Molecular Dynamics (2307.02176v2)

Abstract: While many good textbooks are available on Protein Structure, Molecular Simulations, Thermodynamics and Bioinformatics methods in general, there is no good introductory level book for the field of Structural Bioinformatics. This book aims to give an introduction into Structural Bioinformatics, which is where the previous topics meet to explore three dimensional protein structures through computational analysis. We provide an overview of existing computational techniques, to validate, simulate, predict and analyse protein structures. More importantly, it will aim to provide practical knowledge about how and when to use such techniques. We will consider proteins from three major vantage points: Protein structure quantification, Protein structure prediction, and Protein simulation & dynamics. We know that many proteins have functional motions, and in Chapter "Structure Determination" we already introduced the famous example of the allosteric cooperative binding of oxygen to the haem group in hemoglobin. However, experimentally, such motions are hard to observe. Here, we will introduce MD simulations to investigate the dynamic behaviour of proteins. In a simulation the forces and interactions between particles are used to numerically derive the resulting three-dimensional movement of these particles over a certain time-scale. We will also highlight some applications, and will see how simulation results may be interpreted.

• Molecular dynamics (MD) simulations use classical physics to simulate the dynamic behavior of large biomolecules like proteins and polymers by calculating particle forces over time.
• Accurate MD requires small timesteps (fs), resulting in high computational cost to reach biologically relevant timescales, necessitating experimental validation and efficiency improvements.
• MD relies on detailed force fields and integration schemes like Verlet, with advanced methods such as REMD available to enhance sampling for complex systems or studies like protein model optimization.

This document introduces molecular dynamics (MD) simulations as a tool to investigate the dynamic behavior of proteins and polymers by simulating the forces and interactions between particles to derive their three-dimensional movement over time.

Key concepts discussed include:

• MD simulations utilize classical or Newtonian physics to investigate the dynamics of large biomolecules.
• The importance of verifying mechanistic workings obtained from MD simulations with experimental evidence.
• The computational limitations of simulating biologically relevant timescales due to the small integration time steps required.
• The importance of the water environment to the behavior of proteins.
• Approaches that can be used to improve computational efficiency.

The document highlights the relevant time and length scales in molecular simulations. It notes that accurate simulations require time steps smaller than the fastest motions, typically 1-2 fs. The document points out that an order of $10^{12}$ integration steps are needed to reach biologically relevant timescales.

The force fields used in MD simulations are described in detail. The document specifies that the force on particle $i$ is $F_i = - \frac{\partial U}{\partial r_i}$, where $U$ is the energy and $r_i$ is the position. The total potential energy is:

$$U_{total} = U_{bonded} + U_{non-bonded} + U_{crossterm}$$

• $U_{bonded}$ includes $U_{bond}$, $U_{angle}$, and $U_{dihedral}$.
• $U_{non-bonded}$ includes $U_{Coulomb}$ and $U_{Van der Waals}$.

The document provides equations for each of these energy terms, such as:

• Bonds: $U(r ) = \tfrac{1}{2} k_b (r-r_0)^2$
  • $U(r)$ is the potential energy of the bond.
  • $k_b$ is the force constant of the bond.
  • $r$ is the length of the bond.
  • $r_0$ is the equilibrium length of the bond.
• Coulomb: $$U(r_{i j} ) = \frac{\epsilon_0 ( q_i \cdot q_j ) }{ r_{i j} }$$
  • $U(r_{ij})$ is the electrostatic potential energy between atoms $i$ and $j$.
  • $\epsilon_0$ is Coulomb's constant.
  • $q_i$ and $q_j$ are the partial charges of atoms $i$ and $j$, respectively.
  • $r_{ij}$ is the distance between atoms $i$ and $j$.

The Verlet integration scheme, a numerical method based on a Taylor expansion used to integrate the equations of motion in MD simulations, is presented.

The document also discusses the concept of convergence in molecular simulations, and introduces techniques such as umbrella sampling and replica exchange molecular dynamics (REMD) to enhance sampling. REMD involves running multiple replicas of the system at different temperatures to overcome energy barriers.

A case paper is presented where MD simulations were used to optimize a homology model of the enzyme Styrene mono-oxygenase (SMO). Essential Dynamics (ED) analysis was employed to track the progress of the simulations.