Chemical modelling: applications and theory

Specialist Periodical Reports

Chemical Modelling: Applications and Theory comprises critical literature reviews of all aspects of molecular modelling. Molecular modelling in this context refers to modelling the structure, properties and reactions of atoms, molecules and materials. Each chapter provides a selective review of recent literature, incorporating sufficient historical perspective for the non-specialist to gain an understanding. With chemical modelling covering such a wide range of subjects, this Specialist Periodical Report serves as the first port of call to any chemist, biochemist, materials scientist or molecular physicist needing to acquaint themselves with major developments in the area.

Editor A Hinchliffe

Chemical Modelling: Applications and Theory Volume 5

ISBN 978-0-85404-248-7

www.rsc.org/spr

Hinchliffe

9 780854 042487

Chemical Modelling: Applications and Theory Volume 5

Specialist Periodical Reports Specialist Periodical Reports provide systematic and detailed review coverage in major areas of chemical research. Compiled by teams of leading experts in their specialist fields, this series is designed to help the chemistry community keep current with the latest developments in their field. Each volume in the series is published either annually or biennially and is a superb reference point for researchers.

Specialist Periodical Reports

Chemical Modelling Applications and Theory

Volume 5

A Specialist Periodical Report

Chemical Modelling Applications and Theory Volume 5 A Review of the Literature Published between June 2005 and May 2007 Editor A. Hinchliffe, School of Chemistry, The University of Manchester, Manchester, UK Authors D. J. Evans, Australian National University, Canberra, Australia P. B. Karadakov, University of York, York, UK J. R. Kitchin, Carnegie Mellon University and National Energy Technology Laboratory, Pittsburgh, PA, USA R. A. Lewis, Novartis Institutes for Biomedical Research, Basel, Switzerland S. D. Miller, Carnegie Mellon University, Pittsburgh, PA, USA E. A. Moore, The Open University, Milton Keynes, UK A. J. Mulholland, University of Bristol, Bristol, UK A. H. Pakiari, Shiraz University, Shiraz, Iran D. Pugh, University of Strathclyde, Glasgow, UK D. J. Searles, Griffith University, Brisbane, Australia D. S. Sholl, Carnegie Mellon University and National Energy Technology Laboratory, Pittsburgh, PA, USA T. E. Simos, University of Peloponnese, Tripolis, Greece M. Springborg, University of Saarland, Saarbru¨cken, Germany S. Wilson, University of Oxford, Oxford, UK and Comenius University, Bratislava, Slovakia C. J. Woods, University of Bristol, Bristol, UK

If you buy this title on standing order, you will be given FREE access to the chapters online. Please contact [email protected] with proof of purchase to arrange access to be set up. Thankyou.

ISBN 978-0-85404-248-7 ISSN 1472-0965 A catalogue record for this book is available from the British Library r The Royal Society of Chemistry 2008 All rights reserved Apart from any fair dealing for the purpose of research or private study for non-commercial purposes, or criticism or review as permitted under the terms of the UK Copyright, Designs and Patents Act, 1988 and the Copyright and Related Rights Regulations 2003, this publication may not be reproduced, stored or transmitted, in any form or by any means, without the prior permission in writing of The Royal Society of Chemistry, or in the case of reprographic reproduction only in accordance with the terms of the licences issued by the Copyright Licensing Agency in the UK, or in accordance with the terms of the licences issued by the appropriate Reproduction Rights Organization outside the UK. Enquiries concerning reproduction outside the terms stated here should be sent to The Royal Society of Chemistry at the address printed on this page. Published by The Royal Society of Chemistry, Thomas Graham House, Science Park, Milton Road, Cambridge CB4 0WF, UK Registered Charity Number 207890 For further information see our web site at www.rsc.org Typeset by Macmillan India Ltd, Bangalore, India Printed by Henry Ling Ltd, Dorchester, Dorset, UK

Preface DOI: 10.1039/b801788n Welcome to Volume 5 of the ‘Chemical Modelling’ SPR. Naturally, I want to start by thanking my team of Reporters for the hard work they have put into making this the best and most comprehensive volume so far. We hope you will derive benefit and perhaps even pleasure from our efforts. It seems a long time since I wrote the following in my Preface to Volume 1 (1999) . . . ‘Starting a new SPR is never easy, and there was the problem of where the contributors should start their accounts; since time began? five years ago? An SPR should be the first port of call for an up-to-the-minute account of trends in a specialist subject rather than a dull collection of references. My solution was to ask contributors to include enough historical perspective to bring a non-specialist up to speed, but to include all pertinent references through May 1999. Volume 2 will cover the literature from June 1999 to May 2001 and so on. In subsequent Volumes, I shall ask those Contributors dealing with the topics from Volume 1 to start from there. New topics will be given the same generous historical perspective opportunity as Volume 1 but will have to cover the literature to 2001 + n where n = 0, 2, 4, . . . . This process will continue until equilibrium is reached.’ Equilibrium was reached a couple of Volumes ago and some mature topics don’t need cover every Volume. My final sentence for Volume 1 was ‘I am always willing to listen to convincing ideas for new topics’ as indeed I am. Alan Hinchliffe Manchester 2008 alan.hinchliff[email protected]

RSC Specialist Periodical Report Chemical Modelling: Applications and Theory (Volume 5) Chapter 1 2 3 4 5

Corresponding author Dr Adrian Mulholland Dr Richard Lewis Prof Michael Springborg Dr Elaine A Moore Prof David Sholl

6

Prof Debra Searles

7

Dr Steven Wilson

8

Dr David Pugh

9

Prof Ali Pakiari

10 11

Dr Peter Karadakov Prof Theodore Simos

Topic Multiscale modelling of biological systems Computer-aided drug design 2005–2007 Solvation effects The solid state DFT studies of alloys in heterogeneous catalysis Fluctional relations, free energy calculations and irreversibility Many body perturbation theory and its application to the molecular structure problem Experiment and theory in the determination of molecular hyperpolarizabilities in solution The Floating Spherical Gaussian Orbital (FSGO) method Advances in valence bond theory Numerical methods in chemistry

Chem. Modell., 2008, 5, 7–8 | 7 This journal is

c

The Royal Society of Chemistry 2008

CONTENTS Cover The icosahedral ’golden fullerene’ WAu12 reproduced by permission of Pekka Pyykko¨, Chemistry Department, University of Helsinki, Finland.

Preface Alan Hinchliffe

7

Multiscale modelling of biological systems

13

Christopher J. Woods and Adrian J. Mulholland Introduction Interfacing QM with MM models Interfacing atomistic with coarse grain models Interfacing particle with continuum models Beyond continuum models Conclusion

13 15 26 41 45 46

Computer-aided drug design 2005–2007 Richard A. Lewis Introduction QSAR and ADMET Structure-based drug design Virtual screening De novo structure generation Fragment-based screening

51 51 51 54 56 60 61

Chem. Modell., 2008, 5, 9–12 | 9 This journal is

c

The Royal Society of Chemistry 2008

Target fishing Library design Conclusions

62 63 64

Solvation effects

67

Michael Springborg Introduction Fundamental methods Recent studies Conclusions

67 68 75 114

The solid state E. A. Moore Introduction Interatomic potential methods Ab initio methods QM/MM Molecular dynamics and related methods Properties Applications Minerals Conclusions

119

Density functional theory studies of alloys in heterogeneous catalysis

150

John R. Kitchin, Spencer D. Miller and David S. Sholl Introduction Segregation Adsorption properties on alloy surfaces Reaction kinetics Miscellaneous Electrocatalysis Conclusions

150 153 158 166 170 172 176

Fluctuation relations, free energy calculations and irreversibility

182

Debra J. Searles and Denis J. Evans Introduction Fluctuation relations Free energy relations

182 184 191

10 | Chem. Modell., 2008, 5, 9–12 This journal is

c

The Royal Society of Chemistry 2008

119 121 124 130 131 133 134 141 142

Fluctuation relations and irreversibility Comment on the interpretation of the fluctuation relation Conclusions

201 201 202

Many-body perturbation theory and its application to the molecular structure problem

208

S. Wilson Introduction An overview of previous reports Applications An overview of applications of second order theory Application area 1: periodic systems Application area 2: DNA bases and amino acids Application area 3: DFT benchmarking Application area 4: basis set extrapolation and the calibration of general energy models Summary and future directions

208

Experiment and theory in the determination of molecular hyperpolarizabilities in solution; pNA and MNA in dioxane David Pugh Introduction General theory of the response to frequency-dependent electric fields The sum over states method General theory of the EFISH experiment Ab initio and DFT calculations of the pNA b tensor Gas phase measurement Solution EFISH studies of pNA and MNA in dioxane Theoretical approaches to the calculation of the EFISH nonlinearity of pNA in solution Recent work on pNA/MNA Appendix I. Conversion of units

249

The floating spherical Gaussian orbital (FSGO) method A. H. Pakiari Introduction Part I Part II, Development of original theory Part III, Application of FSGO method

279

208 209 211 230 237 238 241 242 243

249 251 253 254 257 260 261 271 275 276

279 280 282 296

Chem. Modell., 2008, 5, 9–12 | 11 This journal is

c

The Royal Society of Chemistry 2008

Part IV, Using the FSGO concept in other methods Appendix

298 307

Advances in valence bond theory

312

Peter B. Karadakov Introduction Comparison of the MO and VB approaches Developments in VB methodology Applications of VB theory Concluding remarks

312 313 316 328 346

Numerical methods in chemistry T. E. Simos Newton–Cotes formulae for the numerical integration of the Schro¨dinger equation Stabilization of a multistep exponentially-fitted methods and their application to the Schro¨dinger equation Appendix A Appendix B Appendix C Appendix D Appendix E Appendix F Appendix G Appendix H Appendix I

12 | Chem. Modell., 2008, 5, 9–12 This journal is

c

The Royal Society of Chemistry 2008

350 350 380

399 408 421 427 439 444 453 464

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 opening1 or closing2 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 Centre for Computational Chemistry, School of Chemistry, University of Bristol, Bristol, UK BS8 1TS

Chem. Modell., 2008, 5, 13–50 | 13 This journal is

c

The Royal Society of Chemistry 2008

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,3 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 timescales 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,4 and modelling fluid and gas flow.5,6 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. Coveney7 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 14 | Chem. Modell., 2008, 5, 13–50 This journal is

c

The Royal Society of Chemistry 2008

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 exchange8–10 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.11 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,11–13 and QM methods may now be used across a length and timescale that ranges from modelling the femotosecond interactions of infra-red laser light with carbon monoxide,14 to modelling the sub-nanosecond dynamics of a complete protein.15 There are a range of QM methods available with varying degrees of approximation, with a range that includes fast semi-empirical Hamiltonians such as AM116 or PM3,17,18 and highly exact coupled cluster methods such as LCCSD(T).19 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.20,21 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.22–26 Several MM forcefields have been developed for application to biomolecular systems. The most popular of these are the CHARMM,27 AMBER,28,29 Chem. Modell., 2008, 5, 13–50 | 15 This journal is

c

The Royal Society of Chemistry 2008

GROMOS30–32 and OPLS33,34 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, "   6 # sij 12 sij ELJ ðrij Þ ¼ 4eij ; ð1Þ rij rij where ELJ(rij) is the Lennard-Jones energy between atom i and atom j, rij is the distance between the pair of atoms, and sij and eij 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 nonbonded 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 16 | Chem. Modell., 2008, 5, 13–50 This journal is

c

The Royal Society of Chemistry 2008

rapid evaluation allows these forces and energies to be used by statistical conformational sampling methods, such as molecular dynamics (MD)35,36 or Monte Carlo (MC),36,37 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.38,39 Using a broad definition, QM/MM multiscale methods are those that involve a transfer of information across an interface between the QM and MM levels of modelling. There are several different types of interface in use, which fall into four categories: 1. One-way bottom-up methods. These involve a single transfer of information from a QM calculation to a classical simulation, e.g. by using QM calculations to parameterize the classical potentials used in MM force fields. 2. Two-way dynamic parameterisation methods. These involve a dynamic transfer of information between separate classical and quantum calculations, e.g. using successive QM calculations to dynamically re-parameterize the classical potentials the QM atoms of an MM forcefield during a live simulation. 3. Two-way embedded methods. These involve embedding of molecules or parts of molecules modelled using QM into a system of molecules modelled using MM, e.g. using QM to model a substrate, and MM to model the enzyme and solvent. 4. Two-way parallel methods. These involve the running in parallel of classical and quantum simulations, and dynamically sharing information between them at run time. Examples of each of these different types of interface, and recent developments in their methodology, will now be discussed in turn. 2.1 One-way bottom-up QM/MM interfacing methods As described in the last section, molecular mechanics (MM) forcefields use classical potentials to calculate the interaction energy between pairs of atoms. Several types of interaction potential are required to fully describe the MM energy of a set of molecules: 1. Non-bonded potentials. These typically take the form of a Coulomb potential between non-bonded pairs of atoms to describe polarisation and permanent electrostatics, and a Lennard-Jones (LJ) potential between non-bonded atom pairs to describe the vander Waals (vdW) interactions. 2. Bonded potentials. These typically involve harmonic terms that are applied between 1–2 bonded and 1–3 bonded pairs of atoms, and cosine terms between 1–4 bonded pairs. These potentials try to model the effect of chemical bonding. These classical interaction potentials must be parameterized, e.g. the magnitude of the partial charges on each atom in the molecule must be assigned, and the equilibrium bond length and size of the harmonic force constant must be attached to each bond. In the early biomolecular MM forcefields, these parameters were developed to produce molecular models that could reproduce known experimental properties of the bulk system. For example, several MM water models have been developed.26,40,41 One of the earliest successful models, TIP3P,42 was parameterized such that simulations of boxes of TIP3P molecules reproduced known thermodynamic properties of water, such as liquid density and heats of vaporisation. Such a parameterisation scheme is to be applauded, as it ties the molecular model closely to experiment. Indeed many of the common MM models of amino acids were developed by comparison to experiment, e.g. OPLS.33 Indeed it is such a good Chem. Modell., 2008, 5, 13–50 | 17 This journal is

c

The Royal Society of Chemistry 2008

scheme that even some modern water models, like TIP5P,43 are still parameterized in this way. However, it was quickly realised that parameterisation against experiment required large amounts of physical data that was just not available for the novel molecules being conceived during rational drug design. The developers of biomolecular forcefields therefore created recipes that allowed for the parameters of new molecules to be derived from quantum chemical calculations. One popular example of such a recipe is GAFF (generalised AMBER forcefield),44 which is an MM forcefield for small drug-like ligands that is compatible with AMBER. GAFF uses generic atom types that are assigned to each atom in the drug-like molecule, e.g. aliphatic carbon or aromatic hydrogen. These atom types are used to assign LJ, bond, angle and dihedral parameters to the molecule from a large parameter library. The partial charges for the atoms are derived by performing an AM116 semiempirical QM calculation, and calculating BCC45,46 charges. In a very broad sense, this parameterisation is a multiscale method, as information (the charge distribution) from a QM calculation is transferred to an MM simulation via the parameterisation of the atomic partial charges. This multiscale parameterisation therefore represents a one-time, one-way flow of information up from the QM level to the MM level. A similar scheme is available for the OPLS forcefield,47 which uses CM1A charges48 that are also derived from semiempirical AM1 QM calculations. Wang and Sandberg49 used a more complex multiscale parameterisation to derive intramolecular CHARMM parameters for the interaction of DNA bonded to a gold surface. The parameters were calculated by fitting to density functional theory (DFT) QM calculations. 2.2 Two-way dynamic parameterisation methods QM calculations are now used routinely as the source of parameters for MM forcefields. Indeed this application is now so routine that most workers would not consider forcefield parameterisation to be an example of a multiscale method. However, there is a drawback with using QM calculations to provide MM forcefield parameters. The problem is that the information flow is only in one direction, from the QM to the MM level. This means that the QM-derived parameters for a molecule have to be very general, and be able to represent the molecule in a variety of conformations and environments. This is an unreasonable requirement, as it is clear that the charge distribution and polarisation of the molecule depends on both its conformation and its environment, e.g. whether it is in bulk solvent or whether it is bound to the active site of a protein. Because information only flows from the QM to MM level, there is no mechanism that allows information about the environment and conformation of the molecule experienced during the MM simulation to be fed back to the QM calculation. A solution is to modify multiscale parameterisation so that there is a two-way interface between the QM and MM levels of modelling. There have been two recent applications that have made such a modification: one targeted at creating a QM/MM multiscale docking method,50 and another targeted at QM/MM multiscale free energy calculations.47,51,52 Docking is one of the primary tools used during the process of rational drug design.53 The aim of docking is to predict the binding mode of a ligand with a protein. Because docking calculations are typically used to study how libraries of thousands of ligands bind to a protein, the calculations involved must be simple and efficient. This means that the interaction potentials used in docking tend to be based on molecular mechanics forcefields.53 MM forcefields struggle to model the changes in polarisation upon protein-ligand binding, an effect that is thought to account for as much as 10–40% of the binding affinity.54 Multiscale docking methods that attempt to use QM calculations to overcome this problem have therefore been developed.50,55 Cho et al.50 have developed a QM/MM docking method that uses multiscale parameterisation dynamically throughout a docking calculation. The electrostatic interaction energy between the ligand being docked and the protein is 18 | Chem. Modell., 2008, 5, 13–50 This journal is

c

The Royal Society of Chemistry 2008

calculated using Coulomb’s law, with atomic partial charges placed on the protein and ligand atoms. Cho et al. first showed that the docking prediction is improved if the atomic partial charges on the ligand are derived from a QM calculation of the ligand in the bound geometry. They demonstrated this by selecting several test protein-ligand systems whose bound geometries were known via crystal structures available in the protein databank (PDB). They calculated the partial charges by performing a density functional theory (DFT) calculation of the ligand in the bound geometry. The wavefunction was polarised by embedding the partial charges from the protein within the Hamiltonian of the QM calculation. By using the bound geometry, and by embedding the partial charges of the protein, information about the environment of the ligand in the protein active site was made available to the QM calculation. The polarised wavefunction was used to obtain the molecular electrostatic potential (MEP) surface around the ligand. Partial charges were generated using an electrostatic potential fitting procedure to reproduce the QM MEP. By calculating these partial charges from a QM calculation that had information about the bound geometry, it could be argued that these atomistic partial charges were optimised for the bound geometry. Cho et al. demonstrated this50 by running redocking calculations where the ligands were docked using both optimised and generic partial charges. The results demonstrated that docking calculations using the optimised charges were significantly more likely to rediscover the known experimental binding modes. To turn this observation into an effective docking algorithm, Cho et al. created an iterative dynamic reparameterisation algorithm; 1. Dock the ligand using the default partial charges taken from the docking MM forcefield. 2. Perform a QM calculation on the predicted binding mode to obtain optimised partial charges for the ligand. 3. Dock the ligand again, this time using the optimised partial charges. 4. Keep iterating until the charges and predicted binding mode converge to within a set limit. By dynamically reparameterising the ligand throughout the docking calculation, Cho et al. allow information to flow both ways between the QM and MM levels. A similar idea has been developed by Jorgensen and co-workers to create a QM/MM multiscale method for free energy calculations.47,51,52 Jorgensen and co-workers developed a method to obtain atomic partial charges efficiently from a QM calculation that were compatible with the partial charges from the standard OPLS all-atom forcefield.47 The charges were calculated using the charge model 1 (CM1A)48 analysis of an AM1 semiempirical QM calculation. However, as CM1A charges were parameterized to reproduce gas-phase dipole moments,48 they had to be scaled by a factor of 1.2 so that an implicit account could be made for the extra polarising effects of polar solvents. Jorgensen and co-workers first used this method to perform QM/MM hydration free energy calculations. The solute molecule was modelled using QM (AM1), while the solvent molecules were modelled using MM (OPLS). The QM calculation was used to obtain the intramolecular energy of the solute. The QM calculation was also used to obtain the atomic partial charges on the solute using AM1/CM1A. These partial charges were used to calculate the electrostatic interaction energy between the solute and solvent via Coulomb’s law. The LJ equation was used to obtain the vdW interaction between the solute and solvent using pre-assigned OPLS e and s LJ parameters. The QM calculation in this method was used only to dynamically re-parameterize the MM atomic partial charges during the simulation. The only information flow from the MM to QM level was the change in conformation of the solute. The solvent environment around the solute was not passed explicitly, as it was not included within the QM calculation. The effect of the solvent was only felt implicitly in the QM calculation via the application of the scale factor. Despite the lack of explicit modelling of the solvent environment in the QM calculation, Jorgensen and co-workers have successfully used this method to study solution-phase Diels-Alder reactions,56 and to study the enzyme-catalysed Claisen Chem. Modell., 2008, 5, 13–50 | 19 This journal is

c

The Royal Society of Chemistry 2008

rearrangement reaction of chorismate to prephenate.51 Jorgensen and co-workers have since adapted this method57–59 to use the PDDG/PM3 semiempirical QM Hamiltonian,60 using the CM361 method to extract charges, which are then scaled by a factor of 1.14,62 again to provide an implicit account for the extra polarising effects of the solvent. 2.3.

Two-way embedded methods

Periodic multiscale reparameterisation, i.e. periodically during a simulation, provides a two-way conduit for information flow between the QM and MM levels of modelling. However, the coupling between levels is not particularly strong. The interchange of information between levels occurs only periodically, which can lead to the MM level falling out of step with the QM. This problem was experienced during the development of the QM/MM docking method of Cho et al.50 If the ligand was initially docked in a poor configuration, then the partial charges derived for that conformation could bias the subsequent docking runs to rediscover the poor configuration in preference to the correct binding mode. Cho et al. developed a survival of the fittest algorithm50 that ran multiple docking runs in parallel, thereby preventing one poor result from biasing the rest of the calculation. A closer coupling between the QM and MM levels can be achieved using an embedded interace method. A QM region of the biomolecular system is embedded within a larger MM simulation. One of the primary application areas for embedded QM/MM methods is computational enzymology (the computational modelling of enzyme-catalysed reaction mechanisms), where typically a QM model of the substrate and part of the enzyme active site is embedded within an MM model of the rest of the enzyme and solvent.63 Embedding a QM model within an MM simulation creates a dynamic and permanent interface between the two levels, with information flow across that interface having to be managed for each configuration of the simulation. The ONIOM method, developed by Morokuma and co-workers,64 provides such an interface via the use of multilevel corrections. The ONIOM method partitions the system into multiple layers. For example, consider a two-layer system, where a low-level QM region, A, is embedded within a high-level MM region, B. The energy of both regions, A + B, is first calculated using only the MM Hamiltonian, giving EMM(A + B). This total energy is corrected by calculating the difference in energy between the QM and MM energies of the QM region, EQM(A) EMM(A). The total ONIOM energy of the system is therefore EMM(A + B) + EQM(A) EMM(A). The generalisation of this algorithm to multiple levels is straight-forward, e.g. a system can be divided into an ab initio QM region, A, which is embedded within a semiemprical QM region, B, which is itself embedded within an MM system, C. The ONIOM energy in this case would be EMM(A + B + C) + [Esemiemprical(A + B) EMM(A + B)] + [Eab initio(A) Esemiemprical(A)]. The ONIOM method allows the facile combination of QM and MM levels of modelling. However, the electrostatic interaction between the QM and MM regions in the original ONIOM implementation is handled at the MM level only, via EMM(A + B), based on partial atomic charges derived from the QM calculation. The use of this method, called classical or mechanical embedding,65 means that information about the electrostatics of the system flows only from the QM up to the MM level. There is no conduit by which the electrostatic environment of the MM atoms is able to flow down to the QM region, to polarise the QM wavefunction.65 An alternative method of interfacing QM and MM calculations, called electronic embedding, solves this problem. In electronic embedding, the partial charges of the MM atoms are embedded within the QM Hamiltonian. This allows the QM wavefunction to be polarised by the MM atoms, thereby providing a two-way conduit for electrostatic information between the QM and MM regions. The ONIOM method has since been extended to use electronic embedding,65 thereby overcoming one of the fundamental weaknesses of the algorithm. 20 | Chem. Modell., 2008, 5, 13–50 This journal is

c

The Royal Society of Chemistry 2008

Electronic embedding has had a rich and long history of application in other QM/ MM schemes. The first application of electronic embedded QM/MM to a biomolecular system was in the ground-breaking work of Warshel and Levitt in 1976.39 They developed the method to study the reaction mechanism of hen egg-white lysozyme. Warshel has recently produced a detailed, clear and very interesting review63 of the details of the algorithm and the developments in QM/MM methodology since his pioneering work in the 1970s. Acceptance of this approach took a long time,63 but there is now a large body of QM/MM applications that use such methodology (see one of the very many reviews of QM/MM methods available in the literature66–69). The underlying theory of this method has been covered by many different authors68,69 so only a brief description will be given here. The biomolecular system is divided into a QM region, for example a substrate, and an MM region, e.g. an enzyme and surrounding solvent molecules. The total energy of the system is the sum of the energy of the MM region, evaluated using a standard MM forcefield, the energy of the QM region, and the QM/MM interaction energy between the two regions. The QM/MM interaction energy is split into two parts: an electrostatic part and a vdW part. The vdW part is calculated by assigning LJ parameters to all of the QM atoms and calculating the interaction between the QM and MM atoms using the LJ equation. The electrostatic part of the QM/MM interaction is calculated by bundling it together with the calculation of the QM energy of the QM region. This is achieved by embedding within the QM Hamiltonian the locations and partial charges of all of the MM atoms that are within a pre-determined cutoff distance of the QM atoms. Normally the MM atoms are represented as point charges in the QM calculation, but Gaussian charge distributions, with the width of the Gaussian that is similar to the covalent radii of the MM atoms may instead be used. 70 The MM partial charges act to polarise the QM wavefunction of the QM atoms, and therefore the evaluation of the energy of this wavefunction returns both the intramolecular energy of the QM atoms and the electrostatic interaction between the QM and MM atoms. The simple split of the QM/MM interaction into electrostatic and vdW parts is, however, only possible if the interface between regions lies between molecules, i.e. all molecules are either QM or MM, and there are no molecules that sit across the interface. It is desirable (e.g. within computational enzymology) for a single molecule to be able to bridge this interface, e.g. while the majority of an enzyme is modelled at the MM level, it is usually necessary to represent some (e.g. catalytic) active site residues at the QM level. The problem with having a single molecule straddle the interface is that the QM/MM interaction energy, as well as modelling electrostatic polarisation, must now also include terms that account for the chemical bonding between atoms modelled at the QM level and atoms modelled at the MM level. This is a particular problem with the QM side of the calculation, as when the boundary bisects a covalent bond, the electron density is terminated abruptly at the end of the QM region and electrons of the bonded atom are missing, potentially leading to unpaired electrons.66 The three most popular methods for solving this problem are the link atom method,66,71,72 the local selfconsistent field (LSCF) method73,74 and the generalized hybrid orbital (GHO) method.75,76 The ‘‘dummy junction atom’’ or link atom approach introduces so-called link atoms to satisfy the valence of the atoms on the QM side of the QM/MM interface.66,71,72 Usually this atom is a hydrogen, but other atom types have also been used, e.g. halogens such as fluorine or chlorine.77 The link atom method can be used with both the Warshel-type QM/MM methods and ONIOM methods.65,78 The link atom method has been criticised because it introduces extra unphysical atoms to the system, which come with associated extra degrees of freedom. Another problem is that a C–H bond is clearly not chemically exactly equivalent to a C–C bond. Despite these problems, the simplicity of the link atom method means that it is used widely in the QM/MM modelling of proteins and other biological molecules. 79

Chem. Modell., 2008, 5, 13–50 | 21 This journal is

c

The Royal Society of Chemistry 2008

Zhang et al.80,81 solved these problems with their pseudo bond method. If X and Y are the bonded QM and MM boundary atoms, respectively, then the link atom method effectively replaces the Y atom with a hydrogen, thereby changing X–Y into X–H. This has the problem that H is not necessarily chemically similar to Y, and also the X–H bond length may not be the same as the X–Y bond length. In the pseudobond method, Y is instead replaced by Yeff. This is a one-free valence boundary atom that has a parameterized effective core potential that mimics the strength of the X–Y bond. This method has been tested successfully80 with both Hartree Fock (HF), density functional theory (DFT) and MP282 QM/MM calculations. Antes and Thiel83 have also introduced a conceptually similar approach, which they call ‘‘adjusted connection atom’’, that works at the semiemprical QM level. Parameterisation of these effective link atoms is not straight-forward, as they must minimally perturb the electron density compared to a QM calculation of the entire molecule. Ro¨thlisberger and co-workers have developed a sophisticated scheme that derives parameters for these atoms via density functional perturbation theory.84 A second approach to define the bonding interface between the QM and MM regions is the local self-consistent field (LSCF) algorithm developed by Rivail and co-workers.73,74 In the LSCF method the bonds between the QM and MM atoms are represented by strictly localised bond orbitals, which are parameterized by separate QM calculations on small molecules.75 These localised orbitals are assumed to be transferable to the protein system, and are used, and kept constant, throughout the self-consistent field (SCF) QM calculation. An elegant feature of the LSCF method is that it does not require use of link atoms, and a comparison of the LSCF and link atom methods85 showed that both give equivalent energy results. However, the parameters for the localised bond orbitals have to be determined for each new system studied.75 Fonili et al.86 have recently shown that it is possible to use frozen core orbitals on the MM frontier atom within the LSCF scheme. This provides an explicit description of the core electrons of the atoms on the MM frontier atom, thereby improving the physical description of the interface, thus reducing the error on the calculation. While the LSCF method is elegant, and has been shown to work well, the parameters for the localised orbitals are not very portable. Gao et al. addressed this problem by developing generalised hybrid orbitals (GHOs).75,76 The QM boundary atom at the QM/MM interface has the standard valence s and p orbitals as all of the other non-hydrogen atoms in the QM region. These four sp orbitals are transformed into a set of orthogonal hybrid orbitals, which can be defined by the bound geometry of the atom.75 These hybrid orbitals are used, along with the atomic orbitals of the QM region, to determine the QM energy. However, only one hybrid orbital points along the bond between the QM and MM boundary atoms, and it is this active hybrid orbital that needs to be optimised. The complete set of this one active hybrid orbital, plus all of the atomic orbitals from the other QM atoms form the active set that is optimised during the SCF calculation. The remaining three hybrid orbitals, called auxiliary orbitals, act, together with the nucleus charge, to generate an effective core potential for the QM boundary atom. Gao et al. realised that these auxiliary orbitals may be parameterized to mimic the effective core potential for the active electrons from the MM region. Therefore rather than parameterising the charge density of the hybrid orbitals for each specific system, as is the case for the LSCF method, they instead optimise the semiemprical parameters for the boundary atom to reproduce the bonding properties of full QM systems. As a result, the parameters for this GHO method are expected to be general and transferable in the same way as all the semiemprical parameters. 75 The method has since been extended by Gao and co-workers87 to work with the selfconsistent charge density-functional tight binding (SCC-DFTB) method.88 The accuracy and efficiency of SCC-DFTB is making it a popular choice to model the QM region, and its use in computational enzymology has increased greatly in recent 22 | Chem. Modell., 2008, 5, 13–50 This journal is

c

The Royal Society of Chemistry 2008

years.88 Gao and co-workers76,89 have also developed a GHO method that is suitable for ab initio Hartree Fock (HF)89 and density functional theory (DFT) calculations.76 Ko¨nig et al.90 recently compared several different algorithms to model chemical bonds between QM and MM atoms, and concluded that while none of the approaches were perfect, error cancellation meant that enzyme catalysed reaction free energies using an SCC-DFTB QM region were only marginally affected by the choice of algorithm, if total charge was conserved during the reaction. The conclusion contrasts with a comparison study in which we were involved,91 in which we compared the link atom and GHO boundary methods. While we found that both methods, when properly applied, can lead to similar behaviour, the inclusion of conformational sampling amplified the effects of the differences between the methods. This led to the two methods returning different free energy reaction profiles despite being applied to the same enzyme system. It is important to make clear that while the same enzyme system was used, the QM region was not the same for the two boundary methods. This was because of the different constraints on the partitioning of the two methods (the GHO method requires that the QM atom bound to an MM atom must be an sp3 carbon91). The origin of the observed difference may well be in the different QM and MM systems, rather than in the partitioning schemes themselves. Despite these questions over the fine detail of how QM/MM methods are applied, their development and application has now matured to a point where they can provide near-quantitive results for activation enthalpies and free energies of reaction.92 It is now possible to perform electronic structure calculations on large systems approaching chemical accuracy, thus allowing quantitative studies of reaction mechanisms in enzymes.92 2.4.

Two-way parallel QM/MM methods

Quantum mechanics calculations are computationally demanding. This has caused difficulties with their application to the calculation of thermodynamic properties, such as free energies. This is because thermodynamic properties are calculated as an average over a large ensemble of conformations of a system. Sampling methods, such as molecular dynamics (MD)35,36 or Monte Carlo (MC)36,37 must be used for the rigorous generation of such ensembles. The computational expense of QM calculations means that it is only practical to generate small ensembles via standard MD or MC, e.g. current methods are limited to picoseconds of molecular dynamics. Two-way parallel QM/MM interfaces have therefore been developed in an attempt to overcome this problem. We use the term parallel methods to mean those that use both a QM or QM/MM simulation running in parallel with a standard MM simulation, using only periodic exchange of information between the two simulation levels. Warshel and co-workers developed a successful QM/MM parallel method in a set of pioneering papers in the late 1990s.93,94 The aim of this method was to calculate the free energy difference between two systems, A and B. For example, system A could be a substrate bound to an enzyme, while system B could be the transition state. The free energy difference between these two corresponds to the activation free energy of the enzyme catalysed reaction. Warshel and co-workers calculated the relative free energy of A and B by first using a molecular mechanics type potential. Because an MM potential was used, molecular dynamics sampling was efficient, and therefore a large ensemble, and well-converged relative free energy were calculated. This relative free energy, DGMM(A - B), can only be as good as the MM potential used during the calculation. Ideally, this free energy should be calculated using a QM or QM/MM representation of A and B, giving DGQM(A - B). However, the computational expense of the QM calculation prevents the efficient generation of the large ensembles necessary to evaluate converged free energies. Warshel and coworkers solved this by rather than calculating DGQM(A - B) directly, they used the Chem. Modell., 2008, 5, 13–50 | 23 This journal is

c

The Royal Society of Chemistry 2008

Fig. 1 The free energy cycle93,94 used to calculate the QM/MM free energy difference between systems A and B, DGQM/MM(A - B). The free energy difference between A and B is first estimated using an approximate potential (e.g. an MM potential), giving DGMM(A - B). This is then corrected to the QM/MM value by calculating the free energy necessary to perturb system A from MM to QM/MM (DGMM-QM/MM(A)) and the free energy to perturb system B from MM to QM/MM (DGMM-QM/MM(B)).

MM ensembles to calculate the difference in free energy between the QM and MM representations of A and B. In essence, Warshel and co-workers calculated the free energy error associated with using the MM forcefield. By calculating these errors, Warshel and co-workers were able to correct DGMM(A - B) so that it was formally equal to DGQM(A - B)93,94 (see Fig. 1). The correction free energies were calculated by generating ensembles for systems A and B using the MM model. The difference in energy between the QM and MM models was calculated for a subset of each ensemble, and the difference between these energies used as input to a single-step free energy perturbation (FEP)95,96 between the MM model (the FEP reference state) and the QM model (the FEP perturbed state). As long as the MM model is a good approximation of the QM model, i.e. the phase space overlap of the two models is good, then the average calculated via the FEP equation will converge to an accurate estimate of the correction free energy. The key advantage of this method is that all of the thermodynamic sampling is performed using only the MM model of the system. QM or QM/MM calculations are run in parallel with the MM sampling to estimate the correction free energies. The disadvantage of this method is the requirement of good overlap between the QM and MM models. Warshel and co-workers mitigate this disadvantage through the development of the empirical valence bond (EVB)93,94,97 forcefield, which has been designed to give energies that are in good agreement with experiment and QM calculations. In addition, the EVB potential has been developed so that it can be used to study chemical reactions, something that is not possible using most of the biological MM forcefields. The EVB potential and the Warshel parallel QM/MM method have been very successful, and have been used to study a variety of systems.98–102 Warshel and co-workers developed their method to avoid the problem of poor sampling of a QM or QM/MM Hamiltonian. MD methods are currently limited to picoseconds of QM/MM dynamics for typical biomolecular applications, even using relatively low levels of QM theory. Monte Carlo (MC) methods suffer from even greater problems. MC works by performing typically millions of small random moves of the biomolecular system, each of which are tested according to the change in energy associated with that move. MC sampling of a QM/MM Hamiltonian would potentially require millions of QM energy evaluations, which is impractical using current methods and computers. A second class of parallel QM/MM methods attempt to solve this problem. These methods use a novel Monte Carlo algorithm developed by Hastings in 1970.103 This is a multiscale sampling method that uses MC sampling at one modelling level to generate an ensemble which is formally correct for a different modelling level. The algorithm works by creating a new type of Monte Carlo move. The move starts at configuration i. The energy of this configuration is evaluated using both the fast, high-level model, giving Efast(i), and the slower, low-level forcefield, giving Eslow(i). A block of MC moves is then performed using the fast forcefield. This results in a new configuration, j. The energy of this configuration is evaluated using both forcefields, giving Eslow(j) and Efast(j). These 24 | Chem. Modell., 2008, 5, 13–50 This journal is

c

The Royal Society of Chemistry 2008

Fig. 2 Application of the Metropolis-Hastings103 algorithm to accelerate sampling of a system represented using a QM/MM Hamiltonian.104,105 The Monte Carlo move starts at configuration i. The energy of this configuration is evaluated using the target QM/MM Hamiltonian (giving EQM/MM(i)) and on an approximate (MM) Hamiltonian (giving EMM(i)). Standard Metropolis Monte Carlo moves are then attempted from configuration i using only the approximate MM Hamiltonian, until after a set number of moves, the system is in configuration j. The energy of configuration j is evaluated using both the QM/MM and MM Hamiltonians (giving EQM/MM(j) and EMM(j)). Configuration j is then accepted into the QM/ MM ensemble according to the probability min{1,exp( DDE/kBT)} where DDE = (EQM/MM(j) EMM(j)) (EQM/MM(i) EMM(i)).

energies are used to test configuration j according to a new MC acceptance test. Configuration j is accepted into the ensemble if this test is passed (see Fig. 2). Otherwise the whole block of sampling is rejected and the simulation is reset to configuration i. The form of the MC test is such that even though the trial configurations are generated using the fast forcefield, they are accepted into the ensemble of the slow forcefield with the correct Boltzmann probability. This algorithm was popularised for applications to QM and QM/MM systems by Schofield and co-workers,104,106 who coined the phrase ‘‘molecular mechanics-based importance function’’ (MMBIF). One of the problems with this algorithm is that the acceptance ratio of the MC test will be low if there is poor overlap between the fast and slow forcefields. This is similar to the problem encountered in Warshel’s correction free energy method. Effort may therefore need to be spent optimising the fast forcefield such that it is a better match to the slow forcefield. We have developed a parallel QM/MM method105 that combines the advantages of both the Warshel and MMBIF algorithms. The method works by using the MMBIF method to generate QM/MM ensembles from which the Warshel correction free energies can be calculated in full. The correction free energy is calculated using thermodynamic integration (TI)107,108 over a fictional l scaling parameter that maps from the QM model to the MM model. This l parameter allows the QM model to be changed over a series of windows into the MM model. The MMBIF algorithm can then be used to generate ensembles of conformations of the system at different values of l, so that the gradient of the free energy with respect to l can be calculated at several points between the QM and MM models. These gradients can then be integrated across l to return the correction free energy. We overcome the problems associated with potentially poor overlap between the QM and MM models by using replica exchange8,9,109 moves across the l coordinate during the simulation. Replica exchange moves are additional Monte Carlo moves that lightly couple multiple trajectories together. All of the MC simulations at different l values are run in parallel. Neighbouring pairs of simulations are tested periodically according to a replica exchange MC acceptance test. If this test is passed, then the l values of the neighbouring pairs are swapped. This has the effect of allowing each MC simulation to sample multiple l values during the simulation. This enhances convergence of the free energy averages. We have used this method105 to calculate converged relative hydration free energies of water and methane, using an MP2 ab initio QM model of water and methane, solvated by a periodic box of MM waters. While this was a nonbiological application, we are currently using this method to calculate QM/MM relative binding free energies of protein-ligand systems (using a DFT QM model of the ligand and an MM model of the protein and explicit solvent), and are planning to use it to perform some computational enzymology calculations. Chem. Modell., 2008, 5, 13–50 | 25 This journal is

c

The Royal Society of Chemistry 2008

3. Interfacing atomistic with coarse grain models In mid 2007, Leontiadou, Mark and Marrink110 produced a paper in which they used atomistic molecular dynamics simulations to model the effects of ionic concentration on the transport of ionic species across a pore in a lipid membrane. This work involved several large simulations, involving 128 dipalmitoylphosphatidylcholine (DPPC) lipids (each containing 130 atoms) and about 6000 water molecules. This work pushed the limits of what is achievable with current atomistic molecular dynamics, and, by using approximations such as modelling long-range electrostatics using a reaction field, and using bond constraints such that a 5 fs integration timestep could be used, they were able to run several simulations of between 50 ns to 100 ns in length. Despite the impressive size of these simulations, they are still limited to biologically small length and time scales. 128 lipids is merely an 8  8 membrane bilayer, which is too small to model effects such as membrane curvature or membrane waves.111,112 100 ns is also too short a time to capture events such as membrane protein aggregation or lipid raft formation within a membrane.111 Coarse grain models provide a route to longer time and length scales in biomolecular simulations. Coarse grain models are a class of mesoscale model that work by grouping several atoms together and modelling them as a single interaction site. In effect, groups of atoms are smeared together into beads. For example, a coarse grain model could be constructed that represents an amino acid residue as a single bead, and a protein as a string of beads. The use of coarse grain models reduces the computational expense of a simulation, as coarse graining reduces the number of interaction sites. In addition, CG models contain fewer degrees of freedom, and use forcefields that lead to smoother potential energy surfaces. The smoother potential energy surface reduces the problems associated with frustration or non-ergodic trapping, thereby leading to improved sampling and a lower correlation time. Also coarse graining tends to remove the stiffest degrees of freedom from the model (e.g. the C–H bond vibrational modes), thereby allowing a CG model to use a larger timestep. All of these effects mean that CG simulations provide a route to modelling length and time scales that are far beyond that which is practically achievable via atomistic molecular dynamics. Coarse grain modelling is currently undergoing a renaissance, and there is now significant international effort being spent developing and applying coarse grain methods to model biological systems. It is not the purpose of this review to cover all of these recent developments in depth, so the interested reader is directed to several excellent modern reviews of the development and application of CG methods.113–115 Coarse grain models allow simulators to routinely access length and time scales that are not practically possible using atomistic modelling methods. However, in smearing out the atomistic detail, CG models run the possibility of missing out important atomistic effects, much in the same way that molecular mechanics models, in smearing out all of the electronic detail, can fail to model important electronic effects such as polarisation. There is now significant interest in interfacing atomistic and coarse grain models within a multiscale framework, so that this problem may be overcome. Just as there is significant variation in the type and strength of interaction in the different methods developed to interface QM and MM models, so too is there significant variation in the type and strength of interaction of the different methods of interfacing atomistic and CG models. The type of interfaces broadly fall into four categories, which are similar in nature and definition to the interfaces that have been developed for QM/MM interfaces: 1. One-way, bottom-up methods. These involve a single transfer of information from atomistic simulations or calculations to the CG simulation, e.g. by using an atomistic simulation to parameterize a CG model. 2. One-way, top–down methods. These involve a transfer of information from the CG simulation to the atomistic simulation, e.g. using a CG model to enhance

26 | Chem. Modell., 2008, 5, 13–50 This journal is

c

The Royal Society of Chemistry 2008

sampling, and then using an atomistic model to explore interesting conformations that are discovered. 3. Two-way parallel methods. These involve the running, in parallel, of both a CG and atomistic model, and dynamically exchanging information between them at run time. 4. Two-way embedded methods. These involve the running of simulations in which an atomistic region (e.g. a membrane protein) is embedded within a CG model (e.g. a lipid membrane). Examples of each of these different types of interface will now be presented and discussed in turn. 3.1 One-way, bottom-up interfacing methods Coarse grain models can be interfaced using a one-way, bottom-up scheme. In this scheme, an atomistic model of the system (be it MM or QM) is used to parameterize a coarse grain model of the system. This idea is not new, and was the parameterisation method used by Levitt and Warshel116,117 when they first introduced coarse grain models in their pioneering 1970s papers. Levitt and Warshel’s 1975 paper introduced the first coarse grain model of a globular protein. It used two CG particles per residues: one that represented the a-carbon of an amino acid (Ca), and one that represented the side-chain atoms. A torsion potential acted about Ca particles, while a Lennard-Jones type potential acted between pairs of side chain particles. This CG model was built to represent bovine pancreatic trypsin inhibitor (BPTI), and was very successful, being able to correctly refold the protein starting from a completely denatured configuration. This work is interesting not only because it represents the first coarse grain biomolecular model, but also because it represents the first multiscale MM/CG simulation. Levitt and Warshel generated the parameters for their coarse grain model by averaging over interaction energies calculated using an atomistic potential. The parameters for the interaction potential between side chain particles were calculated by assuming that the side chains had spherical symmetry. The effective potential between identical side chain particles was calculated at various distances apart by summing the interaction energies of all of the atoms in one sphere with all of the atoms in the other sphere using a molecular mechanics style potential. This interaction potential between like-particles was then used to parameterize a Lennard-Jones type function, and the parameters between different side chain particles were then obtained using a geometric combining rule. The torsion potential between Ca atoms was also calculated by fitting to atomistic calculations, with the torsion potential between a pair of residues based on a time average of the energy calculated from atomistic simulations of a set of dipeptides. 3.1.1. Coarse grain models of lipid membranes. Levitt and Warshel’s method of obtaining CG parameters from underlying atomistic calculations can now be recognised as an example of a multiscale parameterisation scheme, where there is a one-way, bottom-up, flow of information from the atomistic to the CG calculation. Levitt and Warshel were ahead of their time in using a multiscale parameterisation scheme for their CG model. Coarse grain models only became popular for biomolecular modelling in the 1990s, which saw the beginning of the development of CG representations of lipids. Because the description of the development of coarse grain lipid molecules is outside the scope of this review (excellent reviews of this subject113,115 exist already) only a brief history of CG lipid models will be presented, in particular to highlight how CG lipid models have recently moved to using multiscale parameterisation methods. The first applications of CG lipid models looked for qualitative, rather than quantitative predictions, and therefore did not use any information from atomisticlevel calculations in their parameterisation. One of the first models produced during this time was by Smit et al.118 This model was used to investigate the phases of an oil/ Chem. Modell., 2008, 5, 13–50 | 27 This journal is

c

The Royal Society of Chemistry 2008

water/surfactant system. This was a qualitative model that used just two types of particle; an oil (o) particle and a water (w) particle. An oil molecule was represented using a single o particle, water using a single w, and a surfactant was modelled as a chain of two w particles followed by 5 o particles, all held together with harmonic springs. All particles interacted using a Lennard-Jones potential, with attractive o–o and w–w interactions, and a purely repulsive o–w interaction. This simple representation was qualitatively able to model micelle formation. Goetz et al.119 and Noguchi and Takasu120 also developed qualitative CG models of amphiphiles, and used them to investigate phenomena such as bilayer and vesicle formation. One of the problems with a CG model is that the form of the interaction potential between CG particles is not clear, and there are questions over whether simple potentials are sufficient. Noguchi and Takasu developed their qualitative model of amphiphiles using custom potentials; a pairwise exponential term to give a soft-core repulsive shape to the CG particles, and a many-body attractive potential that aimed to model the hydrophobic effect. Using these potential forms, and a small number of adjustable parameters, they were able to simulate the self-assembly of bilayer vesicles. Brannigan and Brown121 also developed a pure CG model with a complex form. They modelled lipids as soft spherocylinders that interacted via a short-range r 8 repulsive term and an isotropic attractive alignment potential, which was based on an r 2 attractive term that was mediated by an angular dependence. These terms gave the lipids shape and encouraged them to form an aligned lamellar phase. To encourage bilayer formation, they designated one end of the spherocylinder as the lipid tail, and added a r 6 Lennard-Jones type attractive term between tails. This model resulted in a very small number of parameters, which allowed its phase behaviour with respect to a parameter search to be investigated. With the right parameters, this model was observed to self-assemble into a bilayer. Michel and Cleaver122 were still investigating the correct functional form to use for coarse grain amphiphile molecules in 2007, in a paper that investigated the use of the GayBerne123 potential. The Gay-Berne potential is effectively an anisotropic version of the Lennard-Jones potential, where the r 12 repulsive and r 6 attractive terms are attenuated by the angle of interaction between two particles (using the dot product of the particles alignment vectors). This has the effect of elongating the LennardJones sphere along an alignment vector and turning it into an ellipsoidal rod. Michel and Cleaver performed a parameter space search for Gay-Berne particles, and were able to qualitatively observe several different liquid crystal phases. Despite Levitt and Warshel’s demonstration of the effectiveness of using quantitatively parameterized CG models, it took until 2001 before other workers began developing quantitative CG models for biomolecules. The Shelley model, developed by Shelley et al.124,125 was the first quantitative CG biomolecular representation that was used to model lipid membranes. Shelley et al. developed a CG model of the lipid dimyristoylphosphatidylcholine (DMPC). They represented the 46 atoms of DMPC with just 13 spherical interaction sites. Four sites were used for each hydrocarbon chain (three chain SM sites, and one terminating ST site), one site each for the esther link (E1), one site for the glycerol backbone (GL) one site for the choline (CH) and one site for the phosphate (PH) in the headgroup (see Fig. 3). In addition, they also used a spherical W particle, that represented a grouping of three water molecules. The interaction potentials between these spherical groups were based on LennardJones potentials, e.g. interactions between W particles used a LJ 6–4 potential (an r 6 repulsive term coupled with an r 4 attractive term). The parameters for these LJ potentials were obtained by reproducing thermodynamic properties, e.g. the sWW parameter was set to reproduce the experimental density of water at 303.15 K, while calculations of the vapour pressure of a box of W particles, where used to set eWW such that the experimental boiling point of water was reproduced. A Lennard-Jones 9–6 potential was used between non-bonded hydrocarbon particle sites (SM and ST), which the LJ parameters obtained by matching the densities and vapour pressures of boxes of nonane and dodecane. The Shelley CG model was mostly 28 | Chem. Modell., 2008, 5, 13–50 This journal is

c

The Royal Society of Chemistry 2008

Fig. 3 A comparison of different coarse grain lipid models. The Shelley model124,125 of DMPC, and Marrink126 and Essex127 models of DPPC are compared to their atomistic equivalents (for ease of comparison, hydrogen atoms of the atomistic models are not shown). Solid lines represent harmonic bonds connecting CG particles, and the CG particle types for the Shelley and Marrink models are labelled (the labels are the same as those used in the main text). The point charges (represented by + and ) and point dipoles (represented by arrows) are shown for the Essex model (the charges and dipoles are located at the centre of their associated CG particle). The Shelley and Marrink models use LJ particles (represented by spheres), while the Essex model uses a combination of LJ particles (spheres) and Gay-Berne particles (ellipsoids). Finally, the ‘blob’ model proposed by Chao et al.128 is also shown for comparison. This model represents groups of atoms as rigid non-spherical ‘blobs’ that use interaction potentials based on multipole expansions.

parameterized from experimental and thermodynamic data, similarly to how the original atomistic molecular mechanics forcefields were parameterized from experimental and thermodynamic data. However, due to the importance of correctly modelling the interactions between head group particles, Shelley et al. did use a multiscale parameterisation method to generate the parameters for the head group particles from an underlying atomistic simulation. The interactions between head group particles were based on radial distribution functions (RDFs) between head group particles calculated from atomistic simulations. These RDFs were tabulated, and then used to create a potential of mean force (PMF) which was used as the interaction potential between the head group particles. Use of this PMF led to RDFs that disagreed with the atomistic simulation, so corrections to the interaction potential were calculated by comparing the atomistic and CG RDFs, and running an iterative scheme that updated the interaction potential until the differences Chem. Modell., 2008, 5, 13–50 | 29 This journal is

c

The Royal Society of Chemistry 2008

between the atomistic and CG RDFs were minimised. The model semi-quatitatively reproduced the density profile of an aqueous DMPC bilayer, and could be used to simulate the self-assembly of the bilayer. Shelley et al. used this model to investigate diffusion of halothane through the membrane.125 Halothane was modelled using a LJ 6–4 potential, which was parameterized by matching the experimental density and boiling point of the molecule, but a lack of experimental data prevented the detailed parameterisation of the interactions between halothane and the particles in the lipids, and this led to a model which was shown to not be truly predictive. Lopez et al.129 used existing beads from the Shelley DMPC model to model a synthetic antimicrobial polymer, which was inspired by a natural antibiotic. They then qualitatively investigated the interaction between the antimicrobial model and a bilayer of Shelley DMPC molecules. Despite the quantitative foundations of the Shelley model, it has several limiting features which arise from the use of multiscale parameterization:126 the model is optimized for the bilayer phase of DMPC only (as it was an atomistic simulation of this phase that led directly to the head group parameters), the interaction potentials are more complicated, so short time steps must be used in the dynamics integrator, and it evaluates long-range interactions, so it is not possible to reduce the computational expense of the calculation by using a short-range cutoff. These disadvantages of multiscale parameterisation, the lack of portability and the complexity of the interaction potential, led Marrink and co-workers to develop a simple, experimentally parameterized CG lipid model.126 The Marrink model is semi-quantitative, and was developed to target four goals; speed, accuracy, applicability and versatility. Marrink et al. achieved speed by using only short range potentials, which were also very smooth, so that large integration timesteps may be used. Accuracy was maintained by deriving the parameters to match experimental results and by comparing results calculated using their model to results from equivalent atomistic simulations. The simplicity of the forcefield (it is composed as a set of generic particles), together with its flexibility to represent chemical groups achieved the aims of applicability and versatility. The coarse grain model is constructed as follows; on average four atoms are represented by a single interaction site. To keep the model simple, only four main types of interaction site are used; polar (P), nonpolar (N), apolar (C) and charged (Q). The nonpolar and charged groups are then futher divided into subtypes, which depend on whether or not hydrogen bonding is possible to these groups. Fig. 3 shows how a DPPC lipid is represented using this model. The coarse grain particles interact using a LennardJones function. However, only five levels of interaction are defined; attractive, where e = 5 kJ mol 1, semi-attactive, where e = 4.2 kJ mol 1, intermediate (e = 3.4 kJ mol 1), semirepulsive (e = 2.6 kJ mol 1) and repulsive (e = 1.8 kJ mol 1). The level of interaction between a particular pair of coarse grain particles is chosen from a table look-up based on the types of the interacting sites, e.g. a pair of P particles have an attractive level of interaction, while a P and a C have a repulsive level of interaction. The complete lookup table is provided in the original paper describing this model.126 These parameters were obtained by trial and error, running multiple oil and water CG simulations and modifying the small number of parameters to reproduce experimental properties (e.g. density of water and alkanes at room temperature, mutual solubility and diffusion rates). For all interaction levels, the same value of s was used (0.47 nm). To keep the interactions short-ranged, they were force shifted to zero between 0.9 nm and the user-defined parameter rcut using the standard Gromacs130,131 shift function. All coarse grain particles except for nearest neighbours interact through this Lennard-Jones potential. Nearest neighbours are connected via a weak harmonic spring, while next-nearest neighbours are connected via a harmonic angle potential. In addition to these interactions, and unusually for CG lipid models, charged (Q) particles were included, which interact via a Coulomb potential which used a relative dielectric constant of er = 20, to implicitly account for charge screening. This electrostatic interaction is also shifted using the standard 30 | Chem. Modell., 2008, 5, 13–50 This journal is

c

The Royal Society of Chemistry 2008

GROMACS130,131 shift function between 0 nm and rcut. The Q particles were designed to represent groups that have a full charge, and the groups full charge should be used. The exception are small hydrated ions, that use a reduced charge, to take into account the effect of an implicit hydration shell. Energy, density, length, temperature and pressure all have the same meaning for in the Marrink CG model as they do for any atomistic MM model. Time is different though, as the CG interaction potential is very smooth, and therefore conformational sampling is faster as the system is not trapped in local minima. This means that CG time is probably 3–6 times faster than atomistic MD time. To achieve parity, Marrink et al. scale time by four. Part of the reason for the success of the model is that the smooth potential surface allows large timesteps to be used with the dynamics integrator—timesteps up to 50 fs can be used, though 40 fs gives more stable results. The Marrink model is implemented in Gromacs,130,131 and the input parameters for this model can be downloaded from http://md.chem.rug.ml/marrink/ coarsegrain.html. The Marrink model has been used in several applications110,132–134 and has been compared against a united atom atomistic force field (GROMOS)135–137 (a united atom MM forcefield is one in which non-polar hydrogen atoms are not modelled explicitly, but are instead combined into the representation of their bound heavy atom, e.g. ethane is represented as a pair of CH3 particles). This comparison included an estimate of the loss of configurational entropy incurred in moving from the atomistic to coarse grain model.135,136 These comparisons showed that while the CG model compared well, it failed to fully represent some thermodynamic properties, e.g. oil/oil interactions were too weak in the CG model, and water/oil repulsion appeared to be overestimated.137 In addition, the CG model showed less enthalpy/ entropy compensation than an atomistic model for the mixing of oil and water, leading the authors of the comparison to suggest that accurate solvation thermodynamics should be employed to help in the reparameterisation of CG models. 137 Marrink et al. have now performed this reparameterisation and have created the MARTINI forcefield.138 The MARTINI forcefield was parameterized in a consistent manner to reproduce the partitioning free energies between polar and apolar media of a large number of chemical compounds. The advantage of the Marrink model is that it provides a set of basic CG bead building blocks that may be used to create different types of molecules. Bond et al.139,140 used these building blocks to construct CG models of membrane proteins, so as to investigate membrane-protein insertion. They compared their results to fully atomistic simulations, and found qualitative agreement. Marrink and co-workers have also now parameterized CG beads to represent amino acid residues in proteins,134 and have used these parameters to perform a CG study of the selfassembly of G protein-coupled receptors in membrane bilayers.

3.1.2. Multiscale CG parameterisation. Despite the problems of multiscale parameterisation, e.g. the lack of portability of the parameters and the potentially complex form of the interaction potentials, and the success of the experimentally parameterized Marrink CG model, a trend over the last few years is that CG models of lipids have begun to move towards using parameters that are derived systematically from atomistic simulations. This is because multiscale parameterisation provides a route to systematically obtain CG potentials for mixed or complex systems, or for cases where experimental data is difficult to obtain. Parameterisation methods are appearing that use the type of multiscale one-way atomistic to CG information flows that were first used by Levitt and Warshel in the 1970s. This trend parallels the historical development of atomistic MM force fields, which also moved away from complex experiment-based parameterisation schemes towards using wellvalidated recipes (e.g. GAFF44) to get MM parameters from QM calculations. CG parameterisation schemes now seem to be following this trend, and systematic Chem. Modell., 2008, 5, 13–50 | 31 This journal is

c

The Royal Society of Chemistry 2008

methods to obtain CG parameters from atomistic MM simulations are now becoming popular. Izvekov and Voth141,142 have developed a method that automates the generation of CG potentials and parameters from an atomistic MM simulation. Their method, which they term ‘‘multiscale coarse graining’’ (MS-CG), automatically generates pairwise interaction potentials for a defined CG model based on results from an atomistic simulation. They have developed a force matching method, which is an extension of the least-squares force matching approach developed by Ercolessi and Adams.143 The method is used to coarse-grain an underlying atomistic simulation trajectory. As in other coarse grain models, the atoms are combined together into beads, and the system as a whole is therefore represented as a set of interacting beads. The potential of mean force is estimated between each bead based on the average forces experienced during the atomistic trajectory. This is evaluated using the atomistic potential energy surface, and CG potentials are derived that minimise the difference between the CG force and the average atomistic force. An analytical derivation of the method has been published,144,145 and statistial mechanical justification of using pair potentials for the CG particles has been developed.146 If the CG potential depends linearly on the fitting parameters, then the fitting of the parameters can be written as a series of over-determined linear equations, and the least-squares solution to which can be found by orthogonal matrix triangulation (QR decomposition).141 The efficiency of the fitting can be enhanced by breaking the entire atomistic trajectory into small chunks and generating the CG parameters for each chunk independently. The final CG parameters are then found by averaging the parameters over each chunk. This method is very efficient, but it requires that the CG potential depends linearly on the CG parameters. This is achieved by using a series of splines, which are linear in the spline parameters, and combining this with a Coulomb potential for the electrostatics. The effect of this coarse-graining is to return a model of the forces between pairs of CG beads that accurately reflects the average forces experienced between the atoms contained within those beads during the course of the atomistic molecular dynamics trajectory. Izvekov et al. applied this method to generate a CG potential for a DMPC bilayer. They collapsed the forces over a 400 ps atomistic trajectory onto a bead representation of DMPC that was similar to the Shelley125 or Marrink126 models. These forces were input to the force matching equations to yield the CG potentials. Despite the fact that some of the CG beads were charged, they were able to omit the Coulomb term from the fitting, as it was found that due to charge screening, the effect of electrostatics could be handled sufficiently by the short range spline functions of the MS-CG forcefield. This method has since been used to derive a CG model of carbonaceous nanoparticles,147 monosaccharides in water148 and of ionic liquids.149 Because this method systematically derives CG parameters using only an atomistic simulation, it is much more straightforward to develop parameters for mixed systems, for which experimental data is either unavailable or difficult to obtain. Izvekov et al. demonstrated this advantage by deriving a CG model for a mixed DMPC/cholesterol bilayer.150 MS-CG effectively provides a CG model of a system that approximates the free energy surface. However, dynamics simulations using the MS-CG model cannot be related to the true dynamics of the atomistic model. Izvekov et al. have tried to overcome this problem by using the atomistic simulation to parameterize friction coefficients which can be used with a Brownian dynamics integrator.151 While this has only currently been applied to liquid methanol, they plan to apply this method to membrane systems. Lyubartsev152 has also developed a multiscale parameterisation method that has been used to systematically build a CG model of a DMPC bilayer. Lyubartsev uses an inverse Monte Carlo method153 to generate the CG parameters from an underlying atomistic simulation. The atomistic simulation trajectory is analysed to generate the radial distribution functions (RDFs) for the CG bead model. These RDFs can be converted into pairwise interaction potentials between the beads. The 32 | Chem. Modell., 2008, 5, 13–50 This journal is

c

The Royal Society of Chemistry 2008

RDFs calculated from a CG simulation using these initial interaction potentials differ from those calculated from the atomistic simulation. An inverse Monte Carlo algorithm153 is therefore used to iteratively refine these interaction potentials by correcting them by the difference between the CG and atomistic RDFs. This is essentially the same method that was used by Shelley et al.125 to derive the parameters between the CG lipid head group particles, and is also similar to the Boltzmann inversion method,154 which also uses an iterative procedure that uses RDFs measured from atomistic simulations to derive CG interaction potentials. Lyubartsev has used this method to fully parameterize his own CG model of DMPC. Arkhipov et al. have taken multiscale coarse graining to an extreme in their recent work,155 in which they use atomistic models to parameterize CG models of a complete virus capsids. Each CG particle in the simulation represented about 200 atoms. Each CG particle interacted via a Lennard-Jones potential, which was parameterized to match the size of the domain the CG particle represented, as calculated from the radius of gyration of that domain measured from an atomistic model. The resulting CG model was able to simulate complete virus capsids (of dimensions 10 nm to 100 nm) over timescales of 1 ms to 10 ms. Chao et al.128 have developed a CG model that is based on the concept of soft rigid blobs. A blob is a grouping of atoms within a molecule that interacts with other blobs using a non-spherical, soft potential. A molecule is divided into blobs, e.g. see Fig. 3 for a proposed blob model of a phospholipid.128 Each blob represents a rigid grouping of atoms, and the interaction potential between a pair of blobs is calculated based on the underlying atomistic potential energy of interaction between the atoms within those blobs. However, unlike other CG models, which try to approximate this interaction potential using high symmetry beads, the rigid blob model expands this interaction potential as a Taylor series to produce interaction moment tensors between blobs. These are akin to the multipole tensors in a standard multipole expansion. The result is that the blob potential more accurately represents the three dimensional shape of the group of atoms that it represents. A further advantage is that the quality of the model can be increased by adding higher-order moments to the interaction potentials. 3.2 One-way top–down interfacing methods The multiscale atomistic/CG methods discussed so far all involve a single, one-way transfer of information from the atomistic model to the CG model. Essentially the atomistic models are used to parameterize a CG model, so that the CG model can then be used to explore the conformational space of the system more rapidly. However, in building the CG model, the atomistic fine detail of the system has been lost, as it has been smeared out into a set of beads. Recovering the atomistic fine detail would however be very useful, as this would allow the simulator to zoom back in and observe interesting configurations revealed during the CG trajectory at an atomistic level of detail. Knecht and Marrink133 realised the advantages of this approach, and have used it together with the Marrink CG lipid model to model vesicle formation. The CG model is used to self-assemble a vesicle, and this CG structure is then used as a template for the starting structure for an atomistic simulation. The conversion from the CG model to the atomistic model is straightforward. The coordinates from the CG simulation were scaled by a factor of 1.6. Atomistic models of the lipid, taken from an atomistic simulation of the planar bilayer, were fitted on top of their CG counterparts. The CG model had to be scaled to prevent steric clashes caused by the curvature of the CG vesicle bilayer. Molecular dynamics simulations were run on the atomistic model, allowing vesicle membrane fusion to be studied at the atomistic level. It is relatively straight-forward to move from a coarse grain model of a lipid to a fully atomistic model, as the main features of the lipid, namely head group and two tails, are still present in the CG model. It is less straight-forward to reconstruct the Chem. Modell., 2008, 5, 13–50 | 33 This journal is

c

The Royal Society of Chemistry 2008

atomistic coordinates of a protein from a CG model. This is because many CG protein models represent only the backbone of the protein, and if the side chain is represented, it is done so using only one or two beads. There is a lot of atomistic structural information missing from a CG protein model, and reconstructing this fine detail is not trivial. McCammon and co-workers156 attempted in their multiscale exploration of ligand binding to HIV protease to map from a single residue per bead protein CG model back to an atomistic representation. The binding site of HIV protease is covered by two large flaps, which open and close on a timescale that is beyond that is accessible on the atomistic molecular dynamics timescale. McCammon and co-workers have, over a series of papers,157–162 developed a very successful CG model of HIV protease. This model uses a single bead per residue, and was parameterized based on a statistical analysis of the crystal structure of the protein. This CG model is capable of simulating opening and closing motion of the large flaps that gate the entrance to the active site.158,162 McCammon and co-workers used their CG model within a multiscale simulation to investigate the binding pathways of an HIV protease inhibitor. The CG simulation was used to investigate opening and closing of flaps which gate the binding site of the enzyme, to see how the presence of the inhibitor affected flap dynamics, and to test whether the inhibitor could bind with the flaps closed. Because the CG model lacks atomistic detail, McCammon and co-workers selected two configurations from the CG trajectory to act as starting points for fully atomistic, implicit solvent Brownian dynamics simulations of the protease plus inhibitor. While it was straight-forward to rebuild a fully atomistic model of the inhibitor from the CG model, it was not simple to rebuild an atomistic model of the enzyme. This is because the CG model is missing lots of important structural information, e.g. the orientation and location of the amino acid side chains. To avoid the difficulty of rebuilding the atomistic detail of the enzyme, they instead used crystal structure configurations of the protease that had backbone geometries that were similar to those observed in the two snapshots from the CG trajectory. McCammon and co-workers156 used the CG simulation to provide input to a pair of atomistic simulations, thereby performing a pair of one-way, top–down information transfers from the CG to atomistic level. They sidestepped the great challenge of top–down transfers, namely the difficulty of recovering fine atomistic detail, by only using the CG model to locate the binding geometry of the inhibitor, and as a means to select which crystal structures corresponded to that particular binding geometry. Other workers have directly tackled the problem of reconstructing atomistic detail from CG models. The RACOGS algorithm (Reconstruction Algorithm for Coarse Grain Structures)163 is an example of a recently developed algorithm that attempts to solve the reconstruction problem. The algorithm is designed to rebuild an atomistic protein model given only the position of the Ca atoms. The algorithm has several stages; 1. The backbone atoms (C, O, N) are located based on the input positions of the Ca atoms using the algorithm developed by Feig et al.,164 which is itself based on the work by Milik et al.165 The algorithm places the backbone atoms into the average positions based on a statistical analysis of the positions of backbone atoms taken from 4013 non-redundant protein structures from the PDB. 2. The next step is the positioning of side chain atoms. These are positioned using the algorithm developed by Xiang and Honig,166 which uses a rotamer library to initially place the side chain atoms. The algorithm starts by adding the side chain atoms to a residue based on selecting the rotamer for that residue that has the lowest energy of interaction with the backbone atoms of the other residues. The interaction energy is evaluated using the vdW and dihedral energy terms from the AMBER99 forcefield.167 In performing this step, the rotamer library is pruned of any rotamers that involve a steric clash with the backbone. This helps improve the efficiency of the algorithm by removing any structurally unrealistic conformations as quickly as possible. 34 | Chem. Modell., 2008, 5, 13–50 This journal is

c

The Royal Society of Chemistry 2008

3. Once all of the side chains have been constructed, an iterative procedure is employed whereby each residue is considered in turn, and its energy of interaction of all possible rotamers of its side chain with all of the other residues’ side chains and backbone atoms is calculated. If another rotamer in the library has a lower energy, then it is selected in place of the existing rotamer. This iteration is repeated until an entire cycle over all of the residues in the protein is completed without a single residue being replaced. 4. The next step involves a minimisation of any high-energy side chains (defined by whether the energy of interaction of that side chain with the rest of the protein is greater than a specified cutoff). Each high energy side chain is, in turn, energy minimised while holding the rest of the protein fixed. The minimisation is performed using the bond, angle, dihedral and vdW terms from the AMBER99 forcefield. 5. Finally, hydrogen atoms are added using the LEAP module of AMBER 8, 168 and the entire protein is energy minimised using the conjugate gradient method, and the full generalized born/surface area (GB/SA)169 implicit solvent AMBER99 forcefield. Heath et al.163 were able to use this algorithm to hop between CG and atomistic models of the proteins src-SH3, S6wt and S6Alz. One of the problems of hopping from a CG to an atomistic model is that, because different forcefields are used, the free energy surfaces of the two models could be very different. Thus configurations that are sampled by the CG model may not be important for the atomistic model. Heath et al.163 investigated this for the CG model (the Das et al. model170) and atomistic model (AMBER99167) that they used. They found good agreement between the free energy surfaces, and were able to successfully use the CG model to investigate the folding and misfolding of these proteins, and to use their RACOGS algorithm to zoom in to obtain the atomistic detail of the misfolded structures. Ding et al.171 have also developed a method to reconstruct the atomistic fine detail of a protein structure from a CG model. They use a CG model that uses the positions of the Ca and Cb atoms to investigate domain swapping in seven proteins. The CG model enables the running of simulations that cover the timescale on which the domain swapping is seen to occur. A fully atomistic model of the protein is then reconstructed from the configurations generated from the CG simulation. The reconstruction algorithm is composed of the following stages; 1. The positions of the Ca and Cb atoms from the CG model are used to position the N and C backbone atoms so that the correct chirality is maintained (D-amino acids). To restrict the rotational freedom, a strong constraint was used, namely that neighbouring residues must have a planar peptide bond. Harmonic potentials were added between neighbouring backbone atoms, the equilibrium distances of which were taken from average distances calculated from the PDB structures. The Ca atoms were then immobilised by setting their masses to infinity, and a short molecular dynamics simulation was then run to relax the system. 2. Backbone hydrogens were added to create non-specific backbone hydrogen bonds. A CG potential was added between the Cb atoms, and a short molecular dynamics simulation was run to optimise the backbone hydrogen bond network. 3. The side chain and backbone O atoms were added using rotamers and a Monte Carlo simulated annealing algorithm. The scoring function for the simulated annealing used the vdW interactions from the Cedar forcefield,172 the EEF1 implicit solvation model173 and a statistical potential for hydrogen bonds, as proposed by Kortemme et al.174 This reconstruction algorithm was also used by Sharma, Ding and Dokholyan175 to reconstruct the atomistic detail from CG models of nucleosomes. This algorithms is very useful, as, like the RACOGS algorithm, it allows the CG model to be used to sample conformational change on timescale beyond that which is accessible to standard atomistic methods, yet it then allows the simulator to zoom in and reconstruct the atomistic data for interesting configurations.

Chem. Modell., 2008, 5, 13–50 | 35 This journal is

c

The Royal Society of Chemistry 2008

3.3 Two-way parallel interfacing methods Multiscale modelling across the atomistic/CG boundary is not limited to single, oneway transfers of information. Two-way interfacing methods, whereby the CG and atomistic levels exchange information dynamically throughout a simulation, are in active development and use. Two main schemes for linking CG and atomistic simulations have become popular; parallel methods, which use loosely-coupled atomistic and CG simulations running in parallel, and embedded methods, whereby atomistic regions are embedded within a CG model. 3.3.1. Parallel atomistic/CG methods. The replica exchange method, discussed in section 2.4 in terms of atomistic and QM calculations, has also been applied with CG models. For example, Nanias et al.176 have used replica exchange moves over temperature to enhance sampling within a CG protein simulation. Just as there has been interest in developing replica exchange moves that map between QM and MM models,105 so to has there been recent interest in developing replica exchange methodology that can map between atomistic and CG representations. Such replica exchange moves represent a parallel, two-way interface between the atomistic and CG levels, as atomistic and CG simulations are run in parallel, and information is only periodically exchanged between them. Lyman and Zuckerman177 have developed a atomistic/CG replica exchange method that they call ‘‘resolution exchange’’ (ResEx). The aim of this method is to run both an atomistic and CG simulation of a system in parallel, and to periodically attempt replica exchange moves between the simulations. The idea is that the enhanced sampling of the CG model will be shared with the atomistic model via the replica exchange moves. For this method to work, the coarse grain model must use a subset of the coordinates of the atomistic model. In this case, Lyman and Zuckerman achieved this by using a united atom model at the CG level (OPLS united atom forcefield33), and an all-atom model for the atomistic level (OPLS all atom forcefield34). While technically being a CG model, a united atom forcefield is more normally considered to be an atomistic representation. This is because a united atom forcefield does not use beads to represent residues or parts of residues, but instead merely removes aliphatic and aromatic hydrogens from the model by collapsing them into their bonded carbon. Despite this, Lyman and Zuckerman’s method can still be considered as a multiscale method, and this application points to some of the difficulties that more ambitious applications would have to overcome. One major difficulty is that the replica exchange moves between the united atom and all atom representations had a very low acceptance ratio. This is because the CG model uses only a subset of the atomistic coordinates, so replica swap moves can lead to unphysical conformations as this subset is moved out of step with the rest of the atomistic coordinates. To overcome this problem, Lwin and Luo,178 in a similar method, minimised the energy of the atomistic model after each swap, but before the Monte Carlo test that was used to accept the configuration. While this does successfully increase the acceptance ratio of the move, it violates detailed balance, so does not sample the required ensemble correctly. Lyman and Zuckerman’s solution to the problem is to use incremental coarsening. With this method, instead of switching the whole protein between the CG and atomistic models in one go, instead individual residues in the protein are switched. This leads to the problem that CG residues then need to interact with atomistic residues, so CG/atomistic crossterms need to be included in the Hamiltonian. Fortunately the OPLS united atom and all atom forcefields are compatible, and so can be combined in this way, at least for this application. Now multiple replicas of the system may be used, with each replica representing a different percentage of the protein using the CG forcefield. Lyman and Zuckerman tested their method by application to the pentapeptide metenkephalin. They used six replicas, the first replica using a fully atomistic model, then second replica using one CG residue and four atomistic, the second using two 36 | Chem. Modell., 2008, 5, 13–50 This journal is

c

The Royal Society of Chemistry 2008

CG residues and three atomistic, and so on. In addition, they also increased the temperature of each replica, such that while the fully atomistic model was simulated at 298 K, the fully CG model was simulated at 700 K. However, despite the small size of the system, the small difference between the CG and atomistic models (OPLSUA vs. OPLSAA) and the use of incremental coarsening, the acceptance ratio of the swap moves was still very low, running between 2.5% and 5.8%. Applications of this method to larger systems, or using a greater difference between the atomistic and CG models therefore looks problematic. Christen and van Gunsteren10 have developed a novel multiscale method that they call ‘‘multigraining’’, which aims to use the CG model to enable both relaxation of large molecular systems and sampling of slow processes with concurrent atomic detail representation of the results. In this method, both an atomistic and a CG model of a molecule are used simultaneously. Each molecule in the simulation has both a set of atomistic (fine grain) coordinates, rFG, and a set of coarse grain coordinates, rCG. The coarse grain coordinates are constrained to map to the atomistic coordinates. The Hamiltonian of the system is composed of atomistic interaction terms, UFG(rFG), and CG interaction terms, UCG(rCG). Molecular dynamics is then performed using both the atomistic and CG potentials simultaneously. This is referred to as multigraining MD, and the algorithm is as follows; 1. First the coarse grain coordinates are determined by mapping them from the atomistic coordinates (e.g. by placing CG beads at the geometric centre of a residue). 2. Calculate the forces on the CG particles using the CG potential energy terms, and the forces on the atoms using the atomistic potential energy terms. 3. Distribute the forces from the CG particles back onto the atoms. 4. Propagate the coordinates of the atoms using the leap-frog integrator. By running both the CG and atomistic models in parallel, and by mapping the CG coordinates from the atomistic coordinates, this method avoids all of the problems associated with information loss when moving back and forth between the CG and atomistic representations. Also, the atomistic and CG potential energy terms are separate, therefore avoiding the problem of parameterising atomistic/CG crossterms. Also, by keeping the atomistic and CG potentials separate, Christen and van Gunsteren are able to use a l scaling parameter to switch between the atomistic and CG potentials, e.g. at l = 0, only the pure atomistic potential is used, at l = 1 only the pure CG potential is used, while at l = 0.5, a 50–50 mix of the CG and atomistic potentials are used. Christen and van Gunsteren then employ replica exchange moves over this l coordinate, thereby allowing several parallel trajectories to swap dynamically between CG and atomistic representations. They tested the method by using it to simulate liquid octance, but like Lyman and Zuckerman, they experienced problems with a low acceptance ratio for the replica exchange moves, and needed to use 24 replicas across the l coordinate. Using 24 replicas, they achieved an average acceptance ratio of just 24%. This figure masks the very poor acceptance ratio for replicas at l = 0, where the purely atomistic potential was used. Just 8% of the 300 replica exchange moves were accepted at l = 0, which is disappointing, given that it is the sampling at l = 0 which is important for the calculation of correct thermodynamic properties. However this method is useful as a means of performing a rapid equilibration or conformation search at the CG level, and then smoothly switching back to the atomistic model to explore interesting configurations.

3.3.2. Embedded atomistics/CG methods. Parallel atomistic/CG methods still use a simple interface between the CG and atomistic models, with information exchange occurring only during the replica exchange moves. This represents a loose coupling between the atomistic and CG models. An alternative to this loosely-coupled method is to develop a tightly coupled method whereby an atomistic model is linked directly to a CG model. This allows for an atomistic region, e.g. an atomistic model of an ion channel, to be embedded within a CG simulation, e.g. a CG model of a cell Chem. Modell., 2008, 5, 13–50 | 37 This journal is

c

The Royal Society of Chemistry 2008

membrane. This requires that the atomistic and CG models are closely coupled together, with information exchange occurring every timestep via atomistic/CG cross-terms in the Hamiltonian. This is challenging, as it is not immediately obvious how to obtain the parameters that describe the atomistic/CG interaction, nor what the form of the interaction potential should take. In section 3.1.2 the multiscale coarse graining (MS-CG) method, developed by Izvekov and Voth, was presented as a method that could be used to systematically derive the functional form and parameters for MM-CG interactions. Shi,179 working with Izvekov and Voth, has used the MS-CG method to systematically derive the functional form and parameters for the atomistic/CG interaction of an atomistic gramicidin A polypeptide ion channel immersed in a CG membrane. The DMPC membrane and solvent water were modelled using the CG model developed by Izvekov and Voth,150 which was developed using the MS-CG method. This method derives the parameters for the CG interactions from the average forces experienced by the atoms during an atomistic dynamics simulation. These forces are then coarse-grained and a CG potential is fitted. Shi ran a short all-atom simulation of the whole membrane/ion channel system and used this to develop the CG/atomistic forcefield using the MSCG method. The forcefield was obtained by treating the atomistic and CG regions equally within the force matching algorithm. However, to reduce computational cost, some simplifications and approximations were made. First, the forcefield was split into three parts; atom–atom, atom-CG and CG-CG. The atom–atom forces were assumed to be the same as those from the wholly atomistic simulation, while the CG-CG forces were taken from an MS-CG application of a pure lipid bilayer embedded in water. This meant that only the atom-CG forces had to be derived from the fully atomistic simulation of the entire system. The atom-CG forces were obtained by subtracting the atom–atom and CG-CG forces from the reference forces from the fully atomistic simulation, and then fitting the residual forces using the MS-CG method. This method is not as rigorous as applying the full MS-CG method in a single step, but it is much more efficient, and Shi et al. found that this strategy worked quite well for this system. Essex and co-workers have developed a CG model that is designed from the outset to interact with an atomistic forcefield.127,180,181 They have developed a CG lipid model of DMPC127,180 that uses six Gay-Berne123 ellipsoids to represent the hydrocarbon tails, and two Lennard-Jones (LJ) particles to represent the head group. The glycerol region was modelled by 2 further Gay-Berne units, thereby allowing the entire DMPC model to be represented by just ten interaction sites (see Fig. 3). The key advance of the Essex model is that it was designed from the outset to be compatible with atomistic forcefields. The Gay-Berne potential is essentially an anisotropic form of the Lennard-Jones potential, which is the most common potential used to model van der waals (vdW) non-bonded forces between atoms. This means that it is possible to interface the Gay-Berne and LJ potentials so that the vdW interaction energies between Gay-Berne beads and Lennard-Jones (LJ) atoms can be calculated. However, to correctly handle the electrostatic interaction between the CG and atomistic levels, the CG model has to explicitly include a representation of the charge distribution of the lipid. The Essex model achieves this by including point charges in the two head group particles, and point dipoles in the two GayBerne particles that represent the glycerol region. Water was modelled explicitly using the soft sticky dipole model182 (SSD), which despite only using a single interaction site, is capable of accurately reproducing the structural, thermodynamic, dielectric and temperature-dependent properties of water.182–185 The lipid model was parameterized by trial-and-error molecular dynamics, whereby the parameter space was searched and molecular dynamics simulation run to reproduce the experimental structure of DMPC bilayers (to test whether the desired experimental properties were obtained). The point charges and dipoles of the model were chosen to reproduce the net electrostatics (charge and dipole) of each bead from the underlying atomistic level model. To parameterize the strength of interaction between atomistic 38 | Chem. Modell., 2008, 5, 13–50 This journal is

c

The Royal Society of Chemistry 2008

and CG molecules, Essex and co-workers181 calculated solvation free energies of atomistic amino acid side chain analogues in boxes of CG octane, and compared this to experiment. The atomistic and CG interactions were found to be unbalanced, but Essex and co-workers found that simple scaling of the cross terms led to good agreement of comparable quality to atomistic simulations. To calibrate the scaling factor for the interaction of an atomistic inclusion with the CG water model, a similar set of calculations were made of the hydration free energy of atomistic amino acid side chain analogues in the SSD water model. The calculated hydration free energies were also compared with experiment. The calculation of these scaling factors removes the need for the atomistic forcefield to be reparameterized for use with the CG model. This is an interesting approach to the solution of the cross-term parameterisation problem of atomistic/CG multiscale modelling, and has the promise to be transferable across a range of solutes and proteins, thereby removing the need to reparameterize the model for each new system under study. The challenges in creating atomistic/CG multiscale methods parallel those of creating QM/MM methods. The definition of the QM/MM interface is relatively straightforward if the divide is between molecules. However the definition of the interface becomes more complicated if the divide is within a molecule, as then the Hamiltonian has to somehow include bonded terms between QM atoms and MM atoms. In much the same way, the definition of the atomistic/CG interface is more straightforward if it falls between molecules, rather then within a molecule, and that is why the majority of atomistic/CG multiscale simulations avoid this approach. The exceptions are the parallel method developed by Lyman and Zuckerman177 (the ResEx method discussed in the last section) and an embedded CG/atomistic method developed by Neri et al.186 Neri et al. have developed a method where a protein is modelled using a CG representation, but the active site is modelled using an atomistic representation. The aim is to allow efficient sampling of protein conformational change, due to the use of the CG model, but to still capture the fine-detail atomistic motions in the active site, which is the area of interest in protein-ligand binding or computational enzymology calculations. They tested the method by application to two proteins of active pharmaceutical interest: HIV protease, which is a target for anti-HIV medication, and human b-secretase (BASE) which plays a role in the onset of Alzheimer’s disease. The active site atoms were modelled using the GROMOS9630 atomistic forcefield, while the rest of the protein was modelled at the CG level by only considering Ca centroids. Solvent-protein interactions were modelled in terms of viscosity and the addition of random forces in the framework of a stochastic dynamics simulation. The interaction between the atomistic and CG regions was handled by the addition of an interface region. Several residues at the boundary of the atomistic and CG regions were modelled using both a CG and atomistic representation. The atomistic region residues interacted with the interface residues using the standard atomistic potential. To maintain backbone integrity between the interface and CG regions, harmonic bonds are added between the C a atoms of the interface residues with the Ca atoms of the neighbouring CG residues. In addition, an exponential-form non-bonded potential is added between CG and interface residues, and the parameters for the CG model and for the interface were chosen so as to reproduce the root mean square fluctuations (RMSF) of HIV protease as observed during fully atomistic and fully CG simulations. A key parameter is the thickness of the interface region, with the key concerns being that the interface region has to guarantee the correct geometry of the atomistic residues, it has to get the local electrostatics right, and it has to be able to transmit the modes of vibration experienced by the rest of the protein, and which are mimicked by the CG model. Neri et al. found that for both HIV protease and BASE they needed to use an interface region that included all residues that had at least one atom within 6 A˚ of any of the residues in the atomistic region. There is a clear separation in the methods presented so far between the atomistic and CG regions. One molecule, or part of one molecule (the protein), is treated at the Chem. Modell., 2008, 5, 13–50 | 39 This journal is

c

The Royal Society of Chemistry 2008

atomistic level, while the remaining molecules (lipid membrane and solvent) are modelled using CG techniques. An alternative to defining the atomistic and CG levels by molecule is instead to define the levels by region, namely to divide the simulation space into an atomistic region and a coarse grain region. This creates a more dynamic split between the two models, as individual molecules are free to diffuse from one region to another, and therefore algorithms must be developed that account for the interconversion of CG and atomistic models. Praprotnik et al. have, over a series of insightful papers,187–191 developed just such algorithms. They have created a multiscale simulation method whereby the simulation space is divided into an atomistic region and a CG region, and the molecules in the system are free to diffuse between these two regions. The methods developed allow on-the-fly interchange between the molecules’ coarse grain and atomistic representations, enabling large length and time scales to be achieved, while still retaining the atomistic fine detail of parts of the system. The key development in this work is the methodology used to allow molecules to switch dynamically between CG and atomistic representations. They call these methods AdResS (adaptive resolution scheme). The original application of the method was to the multiscale dynamics of liquid methane.187 A CG model of methane was constructed based on the radial distribution function of methane obtained from a simulation of the reference atomistic model. The simulation box was split in half, creating an atomistic region and a CG region, and an identical number of methane molecules were placed in each half of the box. To ensure that the transition between the two regions was smooth, a ‘handshaking zone’ was created between the atomistic and CG regions. To achieve a smooth transition from the atomistic to CG forcefields, the forces are scaled in the handshaking zone. As the switching of the resolution can be considered as a first order phase transition, it is necessary for this scheme to be used in combination with a thermostat. Because the latent heat is generated at the interface region,187 it is important that a local thermostat is used that couples directly to local particle motion, e.g. Langevin or dissipative particle dynamics (DPD) thermostats. The other main challenge of an adaptive resolution method is how to handle the loss and reintroduction of atomic degrees of freedom at the atomistic/CG interface. It is important that the vibrational and rotational degrees of freedom at the atomistic model, which are missing in the CG model, are slowly reintroduced as a molecule diffuses across the interface between the CG and atomistic regions. To achieve this, when a molecule moves from the CG to atomistic region it is mapped onto an atomistic representation that has the same centre of mass and linear momentum as the CG representation. In addition, each atom is given rotational and vibrational velocities that come from a random molecule already in the atomistic region. Praprotnik used AdResS successfully to simulate a multiscale box of liquid methane.187 While the application worked well, there was a small problem with a pressure imbalance at the interface zone between the CG and atomistic regions. Praprotnik et al. solved this problem in a new version of their method,188 in which they also extended the scope of application to include spherical boundaries between the atomistic and CG regions. Since then, they have used the method to run multiresolution simulations of liquid water191 (using the CG parameterisation method of Lyubartsev152 to get the CG model of water) and have investigated the thermodynamic implications of running simulations that have a dynamic, and indeed fractional number of degrees of freedom.189 Ensing et al.192 have also developed a new multiscale method that allows for an atomistic region to be embedded within a CG simulation. Like AdResS, the method uses a spherical atomistic region that is interfaced to the CG simulation via a handshaking buffer zone. To ensure a smooth transition from the CG to atomistic representations, the potential energies, rather than the forces, are smoothly scaled from the CG to atomistic Hamiltonians as the molecule crosses the handshaking zone. Additional terms are also added to the Hamiltonian that account for the kinetic energy that is lost when the number of degrees of freedom is decreased when 40 | Chem. Modell., 2008, 5, 13–50 This journal is

c

The Royal Society of Chemistry 2008

a molecule moves from the atomistic to the CG region. These additional terms ensure that the total energy of the system is conserved even as the number of degrees of freedom change. In addition to using a different method to switch between the CG and atomistic regions, Ensing et al. also use a different method to handle the change in dynamics between the CG and atomistic regions. Instead of destroying and recreating the atomistic degrees of freedom as the molecules move across the boundary, they instead use the RESPA193 multiple timestep molecular dynamics algorithm to freeze out the intramolecular degrees of freedom of molecules as they move into the CG region. RESPA is a molecular dynamics method whereby different timesteps can be used to sample different parts of the Hamiltonian. It has traditionally been used to separate the sampling of ‘‘fast’’ forces, such as bond potentials, from ‘‘slow’’ forces, such as those arising from the non-bonded potentials. Ensing et al. cleverly use RESPA to separate the sampling of the CG and atomistic Hamiltonians by region, and thereby effectively freeze the atomistic representation as it crosses into the CG region. This allows them to tune the molecular dynamics integrator optimally across the entire multiscale system. While this method has been applied only to a model system and liquid methane, it shows great promise as a method that could find widespread use in biomolecular modelling.

4. Interfacing particle with continuum models Continuum methods provide the largest time and space ranges considered in this review. While continuum models have been popular for the modelling of solids194 or of liquid or gas flow,195 they have only recently been applied to the study of biomolecular systems. This is because continuum methods tend to be best suited to periodic or smoothly varying systems, and biomolecular systems are distinctly inhomogenous, and exhibit physical behaviours that require the the modelling of fine-level atomistic detail. By far the most popular continuum models used in biomolecular modelling are those that replace atomistic waters with an implicit solvent model. Many implicit solvent models, which include the Poisson Boltzmann (PB)196,197 and Generalized Born (GB)198,199 methods, model the solvent as a dielectric continuum, and calculate the polarisation of that continuum caused by the charge distribution of the atomistic molecule model. This polarisation leads to an electrostatic reaction field with which the atomistic molecular model then interacts. Implicit solvent models have been used for many years in combination with QM,200,201 MM202,203 and CG204,205 calculations. These represent a large class of truly multiscale methods, and have been reviewed in detail many times.204,206–209 Implicit solvent models have been the dominant class of multiscale continuum methods over recent years. However exciting new classes of multiscale continuum models have recently been developed. These new methods fall into the following categories, which are of similar definition and type to the interfaces used between the QM/MM and atomistic/CG levels: 1. One-way bottom-up interfacing methods. These involve a single transfer of information from the atomistic or CG level to the continuum level, e.g. by using the atomistic level to provide parameters for the continuum model. 2. Two-way embedded interfacing methods. These involve embedding an atomistic or CG model within a continuum representation. Implicit solvent models fall into this category. New multiscale methods, which capture hydrodynamic and mechanical effects have now also been developed. 3. Two-way parallel interfacing methods. These involve the running of an atomistic or coarse grain level simulation in parallel with a continuum calculation, and interfacing the two using a lightly coupled interface, e.g. by using properties calculated from the continuum model to update the boundary conditions of the atomistic or CG simulation. Examples of each of these categories of multiscale continuum interfaces will now be discussed in turn. Chem. Modell., 2008, 5, 13–50 | 41 This journal is

c

The Royal Society of Chemistry 2008

4.1 One-way bottom-up interfacing methods Continuum models can be interfaced with atomistic models using a one-way bottomup multiscale parameterisation scheme. In this scheme, an atomistic level calculation is used to provide the input parameters for the continuum model. This scheme is very simple, as it involves a single transfer of information from the atomistic level to the continuum level. Tang et al.210 have recently employed this scheme to get the parameters for a continuum model of a mechanosensitive ion channel embedded in a lipid membrane. The membrane was modelled as a homogenous elastic sheet of thickness 35 A˚, and area of 400 A˚  400 A˚, using a finite element (FEM) model. The transmembrane helicies were represented by homogenous cylindrical elastic rods of diameter 5 A˚with spherical caps at each end. The helicies were embedded within the membrane, and were also modelled using a FEM model. What makes this a multiscale simulation is that the parameters for this model were calculated using information calculated using atomistic-level models. This represents a one-way, bottom-up information flow from the atomistic level to the continuum level. Elastic parameters for the model were calculated from the mechanical properties of the membrane and helicies measured by experiment and calculated from atomistic molecular dynamics simulations. The interactions between the helicies and the membrane were modelled using a Lennard-Jones style attractive/repulsive potential. The parameters for this potential were determined based on molecular mechanics calculations run using the CHARMM forcefield.27 The channel is comprised of ten helicies. The interaction potential between helicies was parameterized by fitting to CHARMM potential energy calculations. For each pair of helicies, the interaction energy in vacuum was calculated using the CHARMM19 forcefield, with the coordinates set to either X-ray or homology structures. This calculation was performed for different combinations of helix pairs, which effectively samples different orientations, and a range of centre of mass inter-helix separations that varied between 20 A˚ and 20 A˚. The interactions between each helix and the membrane were parameterized based on calculating the insertion energy profile of moving an atomistic model of the helix from an implicit solvent water model into an implicit lipid membrane model. In the implicit membrane models the membrane thickness was taken to be 23.5 A˚, which corresponds to the thickness of the hydrophobic part of the membrane. While this model was constructed as a proof of concept of FEM methods, it was able to reproduce the experimental gating tension of the channel, and the structural variations along the gating pathway as observed from biased atomistic molecular dynamics simulations. This application successfully demonstrated the potential of continuum models of biomolecular systems to reach length (sub-mm) and time (multi ms) scales that are not accessible to atomistic level calculations. 4.2 Two-way embedded interfacing methods Continuum models can be directly interfaced with atomistic or coarse grain models using a two-way embedded interface. In this scheme, the atomistic or CG model is embedded within a continuum model. Implicit solvent methods, in which an atomistic or CG model of a solute is embedded within a continuum model of the solvent, are popular and well-established examples of this type of interface. Implicit solvent models represent the solvent as a dielectric continuum, and allow the electrostatics of the atomistic or CG solute to polarise the continuum, which then results in an electrostatic reaction field that returns to interact with the solute. Implicit solvent models have been reviewed in detail many times before, 204,206–209 and enable the dynamic transfer of electrostatic information across the atomistic/ continuum or CG/continuum interfaces. Recently, new multiscale continuum methods have been developed that allow for the dynamic transfer of mechanical and hydrodynamic information across these interfaces. One example is the work by Villa 42 | Chem. Modell., 2008, 5, 13–50 This journal is

c

The Royal Society of Chemistry 2008

et al.,211,212 who have developed a method that allows the transfer of mechanical information across the atomistic/continuum boundary. They have developed a multiscale model to investigate protein/DNA complexes. The method uses an atomistic model of the lac repressor protein and a continuum model of a large loop of DNA that is bound to the protein. The DNA loop is represented using a continuum elastic rod model, using elasticity theory. Protein MD simulations are used to position the DNA loop end points, and these end points then serve as crucial boundary conditions to elasticity theory to determine the shape of the loop. Elasticity theory then in turn provides the forces acting at the end points, which can be fed back into the protein molecular dynamics, completing the loop. Most atomistic/continuum embedded multiscale methods split the system into continuum and atomistic regions by molecule, e.g. the solvent is defined as being in the continuum, while the solute is atomistic. While being straight-forward to implement, such a split does not allow a complete representation of the physical detail of the system. One problem with continuum solvent models is that by representing all of the solvent as a uniform dielectric, they fail to capture atomic details, such as hydrogen bonding between the solute and solvent, which can be important factors to consider in protein-ligand binding or computational enzymology applications. It is therefore preferable to be able to spit the system spatially into atomistic and continuum regions, e.g. to model the solvent in the active site of the protein using an atomistic model, with the remainder modelled as a continuum. However, such a model must account for the diffusion of solvent between the atomistic and continuum regions. Schemes must be developed that allow for the conservation of momentum, mass and energy across the interface. While we are unaware of any such methods that have been developed for application to protein systems, Coveney et al. have, over a series of papers,6,213–219 developed an ambitious and highly promising multiscale method that allows the embedding of condensed phase atomistic simulations within continuum models for the purpose of studying hydrodynamics. The Coveney method works by dividing space into two main regions; an atomistic (particle) region, called P, and a continuum region, called C, modelled using computational fluid dynamics (CFD). An interface lies between these two regions. This interface is split into two parts; a C to P part, which sits on the continuum side of the interface, and a P to C part, which sits on the particle side. In the P region, the motion of the atomistic particles is solved using molecular dynamics. In the C region the motion of the continuum solvent is solved using standard continuum fluid dynamics (CFD). The hybrid scheme applied at the interface is designed to exchange fluxes of conserved quantities, namely energy, momentum and mass. In the C to P region, the fluxes from the continuum region are imposed on the particles. In the P to C region, the atomistic fluxes are coarse grained in time and space to form boundary conditions for the continuum domain.213 The details of how this is achieved is described in detail elsewhere.213,215,220 In essence, the stress induced in the P region by flow in the C region can be calculated and converted into a local momentum flux at the C to P interface. This flux can be converted into a force which is added to all of the particles in the C to P region. This pressure force from the continuum liquid is enough to stop the escape of particles from the C to P region, and, at a basic level, is responsible for the transfer of information from the continuum to the particle region. The P to C interface, on the other hand, is used to transfer information back to the continuum model. As the continuum model is a mesoscale representation of the solvent, the particle model must be averaged in both space and time. An averaging time of Dtavg is used, and the average momentum flux is calculated. This can then be used to establish the boundary condition of the continuum domain at the C to P interface. In addition to controlling the flow of momentum across the interface, the flow of mass must also be managed to ensure that the total mass in the whole system is conserved. This is achieved by calculating the mass flux at the interface and using that to determine whether particles need to be deleted Chem. Modell., 2008, 5, 13–50 | 43 This journal is

c

The Royal Society of Chemistry 2008

(to represent flow of mass form the particle to continuum region), or whether particles need to be created (to represent flow back from the continuum region). Removing particles is simply a case of choosing those that are closest to the top of the molecular dynamics simulation and thus closest to the continuum region. Inserting particles is more difficult, as sufficient space has to be found for the new particle amongst the existing particles, and atomic fine detail (coordinates and momenta) have to be reconstructed. Coveney et al. have developed the USHER algorithm221 which efficiently finds a location and configuration of the new solvent molecule that releases an energy equal to the mean energy per molecule. The velocity of the new particle is obtained randomly from a Maxwell distribution with a zero mean velocity and temperature equal to the simulation temperature. One of the original applications of this method213 used a simple model system, with a model polymer attached to a wall, solvated by a single-particle model solvent. They were able to use this setup to study the dynamics of this polymer within a solvent which was undergoing shear flow. Coveney et al. refined their method214,219 to correctly couple hydrodynamics, using a fluctuating hydrodynamics (FH) continuum model,222 and were able to show that fluctuations, at both the atomistic and continuum level were thermodynamically consistent. They demonstrated their method by simulating sound waves passing through a continuum, then atomistic model of liquid water, and then being reflected by an atomistic model of a lipid membrane bilayer.214 The light coupling between the different types of simulation (MD and CFD or FH) makes it well-suited to application over a distributed computing cluster, and Coveney et al. have demonstrated the method’s successful deployment over computational GRIDs.6 4.3 Two-way parallel interfacing methods Creating a multiscale method that directly embeds an atomistic region within a continuum model introduces a physical interface between the atomistic and continuum models. This complicates the implementation, as techniques must be developed that allow for the transfer of information across this physical interface, e.g. the mass, momentum and energy flux across the interface must be controlled. Chang, Ayton and Voth223 have developed a multiscale atomistic/continuum method that avoids the creation of a physical interface between the two regions. They achieve this by running both a continuum and an atomistic model of the system in parallel, and only loosely coupling this pair of simulations by periodically transferring structural information between them. Chang et al. developed the method, called multiscale coupling (MSC),112,223,224 originally to study the effect of long range dynamics of a lipid bilayer membrane at the atomistic level. They loosely couple together an atomistic model of a bilayer with a continuum-based mesoscopic model. The continuum model is based on an elastic membrane (EM) implicit solvent model originally proposed by Ayton and Voth.225 This model was parameterized from mechanical properties calculated from atomistic MD simulations of a DMPC bilayer. To better capture the dynamics of a membrane, this model has been updated to include an explicit mesoscale model of water.223 The elastic membrane model was used to simulate a bilayer of dimension, 890 A˚  890 A˚. To make this a multiscale simulation, a point on the elastic sheet representing the membrane is chosen, and an atomistic model of the membrane at that point is constructued. The atomistic model comprised 64 DMPC lipid molecules solvated by 1312 TIP3P waters. A real biological membrane is not subject to external constraints, and therefore it adopts a tensionless configuration.226 However simulations using the canonical (NVT) ensemble use a fixed number of particles and fixed membrane area. This results in a simulation of the membrane with a non-zero interfacial tension.227 This observation has resulted in the development of a new ensemble, whereby the membrane tension, g, is kept constant.226 This new ensemble, NPgT, allows the membrane tension to become part of the definition of the 44 | Chem. Modell., 2008, 5, 13–50 This journal is

c

The Royal Society of Chemistry 2008

thermodynamic state of the system, as g has become a thermodynamic parameter. Chang et al. use this fact to couple their continuum-based mesoscopic membrane model to the atomistic model. They run the atomistic model using the NPgT ensemble, and update the value of g every 1 ps in response to the changes in surface tension measured at the corresponding location in the continuum-based mesoscopic model. This allows a dynamic, one-way transfer of information from the continuum model to the atomistic model throughout the simulation. The advantage of this approach is that the atomistic level simulation does not directly interface with the continuum level simulation, so standard approaches can be used, e.g. periodic boundaries and the use of the particle mesh Ewald sum to model long range electrostatics. Also, while this application only allows the dynamic flow of information from the continuum to atomistic level, in theory, mechanical properties calculated from the atomistic simulation could then be calculated and used to reparameterize the continuum model, thereby completing the cycle. Ayton and Voth228 have employed this multiscale coupling (MSC) method to study the effect of long range membrane motion on the dynamics of an ion-channel. An atomistic ion channel was embedded within an atomistic simulation of the solvated bilayer, and this was coupled to the continuum-based EM model using the MSC method. Because formally an ensemble of atomistic simulations should be connected to the continuum model,228 they ran eight atomistic simulations in parallel, and connected each of these to the same point in the continuum model. They compared the dynamics of the ion channel using both the MSC boundary conditions and standard NPT boundary conditions and found that the orientation of histidine residues in the channel appeared more disordered when the atomistic level simulation was coupled to the continuum. This suggested that the fluctuations in stress caused by the long range dynamics of the membrane modelled at the continuum level, could indeed affect the dynamics of the ion channel at the atomistic level.

5. Beyond continuum models Continuum models allow biological simulators to access extremely large length and timescales, e.g. there are continuum models of a complete human heart.229 There is, however, a level of modelling that addresses even larger length and time scales. At this level, biological network models are used to represent cells, 230,231 organs232–234 or creatures.235,236 Detailed reviews230,237,238 of how to build models at this level are available. In brief, the methods work by constructing differential equations that describe the flow of material through a biological system, e.g. using a differential equation to represent an ionic current.237 Several equations can be coupled together to form a model, e.g. of the electric current flow across the human heart.232 The whole network of equations can be run together to produce an output signal that describes the biological property that the simulator is interested in modelling, e.g. modelling the oscillations in concentration of cellular metabolites.238 Multiple levels of biological network models can be combined together. The Physiome project239 is an attempt to define standard interfaces between these levels so that they can be combined together easily. To make these models fully multiscale, they could also be interfaced with atomistic, CG, or continuum representations. The conduits for information flow are the kinetic parameters that calibrate the differential equations that make up the network.238 For example, QM/MM computational enzymology calculations can be used to obtain the rate constants of enzyme catalysed reactions, or MM/CG calculations can be used to obtain the diffusion constants of ions through membrane pores. In this way, biological multiscale modelling will allow the coupling of simulations of electrons all the way up to models of a human heart. Chem. Modell., 2008, 5, 13–50 | 45 This journal is

c

The Royal Society of Chemistry 2008

6. Conclusion This review has outlined the techniques and methods involved in building computer models of biological systems that bridge between different length and time scales. Biomolecular multiscale modelling is not new: Levitt and Warshel pioneered much of the development of the foundations of the QM/MM and MM/CG interfaces in the 1970s. However, there has been an explosion of interest and activity in this area over the last few years that has been prompted both by the increases in available computer power, and the maturing of methodology at each of the different biomolecular modelling levels. As each level has matured, and become predictive, simulators are now rightly asking how can these methods be combined to answer the bigger questions of biology such as how does a cell maintain membrane cohesion, how do small changes in protein configuration initiate cell signalling pathways, and what is it about this collection of molecules that makes them a living system? While multiscale modelling methods are not yet capable of providing answers to these questions, the huge progress made over the last few years is now providing an unprecedented insight into the interplay between the biological and chemical worlds. Multiscale modelling methods will certainly be an exciting growth area of increasing importance in biology.

References 1 L. E. Bilston and K. Mylvaganam, FEBS Lett., 2002, 512(1–3), 185. 2 P. J. Booth, Curr. Opin. Struct. Biol., 2005, 15(4), 435. 3 A. Ababou, A. van der Wart, V. Gogonea and K. M. Merz, Biophys. Chem., 2007, 125(1), 221. 4 D. D. Vvedensky, J. Phys.-Condes. Matter, 2004, 16(50), R1537. 5 P. Koumoutsakos, Annu. Rev. Fluid Mech., 2005, 37, 457. 6 R. Delgado-Buscalioni, P. V. Coveney, G. D. Riley and R. W. Ford, Philos. Trans. R. Soc. A-Math. Phys. Eng. Sci., 2005, 363(1833), 1975. 7 P. V. Coveney and P. W. Fowler, J. R. Soc. Interface, 2005, 2(4), 267. 8 Y. Sugita, A. Kitao and Y. Okamoto, J. Chem. Phys., 2000, 113(15), 6042. 9 H. Fukunishi, O. Watanabe and S. Takada, J. Chem. Phys., 2002, 116(20), 9058. 10 M. Christen and W. F. van Gunsteren, J. Chem. Phys., 2006, 124(15), 154106. 11 R. J. Bartlett and M. Musia, Rev. Mod. Phys., 2007, 79(1), 291. 12 W. A. Lester and R. Salomon-Ferrer, Theochem.-J. Mol. Struct., 2006, 771(1–3), 51. 13 J. Z. Wu and Z. D. Li, Annu. Rev. Phys. Chem., 2007, 58, 85. 14 S. Y. Zou, G. G. Balint-Kurti and F. R. Manby, J. Chem. Phys., 2007, 127(4), 044107. 15 H. Y. Liu et al., Proteins, 2001, 44(4), 484. 16 M. J. S. Dewar, E. G. Zoebisch, E. F. Healy and J. J. P. Stewart, J. Am. Chem. Soc., 1985, 107(13), 3902. 17 J. J. P. Stewart, J. Comput. Chem., 1989, 10(2), 209. 18 J. J. P. Stewart, J. Comput. Chem., 1989, 10(2), 221. 19 M. Schutz and H. J. Werner, Chem. Phys. Lett., 2000, 318(4–5), 370. 20 S. Patel and C. L. Brooks, Mol. Simul., 2006, 32(3-4), 231. 21 A. Warshel, M. Kato and A. V. Pisliakov, J. Chem. Theory Comput., 2007, 3(6), 2034. 22 D. J. Price and C. L. Brooks, J. Comput. Chem., 2002, 23(11), 1045. 23 Y. G. Mu, D. S. Kosov and G. Stock, J. Phys. Chem. B, 2003, 107(21), 5064. 24 A. D. Mackerell, J. Comput. Chem., 2004, 25(13), 1584. 25 M. Patra and M. Karttunen, J. Comput. Chem., 2004, 25(5), 678. 26 B. Hess and N. F. A. van der Vegt, J. Phys. Chem. B, 2006, 110(35), 17616. 27 A. D. MacKerell et al., J. Phys. Chem. B, 1998, 102(18), 3586. 28 D. A. Case et al., J. Comput. Chem., 2005, 26(16), 1668. 29 L. J. Yang et al., J. Phys. Chem. B, 2006, 110(26), 13166. 30 X. Daura, A. E. Mark and W. F. van Gunsteren, J. Comput. Chem., 1998, 19(5), 535. 31 W. R. P. Scott et al., J. Phys. Chem. A, 1999, 103(19), 3596. 32 L. D. Schuler, X. Daura and W. F. Van Gunsteren, J. Comput. Chem., 2001, 22(11), 1205. 33 W. L. Jorgensen, J. D. Madura and C. J. Swenson, J. Am. Chem. Soc., 1984, 106(22), 6638. 34 W. L. Jorgensen, D. S. Maxwell and J. TiradoRives, J. Am. Chem. Soc., 1996, 118(45), 11225. 35 J. A. McCammon, B. R. Gelin and M. Karplus, Nature, 1977, 267(5612), 585.

46 | Chem. Modell., 2008, 5, 13–50 This journal is

c

The Royal Society of Chemistry 2008

36 M. P. Allen and D. J. Tildesley, Computer Simulation of Liquids, Oxford University Press, 1987. 37 N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller and E. Teller, J. Chem. Phys., 1953, 21(6), 1087. 38 A. Warshel and M. Karplus, J. Am. Chem. Soc., 1972, 94(16), 5612. 39 A. Warshel and M. Levitt, J. Mol. Biol., 1976, 103(2), 227. 40 J. Zielkiewicz, J. Chem. Phys., 2005, 123(10), 104501. 41 Q. Zhang and Z. Z. Yang, Acta Phys.-Chim. Sin., 2007, 23(10), 1565. 42 W. L. Jorgensen, J. Chandrasekhar, J. D. Madura, R. W. Impey and M. L. Klein, J. Chem. Phys., 1983, 79(2), 926. 43 M. W. Mahoney and W. L. Jorgensen, J. Chem. Phys., 2000, 112(20), 8910. 44 J. M. Wang, R. M. Wolf, J. W. Caldwell, P. A. Kollman and D. A. Case, J. Comput. Chem., 2004, 25(9), 1157. 45 A. Jakalian, B. L. Bush, D. B. Jack and C. I. Bayly, J. Comput. Chem., 2000, 21(2), 132. 46 A. Jakalian, D. B. Jack and C. I. Bayly, J. Comput. Chem., 2002, 23(16), 1623. 47 G. A. Kaminski and W. L. Jorgensen, J. Phys. Chem. B, 1998, 102(10), 1787. 48 J. W. Storer, D. J. Giesen, C. J. Cramer and D. G. Truhlar, J. Comput.-Aided Mol. Des., 1995, 9(1), 87. 49 G. M. Wang and A. C. Sandberg, Nanotechnology, 2007, 18(13), 135702. 50 A. E. Cho, V. Guallar, B. J. Berne and R. Friesner, J. Comput. Chem., 2005, 26(9), 915. 51 C. R. W. Guimaraes, M. Udier-Blagovic, I. Tubert-Brohman and W. L. Jorgensen, J. Chem. Theory Comput., 2005, 1(4), 617. 52 C. R. W. Guimaraes, M. Udier-Blagovic and W. L. Jorgensen, J. Am. Chem. Soc., 2005, 127(10), 3577. 53 R. D. Taylor, P. J. Jewsbury and J. W. Essex, J. Comput.-Aided Mol. Des., 2002, 16(3), 151. 54 C. Hensen et al., J. Med. Chem., 2004, 47(27), 6673. 55 F. Beierlein, H. Lanig, G. Schurer, A. H. C. Horn and T. Clark, Mol. Phys., 2003, 101(15), 2469. 56 J. Chandrasekhar, S. Shariffskul and W. L. Jorgensen, J. Phys. Chem. B, 2002, 106(33), 8078. 57 O. Acevedo and W. L. Jorgensen, J. Am. Chem. Soc., 2006, 128(18), 6141. 58 O. Acevedo and W. L. Jorgensen, J. Org. Chem., 2006, 71(13), 4896. 59 O. Acevedo and W. L. Jorgensen, J. Chem. Theory Comput., 2007, 3(4), 1412. 60 M. P. Repasky, J. Chandrasekhar and W. L. Jorgensen, J. Comput. Chem., 2002, 23(16), 1601. 61 J. D. Thompson, C. J. Cramer and D. G. Truhlar, J. Comput. Chem., 2003, 24(11), 1291. 62 M. Udier-Blagovic, P. M. De Tirado, S. A. Pearlman and W. L. Jorgensen, J. Comput. Chem., 2004, 25(11), 1322. 63 A. Warshel, Annu. Rev. Biophys. Biomolec. Struct., 2003, 32, 425. 64 M. Svensson et al., J. Phys. Chem., 1996, 100(50), 19357. 65 T. Vreven et al., J. Chem. Theory Comput., 2006, 2(3), 815. 66 K. P. Eurenius, D. C. Chatfield, B. R. Brooks and M. Hodoscek, Int. J. Quantum Chem., 1996, 60(6), 1189. 67 J. L. Gao and D. G. Truhlar, Annu. Rev. Phys. Chem., 2002, 53, 467. 68 D. Riccardi et al., J. Phys. Chem. B, 2006, 110(13), 6458. 69 H. Lin and D. G. Truhlar, Theor. Chem. Acc., 2007, 117(2), 185. 70 T. Laino, F. Mohamed, A. Laio and M. Parrinello, J. Chem. Theory Comput., 2005, 1(6), 1176. 71 U. C. Singh and P. A. Kollman, J. Comput. Chem., 1986, 7(6), 718. 72 P. A. Bash, M. J. Field and M. Karplus, J. Am. Chem. Soc., 1987, 109(26), 8092. 73 V. Thery, D. Rinaldi, J. L. Rivail, B. Maigret and G. G. Ferenczy, J. Comput. Chem., 1994, 15(3), 269. 74 L. G. Gorb, J. L. Rivail, V. Thery and D. Rinaldi, Int. J. Quantum Chem., 1996, 60(7), 1525. 75 J. L. Gao, P. Amara, C. Alhambra and M. J. Field, J. Phys. Chem. A, 1998, 102(24), 4714. 76 J. Z. Pu, J. L. Gao and D. G. Truhlar, ChemPhysChem, 2005, 6(9), 1853. 77 Hypercube, Inc., Hyperchem Users Manual, Computational Chemistry, Hypercube, Inc., Waterloo, Ontario, Canada, 1994. 78 E. Derat, J. Bouquant and S. Humbel, Theochem.-J. Mol. Struct., 2003, 632, 61. 79 P. D. Lyne, M. Hodoscek and M. Karplus, J. Phys. Chem. A, 1999, 103(18), 3462. 80 Y. K. Zhang, T. S. Lee and W. T. Yang, J. Chem. Phys., 1999, 110(1), 46. 81 Y. K. Zhang, Theor. Chem. Acc., 2006, 116(1–3), 43. 82 Y. K. Zhang, J. Chem. Phys., 2005, 122(2), 024114.

Chem. Modell., 2008, 5, 13–50 | 47 This journal is

c

The Royal Society of Chemistry 2008

83 I. Antes and W. Thiel, J. Phys. Chem. A, 1999, 103(46), 9290. 84 O. A. von Lilienfeld, I. Tavernelli, U. Rothlisberger and D. Sebastiani, J. Chem. Phys., 2005, 122(1), 014113. 85 N. Reuter, A. Dejaegere, B. Maigret and M. Karplus, J. Phys. Chem. A, 2000, 104(8), 1720. 86 A. Fornili, P. F. Loos, M. Sironi and X. Assfeld, Chem. Phys. Lett., 2006, 427(1–3), 236. 87 J. Z. Pu, J. L. Gao and D. G. Truhlar, J. Phys. Chem. A, 2004, 108(25), 5454. 88 M. Elstner, Theor. Chem. Acc., 2006, 116(1–3), 316. 89 J. Z. Pu, J. L. Gao and D. G. Truhlar, J. Phys. Chem. A, 2004, 108(4), 632. 90 P. H. Konig, M. Hoffmann, T. Frauenheim and Q. Cui, J. Phys. Chem. B, 2005, 109(18), 9082. 91 A. Rodriguez, C. Oliva, M. Gonzalez, M. van der Kamp and A. J. Mulholland, J. Phys. Chem. B, 2007, 111(44), 12909. 92 F. Claeyssens et al., Angew. Chem.-Int. Edit., 2006, 45(41), 6856. 93 M. Sˇtrajbl, G. Y. Hong and A. Warshel, J. Phys. Chem. B, 2002, 106(51), 13333. 94 R. P. Muller and A. Warshel, J. Phys. Chem., 1995, 99(49), 17516. 95 R. W. Zwanzig, J. Chem. Phys., 1954, 22(8), 1420. 96 W. L. Jorgensen, J. F. Blake and J. K. Buckner, Chem. Phys., 1989, 129(2), 193. 97 A. Yadav, R. M. Jackson, J. J. Holbrook and A. Warshel, J. Am. Chem. Soc., 1991, 113(13), 4800. 98 R. H. Wood, E. M. Yezdimer, S. Sakane, J. A. Barriocanal and D. J. Doren, J. Chem. Phys., 1999, 110(3), 1329. 99 M. Sˇtrajbl, A. Shurki, M. Kato and A. Warshel, J. Am. Chem. Soc., 2003, 125(34), 10228. 100 Y. Ming, G. L. Lai, C. H. Tong, R. H. Wood and D. J. Doren, J. Chem. Phys., 2004, 121(2), 773. 101 T. H. Rod and U. Ryde, J. Chem. Theory Comput., 2005, 1(6), 1240. 102 E. Rosta, M. Klahn and A. Warshel, J. Phys. Chem. B, 2006, 110(6), 2934. 103 W. K. Hastings, Biometrika, 1970, 57(1), 97. 104 R. Iftimie, D. Salahub and J. Schofield, J. Chem. Phys., 2003, 119(21), 11285. 105 C. J. Woods, F. R. Manby and A. J. Mulholland, J. Chem. Phys., 2008, 128(1), 014109. 106 R. Iftimie, D. Salahub, D. Q. Wei and J. Schofield, J. Chem. Phys., 2000, 113(12), 4852. 107 D. A. Pearlman, J. Phys. Chem., 1994, 98(5), 1487. 108 M. Mezei, J. Chem. Phys., 1987, 86(12), 7084. 109 C. J. Woods, J. W. Essex and M. A. King, J. Phys. Chem. B, 2003, 107(49), 13703. 110 H. Leontiadou, A. E. Mark and S. J. Marrink, Biophys. J., 2007, 92(12), 4209. 111 G. S. Ayton and G. A. Voth, Biophys. J., 2004, 87(5), 3299. 112 J. W. Chu, G. S. Ayton, S. Izvekov and G. A. Voth, Mol. Phys., 2007, 105(2–3), 167. 113 S. O. Nielsen, C. F. Lopez, G. Srinivas and M. L. Klein, J. Phys.-Condes. Matter, 2004, 16(15), R481. 114 V. Tozzini, Curr. Opin. Struct. Biol., 2005, 15(2), 144. 115 M. Venturoli, M. M. Sperotto, M. Kranenburg and B. Smit, Phys. Rep.-Rev. Sec. Phys. Lett., 2006, 437(1–2), 1. 116 M. Levitt and A. Warshel, Nature, 1975, 253(5494), 694. 117 M. Levitt, J. Mol. Biol., 1976, 104(1), 59. 118 B. Smit et al., Nature, 1990, 348(6302), 624. 119 R. Goetz, G. Gompper and R. Lipowsky, Phys. Rev. Lett., 1999, 82(1), 221. 120 H. Noguchi and M. Takasu, Phys. Rev. E, 2001, 6404(4), 041913. 121 G. Brannigan and F. L. H. Brown, J. Chem. Phys., 2004, 120(2), 1059. 122 D. J. Michel and D. J. Cleaver, J. Chem. Phys., 2007, 126(3), 034506. 123 J. G. Gay and B. J. Berne, J. Chem. Phys., 1981, 74(6), 3316. 124 J. C. Shelley, M. Y. Shelley, R. C. Reeder, S. Bandyopadhyay and M. L. Klein, J. Phys. Chem. B, 2001, 105(19), 4464. 125 J. C. Shelley et al., J. Phys. Chem. B, 2001, 105(40), 9785. 126 S. J. Marrink, A. H. de Vries and A. E. Mark, J. Phys. Chem. B, 2004, 108(2), 750. 127 M. Orsi, D. Y. Haubertin, W. Sandersony and J. W. Essex, J. Phys. Chem. B, in press. 128 S. D. Chao, J. D. Kress and A. Redondo, J. Chem. Phys., 2005, 122(23), 234912. 129 C. F. Lopez, S. O. Nielsen, G. Srinivas, W. F. DeGrado and M. L. Klein, J. Chem. Theory Comput., 2006, 2(3), 649. 130 D. Van der Spoel et al., J. Comput. Chem., 2005, 26(16), 1701. 131 E. Lindahl, B. Hess and D. van der Spoel, J. Mol. Model., 2001, 7(8), 306. 132 S. J. Marrink, J. Risselada and A. E. Mark, Chem. Phys. Lipids, 2005, 135(2), 223. 133 V. Knecht and S. J. Marrink, Biophys. J., 2007, 92(12), 4254. 134 X. Periole, T. Huber, S. J. Marrink and T. P. Sakmar, J. Am. Chem. Soc., 2007, 129(33), 10126.

48 | Chem. Modell., 2008, 5, 13–50 This journal is

c

The Royal Society of Chemistry 2008

135 R. Baron, A. H. de Vries, P. H. Hunenberger and W. F. van Gunsteren, J. Phys. Chem. B, 2006, 110(31), 15602. 136 R. Baron, A. H. de Vries, P. H. Hunenberger and W. F. van Gunsteren, J. Phys. Chem. B, 2006, 110(16), 8464. 137 R. Baron et al., ChemPhysChem, 2007, 8(3), 452. 138 S. J. Marrink, H. J. Risselada, S. Yefimov, D. P. Tieleman and A. H. de Vries, J. Phys. Chem. B, 2007, 111(27), 7812. 139 P. J. Bond and M. S. P. Sansom, J. Am. Chem. Soc., 2006, 128(8), 2697. 140 P. J. Bond, J. Holyoake, A. Ivetac, S. Khalid and M. S. P. Sansom, J. Struct. Biol., 2007, 157(3), 593. 141 S. Izvekov and G. A. Voth, J. Phys. Chem. B, 2005, 109(7), 2469. 142 S. Izvekov and G. A. Voth, J. Chem. Phys., 2005, 123(13), 134105. 143 F. Ercolessi and J. B. Adams, Europhys. Lett., 1994, 26(8), 583. 144 S. Izvekov, M. Parrinello, C. J. Burnham and G. A. Voth, J. Chem. Phys., 2004, 120(23), 10896. 145 J. W. Chu, S. Izveko and G. A. Voth, Mol. Simul., 2006, 32(3–4), 211. 146 W. G. Noid, J. W. Chu, G. S. Ayton and G. A. Voth, J. Phys. Chem. B, 2007, 111(16), 4116. 147 S. Izvekov, A. Violi and G. A. Voth, J. Phys. Chem. B, 2005, 109(36), 17019. 148 P. Liu, S. Izvekov and G. A. Voth, J. Phys. Chem. B, 2007, 111(39), 11566. 149 Y. T. Wang, S. Izvekov, T. Y. Yan and G. A. Voth, J. Phys. Chem. B, 2006, 110(8), 3564. 150 S. Izvekov and G. A. Voth, J. Chem. Theory Comput., 2006, 2(3), 637. 151 S. Izvekov and G. A. Voth, J. Chem. Phys., 2006, 125(15), 151101. 152 A. P. Lyubartsev, Eur. Biophys. J. Biophys. Lett., 2005, 35(1), 53. 153 A. P. Lyubartsev and A. Laaksonen, Phys. Rev. E, 1995, 52(4), 3730. 154 D. Reith, M. Putz and F. Muller-Plathe, J. Comput. Chem., 2003, 24(13), 1624. 155 A. Arkhipov, P. L. Freddolino and K. Schulten, Structure, 2006, 14(12), 1767. 156 C. E. A. Chang, J. Trylska, V. Tozzini and J. A. McCammon, Chem. Biol. Drug Des., 2007, 69(1), 5. 157 J. Trylska, V. Tozzini and J. A. McCammon, Biophys. J., 2005, 89(3), 1455. 158 V. Tozzini and J. A. McCammon, Chem. Phys. Lett., 2005, 413(1–3), 123. 159 C. E. Chang, T. Shen, J. Trylska, V. Tozzini and J. A. McCammon, Biophys. J., 2006, 90(11), 3880. 160 V. Tozzini, W. Rocchia and J. A. McCammon, J. Chem. Theory Comput., 2006, 2(3), 667. 161 V. Tozzini, J. Trylska, C. E. Chang and J. A. McCammon, J. Struct. Biol., 2007, 157(3), 606. 162 J. Trylska, V. Tozzini, C. A. Chang and J. A. McCammon, Biophys. J., 2007, 92(12), 4179. 163 A. P. Heath, L. E. Kavraki and C. Clementi, Proteins, 2007, 68(3), 646. 164 M. Feig, P. Rotkiewicz, A. Kolinski, J. Skolnick and C. L. Brooks, Proteins, 2000, 41(1), 86. 165 M. Milik, A. Kolinski and J. Skolnick, J. Comput. Chem., 1997, 18(1), 80. 166 Z. X. Xiang and B. Honig, J. Mol. Biol., 2001, 311(2), 421. 167 J. M. Wang, P. Cieplak and P. A. Kollman, J. Comput. Chem., 2000, 21(12), 1049. 168 D. A. Case et al., AMBER 8, University of California, 2004. 169 A. Onufriev, D. Bashford and D. A. Case, Proteins, 2004, 55(2), 383. 170 P. Das, S. Matysiak and C. Clementi, Proc. Natl. Acad. Sci. USA, 2005, 102(29), 10141. 171 F. Ding, K. C. Prutzman, S. L. Campbell and N. V. Dokholyan, Structure, 2006, 14(1), 5. 172 J. Hermans, H. J. C. Bernendsen, W. F. van Gunsteren and J. P. M. Postma, Biopolymers, 1984, 23(8), 1513. 173 T. Lazaridis and M. Karplus, Proteins, 1999, 35(2), 133. 174 T. Kortemme, A. V. Morozov and D. Baker, J. Mol. Biol., 2003, 326(4), 1239. 175 S. Sharma, F. Ding and N. V. Dokholyan, Biophys. J., 2007, 92(5), 1457. 176 M. Nanias, C. Czaplewski and H. A. Scheraga, J. Chem. Theory Comput., 2006, 2(3), 513. 177 E. Lyman and D. M. Zuckerman, J. Chem. Theory Comput., 2006, 2(3), 656. 178 T. Z. Lwin and R. Luo, J. Chem. Phys., 2005, 123(19), 194904. 179 Q. Shi, S. Izvekov and G. A. Voth, J. Phys. Chem. B, 2006, 110(31), 15045. 180 L. Whitehead, C. M. Edge and J. W. Essex, J. Comput. Chem., 2001, 22(14), 1622. 181 J. Michel, M. Orsi and J. W. Essex, J. Phys. Chem. B, in press. 182 Y. Liu and T. Ichiye, J. Phys. Chem., 1996, 100(7), 2723. 183 A. Chandra and T. Ichiye, J. Chem. Phys., 1999, 111(6), 2701. 184 M. L. Tan, B. R. Brooks and T. Ichiye, Chem. Phys. Lett., 2006, 421(1–3), 166. 185 C. J. Fennell and J. D. Gezelter, J. Chem. Phys., 2004, 120(19), 9175. 186 M. Neri, C. Anselmi, M. Cascella, A. Maritan and P. Carloni, Phys. Rev. Lett., 2005, 95(21), 218102.

Chem. Modell., 2008, 5, 13–50 | 49 This journal is

c

The Royal Society of Chemistry 2008

187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239

M. Praprotnik, L. Delle Site and K. Kremer, J. Chem. Phys., 2005, 123(22), 224106. M. Praprotnik, L. Delle Site and K. Kremer, Phys. Rev. E, 2006, 73(6), 066701. M. Praprotnik, K. Kremer and L. Delle Site, J. Phys. A-Math. Theor., 2007, 40(15), F281. M. Praprotnik, K. Kremer and L. Delle Site, Phys. Rev. E, 2007, 75(1), 017701. M. Praprotnik, S. Matysiak, L. Delle Site, K. Kremer and C. Clementi, J. Phys.-Condes. Matter, 2007, 19(29), 292201. B. Ensing, S. O. Nielsen, P. B. Moore, M. L. Klein and M. Parrinello, J. Chem. Theory Comput., 2007, 3(3), 1100. M. Tuckerman, B. J. Berne and G. J. Marthyna, J. Chem. Phys., 1992, 97(3), 1990. R. E. Rudd and J. Q. Broughton, Phys. Rev. B, 2005, 72(14), 144104. S. A. Orszag and I. Staroselsky, Comput. Phys. Commun., 2000, 127(1), 165. N. A. Baker, J. Comput. Chem., 2005, 21, 349. N. A. Baker, Curr. Opin. Struct. Biol., 2005, 15(2), 137. J. H. Chen, W. P. Im and C. L. Brooks, J. Am. Chem. Soc., 2006, 128(11), 3728. Z. Y. Yu, M. P. Jacobson and R. A. Friesner, J. Comput. Chem., 2006, 27(1), 72. C. Park, M. J. Carlson and W. A. Goddard, J. Phys. Chem. A, 2000, 104(11), 2498. Y. H. Jang et al., J. Phys. Chem. B, 2003, 107(1), 344. N. J. English, J. Mol. Model., 2007, 13(10), 1081. S. P. Brown and S. W. Muchmore, J. Chem. Inf. Model., 2007, 47(4), 1493. G. Brannigan, L. C. L. Lin and F. L. H. Brown, Eur. Biophys. J. Biophys. Lett., 2006, 35(2), 104. I. Lotan and T. Head-Gordon, J. Chem. Theory Comput., 2006, 2(3), 541. W. Im, J. H. Chen and C. L. Brooks, Adv. Protein Chem., 2006, 72, 173. A. Warshel, P. K. Sharma, M. Kato and W. W. Parson, BBA-Proteins Proteomics, 2006, 1764(11), 1647. J. Carlsson, M. Ander, M. Nervall and J. Aqvist, J. Phys. Chem. B, 2006, 110(24), 12034. P. Koehl, Curr. Opin. Struct. Biol., 2006, 16(2), 142. Y. Y. Tang et al., Biophys. J., 2006, 91(4), 1248. E. Villa, A. Balaeff, L. Mahadevan and K. Schulten, Multiscale Model. Simul., 2004, 2(4), 527. E. Villa, A. Balaeff and K. Schulten, Proc. Natl. Acad. Sci. USA, 2005, 102(19), 6783. S. Barsky, R. Delgado-Buscalioni and P. V. Coveney, J. Chem. Phys., 2004, 121(5), 2403. G. De Fabritiis, R. Delgado-Buscalioni and P. V. Coveney, Phys. Rev. Lett., 2006, 97(13), 134501. R. Delgado-Buscalioni and P. V. Coveney, Philos. Trans. R. Soc. Lond. Ser. A-Math. Phys. Eng. Sci., 2004, 362(1821), 1639. R. Delgado-Buscalioni, E. G. Flekkoy and P. V. Coveney, Europhys. Lett., 2005, 69(6), 959. R. Delgado-Buscalioni and P. V. Coveney, Physica A, 2006, 362(1), 30. E. G. Flekkoy, R. Delgado-Buscalioni and P. V. Coveney, Phys. Rev. E, 2005, 72(2), 026703. G. Giupponi, G. De Fabritiis and P. V. Coveney, J. Chem. Phys., 2007, 126(15), 154903. R. Delgado-Buscalioni and P. V. Coveney, Phys. Rev. E, 2003, 67(4), 046704. R. Delgado-Buscalioni and P. V. Coveney, J. Chem. Phys., 2003, 119(2), 978. G. De Fabritiis, M. Serrano, R. Delgado-Buscalioni and P. V. Coveney, Phys. Rev. E, 2007, 75(2), 026307. R. Chang, G. S. Ayton and G. A. Voth, J. Chem. Phys., 2005, 122(24), 244716. G. S. Ayton, W. G. Noid and G. A. Voth, Curr. Opin. Struct. Biol., 2007, 17(2), 192. G. Ayton and G. A. Voth, Biophys. J., 2002, 83(6), 3357. M. Kranenburg, J. P. Nicolas and B. Smit, Phys. Chem. Chem. Phys., 2004, 6(16), 4142. M. Kranenburg, C. Laforge and B. Smit, Phys. Chem. Chem. Phys., 2004, 6(19), 4531. G. S. Ayton and G. A. Voth, J. Struct. Biol., 2007, 157(3), 570. G. Zini, A. Sarti and C. Lamberti, IEEE Trans. Ultrason. Ferroelectr. Freq. Control, 1997, 44(2), 297. D. Noble, Biochem. Soc. Trans., 2003, 31, 156. S. H. G. Khoo and M. Al-Rubeai, Biotechnol. Appl. Biochem., 2007, 47, 71. A. Michailova, J. Saucerman, M. E. Belik and A. D. McCulloch, Biophys. J., 2005, 88(3), 2234. D. Noble, Biochem. Soc. Trans., 2005, 33, 539. D. Noble, Biosystems, 2006, 83(2–3), 75. C. V. Forst, Drug Discov. Today, 2006, 11(5–6), 220. D. B. Kell, IUBMB Life, 2007, 59(11), 689. J. B. Bassingthwaighte, H. J. Chizeck and L. E. Atlas, Proc. IEEE, 2006, 94(4), 819. M. Stein, R. R. Gabdoulline and R. C. Wade, Curr. Opin. Struct. Biol., 2007, 17(2), 166. P. J. Hunter, W. W. Li, A. D. McCulloch and D. Noble, Computer, 2006, 39(11), 48.

50 | Chem. Modell., 2008, 5, 13–50 This journal is

c

The Royal Society of Chemistry 2008

Computer-aided drug design 2005–2007 Richard A. Lewis DOI: 10.1039/b609116b

1. Introduction At the 2005 Gordon Research Conference on computer-aided drug design, the delegates were polled for a series of big questions facing our field. These covered the fields of QSAR, structure-based design, ligand-based design, cheminformatics, library design and virtual screening. These questions are dispersed among the individual topics, but all focus on extending the scope and rigour of our methods. In the last two years, there has been a burst of activity around the fundamental science, reflecting the growing influence of modelling. It is no longer good enough to work only in the traditional areas of strength. Modelling is expected to make an impact now in all aspects of the drug discovery process.

2. QSAR and ADMET For the field of QSAR, the issues were: How global can good QSAR models be? How can we do simultaneous modelling against multiple pharmacological endpoints? How can we close the feedback loop so QSAR models could be used directly for generation of new structures? How can we make good predictive models of ADMET properties from molecular structure, everything from solubility to bioavailability to idiosyncratic toxicity? The valedictory article by Yvonne Martin 1 provides a reminder that modellers have most impact when they concentrate on the specific problems faced by medicinal chemists. Thus, providing models that we know to be flawed will have more effect than a ‘perfect’ model that is irrelevant to the daily experience of the users, or is not applicable to their compounds. The examples described are substructural alerts for reactive functional groups, which the chemists can modify and hence take ownership of, and C log P. The latter is a commonly used descriptor used as a surrogate for bioavailability despite the fact the calculation is in error by more than 1 log unit about a third of the time. Despite this, there is no pressing demand to devise a better method. 2.1 Model quality Guha et al.2 explore the relationship between global QSAR models and local models. A local model is obtained using only the compounds in the neighbourhood of the query molecule. This would be most useful when the global trend is non-linear, as one is approximating the first derivative of the non-linear curve with the local model. This has analogies to the kNN method. Of course, one must be careful how the distance metric and parameter k are chosen. In tests, the local models gave very similar global error predictions, but explained outliers in the global model much better. Where a local model falls down is when there are activity cliffs, that is, larger changes in activity for only small changes in structure, or when the neighbourhood is sparse. The key advantage is that no model needs to be built a priori, so that the method is well suited for analysing large data sets. A new method for making classification models from diverse structure, called line-walking recursive partitioning, has been published, using cyp450 data as the training set. The authors claim 85% accuracy, similar to SVM methods, and only 9 descriptors are used. The method was also able to uncover errors in the biological data, implying that it is Computer-aided Drug Discovery, Novartis Institutes for Biomedical Research, Basel CH-4002, Switzerland. E-mail: [email protected]

Chem. Modell., 2008, 5, 51–66 | 51 This journal is

c

The Royal Society of Chemistry 2008

robust even for noisy data sets.3 Gedeck et al.4 provide another view of QSAR, by looking at the quality of predictions as a function of the data set. Using 1000 datasets and creating QSAR models in identical fashions, they showed that the quality of the predictions correlates with the size of the data set and the dynamic range of the activity values, and that no one descriptor set is the best for all data sets. By performing randomised picking experiments, they also saw a wide variation in q2 values, indicating the instability of many of the data sets. The fraction of good models (q2 4 0.5) was low for all descriptor sets, with 2D descriptors performing better than quasi-3D descriptors such as Almond.

2.2 Matched pairs analysis Matched pair analysis (or Free-Wilson analysis) has come back into vogue with a number of papers employing the technique. As one is looking for changes in activity with small changes in structure, there is some tension with the similarity principle, but it is a valid approach. The argument is that classical QSAR models are subject to issues with descriptor validity, extrapolation and so on. By analysing a dataset of matched pairs, one can derive fragment-based rules for activity, with the hope that the rules are independent of the larger context of the rest of the molecule. Leach et al.5 have built a tool for extracting all matched pairs from a dataset, and for evaluating the statistical significance of each rule. They exemplify using solubility, plasma-protein binding and oral exposure in rats. Thus adding Br decreases solubility by 1 log unit in 98% of the cases investigated. Conversely, analysis of outliers can give hints about why the usual trend is bucked: adding a orthosubstituent may cause some change in conformational preference, i.e. the wider environment is affected and the change of fragment is not independent of the neighbourhood. Haubertin and Bruneau6 also use matched pairs in a similar fashion. Sheridan et al.7 employ the idea of a molecular transformation, that is the difference between two molecules, as a substructure difference vector. The key is to look at molecules and their activities in pairs, rather than as a whole data set. This allows one to pick up small signals that might otherwise be masked by other larger signals and errors. The substructure vector is based on the atoms left after subtraction of the maximal common substructure. By analysing all pairwise transformations, clustering them by similarity, then adding in the change in activity most strongly associated with the cluster, one can quickly see if a transformation was beneficial or deleterious, even if the original absolute activity was weak. A similar strategy can be applied to find which (beneficial) transformations derived from the training set could be applied to a query molecule. In studies using D2 agonists, DHFR inhibitors, ACE inhibitors, they showed that, in comparison with some global QSAR methods, their local method could pick up local trends that the global models missed, and vice versa. In addition, as the transformations have a well-defined neighbourhood, it is not generally possible to apply them multiple times, thus avoiding the unreasonable optimisation issues caused by applying global models to morph query structures in a de novo design paradigm. To provide balance, one can also look for matched pairs that do not show changes in activity at a position which has been shown to be important to the SAR. If this is combined by an analysis of the X-ray complexes, one can assume that there is bioisosterism. The work by Kennewell et al.8 does just that starting with an alignment of the relevant binding sites. The ligands are broken down into fragments by splitting at single bonds and the fragment pairs identified by measuring the volume overlap with a reference molecule. The analysis was performed across several targets and bioisosteric pairs are found. However the authors are keen to stress that bioisosteres are particular to the target and not necessarily universal. 52 | Chem. Modell., 2008, 5, 51–66 This journal is

c

The Royal Society of Chemistry 2008

2.3 Multiobjective optimisation Inverse QSAR and multiobjective optimisation have been combined into a single workflow by Brown, McKay and Gasteiger,9 using a GA-based compound generator, and two QSAR models for solubility and polarizability. The issue of model knowledge domain is handled by Hotelling’s T2 statistic, and the multiobjective optimisation by Pareto ranking. They were able to show that the evolved molecules were consistent with the training sets. 2.4 Predictive models of ADMET The focus on producing general models for ADME phenomena can produce an over-reliance on the many robust methods for deriving QSAR models, while ignoring the underlying structure of the data. Good and Hermsmeier10 have re-examined the problem of predicting druglikeness, and show that there is considerable bias in the data, which can make the models look better than they really are. They argue for training set selection based on target class and ensuring that the decoy set is matched to the true actives by molecular weight as a minimal requirement. Random selection can fall foul of analogue bias, whereby certain classes of molecule are overrepresented, leading to recognition rather than prediction. Mutagenicity is an observation that will often halt the progress of a compound through the clinic. Based on the assumption that the features that give rise to mutagenic effects are quite localised, Kazius et al.11 have created a set of toxicophores based on a dataset of 4337 structures with Ames data. 29 toxicophores could classify the actives and inactives with an error of 18%, around the interlaboratory reproducibility level for the AMES test. An interesting case study is the aromatic nitro group. While is it strongly associated with mutagenic effects, its influence can be mitigated by other substituents on the ring: this observation lead to more specific rules, all of which are published as supplementary information in SMARTS format. This could form part of any assessment for compound purchasing or library assessment. Although it is included in the ADMET section, the review by Kalgutkar et al.,12 listing the bioactivation pathways of organic functional groups, has much wider relevance. Although the authors are careful to stress that no functional group should be ‘black-listed’, this survey could be used to bias away from certain groups during compound selection or library design or de novo design. Two new models for volume of distribution in humans have been published. This involved a very careful assembly of data, coming as it did from different data sources. In the Lombardo model,13 384 compounds were used in the study, most with quite diverse structures. As expected, C log P was an important descriptor, but also terms involving the fraction of the molecule ionised. It is clear that a good predictor for pKa, especially around physiological pH, will be key to future success in the modelling of pharmacokinetics. A model that predicts the pKa of amines accurately is not that helpful, if all the predictions are 48.5, as the compounds will be fully ionised. A program that could distinguish between a pKa of 6 and 7 would be immensely valuable. The model for VDss is accurate to 2-fold, as accurate as equivalent models based on animal data, and the underlying, curated data has been made available. The Gleeson model14 also includes rat data and is quoted as having an error of 0.37 and a r2 of 0.53. A curated data set has been put together using drugs marketed in Sweden.15 The total set of 673 pharmaceutical agents was reduced to 24 diverse compounds using multivariate analysis. Experimental data for these compounds was measured in a consistent way, to allow appropriate use in models (pKa, log P, solubility and CACO-2 permeability). Data on VDss, half-life and clearance were obtained from the literature. As all the compounds are commercially available at reasonable prices, this data set could be expanded: even in its present form, it will be a good standard for ADME modelling. Chem. Modell., 2008, 5, 51–66 | 53 This journal is

c

The Royal Society of Chemistry 2008

Herg blockade is mesmerising the pharmaceutical industry, not just due to the implications for drug discovery, but also because the difficulty is understanding the interplay between displacement of dofetilide, patch clamp assays, dog telemetry and the occurrence of toursade des pointes in humans. Several groups have developed pharmacophore models, including one for non-basic molecules.16 However, as Pearlstein has pointed out,17 Herg does not have an endogenous ligand, so that the Herg binding site is under-evolved and promiscuous, and might be better viewed as a host–guest complex. 2.5 Metabolism There have been continued studies into the energetics of hydrogen abstraction as a key factor in both regioselectivity and relative rates of metabolism by Cytochrome P450s. Olsen et al.18 performed studies with B3LYP and AM1 Hamiltonians using 24 substrates covering most commonly found atom types, and a haem surrogate. The surrogates ranged from a full iron-porphyrine to a methoxy radical. By performing full comparisons at the different levels of theory, they showed that the error in a simple AM1 approach using PhOdwas under 4 kJ/mol, a quite acceptable margin. Bond Dissociation Energy was found to be the best correlate for activation energy. All of the results could be summarized into 5 rules based on atom type and local environment. The results were applied to two steroids, and the authors could predict, with good accuracy, the preferred sites of metabolism in both, even without taking into account the accessibility of the atoms to the iron in the cyp450 site. Sheridan et al.19 eschew the energetic approaches to a purely QSAR-based model of regioselectivity for metabolism by Cyp450’s. The best models use only three substructure descriptors and two property descriptors. In comparison with other methods, specifically Metasite,20 the simple QSARs seemed to do as well or better, although the authors acknowledge this could be an artefact, and true comparison of performance is not easy. The authors also not that the models only predict where a molecule might be oxidised, and not how fast or extensively, something the abstraction energies do address.

3. Structure-based drug design The key issues for structure-based drug design are: adequate sampling of protein conformational space; high quality but practical potentials for predicting protein structures and protein/ligand interactions; generation of homology models good enough for hit identification and lead optimization, and to what extent can we refine a homology model based on knowledge of the ligands known to bind? 3.1 Protein flexibility Protein plasticity is a common phenomenon, but not often explicitly addressed except via MD, and even then we cannot be sure that the key conformations have been sampled. The zero-order approximation is to assume rigidity, followed by the use of several rigid snapshots, or side chain flexibility (e.g. GOLD), or even some degree of induced fit is allowed (see below). What is lacking are methods that are fast and cover all the key conformations. To obtain this, we need to understand more about the important motions of protein structures. Kinases are a good test case, as they are plastic, but we have X-ray structures of many of the active and inactive states of the proteins, plus ligands, so we can try to assess the interplay between protein conformation and ligand structure. Subramanian et al.21 have used CDK kinases as a test case: the proteins were aligned to a standard, using all atoms. The size of the binding site was defined using 4 residues common to all the CDKs; this variable was correlated against standard descriptors generated from the ligand structures. The degree of stabilisation of active and inactive forms was found to 54 | Chem. Modell., 2008, 5, 51–66 This journal is

c

The Royal Society of Chemistry 2008

depend strongly on the local hydrophobic character of the ligands, in addition to the usual size and shape constraints. Using the QSAR equations, one can predict which protein conformation is best suited for docking studies with a molecule. Another way to tackle this is to look at the formation of transient pockets. Eyrisch and Helms22 have done this specifically for protein–protein interactions. Starting with a crystal structure and running 10 ns of dynamics, they created multiple snapshots from the trajectories, each of which was examined using PASS, to detect cavities. The cavities were clustered to determine the location of more substantial pockets. The validity of the pockets (and the method) was assessed by superimposing the inhibitor structure: in all cases the pockets opened up in the regions one would expect ligand atoms to bind. The pockets also seemed to be the most non-polar. In docking studies, using the transient pockets gave better results compared to using the original apo-form from the X-ray. Apart from the formation of pockets, one also should consider if they are energetically favourable for binding. Landon et al.23 using computational solvent-mapping with small organic probes to identify energetic hotspots. Using renin as an example, they identified a pocket that is used by aliskiren, but not by other peptidomimetics. Using the Fesik24 training set of druggable and non-druggable proteins, they could shown that targets with a low ranked consensus site (among the probes used) were more likely to be the nondruggable ones. The exceptions to this were small confined sites such as PTB1B. Another way of looking at localised plasticity is to consider the structure ensembles generated by NMR methods.25 X-ray structures of complexes may have higher accuracy, but the protein is collapsed around the ligand, leading to the crossdocking problem, that one will not find other inhibitors of the same target, because the binding site is too closely modelled on one ligand. Using HIV protease as the exemplar, multiple pharmacophore models were built using benzene, ethane and methanol probes within an MD simulation, followed by clustering to identify hotspots of interaction. These models were used to query a conformer database. The effect of using multiple models is to increase the uncertainty (radius) of each feature, compared to using a model from an average structure. This is important when the pharmacophore contains more elements. The confusion matrices for identifying known hits from a database for presumed inactives (renin inhibitors were also found) were very good, with 90% true positive rate, 10% false negative. The study also compared the use of NMR structures to models derived from multiple X-ray structures. The recommendation is that NMR structures give better models, as the X-ray structures contain too little variation. The same group have also used conformations from MD sampling26 with similar findings. 3.2 Prediction of affinity The prediction of free energy, or at least consistent ranking of ligands according to free energy, has attracted much attention over the last few years, partly as a result of better hardware, but mainly due to the availability of more X-ray structures. The usual questions about sampling, the effect of solvent, force field accuracy have all been examined. Michel, Verdonk and Essex27 compared implicit and explicit solvent models for a series of inhibitors of neuraminidase, COX2 and CDK2. While they found that implicit solvent models are as good as explicit models in ranking ligands, while being much quicker to run, they urge caution, as changes in the conformation of the binding site can cause significant errors, so the methods should always be allied to X-rays or reliable docking predictions. Weis et al.28 used a MM/PBSA method to examine the effect of the force field and the starting geometries affected the calculated binding energies. While the force field didn’t matter much overall, mixing the force field for geometry and energy calculations did. They concluded, in contrast with Essex, that for their system explicit water should be used. They also conclude that simulations should be at least 450 ps long and that entropy is the most poorly handled term. Wiseman et al.29 describe their grand canonical Monte Carlo Chem. Modell., 2008, 5, 51–66 | 55 This journal is

c

The Royal Society of Chemistry 2008

method for computing relative affinity. The ensemble is annealed with respect to free energy and the protein and target are treated as rigid entities, and conformational freedom is introduced by means of snapshots. This is less of an issue for fragments or small molecules. For the test case of T4-lysozyme complexes, the method was comparable with much more complex simulations. The goal of this method is to dock fragments, which can then be assembled into larger entities (see sec. 6). Finally, the empirical scoring function used in Glide has been upgraded with new terms, 30 for desolvation, for hydrophobic collapse, and for h-bonding. The force field was parameterised on 198 complexes and 15 screening sets. They claim an accuracy of 2–3 kcal/mol, depending on how well-behaved the system is (that is, can the ligand be docked into the target without substantial reorganisation). 3.3 Selectivity Methods for assessing cross-reactivity a priori are of interest for several classes of targets with similar binding sites, most notable the ATP-binding sites of kinases. Using their dehydron approach, Fernadez and Maddipati31 generate a descriptor for the ATP site that correlates very strongly with the pharmacological activity vector across a panel of 32 kinases. A dehydron is defined by the degree of non-polarity around an h-bonding group. Excessive non-polarity promoted dehydration as a means to enhance the electrostatic contribution of an h-bond to a ligand. A matrix of dehydron patterns can be built by assigning a score if two residues share a dehydron, 0 otherwise. This matrix can be compared between kinases using standard metrics. The descriptor can be sharpened by only looking at dehydrons that are known to interact with inhibitors. To extend the approach, the authors use the fact that there is a strong relation between dehydron wrapping and the disorder score,32 allowing the a priori prediction of cross-reactivity. There is a continuing debate about the ability of F bound to C as an h-bonding group, or rather the extent to which this phenomenon can be observed, and its contribution to affinity. There is also the other effect of F on pKa which can affect affinity. Razgulin and Mecozzi33 have performed a comprehensive analysis using high level ab initio methods of the energetics of C–F


H interactions. They find that fluoroaromatics can h-bond to water with an energy of 2.6 kcal/mol; this is weaker than h-bonds involving O or N, but in the context of structure-based drug design, displacement on water could therefore be much more energetically favourable than was first thought.

4. Virtual screening Virtual screening covers both structure- and ligand-based design, in that the output from these fields is used to search for new ligands. The key issues are around scoring, or enriching actives over inactives in the final ranking. 4.1 Docking A significant advance in docking has been made by Sherman et al.34 by iteratively combining the usual rigid protein/flexible ligand approach (in this case GLIDE) with a program for protein structure prediction (PRIME). This allows the protein to be induced to fit a given ligand. Thus one can start with an inappropriate protein conformation (for example the apo-form) and mould the site around the ligand to produce a better model for docking. In self-docking experiments, the average RMSD could be decreased from 5.5 A˚ to under 1.8 A˚ for the 21 cases considered (chosen as they were challenges for the usual paradigm). The first docking starts with a softened steric term, and the induced fit dockings add in an extra term for the conformational strain of the protein. The induced fit method in the paper is restricted to side chain motions; there are obviously cases where there is much more extensive motion of 56 | Chem. Modell., 2008, 5, 51–66 This journal is

c

The Royal Society of Chemistry 2008

loops and secondary structures. This is not an insuperable difficulty, as loop prediction can be added into the protocol. 4.2 Scoring The well-known bias of scoring functions with size is investigated in detail by Jacobsson and Karlen.35 Using a training set of 945 actives and 10k decoys, with 8 different scoring functions, three methods for correcting or improving the ranking of the actives were tried. MASC36 did not generally make the enrichments better. Normalisation by the square root of molecular weight or number of heavy atoms sometimes improved the enrichment rate. As the authors point out, using structural knowledge about the target is generally far more powerful than relying on the score. The other point to note that this study is concerned with improving the enrichment rates of a docking run, rather than improving the prediction of affinity. Cho et al.37,38 have performed an important study into the effect of ligand atomic charges on the accuracy of docking runs. In most protocols, it is not feasible to recalculate charges on the fly, so that standard force field charges are set once and maintained. Apart from the questionable accuracy for some atom types, this approximation neglects the effects of polarisation. In this study, a mixed QM/MM simulation was used to recalculate the ligand charges during several cycles of docking and minimisation. In the test set of redocking, using induced QM charges to seed the next docking, did not cause any harm, and in most cases gave improvements. However, in subsequent studies, if a local minimum is encountered, using QM charges can reinforce the ‘incorrect’ binding mode. This is countered by running the docking in a stochastic parallel mode to increase the sampling. 4.3 Validation The Astex test set for docking studies was updated in 2007.39 This valuable resource has been refined to just 85 structures of high quality, representing an unbiased sampling of available structures, to compensate for the large fraction of kinase and protease structures in the PDB. The Astex set also requires deposited structure factors, so that ligand positioning can be independently regenerated and assessed. Issues arising from steric clashes or symmetry have been addressed directly. The set was also subject to extensive manual curation around tautomer and protonation states. In a experiment using GOLD, it was shown that performance was improved, not due to better docking and scoring, more that the test set was freer from experimental artefacts that would have caused docking programs to fail. Interestingly, using this high quality test set can also reveal other, more subtle, effects that can impact performance, for example the method used to generate the starting conformation. Using Corina rather than the native conformation caused a degradation of 5% in redocking experiments. Huang, Shoichet and Irwin40 have provided another benchmarking set of ligands and decoys for 40 different targets. Given the correlation of scoring functions with size of lipophilicity, the decoy set has been chosen to resemble the active ligands physically while have distinct chemotypes. Each ligand has its own set of decoys. They showed that using uncorrected decoy sets, such as the MDDR, gave enrichments 0.5 log units better, which is clear evidence of bias. One of the key tools in evaluating virtual screening performance is some measure of enrichment, whether by ROC curves, enrichment factor and so on. What one wants to know is whether a screening method identifies the actives early in the overall prioritised list of compounds. Truchon and Bayley41 have performed a theoretical analysis to show that ROC curves may not be ideal for this purpose, despite their widespread use (and advocacy). A method can get a good ROC score, yet still fail to identify any actives in the top 20%. They propose a new metric based on an enhanced form of the ROC curve called BEDROC, which, via a parameter Chem. Modell., 2008, 5, 51–66 | 57 This journal is

c

The Royal Society of Chemistry 2008

alpha, can be used to emphasis, or weight, the influence of the early part of the curve. Other effects, such as statistical error and saturation effects (the variation of a metric with the number of actives), are also discussed, leading to some practical recommendations for the conditions needed to perform a good evaluation of a new method. 4.4 Pharmacophores The old trick of using pharmacophores to pre-screen before performing expensive docking studies42 is enjoying a renaissance. Lu et al. employed this strategy to find leads against the MDM2-p53 interface.43 The pharmacophore constraints screened out all but 2599 structures from a starting set of 11k. Docking gave further enrichment of 10 confirmed hits out of 67 compounds assayed. As a pharmacophore, if properly constructed, will give very few false negatives (false negatives might arise out of different binding modes or poor feature definition), the scoring of docked hits becomes more useful. The strategy was also validated using known inhibitors, all of which passed the filters, and scored well. Baroni et al.44 combine 4-point pharmacophore fingerprints with a shape-based description of the binding site in the FLAP program. Molecular interaction fields (MIFs) are used the locate maxima and minima interaction regions within the site, which describe the pharmacophore points (which are hence independent of substructure). The same procedure can be applied in reverse to small molecules, leading to a fast fingerprint-based method for prefiltering large databases. The FLAP fingerprints can also be used for alignment-free comparison of binding sites to highlight regions of similarity and selectivity. Protein flexibility is handled at the side-chain level during the generation of the MIFs, and ligand flexibility by conformer sampling. FLAP shows good performance when evaluated against other docking programs (even though that it is not its primary purpose). A new methodology for combining pharmacophore detection and 3D QSAR, PHASE, has been described by Dixon et al.45 The workflow is similar to other procedures in the generation of conformers followed by mapping of pharmacophore features. These features can be the standard h-bond donors, acceptors, hydrophobes, charged groups plus other custom features. For hydrogen bonds, either vectors or projected binding points can be used, obviating to some extent the need to have the heavy atoms of the ligand feature superimposed, a limitation in many packages. Detection of a pharmacophore is also made more likely by reducing the restriction that all active molecules in the training set must display the same common pharmacophore. Normally several candidate pharmacophores can be proposed, so the next step is to score the models, which can be done using the fit of the actives to the model, which actives are fitted, conformational strain and volume overlap. Inactives are also used to penalise a model, but as a separate step. The selected model can, as in other cases, be used to define the alignment for a 3D QSAR model. PHASE uses an occupancy grid (that is, is the grid cube occupied by a feature from the ligand) rather than a probe energy grid as the descriptor for the QSAR; this leads to volume exclusion maps in a very facile manner. In a comparison with CATALYST (and with the caveat that this paper was written by the vendors of PHASE), PHASE seems to do better on the test set used by Patel et al.46 in their comparative study of pharmacophore generation methods. Another new tool for pharmacophore alignment, GALAHAD, has been described.47 This works by iterative construction of hypermolecules. A hypermolecule is generated by aligning successive pairs of molecules; it retains common features and individual atoms. It is therefore also dependent on the order in which the alignment pairs are processed, so the strategy is to align the most similar pairs first. The alignment is scored by the overlap of the pharmacophore features rather than just the atoms. Again, on the test set of Patel et al.,46 this method appears to find better pharmacophore models, but with the caveat that frozen conformations were used. 58 | Chem. Modell., 2008, 5, 51–66 This journal is

c

The Royal Society of Chemistry 2008

Rules-based conformer generators often have fairly basic torsional parameters. Sadowski and Bostrom48 have developed a method based on MIMUMBA, to generate all possible for-atom torsional fragments, to search the Cambridge Crystallographic database with these fragments and to use the results to define torsional energy parameters. These more extensive rules improve the ability of OMEGA to find conformations closer to X-ray conformations. Interestingly, if the conformers are subsequently refined using an MMFF-like force field, the improved performance is removed. Agrafiotis et al.49 have performed a comparative study of conformational coverage provided by several common packages as well as their own stochastic proximity embedding method (SPE). They conclude that SPE, along with a poling method such as catCONF, provides better sampling, whereas the others produce more extended or compact geometries. Coverage was measured using pharmacophore triplets and radius of gyration. As the authors point out, the measure affects the relative performance, so that their conclusions are not a universal endorsement of a particular method. Stochastic methods are not perfect, and still produce horrors such as bonds penetrating through the centres of rings. 4.5 Scaffold hopping Scaffold hopping is the technique of identifying molecules that are structurally different but which might form similar interactions with a receptor. In its own way, scaffold hopping is a form of similarity search, but with more abstract, less connectivity-dependent descriptors. Gillet et al.50 describe the use of reduced graphs, which combine essential pharmacophore information within a graph; these graphs can then be compared using classical means. The methods seem to marginally outperform classical fingerprint similarity in enrichment experiments, as they pick up both the structurally similar and dissimilar actives. Stiefl et al.51 also use a reducedgraph formulation in their ErG program. ErG handles h-bond acceptor/donor groups as two distinct states, allowing for exact matches with structures that do not have a matching donor/acceptor, rather just, say, a donor. A new descriptor is generated for each state, leading to 2N descriptors, where N is the number of donor/ acceptor groups. The paper is also noteworthy for its analysis of why different levels of performance are seen, compared to classical 2D fingerprints, for enrichment experiments. Methods that are strongly dependent on pharmacophore features do well in cases where there is a clear pharmacophore exposed, and less well when the pattern is hidden but the class is structurally homogenous. The Cresset field point method52 uses molecular field extrema to create a representation of molecules. These field patterns can be compared to produce lists of similar molecules. This algorithm is effective at finding scaffold hops, particularly for ring systems. The field points convey the hydrophobic and electrostatic features in the neighbourhood of a molecule in one conformation. The overlay is performed using a rigid-body simplex optimisation, to optimise the fit of the extrema of one molecule onto the full field of another (and vice versa). The starting point is derived by matching the extrema patterns using coloured graphs, very much as in the reduced graph approach. One of the harder aspects is when molecules could be formally charged, as this will drastically affect the fields. The power of the method is illustrated using thrombin inhibitors from different classes and trying to regenerate the X-ray overlay without using any information from the X-ray complexes, which they were able to do. Maass et al.53 offer an update on the CAVEAT methodology with the RECORE program. As before, a database of compounds is shredded to create a fragment database. Fragments are created using the notion of cut points, that is, places at which the removal of a bond splits the structure into two. The cut points are further limited to preserve conformational restraints. Fragments are filtered by size and indexed by the vectors created by two or more cuts, as well as by pharmacophore features. Scaffold hopping is therefore performed by searching the database for Chem. Modell., 2008, 5, 51–66 | 59 This journal is

c

The Royal Society of Chemistry 2008

defined vector geometries plus addition features. In three test cases, viable, unstrained replacements for query structures could be obtained, even for multiple vector queries, where one expect the density of suitable fragments to be much sparser. Bergmann et al.54 use a GRID-based method to combine the Cresset and CAVEAT strategies. The anchor points (as for CAVEAT) are provided by the user, and then a database is searched for similar scaffolds, which match the geometry of the anchor points and the molecular interaction field profile. Using protease test cases, the central core could be used as a query, and known active alternative chemotypes found. The author suggest that the other highly ranked scaffolds could well be bioisosteres. 4.6 Ligand preparation Successful virtual screening relies on an appropriate and correct description of the molecules being screened; two particular sources of error are protonation state and tautomeric form. Oellien et al.55 have developed an exhaustive tautomer enumeration approach based on a set of 21 predefined transforms. The group do not attempt to say which tautomer is favoured, only that every form is represented. Databases built using all tautomers show better performance especially when screened with pharmacophore queries, as the bias introduced by most compound registration tools as to the ‘correct’ tautomer is removed.

5. De novo structure generation Fink and Reymond56 have continued their exhaustive exploration of chemical space for molecules with up to 11 atoms, using the elements C, N, O, F. They could enumerate 26.4 million structures, in comparison with the 63k structures with similar constraints described in the literature, including 538 novel ring structures. The universe has been somewhat pruned by rules around ring strain and obvious instability. The druglikeness of the structures was assessed using Bayesian models to predict for GPCR, kinase and ion channel activity, and a surprising number of novel virtual hits were found. Fechner and Schneider57 have also published a de novo design program. It is based on the assembly of molecular building blocks derived from a dataset using a RECAP-like set of disconnections.58 The starting molecule is also disconnected using the same rules, and rebuilt. A population of structures can exchange fragments, leading to a GA implementation with cross-overs and mutations. The fitness function was defined as the similarity, using topological torsions, to a query template structure. There are also various filters, based on upper and lower bounds on defined properties or descriptors, for example, molecular weight. Some of the queries could be generated, but compounds that could not be well handled by the disconnection rules were not. Lameijer et al.59,60 used de novo methods to generate scaffolds that were subsequently hand-modified by chemists. The protocol is to generate molecules from atoms and some fragments, then quickly stop and seek input from the medicinal chemist, who selects and edits the suggestions. This cycle can be iterated. In the paper, 8 examples of evolved molecules were made, and 4 showed some activity against bioamine targets. The generation process was highly constrained to produce molecules with more leadlike properties. While the input from a chemist resolves questions about synthetic feasibility, it raises others around bias and prior knowledge of pharmacophoric requirements. However, the authors argue that it is easier to evaluate a suggestion, than to invent something new, hence the close coupling of the computer program with the human end-user. Boda and Johnson61,62 take a different tack, suggesting that complexity (as a surrogate for synthetic accessibility) can be measured by precedent. A database of fragments was created using standard procedures from the MDDR set. Query 60 | Chem. Modell., 2008, 5, 51–66 This journal is

c

The Royal Society of Chemistry 2008

molecules from de novo generation can be fragmented in the same way. The frequency of occurrence of the fragments (together with terms for stereocentres and rotatable bonds) can be combined as penalty scores to give an index of complexity. Unusual fragments, or unknown fragments, will incur high penalties (for example pentasubstituted phenyl rings) whereas common fragments (3-substituted indoles) will not. Naturally, this does depend on the dataset and the fragmentation rules used to define the chemical precedent. Some well known drugs do have high complexity scores, reflecting this issue.

6. Fragment-based screening Fragment-based screening and drug design is intended to bridge the gap between the size of the druggable chemical space, and the relatively small numbers of compounds we have for screening. The argument is that good ligands can be built up from relatively small numbers of fragments, so by performing more sensitive screens with very small numbers (1000’s) of fragments, we can identify the key interactions, and then assemble the fragments back into more traditional ligands. One unresolved question is how to construct an optimal screening set of fragments, looking at diversity and screening set size as the parameters. Makara63 has conducted a retrospective study of 6 datasets, comparing an HTS deck, and virtual amide library and fragment libraries to look at these questions. There are other limiting factors to screening set size, based on the ability to follow up on any hits. The theoretical conclusion was that a few thousand (20k) compounds are required to represent the chemical space of commercially available amide space, and the hit rate from this library was comparable to screening the whole HTS deck. Mauser and Stahl64 also have looked at fragment selection using a backward approach, in contrast to the forward approach of Makara. A set of fragments was derived from bioactive compounds using RECAP rules,58 and filtered using some general leadlike criteria (ring size, rotatable bonds etc.). The fragments were clustered on the basis of maximal common substructure. Using different subsets of fragments of size 2000–4000 members, the chemical space that could be covered by assembling the fragments was also assessed using the Ftrees descriptors65 in comparison to the original database of bioactives. Each subset could represent about 20% of the original database of 85k compounds. Rings have long fascinated cheminformaticians, from determining the smallest set of rings in one structure, to determining the frequency of rings used in drug discovery,66 to try to understand which rings are acceptable, which are not. Much of this argumentation is, like the rings themselves, circular; rings may be favoured not for their biological properties but instead their synthetic accessibility, both historically and today in the cost of goods for synthesis of a drug in clinical development. Ertl et al.67 analysed the World Drug Index (bioactives) and compared this to compounds for sale, and a database of natural products. The diversity of simple aromatic rings was remarkably low, with only 780 scaffolds found from 150k compounds, including 216 singletons. Using exhaustive enumeration on a small number of frameworks, 580k possible scaffolds were created, and described using simple terms like C log P and PSA, as well as parameters derived from QM calculations. Using a Kohenen network as a classifier between the bioactive scaffolds and the virtual ones, it was observed that the bioactive scaffolds did indeed cluster into discrete regions: the descriptors most important to the classifications seemed to be heteroatom count and HOMO/LUMO energies. The authors conclude that we are only scratching the surface of structural chemistry space, even though bioactivity may be limited to small regions of that space. Mapping of hot spots of interaction has been a key concept in fragment-based design. Most scoring functions (for example GRID68) are based on reproducing crystal structures. In a paper by Ciulli et al.,69 the energetic contributions of fragments are measured experimentally using isothermal calorimetry and NMR. Chem. Modell., 2008, 5, 51–66 | 61 This journal is

c

The Royal Society of Chemistry 2008

Using ketopantoate reductase/NAPHH as the model system, it was shown that the change in affinity per heavy atom added was below the figure proposed by Kuntz for drug-like molecules.70 Looking at individual components, adding ribose did not improve average ligand efficiency, but adding phosphate groups did. Thus two hot spots in the ligand-receptor complex could be identified, and targeted for a fragmentbased strategy. Hajduk71 returns to the question of lead-likeness and the minimal potency required for a fragment to be useful. By examining 18 highly optimised inhibitors, and deconstructing them stepwise back to fragments, he claims a linear relationship between weight and potency, i.e. the binding efficiency is almost constant. As a rule of thumb, one should expect 1 log unit of potency for every 64 Da added, if one is to proceed along a successful optimisation. A drop of Binding Energy Index of more than 10% is probably indicative that a plateau has been reached of the modification is not ideal. Working this through, a starting fragment of 250 Da and an affinity of 1 mM would need to grow to 750 Da to achieve a potency of 1 nM. To stay under 500 Da, the starting fragment would need to be less than 200 Da. Finally, Babaoglu and Shoichet72 have performed experiments to assess the fundamental principle of fragment-based design. If we assume that we can build a ligand from fragments, what happens if we deconstruct a ligand into fragments? Will the individual binding modes of the fragments be maintained? Unfortunately, the experiments suggests that this is not always the case, and that fragments can be too small to bind unambiguously and/or correctly. For the original binding mode to be favoured, a larger fraction of the original inhibitor needs to be in place.

7. Target fishing There is considerable interest in the new field of target fishing. This is basically the identification of the target to which a ligand binds. This can be at the level of finding an active compound in a whole cell assay, then driving down to the molecular target, or having a collection of structures, or combinatorial library, and estimating in silico which proteins are these compounds most likely to bind to, before conducting the in vitro assay. Rognan et al.73 have performed the latter, proposing five targets for a set of triazepandiones, and finding significant activity in one screen, sPLA2. The first step is to prepare an annotated library of binding sites. 5 representatives from the screening library were docked into the 2148 binding sites, and the sites that got the highest score were retained. The target proteins from which the sites were derived were assayed experimentally with members of the screening set, to give micromolar inhibitors. It can be argued that the control experiment, of assaying against all the proteins, was not done, but this was impractical. Also, the most potent compounds in the assay were not the one predicted by docking, again highlighting the errors in the virtual screen. However, it is now possible to deorphanise compounds in silico relatively quickly (docking is slow, but it can be performed in parallel), and this result will have major impact in library design and compound purchasing for enrichment of corporate compound archives. Langer et al.74 tackle the issue using pharmacophore models instead. They have built up a library of models against a variety of targets. By screening a set of compounds against these models, they try to predict the biological profile of the compounds, which they managed for 90 out of a set of 100. The models can be generated either from a set of ligands, or from the protein structure itself, as described elsewhere in this review. Using more extensive conformational analysis leads to more false positives and true positives. The quality of the model was critical (how many features, tolerances etc.), as was the stickiness of the site (how many models could be derived), as to the confidence for which a target could be linked with the query structure. Jenkins et al.75,76 have used ligand activity data from the well-curated WOMBAT database to build a series of 2D models to predict the most likely protein targets for 62 | Chem. Modell., 2008, 5, 51–66 This journal is

c

The Royal Society of Chemistry 2008

the compounds in the MDDR database, as well as using a 3D pharmacophore-like descriptor for the same purpose. On average, the correct association of ligand to target could be found 90% of the time by one of the 2D methods, and some of the broader activity categories (for example, ‘‘kinase inhibitor’’) in the MDDR could be disambiguated to more precise targets. The 2D models could also be used for compound selection and library design, with the caveat that, as they are based on 2D fingerprints, they might bias against novel chemotypes (where 3D methods come more into their own). Indeed, the authors ascribe the very high success rate to the congeneric nature of the data. Cleves and Jain77 have used ligand-based methods to look at the link between similarity, pharmacological profile and target similarity for about 1000 therapeutics. The goal of the study was to rationalize, or even predict, off-target effects of compounds. The first stage was to curate the data, so that pharmacological target was named consistently. Their first result was to show that the similarity principle was generally true for the dataset. When ligand-based models were built for 22 targets, recovery rates of true ligands were between 60–70% with only 1–5% of false positives. By looking at the non-cognate ligands returned by the models, one could hop to a new target family, to hint at possible off-target pharmacology; the example given being the relationship between b-lactam antibiotics and beta2-adrenoreceptors. There is crossover between this topic and library design. Increasingly, groups are trying to introduce a target-family bias into new designs, or to select between designs on this basis. The SIFt methodology of Singh et al.78 encodes the pharmacophoric information within a site as a fingerprint that can be used for fast screening of large libraries. The first step is to dock a training library, followed by R-group analysis. Using the properties of the R-group as descriptors, and a SIFt analysis of the docking modes, more refined fingerprint patterns could be defined (describing which R groups make which interactions) which are used to build predictive QSAR models. These models were shown to be more sensitive to detecting compounds that could bind favourably than just using docking score alone. A related study by Fukunishi79 used dockings between known binders and inactive compounds, and set of pockets derived from protein active sites. Using PCA, they could show that the actives clustered together, and that using proximity to the active cluster for a given protein pocket was a good method for finding actives in a larger test set. Also they could cluster proteins with similar pockets together. This does require some fairly extensive calculations, and a large compound/target activity matrix, but only needs to be performed infrequently to generate the scoring model.

8. Library design Industry groups are now starting to publish their in-house tools for library design. Most of the innovation has occurred within Pharma research, as off-the-shelf packages often do not fit well into local workflows. Mosley et al.80 from Merck start us off with a tool for selecting reagents. The user selects the synthon, for example primary amine, and then refines the search using physicochemical properties. An important component is that the initial search is from a set of well-curated and readily available building blocks. One can also specify undesired functional groups which might interfere with the reaction scheme of require extra protection/ deprotection steps. Automatic exclusions are also triggered (i.e. choosing a secondary amine excludes the more reactive primary amine). The property refinement is performed with the true synthon, to remove effects of capping groups e.g. tosylates. The user can select the final reagents after a clustering process based on size and features. Truchon and Bayly81 propose a new algorithm, GLARE, for library subsetting, claiming very fast performance on libraries with up to 1012 members. As one of the slow steps has traditionally been the enumeration and scoring of the full product matrix, they use reagent-based approximate scores, that are 90% correlated with the Chem. Modell., 2008, 5, 51–66 | 63 This journal is

c

The Royal Society of Chemistry 2008

product scores. This is well within the accuracy of most scoring metrics anyway, so the approach seems justified. At each iteration of the subsetting process (dimension reduction), the reagent fitness score is incremented by 1 if the product is scored is as good. This is basically reagent frequency filtering, and the reagents with the lower scores are pruned out, and the cycle continues until the product subset has achieved a ‘goodness’ threshold, as defined by the user. The number of reagents to prune at each stage is the novel twist. In methods like PLUMS82 it is one by one. Here, using a formula based on the approach to the final set, many reagents can be pruned at each iteration without having false negatives. Also reagents coming from a larger pool (e.g. amines) are pruned faster than those coming from a small pool (boronic acids). However, the real speed-up comes from the sampling of the whole library space by smaller partitions. The assumption is that the fitness of the reagents should have the same distribution in the small partitions as for the whole library. The ability to handle very large library at interactive speed makes this very attractive for webbased tools intended for use by chemists. The concept of privileged substructures is frequently invoked in library design, with the assumption that addition of such a motif to a structure will confer some specific activity or selectivity. Schnur et al.83 challenge this notion. Substructural analysis was used to generate potential privileged substructures, and the relative frequency of occurrence of these fragments was examined over a number of target sets. In most cases, no selectivity is observed, leading to the conclusion that the motifs are generally conducive to binding. This may in turn lead to undesired sideeffects or even promiscuity. Promiscuity in HTS screening decks is a perceived issue, in that years of experience looking at HTS results tells us that there are some compounds that are ‘old favourites’, but it is not clear how to quantify the extent, or how to assign guilt to scaffold or fragments, except by anecdote. Pearce et al.84 have developed two methods for mapping promiscuity based on linking 180 substructural queries with HTS data. The absolute number of promiscuous compounds (defined as active in 7 or more assays) is relatively small, but their representation in hit lists is significant. Also the polyhalophenol fragment is over represented in the promiscuous compound set. In contrast, property-based filters showed no association with promiscuity, whereas several substructural queries did. The trend is that promiscuous compounds tend to be smaller and more soluble. Using two measures of promiscuity, compounds can be ranked by consensus for potential promiscuity. In practise, this can reduce the number of compounds considered for triage by 12%.

9. Conclusions The impetus to generate models in the field of ADMET has been sustained, as more high quality data becomes available. There is a price to pay, and now more care is being taken to ensure that these models, which can be widely deployed to unsuspecting users, are robust and give confidence estimates with the predictions. As methods for virtual screening are also becoming more robust, there are increasing numbers of applications for target fishing—rather than use the target to find the ligands, structures are being used to deorphanise targets; this is a field of increasing importance as we explore the effects of compounds in whole cell assays. There have been significant advances in docking, scoring, and the handling of protein flexibility. Many of the big questions posed in 2005 remain unresolved, but the good news is that the amount of new methodological research suggests that we think that one day, we will have the answers.

References 1 2 3 4

Y. C. Martin, QSAR Comb. Sci., 2006, 25, 1192. R. Guha, D. Dutta, P. C. Jurs and T. Chen, J. Chem. Inf. Model., 2006, 46, 1836. M. G. Hudelson and J. P. Jones, J. Med. Chem., 2006, 49, 4367. P. Gedeck, B. Rohde and C. Bartels, J. Chem. Inf. Model., 2006, 46, 1924.

64 | Chem. Modell., 2008, 5, 51–66 This journal is

c

The Royal Society of Chemistry 2008

5 A. G. Leach, H. D. Jones, D. A. Cosgrove, P. W. Kenny, L. Ruston, P. MacFaul, J. M. Wood, N. Colclough and B. Law, J. Med. Chem., 2006, 49, 6672. 6 D. Y. Haubertin and P. Bruneau, J. Chem. Inf. Model., 2007, 47, 1294. 7 R. P. Sheridan, P. Hunt and J. C. Culberson, J. Chem. Inf. Model., 2006, 46, 180. 8 E. A. Kennewell, P. Willett, P. Ducrot and C. Luttmann, J. Comput.-Aided Mol. Des., 2006, 20, 385. 9 N. Brown, B. McKay and J. Gasteiger, J. Comput.-Aided. Mol. Des., 2006, 20, 333. 10 A. C. Good and M. A. Hermsmeier, J. Chem. Inf. Model., 2007, 47, 110. 11 J. Kazius, R. McGuire and R. Bursi, J. Med. Chem., 2005, 48, 312. 12 A. S. Kalgutkar, I. Garner, R. S. Orbach, C. L. Shaffer, E. Callegari, K. R. Henne, A. E. Mutlib, D. K. Dalvie, J. S. Lee, Y. Nakai, J. P. O’Donnell, J. Boer and S. P. Harriman, Curr. Drug Metab., 2005, 6, 161. 13 F. Lombardo, R. S. Obach, F. M. DiCapua, G. A. Bakken, J. Lu, D. M. Potter, F. Gao, M. D. Miller and Y. Zhang, J. Med. Chem., 2006, 49, 2262. 14 M. P. Gleeson, N. J. Waters, S. W. Paine and A. M. Davis, J. Med. Chem., 2006, 49, 1953. 15 C. Skold, S. Winiwarter, J. Wernevik, F. Bergstrom, L. Engstrom, R. Allen, K. Box, J. Comer, J. Mole, A. Hallberg, H. Lennernas, T. Lundstedt, A. L. Ungell and A. Karlen, J. Med. Chem., 2006, 49, 6660. 16 A. M. Aronov, J. Med. Chem., 2006, 49, 6917. 17 R. Farid, T. Day, R. A. Friesner and R. A. Pearlstein, Bioorg. Med. Chem., 2006, 14, 3160. 18 L. Olsen, P. Rydberg, T. H. Rod and U. Ryde, J. Med. Chem., 2006, 49, 6489. 19 R. P. Sheridan, K. R. Korzekwa, R. A. Torres and M. J. Walker, J. Med. Chem., 2007, 50, 3173. 20 G. Cruciani, E. Carosati, B. B. De, K. Ethirajulu, C. Mackie, T. Howe and R. Vianello, J. Med. Chem., 2005, 48, 6970. 21 J. Subramanian, S. Sharma and C. Rao, J. Med. Chem., 2006, 49, 5434. 22 S. Eyrisch and V. Helms, J. Med. Chem., 2007, 50, 3457. 23 M. R. Landon, D.R. Lancia, J. Yu, S. C. Thiel and S. Vajda, J. Med. Chem., 2007, 50, 1231. 24 P. J. Hajduk, J. R. Huth and S. W. Fesik, J. Med. Chem., 2005, 48, 2518. 25 K. L. Damm and H. A. Carlson, J. Am. Chem. Soc., 2007, 129, 8225. 26 K. L. Meagher, M. G. Lerner and H. A. Carlson, J. Med. Chem., 2006, 49, 3478. 27 J. Michel, M. L. Verdonk and J. W. Essex, J. Med. Chem., 2006, 49, 7427. 28 A. Weis, K. Katebzadeh, P. Soderhjelm, I. Nilsson and U. Ryde, J. Med. Chem., 2006, 49, 6596. 29 M. Clark, F. Guarnieri, I. Shkurko and J. Wiseman, J. Chem. Inf. Model., 2006, 46, 231. 30 R. A. Friesner, R. B. Murphy, M. P. Repasky, L. L. Frye, J. R. Greenwood, T. A. Halgren, P. C. Sanschagrin and D. T. Mainz, J. Med. Chem., 2006, 49, 6177. 31 A. Fernandez and S. Maddipati, J. Med. Chem., 2006, 49, 3092. 32 C. Bracken, L. M. Iakoucheva, P. R. Rorner and A. K. Dunker, Curr. Op. Struct. Bio., 2004, 14, 570. 33 A. V. Razgulin and S. Mecozzi, J. Med. Chem., 2006, 49, 7902. 34 W. Sherman, T. Day, M. P. Jacobson, R. A. Friesner and R. Farid, J. Med. Chem., 2006, 49, 534. 35 M. Jacobsson and A. Karlen, J. Chem. Inf. Model., 2006, 46, 1334. 36 G. P. A. Vigers and J. P. Rizzi, J. Med. Chem., 2004, 47, 80. 37 A. E. Cho, V. Guallar, B. J. Berne and R. Friesner, J. Comp. Chem., 2005, 26, 915. 38 V. Guallar, B. F. Gherman, S. J. Lippard and R. A. Friesner, Curr. Op. Chem. Bio., Vol. 6, 01. 39 M. J. Hartshorn, M. L. Verdonk, G. Chessari, S. C. Brewerton, W. T. M. Mooij, P. N. Mortenson and C. W. Murray, J. Med. Chem., 2007, 50, 726. 40 N. Huang, B. K. Shoichet and J. J. Irwin, J. Med. Chem., 2006, 49, 6789. 41 J. F. Truchon and C. I. Bayly, J. Chem. Inf. Model., 2007, 47, 488. 42 P. R. Menard, J. S. Mason, M. I. and B. S., J. Chem. Inf. Comp. Sci., 1998, 38, 1204. 43 Y. P. Lu, Z. Nikolovska-Coleska, X. L. Fang, W. Gao, S. Shangary, S. Qiu, D. G. Qin and S. M. Wang, J. Med. Chem., 2006, 49, 3759. 44 M. Baroni, G. Cruciani, S. Sciabola, F. Perruccio and J. S. Mason, J. Chem. Inf. Model., 2007, 47, 279. 45 S. L. Dixon, A. M. Smondyrev, E. H. Knoll, S. N. Rao, D. E. Shaw and R. A. Friesner, J. Comput.-Aided Mol. Des., 2006, 20, 647. 46 Y. Patel, V. J. Gillet, G. Bravi and A. R. Leach, J. Comput.-Aided Mol. Des., 2002, 16, 653. 47 N. J. Richmond, C. A. Abrams, P. R. N. Wolohan, E. Abrahamian, P. Willett and R. D. Clark, J. Comput.-Aided Mol. Des., 2006, 20, 567. 48 J. Sadowski and J. Bostrom, J. Chem. Inf. Model., 2006, 46, 2305.

Chem. Modell., 2008, 5, 51–66 | 65 This journal is

c

The Royal Society of Chemistry 2008

49 D. K. Agrafiotis, A. C. Gibbs, F. Zhu, S. Izrailev and E. Martin, J. Chem. Inf. Model., 2007, 47, 1067. 50 E. J. Barker, D. Buttar, D. A. Cosgrove, E. J. Gardiner, P. Kitts, P. Willett and V. J. Gillett, J. Chem. Inf. Model., 2006, 46, 503. 51 N. Stiefl, T. A. Watson, K. Baumann and A. Zaliani, J. Chem. Inf. Model., 2006, 46, 208. 52 T. Cheeseright, M. Mackey, S. Rose and A. Vinter, J. Chem. Inf. Model., 2006, 46, 665. 53 P. Maass, T. Schulz-Gasch, M. Stahl and M. Rarey, J. Chem. Inf. Model., 2007, 47, 390. 54 R. Bergmann, A. Linusson and I. Zamora, J. Med. Chem., 2007, 50, 2708. 55 F. Oellien, J. Cramer, C. Beyer, W. D. Ihlenfeldt and P. M. Selzer, J. Chem. Inf. Model., 2006, 46, 2342. 56 T. Fink and J. L. Reymond, J. Chem. Inf. Model., 2007, 47, 342. 57 U. Fechner and G. Schneider, J. Chem. Inf. Model., 2007, 47, 656. 58 X. Q. Lewell, D. B. Judd, S. P. Watson and M. M. Hann, J. Chem. Inf. Comput. Sci., 1998, 38, 511. 59 E. W. Lameijer, J. N. Kok, T. Back and A. P. Ijzerman, J. Chem. Inf. Model., 2006, 46, 545. 60 E. W. Lameijer, R. A. Tromp, R. F. Spanjersberg, J. Brussee and A. P. Ijzerman, J. Med. Chem., 2007, 50, 1925. 61 K. Boda and A. P. Johnson, J. Med. Chem., 2006, 49, 5869. 62 K. Boda, T. Seidel and J. Gasteiger, J. Comput.-Aided Mol. Des., 2007, 21, 311. 63 G. M. Makara, J. Med. Chem., 2007, 50, 3214. 64 H. Mauser and M. Stahl, J. Chem. Inf. Model., 2007, 47, 318. 65 M. Rarey and J. S. Dixon, J. Comput.-Aided Mol. Des., 1998, 12, 471. 66 G. W. Bemis and M. A. Murko, J. Med. Chem., 1999, 42, 5095. 67 P. Ertl, S. Jelfs, J. Muhlbacher, A. Schuffenhauer and P. Selzer, J. Med. Chem., 2006, 49, 4568. 68 P. J. Goodford, J. Med. Chem., 1985, 28, 849. 69 A. Ciulli, G. Williams, A. G. Smith, T. L. Blundell and C. Abell, J. Med. Chem., 2006, 49, 4992. 70 I. D. Kuntz, K. Chen, K. A. Sharp and P. A. Kollman, Proc. Nat. Acad. Sci., 1999, 96, 9997. 71 P. J. Hajduk, J. Med. Chem., 2006, 49, 6972. 72 K. Babaoglu and B. K. Shoichet, Nature Chem. Bio., 2006, 2, 720. 73 P. Muller, G. Lena, E. Boilard, S. Bezzine, G. Lambeau, G. Guichard and D. Rognan, J. Med. Chem., 2006, 49, 6768. 74 T. M. Steindl, D. Schuster, C. Laggner and T. Langer, J. Chem. Inf. Model., 2006, 46, 2146. 75 Nidhi, M. Glick, J. W. Davies and J. L. Jenkins, J. Chem. Inf. Model., 2006, 46, 1124. 76 J. H. Nettles, J. L. Jenkins, A. Bender, Z. Deng, J. W. Davies and M. Glick, J. Med. Chem., 2006, 49, 6802. 77 A. E. Cleves and A. N. Jain, J. Med. Chem., 2006, 49, 2921. 78 Z. Deng, C. Chuaqui and J. Singh, J. Med. Chem., 2006, 49, 490. 79 Y. Fukunishi, S. Hojo and H. Nakamura, J. Chem. Inf. Model., 2006, 46, 2610. 80 R. T. Mosley, J. C. Culberson, B. Kraker, B. P. Feuston, R. P. Sheridan, J. F. Conway, J. K. Forbes, S. J. Chakravorty and S. K. Kearsley, J. Chem. Inf. Model., 2005, 45, 1439. 81 J. F. Truchon and C. I. Bayly, J. Chem. Inf. Model., 2006, 46, 1536. 82 G. Bravi, D. V. S. Green, M. M. Hann and A. R. Leach, J. Chem. Inf. Comp. Sci.s, 2000, 40, 1441. 83 D. M. Schnur, M. A. Hermsmeier and A. J. Tebben, J. Med. Chem., 2006, 49, 2000. 84 B. C. Pearce, M. J. Sofia, A. C. Good, D. M. Drexler and D. A. Stock, J. Chem. Inf. Model., 2006, 46, 1060.

66 | Chem. Modell., 2008, 5, 51–66 This journal is

c

The Royal Society of Chemistry 2008

Solvation effects Michael Springborg* DOI: 10.1039/b608783n

I.

Introduction

The goal of electronic-structure calculations is to provide information on real materials. The field of electronic-structure calculations is therefore placed somewhere between experiment and theory. I.e., by carrying through such calculations on specific systems one obtains results that cannot be manipulated but instead have to be interpreted. This resembles the situation of experiment. On the other hand, by modifying the systems one is able to follow the changes in the properties also to idealized situations that lie outside the range of experiment, which is a typical approach within theory. However, electronic-structure calculations are only then useful, if they can provide results that are in qualitative and quantitative agreement with experimental results, whenever such ones are available. This means that not only idealized situations shall be accessible with such calculations, but also realistic situations that are as close as possible to those found in experiment. Nevertheless, the fact that electronic-structure calculations have to rely on a number of approximations (see, e.g., ref. 1) often puts serious limitations on the systems and situations that can be treated with such calculations. These limitations exist first of all concerning the size of the systems that can be treated, i.e., the number of symmetrically inequivalent atoms and/or the number of structural degrees of freedom. The reason for their occurrence is twofold: At first, the computational requirements of a typical electronic-structure calculation scale as the size of the system to some power that typically is 3 or larger. And second, if the system has no well-known symmetry, the number of independent local total-energy-minimum structures grows faster than any power in the size of the system (see, e.g., refs. 1 and 2). This means that only systems of not too large size can be treated computationally, although, due to the continuous developments in computer technology and computer programs, this limit is continuously been pushed towards larger systems. Nevertheless, there are finite limits for what is computationally possible. Therefore, finite systems without any interactions with any surroundings are the ideal systems for theoretical studies. On the other hand, experimental studies are often carried through under quite different conditions. This includes that the systems of interest may interact with a solvent or with one or more surfaces (for instance of a catalyst). As we shall see below, the presence of such extra media can modify the properties of the system of interest. Therefore, in order to obtain a proper connection to the experimental studies, it is important that the presence of these media are included in the calculations ‘somehow’. In the present contribution we shall discuss how the effects of a solvent can be included. The challenges are, however, huge. A solvent is a macroscopic system containing an enormous number of atoms and most often having essentially no symmetry. Therefore, a standard electronic-structure calculation on the complete system would require more computational resources than ever will be available. Instead, one has to apply special-purpose methods that have been developed with exactly the goal of studying the properties of a single molecule that is moving under the influence of the solvent molecules. Physical and Theoretical Chemistry, University of Saarland, D–66123, Saarbru¨cken, Germany. E-mail: [email protected]

Chem. Modell., 2008, 5, 67–118 | 67 This journal is

 c

The Royal Society of Chemistry 2008

There are more different ways one can model this situation, depending on the precise type of interaction between the solvated molecule and the solvent. In many cases, these interactions are (partly or purely) of electrostatic nature, meaning that one may model the effects of the solvent as that of creating an external, electrostatic field in which the molecule of interest exists. But, there is also a feed-back between the solute and the solvent. I.e., the solute may in return influence the charge distribution of the solvent, thus ultimately leading to a modified effect of the influence of the solvent. Furthermore, in some cases (that are not at all atypical) hydrogen bonds between the solvent and the solute are important and should, therefore, also be included. It is the purpose of the present overview to discuss the various types of methods. In the next section we shall describe their theoretical foundations, and in the subsequent section we shall give some few selected examples of their applications. We add that the theoretical treatment of solvent effects is a currently very active research field, which can be seen both in the recent review by Tomasi et al.3 as well as from the fact that even special issues of certain journals have been devoted solely to this topic.4 Finally, we emphasize that we have not at all tried to be exhaustive in the choice of recent studies that shall be briefly mentioned below. Rather, they are meant as representing typical studies. Many others, not at all worse (or better), could have been selected.

II. Fundamental methods In this section we shall discuss the basic principles behind the different methods that have been developed for including solvation effects in electronic-structure calculations. Starting with the isolated molecules of interest, without the solvent, we shall subsequently see how the effects of the solvent can be included. We shall also briefly discuss the advantages and disadvantages of the various approaches. In the next section we shall then present different recent studies where the methods of the present section have been applied in calculating a number of different properties of different systems. A.

Without the solvent

In a typical electronic-structure calculation, the Born-Oppenheimer approximation is invoked. This means that for a system of M nuclei and N electrons one first seeks the electronic energy Ee by solving the electronic Schro¨dinger equation Hˆe Ce = EeCe

(1)

with ^e ¼  1 H 2

N X i¼1

r2! þ ri

N N X M N X X 1X 1 Zk ! þ  Vext ðr i Þ: ! ! ! 2 i6¼j¼1 j r i  r j j i¼1 k¼1 j !r  R j i¼1 i k

ð2Þ

h  1. Moreover, ~ ri is the Here, we have used atomic units, me = |e| = 4pe0 =  ~k that of the kth nucleus. Zk is the charge of the kth position of the ith electron, and R nucleus and, finally, we have assumed that the system is under the influence of an external, static field, Vext(~ r). Once Ee has been calculated, the total energy is obtained from Etot ¼ Ee þ

M M X ! 1 X Zk Zl þ Vext ðRk Þ: ! ! 2 k6¼l¼1 j R  R j k¼1 k l

ð3Þ

~k) may actually also depend on the charge or mass of the kth nucleus (e.g., Vext(R when being of electrostatic or gravitational nature), but we shall not specify that dependence explicitly. 68 | Chem. Modell., 2008, 5, 67–118 This journal is

 c

The Royal Society of Chemistry 2008

In practical calculations, it is never possible to obtain the exact solution to eqn (1). Instead, approximate methods are used that are based on either the Hartree-Fock approximation or the density-functional formalism in the Kohn-Sham formulation. In the case of the Hartree-Fock approximation one may add correlation effects, which, however, is beyond the scope of the present discussion (for details, see, e.g., ref. 1). Then, in both cases the problem of calculating the best approximation to the ground-state electronic energy is transformed into that of solving a set of singleparticle equations,   1 ! ! ! ! !  r2 þ VC ðr Þ þ Vn ðr Þ þ V^ xc þVext ðr Þ ci ðr Þ ¼ ei ci ðr Þ: ð4Þ 2 Here, VC is the Coulomb potential from the electrons, Z ! rðr 0 Þ !0 ! dr VC ðr Þ ¼ ! ! jr 0  r j

ð5Þ

with the electron density being !

rðr Þ ¼

N X

jci ðr Þj2 : !

ð6Þ

i¼1

Moreover, Vn is the Coulomb potential from the nuclei !

Vn ðr Þ ¼

M X

Zk !

!

ð7Þ

k¼1 jr  Rk j

and Vˆxc is either (in the Hartree-Fock approximation) the exchange operator that depends on all the occupied orbitals or (in the density-functional theory) the exchange-correlation potential that depends on the electron density. Independently of which approach is used, the (approximate) total energy of eqn (3) is a function of the nuclear coordinates, as is the (approximate) electronic energy, ~1, R ~2, . . . , R ~M), Etot = Etot(R

(8)

and the equilibrium structure is that for which Etot has its minimum. For isolated, finite molecules in the gas phase, the external potential Vext(~ r) = 0, and the computational requirements for solving eqn (4) scales as N to some power, typically 3, whereas the determination of the minimum of Etot as a function of the nuclear coordinates scales essentially exponentially with M. This means that exhaustive calculations are only for smaller systems possible. B. Supermolecule approaches When the molecule of interest is solvated in some solvent, interactions between the solvent and the solute have to be included. The most accurate approach would be to treat the complete system with the methods of the preceding subsection, which, however, due to the above mentioned scaling behaviours is computationally prohibitively demanding. One has, therefore, to restore to simpler approaches. In most cases, the solute consists of well-identified, neutral molecules that interact only weakly with each other. This means that reducing the solvent to consisting of just some few molecules around the solute may provide a reasonable approximation. In this case, the weak bonds (i.e., hydrogen bonds or van der Waals bonds) to more distant solvent molecules are broken, but it is expected that this will lead to only insignificant electronic redistributions. Ultimately, the finite system consisting of the solute and the smaller number of solvent molecules can be treated with the accurate methods of the preceding subsection. This is the supermolecule approach. Chem. Modell., 2008, 5, 67–118 | 69 This journal is

 c

The Royal Society of Chemistry 2008

The advantage of this approach is clearly that arbitrarily accurate methods can be applied. Thereby, also the weak interactions (first of all hydrogen bonds) can be treated accurately. On the other hand, when including more than those solvent molecules that are absolutely closest to the solute, the size of the system that shall be treated easily becomes so large that finite computational resources puts serious limits on the calculations. Thus, the computational requirements represent one disadvantage of this approach. Another is that longer-range interactions at first are not considered. For instance, if the solvent molecules possess a dipole moment, the electrostatic potential from more distant solvent molecules may be felt by the solute. Notice that since the number of dipoles scales essentially as r2 (with r being the distance to the solute) whereas the dipole potential scales as r2, long-range effects may be important although screening effects and averaging over different relative orientations lead to a faster convergence with respect to r. One way of introducing the interactions with more distant solvent molecules is to use a supercell approach (see, e.g., ref. 1). Then, a finite system including the solute and a smaller number of solvent molecules is repeated periodically in all three directions. Due to the periodicity, electronic-structure calculations can be carried through, but the drawback is that also the solute is repeated periodically. This means that if the repeated unit is too small, interactions between the solute molecules become non-negligible and may affect the results. C.

QM/MM approaches

A full quantum-mechanical treatment of the solute plus the solvent molecules may appear to be a too complicated approach. It would yield detailed information about the internal properties of the solvent molecules, that is not sought, and it would do so at the cost of large computational requirements. Only the properties of the solute are of interest as well as to which extent they are influenced by the presence of the solvent. Thus, only the effects of the solvent molecules outside their inner parts are of relevance. Therefore, simplified approaches that describe only those effects may be useful. This is the basic idea behind the QM/MM (quantum-mechanical/molecularmechanical) approaches. The solute is treated at the quantum-mechanical level, but the presence of the solvent is modeled through an external potential in which the solute moves. This external potential takes the role of Vext(~ r) discussed in section IIA and, therefore, the solute molecule can be treated using the same methods as for the isolated molecule in the gas phase, except that Vext(~ r) has to be included. The success of such an approach relies on the ability to devise realistic potentials Vext(~ r). One possibility is to distribute a set of solvent molecules at random and with random orientations in a finite, sufficiently large volume under a set of constraints. I.e., the density of the solvent molecules should be realistic, no two solvent molecules should come two close to each other, and no solvent molecule should come too close to the solute. Most often it is then assumed that the external potential Vext(~ r) is of electrostatic nature and can be modeled as a superposition of those of the solvent molecules. These can, e.g., in turn be approximated as a superposition of potentials from point charges placed at the positions of the nuclei. Improvements will allow the point charges to be placed at other positions, and also higher-order multipoles can be included. These potentials are devised so that selected properties of the solvent are reproduced accurately. Since the results may be biased by the randomly chosen (fixed) positions and orientations of the solvent molecules, one may repeat the calculation for more such arrangements and ultimately average the calculated properties of the solute. In refined models there may be a feedback between solute and solvent molecules. Thus, through the charge distribution of the solute, the solvent molecules feel the presence of the former. By calculating the forces acting on the solvent molecules, these may be allowed to move, ultimately leading to a molecular-dynamics simulation for the solvent. In turn, the shifted positions of those will modify the external 70 | Chem. Modell., 2008, 5, 67–118 This journal is

 c

The Royal Society of Chemistry 2008

potential acting on the solute, which, therefore, may respond with a changed charge distribution. Ultimately, the structure of the lowest total-energy for the complete system may be identified. In such simulations it is important that the interatomic interactions go beyond those of point charges (and, eventually, multipoles) in order to avoid that the atoms come too close. At least, short-ranged repulsions shall be included. Moreover, one may also include the internal degrees of freedom of the solvent molecules and, accordingly, not consider them solely as rigid bodies. Since the solvent molecules are included explicitly in the calculations, such approaches are called ‘explicit’. They may be quite accurate of the simple reason that, in many cases, there is a natural separation of the system into entities that interact only weakly (i.e., the solute and solvent molecules). Because the force fields that are used in the molecular-mechanics part are not unique, different models may differ in these force fields. Also differences in the quantum-mechanical approach for treating the solute may lead to different models. The advantage is that quite large systems can be treated. Disadvantages include that relatively stronger interactions between the molecules (first of all hydrogen bonds but also van der Waals interactions) are seldom described accurately. Moreover, there is no electrostatic feedback between solute and solvent molecules at this level of theory. I.e., the charge distribution of the individual solvent molecules remains largely unaffected by the solute. More recent developments aim to at improve this situation, as we shall discuss in the next subsection. D.

Polarizable-molecules approaches

Since the main parts of the solute–solvent interactions are of electrostatic nature, one may improve the models mentioned in the previous subsection by including the response of the solvent molecules to the electric field generated by the solvent and the solute. In the general case, the dipole moment of a (solvent) molecule changes under the influence of an electrostatic field, !

!

!

m ¼ m0 þa  E DC þ    ;

ð9Þ

~DC is the field vector, a is the where ~ m0 is the dipole moment in the field-free case, E polarizability tensor, and higher-order terms have been ignored. By knowing the polarizability (e.g., from experiment or from calculations), one can calculate the dipole moment of the solvent molecules when these are influenced by the electrostatic field generated by the other solvent molecules and by the solute. This leads to a modified dipole moment that in terms leads to a new electrostatic potential created by the solvent molecule, etc. Ultimately this leads to improved QM/MM models compared to those described in the preceding subsection. Although these approaches go back to the end of the 1970s,5–9 they are still widely used as we shall see in the next section. Finally, as was the case for the approaches of the preceding subsection, different descriptions of the interatomic interactions for the MM part lead to different models, and further differences in the results of such calculations may be due to differences in the approach used for the quantum-mechanical treatment of the solute. E. Continuum approaches The approach of the preceding section was improved over the one of section IIC by including an electrostatic response of the solvent to the presence of the solute (and of the other solvent molecules). It is thereby taken into account that the most important inter-molecular interactions in such a system most often are of electrostatic nature and an accurate description of those is essential. Since the approach considers the individual molecules, it is an explicit approach. Chem. Modell., 2008, 5, 67–118 | 71 This journal is

 c

The Royal Society of Chemistry 2008

A similar rationale is behind the polarizable-continuum approaches. Here, the solvent is modeled by a homogeneous continuum, i.e., the individual solvent molecules are not treated, and, therefore, this corresponds to a so called implicit approach. There are two central features of the model that can influence the outcomes of the calculations critically. At first, the solute is supposed to occupy some cavity in the solvent, whereby the precise definition of the size and shape of the cavity is not unique. Second, the continuum is able to respond to the charge distribution of the solute and, thereby, in turn by creating an electrostatic potential in which the solute exists to influence the properties of the solute. Again, the precise description of this response can differ among different approaches. We shall here briefly outline the main ideas behind this class of models. For more details, the interested reader is referred to the recent review by Tomasi et al.3 The size and shape of the cavity is one critical issue. The calculations become easier when simple shapes like spheres and ellipsoids are used, but in very many cases it is unrealistic to assume that the solute has such a simple shape. Instead, it is common practice to construct the cavity by superposing spheres placed at the positions of the atoms. Thus, any point that is inside at least one such sphere is considered a part of the cavity, and all other points are part of the polarizable continuum. Often, the radii of the spheres around each atom are chosen as van der Waals radii as defined, e.g., by Bondi10 or by Pauling,11 but other approaches exist, too (see, e.g., ref. 3). The surface of a cavity constructed as the overlap of atom-centered spheres has kinks where two spheres meet. Connolly12,13 has proposed a method that can lead to a more smooth surface and that very often is used nowadays. The idea is to let a smaller sphere ‘roll’ on the outer surface of the cavity. In most cases, this smaller sphere touches the cavity at exactly one point, but close to the regions where two (or more) spheres meet, the smaller sphere will touch all these spheres. In that case, that part of the smaller sphere that is closest to the cavity is used as a part of the cavity surface replacing the parts where the kink occurs. The presence of the continuum modifies the electrostatic potential. Formally, one may split this into one part from the solute and one part from the continuum, V(~ r) = Vm(~ r) + Vc(~ r),

(10)

but only for the total potential, one may set up the Poisson equation, r[e(~ r)rV(~ r)] = 4pr(~ r),

(11)

r(~ r) = rm(~ r) + rc(~ r),

(12)

where

is the total charge distribution from the molecule (rm) and from the continuum (rc). In the electronic-structure calculation for the solute, we replace in eqn (4) V(~ r) = VC(~ r) + Vn(~ r) + Vext(~ r).

(13)

In order to solve eqn (11) we also need the boundary conditions. First of all, very far from the system, r - N, we require that limr-N rV(~ r) and limr-N r2V(~ r) remain finite. More important is it that V(~ r) is continuous across the cavity/continuum interface, !

!

lim Vðr Þ  limþ Vðr Þ ¼ 0;

!

!

!

r!rC

!

ð14Þ

r!rC

rC (~ rC+) is a point just inside (outside) the where ~ rC is a point on the interface, and ~ cavity next to ~ rC. Moreover, the derivative of the potential experiences a discontinuity !

!

@Vðr Þ @Vðr Þ  e limþ ¼ 0: ! ! ! ! @n @n r!rC r!r lim

C

72 | Chem. Modell., 2008, 5, 67–118 This journal is

 c

The Royal Society of Chemistry 2008

ð15Þ

Here, ~ n is an outward pointing normal to the interface. Moreover, we have assumed that inside the cavity the relative dielectric constant takes the value 1 (as in vacuum) and in the continuum the constant value e. The systems of eqns (11)–(15) is under-determined because we know neither the potential V, nor the charge density rc. However, ultimately we are only interested in the potential inside the cavity and can therefore choose V and rc arbitrarily as long as they satisfy all equations above. One suggestion is then to assume that rc vanishes except that we have a charge density on the cavity/continuum interface that is determined so that it satisfies the Poisson equation with boundary conditions. This is the strategy behind the polarizable continuum model that goes back to the beginning of the 1980s.14 Then e is given the value that is typical for the macroscopic solvent. A modified version, the conductor-like screening model (COSMO), was later proposed by Klamt and Schu¨u¨rmann,15 where the equations above were solved for the continuum being a conductor with e - N. Subsequently, the potential Vc was scaled in order to take the finite value of e into account. Since the shape of the cavity is that of overlapping spheres, an alternative, natural strategy could be to expand the potential V(~ r) in harmonics (i.e., potentials from monopoles, dipoles, quadropoles, . . . ). Also here various strategies have been proposed.16–19 F. Hybrid approaches None of the methods we have mentioned is new, and all of them are still the subject of improvements. Here we shall not discuss those, as we consider that beyond the scope of this presentation. However, as a single example of recent developments we mention the work of Li and Gordon.20 These authors presented a hybrid method that aims at merging the two approaches of sections IID and IIE into one hybrid method. G.

Frozen-density approaches

Alternatively, there have been other approaches earlier that, currently, seem to be less frequently used than those discussed above. Among those is the frozen-densityfunctional approach of Wesolowski and Warshel.21,22 This is an explicit approach where solvent molecules are directly included in the calculations. It is based on density-functional theory and a separation of the total electron density into that of the solute, r1, and that of the solvent, r2. According to the density-functional theory, we may write the total electronic energy as a functional of r(~ r) = r1(~ r) + r2(~ r),

(16)

Ee[r] = Ee[r1 + r2] = Ee[r1] + Ee[r2] + Ee,non-add[r1,r2],

(17)

i.e., as

where the last term is the non-additive part of the total electronic energy. Since we are mainly interested in the properties of the solute, we may choose to use less accurate approaches for describing those part of the total electronic energy that originate from the solvent. Doing so we end up with a method that scales much more favourable with system size than the supermolecule approaches of section IIB. H.

Recent developments

As may have become clear, methods for the quantum-mechanical treatment of molecules in solutions were originally developed more than 20 years ago. But as in any other active field, the methods have continuously been improved, partly in order to increase accuracy and computational speed and partly to make possible the Chem. Modell., 2008, 5, 67–118 | 73 This journal is

 c

The Royal Society of Chemistry 2008

calculation of further properties, for instance such that are experimentally accessible or relevant. Many of the developments are directly tied to specific scientific questions and will in some cases be discussed further in the next section where we focus on applications, but here we shall briefly discuss some few recent works where methodological aspects have been put at the foreground. Iozzi et al.23 considered systems where the solute was experiencing surroundings that had some kind of structure. When modifying the polarizable continuum model of section IIE, this means in effect that the continuum was heterogeneous, i.e., e takes different values in different parts of space. As examples the authors mention different biological systems like water-soluble proteins where some residue interact both with the water and with the bulk proteins. Furthermore, molecules in lipid bilayers interact both with the water solvent and with the lipid bilayer. In order to treat such situations the authors suggested an extension of the continuum method where different parts of the surroundings have different relative dielectric constants. This leads to some complications that, however, could be solved. Finally, the authors demonstrated the applicability of the approach by addressing two biological issues, i.e., the protonation energies of the histidine residue, and the calculation of the nitrogen hydrogen-fine-splitting of the tempo radical in biological membranes. Here, we do not consider it relevant to discuss these issues further. In another recent development, Mennucci24 presented an extension of the polarizable continuum method that allows for studying time-dependent phenomena. Such approaches are, e.g., relevant when studying fast processes like electronic excitations or transfers. Due to the changed charge distribution of the solute following electronic excitations or transfers, the interaction with the solvent will change which in turn may change the properties of the solute, as we have discussed in this section. Since these are dynamical processes, a theoretical study of the time evolution is highly relevant. In particular when the time scale of the processes in the solute is comparable with that of the response of the solvent, resonance phenomena beyond those of a static treatment may be important. Mennucci presented an approach for such studies. Subsequently, she applied it to two processes, a charge transfer from a donor to an acceptor in a smaller molecule, as well as to the calculation of the time dependence of a Stokes shift which is used spectroscopically to measure polar solvation dynamics. An interesting approach was proposed by Yamamoto and Kato.25 They observed that calculations for a solute in some solvent bear a close resemblance to a standard electronic-structure calculations within the Born-Oppenheimer approximation. In ~1, R ~2, . . . , R ~M)  R ~ [together with the the latter case, the nuclear coordinates, (R ~ define some external parameters on which nuclear charges, (Z1, Z2, . . . , ZM)  Z], the solution to the (electronic) Schro¨dinger depends. When including the solvent, one in essence introduces one further dependence of the solution to the electronic Schro¨dinger equation, i.e., on the potential generated by the solvent Vext(~ r). In their approach, Yamamoto and Kato25 let Vext(~ r) be determined through a non-polarizable model for the solvent so that Vext(~ r) can be written as a superposition of Coulomb potentials from point charges representing the solvent. They showed, subsequently, how electron-transfer processes could be treated within their approach, even in the case when quantum effects of light nuclei (most notably, protons) need to be included. Finally, they presented results for a specific electrontransfer process. Tan and Luo26 have presented an improved implicit model for solvation effects. In the discussion above, in section IIE, the continuum is considered being a medium with a certain charge density. This charge density may be vanishing in large parts of space, except in those parts where the presence of the solute is being felt. The solute is in most cases treated quantum-mechanically, which for larger (e.g., biologically relevant) systems may provide a computationally too large challenge. Therefore, simpler treatments of the solute could be useful. Instead of replacing the continuum 74 | Chem. Modell., 2008, 5, 67–118 This journal is

 c

The Royal Society of Chemistry 2008

description of the solvent with a force field approach that, then, could include the solute, too, Tan and Luo26 suggested to use a polarizable model for describing the properties of the solute. The interior of the solute is modeled through a dipolemomentum density that then replaces the quantum-mechanically treatment of the solute. A feed-back between the solvent and the solute adjusts the densities of both media. This approach is significantly simpler than that of section IIE and could, therefore, be relevant for more complex systems. Tan and Luo carried through a set of test calculations using this proposal. They found that the model was robust, i.e., model parameters could be transferred between different structures and surroundings of the solute. In some cases, the performance was even better than that of a force field, although the computational requirements of the latter are larger. Not only the development of theoretical approaches and accompanying computer programs has led to an increase in the accuracy that can be achieved and the sizes of the systems that can be treated, i.e., number of atoms in the system as well as the length of the time for which a molecular-dynamics simulation can be carried through. Also developments in the computer powers are important. This was recently demonstrated impressively by Saito and Okazaki.27 They carried through a molecular-dynamics simulation over 45 ns for hemoglobin in water using the ‘Earth Simulator’ computer that 2002–2004 was the worldwide fastest supercomputer. They considered a system consisting of the human adult hemoglobin molecule surrounded by water molecules so that the complete system occupied a sphere with a radius of 66 A˚. This system, consisting of 119 421 atoms, was treated with a force-field approach, and longer-range solvent effects were treated using a polarizable continuum approach. From the simulation the authors were able to study the dynamics of even the tertiary and quaternary structures of the hemoglobin molecule.

III.

Recent studies

In the preceding section we outlined briefly the main ideas behind the different methods/models that have been introduced for the theoretical treatment of solvated molecules. We emphasize that we concentrated on presenting the foundations and not on discussing the many, also recent, improvements that have been introduced in order to increase both the accuracy and the efficiency of the different approaches, which has led to a large variety of methods, each with its advantages and disadvantages. Thus, if the reader gets the impression that all currently applied methods were developed long time ago, this is only partly the case. However, the reader who is interested in the recent technical and methodological developments is referred to the original literature. Here, we shall instead present results of recent studies on various systems and in the discussion use the rough classification of the methods from the preceding section. We shall briefly discuss the results of various, subjectively chosen, recent studies, emphasizing that these were chosen not as an attempt of identifying the scientifically best studies, but more as representing typical studies at the time of writing. A.

Comparing the methods

None of the approaches that has been proposed is exact. Instead, each approach has its advantages and disadvantages, and each approach may be more accurate for certain systems and/or properties than for others. Thus, one of the first questions to address is, how well the various approaches perform. This has been the topic of some recent studies. Jacob et al.28 compared a frozen-density and a polarizable-molecules approach for the calculation of various molecular properties. In both approaches they used a density-functional method for the quantum-mechanical treatment of the solute. Chem. Modell., 2008, 5, 67–118 | 75 This journal is

 c

The Royal Society of Chemistry 2008

Table 1 Various properties for a water molecule in the gas phase and in an water solvent. The FD results were obtained with a frozen-density approximation, and the QM with a polarizablemolecules approximation. Listed are the size of the dipole moment, the three independent components of the quadrupole moment, excitation energies, static polarizability, and static hyperpolarizability. All results are from ref. 28 Property

Quantity

Dipole moment (D) Quadrupole moment (a.u.)

m 1.80 Qxx 1.79 Qyy 1.86 Qzz 0.07 7.76 11A1 - 11B1 11A1 - 11A2 9.61 11A1 - 21A1 9.72 a 9.40 g 0.91

Excitation energy (eV)

Static polarizability (a.u.) Static hyperpolarizability (a.u.)

Gas phase

Solution (FD) Solution (QM) 2.66 2.05 2.15 0.11 8.41 10.38 10.40 9.62 0.72

2.71 2.09 2.17 0.08 8.88 10.69 11.01 8.67 0.50

Their test system was a single water molecule embedded inside a solvent consisting of 127 further water molecules. For the latter a fixed structure was used. In Table 1 we summarize their main findings. For the frozen-density calculations they considered two different approaches, one where the solvent-molecule density was kept fixed and one where it was allowed to relax. In the table we have only shown the results for the latter, which according to the authors led to an improved accuracy. The table shows that the dipole and the quadrupole moments are very similar for both approaches, which is to a lesser extent the case for the excitation energies and the static (hyper)polarizabilities. The latter were calculated using timedependent density-functional theory. In order to understand this discrepancy the authors used also a supermolecule approach with just two solvent molecules. By comparing with results from calculations with the frozen-density and the polarizable-molecule approaches on the same system they concluded that the frozen-density approach was the more accurate one in calculating the responses to electromagnetic fields. Although we in this report mainly is concerned with the behaviour of a single solute molecule in some solvent, the study of mixed systems can give some insight into the accuracy of the approaches. Therefore, we shall next discuss the work of Chevrot et al.29 who reported a study of the interface between water and the ionic liquid [BMI][PF6] (1-butyl-3-methyl-imidazolium hexafluorophosphate). They used molecular-dynamics simulation with different force fields to describe the interactions between the atoms of the system. They constructed an interface by joining a box of 2119 water molecules with a box of 181 BMI+PF6 molecules. In addition, they studied demixing of the two systems by considering a system that initially consisted of a mixture of water and the ionic liquid. In Table 2 we summarize some of their findings. The table shows the amount of water in the ionic liquid, the amount of ionic liquid in the water, and the width of the interface separating the two phases at the end of the simulation for the different force fields. It is seen that the popular TIP3P model gives quite different results from those of the other force fields and in the simulation of the mixing process there is not even a well defined interface. The other two force fields give similar results. All simulations support that the ionic liquid is essentially water-free, as should be. In order to understand the differences between the results of the different models, the authors also calculated the interaction energies between water and the ions of the ionic liquid. The results are reproduced in Table 3. The table shows at first that the variation in the interaction energies for the different computational approaches is small. This demonstrates that even small inaccuracies can have significant effects on the output of a simulation. Moreover, the table shows also that the TIP5P model 76 | Chem. Modell., 2008, 5, 67–118 This journal is

 c

The Royal Society of Chemistry 2008

Table 2 Molar fraction of water in the ionic liquid and of the ionic liquid in water, as well as width of the interface as found in the molecular-dynamics simulations with different force fields. q(IL) describes the effective charge of the ions in the ionic liquid used in the force field for this system, and FF(H2O) describes the force field for the water. The results for the ‘mixing’ simulations were obtained by considering a system where the ionic liquid and the water initially were separated, whereas ‘demixing’ marks results where the two systems initially formed a mixture. All results are from ref. 29 Simulation

q(IL)

FF(H2O)

xH2O

xIL

Width (A˚)

Mixing

1 1 0.9 0.9 0.9 0.9 0.9 0.9

SPC TIP5P TIP3P SPC TIP5P TIP3P SPC TIP5P

0.65 0.45 0.66 0.23 0.27 0.71 0.29 0.25

0.023 0.004 0.008 0.004 0.001 0.009 0.002 0.001

— 14.3 18.9 10.9 9.3 22.1 10.6 9.4

Demixing

with scaled effective charges of the ionic liquid is the model that at best reproduces the results of the parameter-free calculations. In section II we focused on an accurate description of electrostatic interactions between solute and solvent. Although these interactions account for the largest parts of the free energy of solvent, other interactions may contribute as well. These interactions, due to dispersion and repulsion, were studied by Curutchet et al.30 Amoville and Mennucci31,32 had earlier developed a formalism for including these interactions in the polarizable continuum model and Curutchet et al. tested it on a larger set of solvent-solute systems for which they also calculated these interactions with force-field methods. Since these interactions only rarely are discussed, we list typical results from their work in Table 4. It is seen that fairly large basis sets are needed in order to obtain results that can be considered close to converged. Moreover, the density-functional B3LYP method yields larger absolute values than does the Hartree-Fock method. Finally, the repulsion contribution amounts to only some 10–20% of the dispersion contribution. Using the B3LYP method for reference they compared subsequently the performance of four force-field methods for reproducing the B3LYP results for the dispersion and repulsive contributions to the free energy of solvation for 22 neutral molecules in four solvents. The results, Table 5, show that in most cases the force fields do not perform very well with the MST being an exception. Aidas et al.33 calculated NMR parameters for formaldehyde and acetone solvated in water using different models. Their main findings are summarized in Table 6. In Table 3 The interaction energies (in kcal/mol) between a water molecule and an ion from the ionic liquid, as obtained from calculations with Hartree-Fock (HF), density-functional (DFT), or force-fields. All results are from ref. 29 Method

q

PF6q  H2O

BMI+q  H2O

HF DFT TIP3P SPC TIP5P TIP3P SPC TIP5P

1 1 1 1 1 0.9 0.9 0.9

9.1 9.5 11.5 11.4 10.5 10.2 10.1 9.3

9.7 9.5 10.2 10.0 10.3 8.8 9.1 9.3

Chem. Modell., 2008, 5, 67–118 | 77 This journal is

 c

The Royal Society of Chemistry 2008

Table 4 The dispersion and the repulsion contributions (in kcal/mol) to the free energy of solvation for C2H4O solvated in water or chloroform. The table shows the dependence of the results on the basis set and on the quantum-mechanical method for treating the solute. All results are from ref. 30 Solvent

Method

Basis set

DGdis

DGrep

Water

HF

6-31+G(d) cc-pVDZ aug-cc-pVDZ cc-pVTZ aug-cc-pVTZ 6-31+G(d) cc-pVDZ aug-cc-pVDZ cc-pVTZ aug-cc-pVTZ cc-pVDZ aug-cc-pVDZ cc-pVTZ aug-cc-pVTZ cc-pVDZ aug-cc-pVDZ cc-pVTZ aug-cc-pVTZ

7.4 7.3 9.2 9.1 10.2 8.4 8.5 10.9 10.3 12.0 8.6 10.8 10.6 11.9 10.0 12.8 12.0 14.0

3.1 2.6 3.1 2.9 3.1 3.4 2.7 3.5 3.2 3.4 1.7 2.0 2.0 2.0 1.8 2.4 2.1 2.3

B3LYP

Chloroform

HF

B3LYP

their calculations they compared the polarizable continuum models with QM/MM models and, moreover, they examined the effects of including two water molecules in the quantum-mechanical calculations. The results of Table 6 show that both longand short-range interactions are important. Thus, a quantum-mechanical treatment of the finite C3H6O + 2H2O system is not capable of reproducing the experimental results, showing the importance of long-range interactions. But without the two water molecules, the polarizable continuum model does not yield satisfactory results, which indicates that also short-range interactions are important. In this context it is interesting to notice that including the two water molecules in the part that is treated quantum-mechanically does not seem to be very important for the results with the QM/MM simulations. Sun et al.34 studied the properties of thirteen smaller peptides in an aqueous solution using both explicit models with different force fields for the solvent Table 5 A comparison between quantum-mechanical and force-field treatment of dispersion and repulsive contributions to the free energy of solvation. The force field is given under method, and for each solvent the first line gives the root mean square deviation (rmsd, in kcal/ mol) and the second line gives the coefficient c that gives the best fit of y = cx with y being the force-field results and x being the quantum-mechanical results. All results are from ref. 30 Method

Property

Water

Octanol

Chloroform

Carbon tetrachloride

Claverie

Rmsd c Rmsd c Rmsd c Rmsd c Rmsd c

1.9 0.87 4.0 0.71 5.7 0.59 5.6 0.59 1.2 1.01

5.4 0.68 7.5 0.56 5.4 0.68 5.6 0.67 3.1 1.17

9.8 0.48 8.8 0.54 9.2 0.51 9.1 0.52 3.6 0.82

10.3 0.47 8.9 0.54 9.6 0.50 9.6 0.50 3.8 0.81

MM3 AMBER OPLS MST

78 | Chem. Modell., 2008, 5, 67–118 This journal is

 c

The Royal Society of Chemistry 2008

Table 6 Isotropic shielding constant (in ppm) sX for X being oxygen or carbon and the shift dX compared to the gas-phase value. QM denotes the system that is treated quantum mechanically, and the water part describes the approach for treating the solvent. For the latter, PCM1 and PCM2 denotes polarizable continuum models and TIP3P and SPC two different force fields.  marks gas-phase calculations. All results are from ref. 33 QM part

Water part

sO

dO

sC

dC

C3H6O C3H6O C3H6O C3H6O + 2H2O C3H6O C3H6O + 2H2O C3H6O + 2H2O C3H6O + 2H2O C3H6O + 2H2O Experiment

— PCM1 SPC SPC TIP3P TIP3P — PCM1 PCM2

339.2 285.6 259.3 258.7 250.6 251.0 297.3 262.7 250.1

— 53.6 79.9 80.5 88.6 88.2 41.9 76.5 89.1 75.5

21.5 33.2 37.0 37.7 38.2 38.9 31.1 36.6 38.2

— 11.7 15.5 16.2 16.7 17.4 9.6 15.1 16.7 18.9

      

0.7 0.8 0.6 0.6 0.5 0.5 0.5

      

0.7 0.8 0.6 0.6 0.5 0.5 0.5

      

0.1 0.1 0.1 0.1 0.1 0.1 0.1

      

0.1 0.1 0.1 0.1 0.1 0.1 0.1

molecules and implicit models. Their interest focused on the interactions of the peptide with a hydrophobic surface, and to this end they studied the so called potential of mean force (PMF) for the solvated peptides using the different solvation methods. They calculated the PMF by considering the interaction between a selfassembled monolayer of some molecules on some surface. The important point is that the molecules were terminated by CH3 groups, which then are the ones that the peptide will interact with. The interactions will, of course, depend on the distance of the peptide to the CH3 group, but also on the relative orientation of the two as well as on the internal properties of the peptide. Here, the latter depends on the changes induced by the solvent, i.e., on the model for the solvent. Sun et al. found significant differences between the implicit and the explicit solvent models. However, lacking further information they were not able to conclude which of the models gave more accurate results. In a recent study, Kongsted and Mennucci35 compared explicit and implicit solvation models for the calculation of various properties of aqueous solutions of pyrazine, pyrimidine, and pyridazine, i.e., three diazines that can be obtained from benzene upon replacing two CH groups by two N atoms (cf. Fig. 1). For the quantum-mechanical part they used a density-functional approach and, moreover, they also considered cases where 2 or 4 H2O molecules of the solvent were included in the QM treatment. Some of their results are summarized in Table 7. Some spread in the results depending on the model is observed, and in this case we also notice that the results that are obtained with the continuum approaches depend on the size of the spheres that are used in the construction of the cavity. Ultimately, however, the authors showed that when including 2 water molecules in the quantum-mechanical treatment of the system (thereby allowing for hydrogen bonds) and allowing the complete quantum-mechanical part of the system to relax to the structure of the lowest total energy, a good agreement with experimental values could be obtained with the unscaled polarized-continuum model (Table 8).

Fig. 1 Schematic representation of (from left to right) pyrazine, pyrimidine, and pyridazine. Open and closed circles represent CH groups and N atoms, respectively.

Chem. Modell., 2008, 5, 67–118 | 79 This journal is

 c

The Royal Society of Chemistry 2008

Table 7 Lowest electronic excitation energy (in eV) and isotropic shielding constant (in ppm) sN. n gives the number of water molecules that are explicitly treated quantum theoretically. PCM denotes the standard polarized continuum method, whereas in PCM* the radii spheres surrounding each atom in the construction of the cavity have been scaled by 91.7%. Finally, MM marks results obtained with an explicit description of the solvent. All results are from ref. 35 Property

n

Model

Excitation energy (eV)

0 0 0 2 2 0 2 4 0 0 0 2 2 0 2 4

Vacuum PCM PCM* Vacuum PCM MM MM MM Vacuum PCM PCM* Vacuum PCM MM MM MM

sN (ppm)

Pyrazine 3.95 4.00 4.00 3.95 4.00 4.05 4.03 113.3 101.8 97.9 110.4 99.7 98.9 100.3

Pyrimidine

   

   

0.01 0.01 0.01 0.01

0.8 0.3 0.8 0.8

4.28 4.45 4.49 4.30  0.01 4.48  0.01 4.52  0.01 4.49  0.01 4.48  0.01 72.1 59.9 55.7 67.8  0.5 57.4  0.4 56.4  0.5 58.4  0.5 58.8  0.5

Pyridazine 3.55 3.89 3.95 3.65 3.92 3.85 4.03 210.4 176.1 168.8 198.5 171.0 178.5 179.7

   

   

0.01 0.01 0.01 0.01

1.0 0.6 1.0 1.0

As the last example we mention the study of Takemura and Kitao.36 They studied different models for molecular-mechanics simulations on water. To this end, they studied the dynamics of a ubiquitin molecule solvated in water. They performed molecular-dynamics simulations for a system with periodic boundary conditions. At first they considered pure water without the solute and studied boxes with 360, 720, 1080, and 2160 water molecules. It turned out that even for these fairly large systems, finite-size effects could be recognized. Thus, the translational diffusion constant was found to depend linearly on V1/3, where V is the volume of the repeated unit. Subsequently, they studied a periodically repeated volume containing the ubiquitin molecule as well as 4908, 9133, 13 200, or 29 182 water molecules. Again, the linear dependence of the translational diffusion constant of the protein on V1/3 was observed, which is a confirmation of well-known finite-size effects.37–39 Correcting the results for the finite-size effects the authors obtained the results of Table 9. A clear scatter in the calculated values is observed, making it non-trivial to determine the correct model for molecular-dynamics simulations for aqueous solutions. B. Aqueous solutions of atoms The transport of ions in aqueous solutions is of fundamental importance in chemistry and biology. However, despite its apparent simplicity, the transport of Table 8 Change in lowest electronic excitation energy (in eV) and in isotropic shielding constant (in ppm) sN due to solvation for the molecules of Fig. 1. PCM denotes results with the polarized-continuum model and Exp marks experimental results. Two water molecules have been included in the quantum-mechanical treatment. All results are from ref. 35 Property

Model

Pyrazine

Pyrimidine

Pyridazine

Excitation energy (eV)

PCM Exp. PCM Exp.

0.16 0.19 20.8 16.9

0.31 0.33 20.2 16.8

0.55 0.48 59.2 41.6

sN (ppm)

80 | Chem. Modell., 2008, 5, 67–118 This journal is

 c

The Royal Society of Chemistry 2008

Table 9 The self-diffusion constant for water Dtw and the ubiquitin molecule Dtp in aqueous solutions as calculated with different force fields for the water. In addition, t1 and t2 are rotational relaxation times for the ubiquitin molecule. Where possible, the results are compared with experimental values. All results are from ref. 36 Model

Dtw (105 cm2/s)

Dtp (105 cm2/s)

t1 (ns)

t2 (ns)

TIP3P SPC SPC/E SPC/Fw Exp.

6.35 4.96 3.06 3.28

36.9 30.6 17.1 14.2 14.3, 14.9

4.09 4.88 9.13 8.18

1.33 1.78 2.97 2.69 4.09, 4.1

ions in water is a non-trivial process, where not only the ions but also a certain cloud of water molecules surrounding the ions is being transported. The study of these water shells around the ions is, therefore, still being pursued. Here, we shall discuss two such recent studies.40,41 Using a molecular-mechanics force-field model in combination with moleculardynamics simulations, Glezakou et al.40 studied the structure of K+ ions solvated in water. The simulation considered the potassium ion as well as 550 water molecules and was carried through at different temperatures. Among the results is the (timeaveraged) pair distribution function. Due to the finite temperature, such a function does not contain sharp peaks but more broad features from which one may try to extract the number of nearest neighbours. Using two different approaches to this end, Glezakou et al.40 obtained two different values, cf. Table 10. However, it is clear that an increase in the temperature results in a slight decrease in the number of water molecules in the first shell and a small increase in the distance between the ion and the water molecules. Glezakou et al.40 also performed experiment at 300 K, resulting in the values given in the table, too. With a supermolecule approach, Bock et al.41 studied the first and second shell of water molecules surrounding various metal ions. They studied M+q(H2O)18 clusters using parameter-free density-functional calculations and trying to optimize the structures in largely unbiased calculations. Due to the size of the system (55 atoms, or 19 building blocks when assuming that the H2O units stay intact), this is a far from trivial endeavour (see, e.g., ref. 2). Their results are summarized in Table 11. For all systems, a structure with one shell containing 6 water molecules and another shell with 12 water molecules is found, but for some cases (Be, Li, and Na) another structure with three shells of water with, respectively, 4, 8, and 6 molecules is also stable. However, the fact that the metal-oxygen distances are not constant for each shell shows that the structure is one of a low symmetry. As the last example we mention the study of Blumberger and Sprik42 who were interested in the electron-transfer redox reaction Ru2+ " Ru3+ in an aqueous solution. This reaction can be studied in the framework of the Marcus theory of electron transfer43–46 which provides a simple and effective treatment of the polar solvent. Blumberger and Sprik demonstrated how the ingredients of the Marcus theory could be extracted from electronic-structure calculations on Ru2+ and Ru3+ ions in a water solvent. To this end, they carried through density-functional calculations on a periodically repeated box containing 32 H2O molecules and one metal ion using the Car-Parrinello approach (see, e.g., ref. 1). In order to keep the system neutral, a compensating constant background density was added. Their results indicated that the Ru–O distance for the nearest water shell around the metal ion decreases from 2.18 A˚ for Ru2+ to 2.10 A˚ for Ru3+. These values compare reasonable well with the experimental values of 2.11 and 2.03 A˚, respectively. Ultimately, the authors estimated that the free-energy barrier for the Ru2+ " Ru3+ reaction is 0.38 eV, whereas the experimental value is 0.55 eV. The authors Chem. Modell., 2008, 5, 67–118 | 81 This journal is

 c

The Royal Society of Chemistry 2008

Table 10 The number of water molecules in the first shell around the K+ ions, n1, estimated in two different ways, as well as the average distance between the potassium ion and the oxygen atom of the water molecules, all quantities as a function of temperature. The experimental values were obtained at 300 K. All results are from ref. 40 T (K)

n1

d1 (A˚)

300 350 400 Exp.

6.6/5.7 6.6/5.6 6.5/5.3 5.7

2.770 2.777 2.782 2.732

Table 11 The number of water molecules around a metal M+q ion. n1, n2, and n3 give the number of molecules in the first, second, and third shell, respectively, and d1, d2, and d3 the distances between the metal ion and the oxygen atoms of the water molecules. All results are from ref. 41 M

q

n1

d1 (A˚)

Al Be Mg Na Ti Li Zn Be Li Na

3 2 2 1 4 1 2 2 1 1

6 6 6 6 6 6 6 4 4 4

1.922 1.858 2.099 2.415 1.993 2.191 2.120 1.638 1.956 2.279

         

0.001 0.010 0.001 0.16 0.28 0.10 0.006 0.018 0.05 0.06

n2

d2 (A˚)

12 12 12 12 12 12 12 8 8 8

3.988 4.103 4.130 4.184 4.092 4.075 4.123 3.672 3.955 4.086

         

0.010 0.4 0.017 0.19 0.4 0.13 0.07 0.15 0.18 0.28

n3

d3 (A˚)

6 6 6

4.826  0.18 4.808  0.16 4