Chemical Modelling Applications and Theory Volume 5

A Review of the Literature Published Between June 2005 and May 2007

By A. Hinchliffe

The Royal Society of Chemistry

Copyright © 2008 The Royal Society of Chemistry
All rights reserved.
ISBN: 978-0-85404-248-7

Contents

Preface Alan Hinchliffe, 7,
Multiscale modelling of biological systems Christopher J. Woods and Adrian J. Mulholland, 13,
Computer-aided drug design 2005–2007 Richard A. Lewis, 51,
Solvation effects Michael Springborg, 67,
The solid state E. A. Moore, 119,
Density functional theory studies of alloys in heterogeneous catalysis John R. Kitchin, Spencer D. Miller and David S. Sholl, 50,
Fluctuation relations, free energy calculations and irreversibility Debra J. Searles and Denis J. Evans, 182,
Many-body perturbation theory and its application to the molecular tructure problem S. Wilson, 208,
Experiment and theory in the determination of molecular hyperpolarizabilities in solution; pNA and MNA in dioxane David Pugh, 249,
The floating spherical Gaussian orbital (FSGO) method A. H. Pakiari, 279,
Advances in valence bond theory Peter B. Karadakov, 312,
Numerical methods in chemistry T. E. Simos, 350,

CHAPTER 1

Multiscale modelling of biological systems

Christopher J. Woods and Adrian J. Mulholland

DOI:10.1039/b608778g

1. Introduction

At what point does a collection of molecules become a biomolecular system? At what length scale does biology begin, and chemistry end? Biological phenomena involve the flow of information across a range of length and timescales. For example, a cell may be placed under physical stress at the macroscopic level, which causes an increase in pressure within its protective membrane. This pressure has the effect of opening or closing mechanosensitive ion channels, thereby changing the flow of individual ions into the cell. This changes the ionic concentration within the cell, which then acts as the trigger for a signal sent via a protein signalling pathway. A chemist would look at this as a molecular system that was capable of converting mechanical forces into electrical signals. A biologist would however look at this as the mechanism a cell uses to adapt to stress, and thereby stay alive. Biology is full of such examples. Every thought we have involves the passage of signals between neurons, which itself requires the conversion of electrical signals into flows of ions. These ions trigger the release of neurotransmitter molecules, which cross the synaptic gap between neurons, and bind to individual receptor proteins at the synapse. This causes a change in protein conformation, which open nearby ion channels, causing ions to rush in or out of the neuron, thereby continuing the signal. Information is constantly flowing between the macroscopic world and the atomic, chemical world. Indeed it is this interplay between the chemical and macroscopic worlds that is a real beauty of biology, and it is the recent advances made by the science of biochemistry that has revealed the elegance of the chemicals of life to all. However, while it is possible to use a microscope to watch how an individual cell responds to external stimuli, it is not possible to ‘zoom in’ further and observe what is occurring at the chemical level. Experiments can infer what is happening, and can provide supporting evidence for a particular hypothesis, but there is no experimental technique or microscope that allows us to watch a chemical reaction within an enzyme active site. Until such techniques are developed, the most appealing route that currently exists is to use computers to create models of the biochemical world. Computational scientists can create virtual enzymes, and models of cell membranes, and then use these to provide a window through which the interactions of biomolecules can be observed. If the models are constructed on the firm foundations of physics and chemistry, and if their predictions are carefully compared and validated against experiment, then simulations using these models can provide the valuable insight necessary to link the chemical and biological worlds.

Computational scientists have developed many tools for modelling molecules. Computer models are not perfect recreations of reality. Instead, approximations and assumptions have to made, and the model compromised for the sake of computational efficiency. As the size of the system gets larger, and so the size and number of molecules increases, so to does the computational expense of the calculation. This means that the larger the system, the more compromises and approximations must be made. This act of compromise has led computational scientists to develop four main levels of biomolecular modelling:

1. Quantum mechanics (QM). Quantum chemical calculations model the fine detail of the electrons in the molecule. They achieve this by modelling the electrons as a quantum mechanics wavefunction that interacts with the electrostatic potential field generated by the atomic nuclei. Quantum chemical calculations provide the most physically realistic and accurate models of molecules, but this accuracy comes at a cost. While methods have been developed that allow QM calculations on complete proteins, in general the high computational expense of QM methods limits their application to small molecular systems.

2. Molecular mechanics (MM). Atomistic molecular mechanics calculations apply the assumption that the fine detail information about the behaviour of the electrons can be ignored, and instead they are approximated by representing their effects using simple descriptors such as atomic partial charges or polarisabilities. By modelling the electrons implicitly, MM methods are much less expensive than QM methods, and so they are able to model significantly larger systems. By including atomic detail, MM models are still limited to the molecular level, and even today’s largest applications can only achieve the modelling of hundreds of thousands of atoms over hundreds of nanoseconds.

3. Coarse grain (CG). Coarse grain (or coarse grained) calculations apply the assumption that the fine detail information about the position of each atom in the molecule can be ignored, and instead groups of atoms are approximated by smearing them out into single ‘beads’. So, for example, rather than modelling each atom in a protein, a CG representation would portray each residue as a single bead. This approximation allows CG simulations to achieve length and timescales that are far beyond those possible using atomistic MM models.

4. Continuum. Continuum models apply the assumption that the fine detail information about the location of any particles or groups can be ignored, and instead systems are modelled as continuum regions. For example, implicit solvent models ignore the location of each individual solvent molecule, but instead represent the complete solvent as a fuzzy dielectric continuum. Equally, continuum models of a cell membrane ignore the individual locations of each lipid molecule, and instead model the membrane as a homogenous elastic sheet. By ignoring particles, and instead modelling biological systems as continuous fields or homogenous assemblies, continuum models are able to simulate the largest length scales and longest time-scales of any of the four levels.

These four levels of biomolecular modelling are each well-suited to modelling phenomena at the length and timescales for which they were designed. However, what makes biology work, and what makes it scientifically interesting, is the interplay and flow of information across the different length and timescales. It is not possible for simulations at any one of these biomolecular modelling levels to represent these complex, multiscale biological phenomena on their own, and so methods that allow the combination of different levels of biomolecular model together must therefore be sought. Multiscale modelling, in which calculations at multiple length and/or timescales are combined together into a single simulation, is now becoming popular, and its development is now the focus of significant research effort. Multiscale modelling is not new, for example combined QM/MM methods, and MM/continuum implicit solvent methods have been used for over 30 years, and multiscale methods have a rich heritage of applications in the fields of materials modelling and nanomaterials, and modelling fluid and gas flow. Recently, there has been a huge increase in the development and application of multiscale methods for biomolecular modelling. This review focuses on these developments, in particular the application of multiscale methods to biomolecules covering the period from 2005 to 2007. Coveney has produced a review of biological multiscale modelling that covers the period up to 2005.

Before starting this review, there first needs to be a definition of what is meant by a multiscale method. There are several different definitions that vary depending on the type of coupling between the different modelling levels. This review will adopt perhaps the most broad definition of a multiscale method, namely that it is any method that involves a flow of information from one modelling level to another. By definition, if there is a flow of information from one level to another, then there must be an interface between the levels through which this information will flow. Throughout this review it will become clear that there are four main classes of interface;

1. One-way, bottom-up interfaces. These involve a one-way, often one-time transfer of information from a lower level of modelling to a higher level. Examples include using a QM calculation to parameterize an MM forcefield, or using an MM simulation to parameterize a CG potential.

2. One-way, top-down interfaces. These involve a one-way transfer of information from a higher level of modelling to a lower level. Examples include using a CG model to reconstruct an atomistic model of a protein, or using a continuum model to provide the boundary conditions for an atomistic simulation.

3. Two-way parallel interfaces. These involve a two-way dynamic transfer of information between two simulations running in parallel at two different modelling levels. An example includes running both an MM and CG simulation of a system and using replica exchange moves to exchange coordinates between the two levels.

4. Two-way embedded interfaces. These involve embedding a low modelling level region within a simulation at a higher level, e.g. embedding a QM model of a substrate and active site within an MM model of the enzyme, or embedding an MM model of an ion channel within a CG model of a membrane.

This review is therefore organised according to the different interfaces between levels (QM/MM, atomistic/CG, particle/continuum), and then by the different classes of interface that are used between these levels.

2. Interfacing QM with MM models

The most accurate physical description of atoms and molecules is provided by quantum chemical calculations. Quantum chemical calculations are capable of correctly predicting the energetics and conformations of small molecules from first principles, using broadly applicable approximations (e.g. the Born-Oppenheimer approximation) and nothing more than fundamental physical constants as input. Quantum chemical calculations model electrons as a quantum mechanics (QM) wavefunction that interacts with the electrostatic potential field created by the atomic nuclei of the molecule. QM provides the most exact physical model of matter at the atomic scale, and QM calculations are capable of predicting chemical bonding and chemical reactivity. There are several recent reviews of quantum chemical methods, and QM methods may now be used across a length and timescale that ranges from modelling the femotosecond interactions of infrared laser light with carbon monoxide, to modelling the sub-nanosecond dynamics of a complete protein. There are a range of QM methods available with varying degrees of approximation, with a range that includes fast semi-empirical Hamiltonians such as AM1 or PM3, and highly exact coupled cluster methods such as LCCSD(T). Because QM methods include an explicit representation of electrons, they are able to model chemical processes such as charge transfer, bond breaking and formation, and changes of molecular polarisation. However, the high computational expense of QM methods prevents their application to the large length and timescales that are required to understand complex biomolecular processes.

MM methods provide a simpler representation of molecules, in which the fine detail of the electrons represented implicitly via partial charges and, is some cases, molecular polarisabilities. MM models represent molecules as a collection of atoms interacting through classical potentials. There are several MM models (or forcefields), and they differ in the functional forms of the interaction potential used between atoms, and in the means by which these interaction potentials are parameterized. Several good recent reviews of MM forcefields have been produced. Several MM forcefields have been developed for application to biomolecular systems. The most popular of these are the CHARMM, AMBER, GROMOS and OPLS forcefields. Each of these forcefields has evolved over time, with different versions produced periodically. However, despite the proliferation and evolution of MM forcefields for biomolecular modelling, their functional forms all remain broadly similar. Each atom is modelled as a single point in space. Pairs of atoms in separate molecules interact through a pairwise non-bonded potential, Enb, which depends only on the distance between the atoms, r. The electrostatic part of this non-bonded potential, Eelec, is modelled using Coulomb’s law, assigning a fixed partial charge to each atom in the molecule. The non-bonded potential must also model the van der Waals (vdW) forces between the molecules, which result from the combination of the Pauli repulsion that results from the inability of two electrons to occupy the same space with the same set of quantum numbers, and the attractive dispersion (London) forces whose physical basis lies in the ‘instantaneous dipoles’ that result from the wavefunctions of close atoms moving in phase. These vdW forces have their origin in the behaviour of electrons, which are not explicitly modelled in MM forcefields. These forces must therefore be approximated. The most common approximation used for biomolecular applications is the Lennard-Jones (LJ) potential. This approximates the vdW interactions using a 12–6 repulsive-attractive potential, ELJ,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

where ELJ(rij) is the Lennard-Jones energy between atom i and atom j, rij is the distance between the pair of atoms, and σij and εij are parameters that are tuned to reproduce the strength of the vdW forces between this pair of atoms, often by fitting to macroscopic properties. Note that this is a pairwise potential that acts only between pairs of atoms. This is despite the fact that unlike the permanent electrostatic forces, vdW forces are not pairwise in nature. Indeed, while permanent electrostatic forces are pairwise, MM forcefields use Coulomb’s law and fixed atomic partial charges to model both the permanent electrostatics of the molecule and, implicitly, its polarisation. Charge polarisation is also not a pairwise phenomenon. The non-bonded potential energy between two molecules is given by the sum of the Coulomb and LJ energies between all pairs of atoms in the molecules. It is therefore an effective pair potential, as the derivation of the partial charges and LJ parameters must account for the errors implicit in only using a pairwise sum over atoms, and must therefore include 3-, 4- to n-body effects implicitly.

Modelling the electronic detail of a molecule, as well as providing an explicit representation of polarisation and vdW forces, is also responsible for giving a correct representation of chemical bonding. As MM forcefields do not explicitly model electrons, they must include classical interaction potentials that mimic the effects of chemical bonding. MM forcefields include classical intramolecular interaction potentials, e.g. a harmonic bond potential, Ebond that acts between bonded atoms (called 1–2 atoms), a harmonic angle potential, Eangle that acts on the angle between a series of three bonded atoms (1–3 atoms), and a torsional potential, Etorsion, that acts about the dihedral formed by four bonded atoms (1–4 atoms). Atoms that are separated by more than three bonds (1–5 + atoms) are treated as being non-bonded, and so their interaction energy is calculated using the sum of their Coulomb and LJ interaction energies. The total intramolecular energy of a molecule is then given by the sum of the bond energy between all 1–2 atoms, the sum of the angle energy between all 1–3 atoms, the sum of torsion energy between all 1–4 atoms, and the sum of the non-bonded Coulomb and LJ energies between all pairs of 1–5+ atoms. The total energy of a system of molecules can then be calculated as the sum of the intramolecular energies of all of the molecules, together with the sum of the non-bonded potential energies between all pairs of molecules.

By using classical potentials, MM models allow for a very rapid evaluation of the energy and forces acting on each atom within a large biomolecular system. This rapid evaluation allows these forces and energies to be used by statistical conformational sampling methods, such as molecular dynamics (MD) or Monte Carlo (MC), to generate large ensembles of configurations of the system, from which macroscopic (thermodynamic) properties may be evaluated. However, by not explicitly modelling electrons, MM models struggle to model many chemically important phenomena, such as chemical bond breaking and formation, electronic polarisation and charge transfer. There is therefore a strong motivation to combine MM models with quantum mechanics (QM) calculations within a multiscale modelling framework, and combined QM/MM biomolecular simulation methods have a rich history of application and evolution since their original inception in the early 1970s.

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