reactivity and Dynamics at Liquid Interfaces

Chapter 1
Free-Energy Calculations with Metadynamics: Theory and Practice

Giovanni Bussi,^a

Davide Branduardi^b

^aStatistical and Molecular Biophysics Group, International School for Advanced Studies (SISSA), Trieste, IT 34136, Italy

^bTheoretical Molecular Biophysics Group, Max Planck Institute of Biophysics, Frankfurt am Main D-60438, Germany

INTRODUCTION

Molecular dynamics (MD) is a powerful tool in modern chemistry that allows one to describe the time evolution of a computational model for a complex molecular system.^1–3 Typical models range from being highly accurate where energy and forces are computed with advanced and expensive quantum chemistry methods to faster but less accurate empirically parameterized force fields at atomistic or coarser resolution. The power of these techniques lies in their ability to reproduce experimental observable quantities accurately while, at the same time, giving access to the mechanistic details of chemical reactions or conformational changes at very high spatial resolution - typically at atomistic scale. For this reason, MD is often used to complement experimental investigations and to help in interpreting experiments and in designing new ones. Moreover, thanks to new parallelization algorithms and to the continuous improvements in computer hardware driven by Moore’s law, the range of application of these techniques has grown exponentially in the past decades and can be expected to continue growing.

In spite of its success, however, MD is still limited to the study of events on a very short timescale. Indeed, depending on the required accuracy and on the available computational resources, MD can provide trajectories for events happening on the timescale of picoseconds (quantum chemistry) to microseconds (empirical force fields). Thus, many interesting phenomena, namely, chemical reactions, protein folding and aggregation, and macromolecular rearrangement are still out of reach of direct investigation using straightforward MD trajectories. Besides the optimization of computer software (e.g., Ref. 4) and/or hardware (e.g. Refs. 5, 6), it is a possible complementary strategy to alleviate this issue by using algorithms where the time evolution is modified to sample more frequently the event under investigation. Then, appropriate postprocessing techniques are necessary to recover unbiased properties from the accelerated trajectories.

Many algorithms to accelerate MD simulations have been designed in the past decades, and a discussion of all of them is out of the scope of this chapter. Some of these algorithms are based on increasing the temperature of the simulated system (e.g., parallel tempering⁷ and solute tempering⁸), while others are based on exploiting an a priori knowledge of the investigated transition to design a proper order parameter to both describe and accelerate it. This last class includes umbrella sampling,⁹ adaptive biasing force,¹⁰ metadynamics,¹¹ self-healing umbrella sampling,¹² and other methods that keep the selected order parameters at an artificially high temperature.^13–15 This chapter focuses on metadynamics, which was first introduced in 2002¹¹ and then improved with several variants in the past decade. Metadynamics has been employed successfully in several fields, ranging from chemical reactions¹⁶ to protein folding¹⁷ and aggregation,¹⁸ molecular docking,¹⁹ crystal structure prediction,²⁰ and nucleation.²¹ A further push in the diffusion of metadynamics application has been its availability in a few widespread molecular dynamics codes^22–24 and in open-source plugins.^25–27

The main goal of this chapter is to provide an entry-level tutorial for metadynamics. In Section “Molecular Dynamics and Free-Energy Estimation” we provide an introduction to the basic concepts of molecular dynamics and of free-energy calculations. In Section “A Toy Model: Alanine Dipeptide” we introduce a toy model that will then be used for subsequent examples. Section “Biased Sampling” is devoted to the introduction of biased sampling. In Sections “Adaptive Biasing with Metadynamics” and “Well-Tempered Metadynamics” metadynamics is introduced, and Section “Metadynamics How-To” provides a practical how-to for performing a free-energy calculation with metadynamics. For all the simulations described in that section a sample input file for the open-source package PLUMED 2²⁶ is given in the Appendix. In the remaining sections, a quick overview of some of the latest improvements in the field is given, followed by a concluding section.

MOLECULAR DYNAMICS AND FREE-ENERGY ESTIMATION

Molecular Dynamics

In classical MD,^1–3 the Hamilton equations of motion are solved numerically to follow in real time the propagation of a collection of atoms. For a system of N_at atoms with coordinates q_i, momenta p_i, and masses m_i, a potential energy U(q) should be defined. Notice that we use here q, without subscript, meaning the full 3N_at-dimensional vector containing all the atomic positions. The Hamilton equations of motion will then read

1a

1b

Here with ẋ we mean the derivative with respect to the time of the variable x. The potential energy function U(q) describes the interatomic interactions. These interactions are sometimes defined in terms of empirically parameterized force fields, which provide a cheap and reasonably accurate approximation for U(q), and sometimes the interactions are obtained by solving the Schrödinger equation for the electrons (ab initio calculations), to allow studying phenomena su...

Table of contents

• Cover
• Title Page
• Copyright
• Table of Contents
• Preface
• List of Contributors
• Contributors to Previous Volumes
• Chapter 1: Free-Energy Calculations with Metadynamics
• Chapter 2: Polarizable Force Fields for Biomolecular Modeling
• Chapter 3: Modeling Protein Folding Pathways
• Chapter 4: Assessing Structural Predictions of Protein-Protein Recognition
• Chapter 5: Kinetic Monte Carlo Simulation of Electrochemical Systems
• Chapter 6: Reactivity and Dynamics at Liquid Interfaces
• Chapter 7: Computational Techniques in the Study of the Properties
• Chapter 8: The Quantum Chemistry of Loosely-Bound Electrons
• Index
• End User License Agreement