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CCA26- Combined Quantum Mechanical and Molecular Mechanical Potentials Patricia Amara and Martin J. Field Institut de Biologie Structurale Jean-Pierre Ebel, Grenoble, France 1 2 3 4 5 6 Introduction Methodology Applications Conclusion Related Articles References Abbreviations EVB D empirical valence bond; MM D molecular mechanical; QM D quantum mechanical. 1 INTRODUCTION Molecular systems, such as organic molecules in solution and biological macromolecules, have been the subject of numerous theoretical studies. Molecular dynamics and Monte Carlo simulations are two techniques that can provide useful insights into the structural, thermodynamic, and, for molecular dynamics, dynamical properties of systems in the condensed phase.1 An essential part of simulation methodology is to have a potential energy function which is accurate enough to model the properties of the system of interest. A number of such functions, or force fields, have been developed for the simulation of biomolecules and these appear to give reliable results in many cases. However, in many standard force fields, the arrangement of bonds between the atoms in the system must be specified in advance and so they will not be appropriate for modeling processes in which chemical bonds are broken or created.2 One way to overcome this problem is to modify the force fields in some fashion. However, this is difficult to do consistently, particularly for atoms such as the transition metals which show a rich electronic behavior. Another approach is to use one of the many quantum mechanical (QM) methods, based on the Born Oppenheimer approximation, that have been developed to determine the electronic structure of molecular systems. In these methods the number of electrons and nuclei for each system is specified and the electronic density and the energy are calculated for each different configuration of the nuclei. Thus, it is possible to determine the redistribution of electrons among atoms as a reaction proceeds. The drawback with QM calculations is that they are much more expensive than force field calculations and can only be applied to the simulation of systems comprising a small number of atoms. Although recent advances in algorithmic techniques suggest that much larger systems could be treated,3 it is clear that possible applications are still limited by existing computational technology. In 1976 Warshel and Levitt introduced the idea of a hybrid QM/MM (molecular mechanical) method4 that treated a small portion of the system using a quantum mechanical representation while the rest of the system, which did not need such a detailed description, was represented by an empirical force field. They used their potential to study the reaction catalyzed by the enzyme, lysozyme. Since then, Warshel has continued to use such hybrid methods, most notably his empirical valence bond (EVB) approach, to study a wide variety of reactions in solution and in enzymes. It is undoubtedly he who has made the major contribution, both in terms of method development and in applications, to this area. Warshel has summarized his work on studying reactions in enzymes and interested readers are recommended to look at that for further details.5 The next workers to use these methods were Singh and Kollman, who devised a combined QM/MM potential6 using an ab initio molecular orbital method and the AMBER MM force field.7 They applied their method to studying the methyl chloride/chloride exchange reaction in solution and to the protonation of polyethers in the gas phase6 and, in a separate study, to the reaction catalyzed by the serine protease, trypsin, in solution and in the enzyme.8,9 The limitation of the potential developed by Kollman and co-workers was that it used an expensive ab initio QM method. While accurate, this precluded the use of the potential in molecular dynamics and Monte Carlo simulations and limited it to the static exploration of the potential energy surface. To overcome this problem, Karplus and co-workers10 developed a combined potential of the same basic form as that of Singh and Kollman but using the AM1/MNDO family of semiempirical QM methods11 and the CHARMM MM force field.12 Since these initial studies, potentials with both ab initio and semiempirical QM methods have been developed and applied by a large number of groups to a wide range of problems in condensed phase chemistry. The aim of this review is to give an overview of the form and the use of the type of hybrid QM/MM potentials mentioned above and to describe recent developments and applications. Section 2 discusses the methodology of the hybrid potentials, Section 3 discusses their application, and Section 4 summarizes the scope for future developments. 2 METHODOLOGY 2.1 General Principles The fundamental idea behind the use of hybrid potentials is that a system is partitioned into several regions which are then modeled with different levels of approximation. The atoms from the residues or molecules that are participating in the reaction process, or atoms that are rich in electrons, such as metals, are treated quantum mechanically. The rest of the system is described by standard molecular mechanics force fields. Finally a boundary region is defined to account for the finite size of the system. Different strategies can be adopted to partition the system. The simplest approximation is to divide the system into three regions a QM region and an MM region surrounded by CCA26- 2 COMBINED QUANTUM MECHANICAL AND MOLECULAR MECHANICAL POTENTIALS a boundary. However, it is possible to envisage a system partitioned into multiple regions in which less sophisticated potentials are used as the distance from the central core increases (see Figure 1). Such ‘onion’ models have been proposed by a number of groups, including Morokuma and co-workers.13 The most important point when employing any hybrid scheme is to test how the results of calculations change as different partitionings are tried. In general, the aim is to have as few atoms as possible treated with a high level of theory (which is expensive) but enough so that the results obtained are of the same quality as if all the system were being treated at the highest level. In other words, the minimum number of QM atoms should be chosen for a reasonable simulation time without losing too much accuracy in the model. A convenient way to describe the behavior of a system being studied with a hybrid potential is to use a notation O eff .10 In this formulation, involving an effective Hamiltonian, H the energy for the system at a given configuration of the atoms, E, is obtained by solving the time-independent Schrödinger equation, which has the form: Ĥeff .ri , R˛ , RM / D E.R˛ , RM /.ri , R˛ , RM / .1/ where ri and R˛ stand for the coordinates of the electrons and the nuclei in the QM region and RM for the coordinates of the atoms in the MM region, respectively. is the wavefunction whose square gives the electron density in the system. For the simplest case of a system partitioned into three regions, the effective hybrid Hamiltonian is composed of four terms: O QM C H O MM C H O QM/MM C H O boundary Ĥeff D H .2/ O QM , has the standard form for The electronic Hamiltonian, H the quantum mechanical method being used. For an ab initio method, for example, this would be: ĤQM D 1X 2 X 1 r C 2 i i rij ij X Z˛ X Z˛ Zˇ C ri˛ r˛ˇ i˛ ˛ˇ .3/ where Z˛ is the charge on nucleus ˛ and rxy is the distance between particles x and y. The subscripts (i, j) and (˛, ˇ) refer to electrons and nuclei, respectively. O MM , has interactions between The MM Hamiltonian, H MM atoms only and can be equated directly to the standard molecular mechanics energy, EMM . This will itself consist of a sum of independent energies, such as the ‘bonded’ bond, angle, and dihedral terms and the ‘nonbonded’ electrostatic and Lennard-Jones interactions, i.e. ĤMM D EMM D Ebond C Eangle C Edihedral C ECoulomb C ELennard The form of the QM/MM Hamiltonian will depend upon the forms of the separate QM and MM Hamiltonians. However, for the case where there are electrostatic and Lennard-Jones interactions between the QM and MM regions, a possible Hamiltonian is Ĥ0QM/MM D X qM X Z˛ qM X A˛M C C 12 r r˛M R˛M iM iM ˛M ˛M B˛M 6 R˛M .5/ There are three terms. The first two terms are electrostatic interactions which consist of two Coulombic terms, one between the electrons and the MM atoms and one between the nuclei and the MM atoms. The third term is a Lennard-Jones term which takes care of close-range exchange repulsion terms and the longer-range dispersion contributions which cannot be explained using a pure Coulombic interaction. This term is particularly important when the charge on an MM atom (and, hence, its electrostatic interaction) is zero. A number of extensions to the simple interaction Hamiltonian given above have been proposed. An example is for the case where polarization effects on the MM atoms are important. In general, there will be some polarization of the atoms in the QM region because the electron cloud will be distorted by interaction with the surrounding MM charges. There are several ways to include equivalent effects on the MM atoms but a common approach is to give to each MM atom an isotropic dipole polarizability, ˛M , which means that a dipole, mM , can be induced on each atom that is proportional to the electric field at the atom, EM : mM D ˛M EM Figure 1 Partitioning of a system into different regions for use with a hybrid potential. There are four regions in this representation. The yellow region, QM1, corresponds to high-level QM calculations while the dark green area, MM, is modeled with MM force fields. The light green region, QM2, could be treated with a lower level of QM theory or by a more complicated MM force field (which includes polarization, for example). The hatched area is the boundary region .4/ Jones .6/ Because the electric field at an atom has contributions arising from the charges and induced dipoles on the other MM atoms as well as the QM electrons and nuclei, the calculations of the QM energy and the MM electrostatic energy are coupled when polarization is included on the MM atoms and the time required for the evaluation of the hybrid energy is substantially increased. O boundary in equation (2) is the boundary Hamiltonian which H accounts for the truncation of the system. The same boundary approximations that are commonly used for simulations with MM potentials1 can be used with hybrid potentials. Typical examples include periodic boundary conditions in which the system is replicated in each direction in space, boundary force methods which introduce a force in the boundary region to constrain the atoms in the central region, and methods which treat the neglected part of the system as a continuum dielectric. The term in the Hamiltonian will typically consist of two O boundary.QM/ , and an MM contribution, terms, a QM part, H Eboundary.MM/ . CCA26- COMBINED QUANTUM MECHANICAL AND MOLECULAR MECHANICAL POTENTIALS 3 To obtain the total energy for the system, the expectation value of the wavefunction over the effective Hamiltonian is calculated. For the effective Hamiltonian described above (without polarization on the MM atoms), the energy is: O QM C H O QM/MM C H O boundary.QM/ /j i/h j i Etot D h j.H C EMM C Eboundary.MM/ .7/ Once the total energy is known, the forces on the atoms are easily derived as derivatives of the energy with respect to the atomic coordinates: F˛ D ∂Etot ∂R˛ FM D ∂Etot ∂RM .8/ The resulting energy and the associated forces can be employed for geometry optimizations and molecular dynamics and Monte Carlo simulations on the hybrid potential surface in exactly the same way as for calculations with purely QM or MM potentials. 2.2 The Junction Between the QM and MM Regions For a reaction involving relatively small molecules in solution, the partitioning of the atoms in a system between different regions is simple. The molecules implicated in the reaction are put in the QM region and the solvent molecules in the MM region. The situation is more complicated though when the region of interest is part of a large molecule, e.g., the active site of an enzyme. In that case, parts of the same molecule will be in different regions and there will be covalent bonds between the QM and MM atoms. It is not possible to simply truncate these bonds as this would leave lots of half-filled orbitals and give an inaccurate description of the electronic state of the QM region. Therefore, some means must be found to terminate these bonds reliably. In the original QM/MM study of Warshel and Levitt,4 this problem was overcome by including only a single hybrid sp2 orbital with a single electron for each of the QM atoms (carbon or nitrogen) at the junction. The remaining interactions for these atoms were treated using MM terms. A simpler, although less rigorous approach was introduced by Singh and Kollman6 and also used by Karplus and co-workers.10 They included ‘dummy junction’ or ‘link’ atoms in the QM region. These were QM atoms (usually hydrogens although other elements have been used) which were placed along the QM MM bond at an appropriate distance. They were included in the QM calculation but the interactions with the MM atoms were reduced in magnitude or even neglected altogether. In these cases, the interactions between the QM and the MM atoms were accounted for solely using MM force field terms. A schematic of both these types of approach is shown in Figure 2. Both the above schemes were not entirely satisfactory and several modifications have been suggested. Rivail and coworkers have developed a hybrid orbital method,14 similar in spirit to that of Warshel and Levitt, in which the QM atom at the junction is taken to have a normal complement of orbitals but the hybrid orbital which points towards the MM atom is kept frozen. The form of this orbital is not optimized in the QM calculation although it is counted in the evaluation of the QM energy terms and its interaction with the MM atoms is carefully parametrized. The hybrid orbital method is relatively Figure 2 Two alternative methods to treat the junction between the QM and MM regions. On the left, the ‘link atom’ method is represented where the dummy atom is a QM hydrogen placed along the QM MM bond. On the right is the ‘single hybrid orbital’ model where there is a single hybrid orbital along the QM MM bond easy to implement for semiempirical QM methods (as done by Warshel and Rivail et al.) but becomes more complicated for ab initio HF wavefunctions. An equivalent approach is possible for DFT wavefunctions and this is to define a frozen electron density on one of the atoms at the junction whose form is not included in the variational QM calculation but which interacts with the QM and MM atoms in the appropriate fashion. This method was first introduced, but in a different context, by Wesolowski and Warshel.15 An alternative scheme deserving mention is that devised by Morokuma and co-workers,13 which resembles the ‘link atom’ scheme. In their method, the system is defined as being composed of several superposed systems each of which is treated with a potential of a different quality. Thus, for a system with two partitions, there is a small central QM region and an MM region which includes all the atoms of the system, including those of the QM region. To avoid overcounting of atoms, the total energy for the system has to be redefined so as to subtract out the MM energy of the QM atoms. The problem with this method is that it is essentially steric in character and it does not, in its present incarnation, allow interactions between the different superposed systems. 2.3 Implementations of the QM/MM Approach As mentioned in the introduction a number of QM/MM methods have been implemented using different QM and MM approximations and different QM/MM interaction schemes. In principle, there is no restriction on the types of potential that can be coupled. In practice, certain combinations have proved more popular than others, in large part due to the computational expense of using ab initio QM methods. Thus, it is true that while combined potentials with ab initio HF or DFT methods would be preferable due to their greater reliability than semiempirical methods, the semiempirical methods have proved more popular because they can be used to study relatively large systems, with up to 100 QM atoms, using molecular dynamics or Monte Carlo simulation techniques. While, even for semiempirical QM methods, the QM portion will be the most time-consuming part of the calculation, the most difficult part of the implementation of a hybrid potential is the parametrization of the QM/MM Hamiltonian which determines the precision of the interactions between the QM and MM regions. The parametrization has to be done for the CCA26- 4 COMBINED QUANTUM MECHANICAL AND MOLECULAR MECHANICAL POTENTIALS nonbonded interactions between the atoms in the QM and MM regions and also for the covalent or ‘link’ atom interactions. For the covalent interactions it is probably true to say that no fully satisfactory parametrization has been performed. However, for the nonbonded interactions a number of extensive parametrization studies exist. For hybrid potentials with ab initio methods, only the Lennard-Jones interactions need to be parametrized as the electrostatic interactions between the QM and MM atoms can be evaluated exactly. For semiempirical QM methods it is necessary also to parametrize the electrostatic interactions as semiempirical methods, such as AM1 and MNDO, use a modified form for the Coulomb interaction which is softer at short range. Likewise, when polarization is included in the MM part of the force field, no parametrization is needed with ab initio methods but the form of the QM/MM dipole interaction needs to be parametrized with semiempirical QM methods. To conclude this section, it is worth mentioning the EVB approach that has been extensively used by Warshel and his collaborators5 and is being increasingly used by other workers (see, for example, Ref. 16). This method is essentially similar in conception to the types of potential described here but it is different in that it uses a much simpler valence bond QM model of the reacting region. The method is much faster than standard hybrid potentials but the potential is less easy to use as it must be conceived and parametrized for each different problem studied. 3 APPLICATIONS In this section we review some of the simulation studies that have been performed using hybrid QM/MM potentials. The most extensive set of applications has been to chemical processes in solution. We consider these first and then studies of reactions in biomolecules. It should be noted that the study of a reaction in solution is often a useful adjunct to the study of an equivalent reaction in some other environment, such as a protein. 3.1 Chemical Processes in Solution The study of many chemical processes in solution is relatively straightforward using hybrid potentials as the partitioning of the system into different QM and MM regions is often obvious the atoms of the reacting molecules are in the QM region and the solvent molecules in the MM region and there is no necessity to divide a single molecule into separate regions. Of course, in some cases the partitioning might not be evident, notably for reactions which involve the solvent molecules themselves (water being the most important example). Both Singh and Kollman5 and Karplus and co-workers17 in their early works investigated the methyl chloride/chloride exchange reaction, CH3 Cl C Cl , which is a standard test case for theoretical studies of reaction mechanisms in solution. In the gas phase, stable symmetric chloride/methyl chloride complexes are formed with complexation energies of about 36 kJ mol 1 between which there is a barrier to reaction of about 50 kJ mol 1 . In solution, however, the profile is simpler with no stable minima and a single barrier with a height of about 113 kJ mol 1 . In both Refs. 5 and 16 the reaction was studied in the gas phase with all seven atoms treated quantum mechanically and then in solution with the water molecules treated as the MM region. Singh and Kollman with their ab initio potential carried out minimizations of various structures along the reaction path and found a higher barrier to reaction and two complexes which were less stable than in the gas phase. Bash et al.,17 because they used a semiempirical method, were able to perform a full calculation of the free energies along the path and obtained results which agreed very closely with experiment. One of the advantages of studies with hybrid potentials and which was illustrated well here was that full information about the charge distribution for the reacting atoms was obtained along the reaction path with the result that a detailed analysis of the effect of the presence of solvent on the reaction profile was possible. The work described in Refs. 5 and 17 indicated that hybrid potentials could describe reaction processes in solution. However, it is only with the work of Gao that the real utility of this approach has become apparent. He and his co-workers have developed a hybrid potential that combines the AM1/MNDO semiempirical method with a force field similar to the OPLS force field of Jorgensen18 and with it they have studied a very wide range of solution phenomena in combination with Monte Carlo simulation techniques. As Gao has recently published two excellent reviews of hybrid potentials which include discussion of his own work19,20 only a few brief details will be given here. One area in which Gao has made specific contributions to the development of hybrid potentials is in their parametrization and, in particular, of the Lennard-Jones interactions. The strategy that he has adopted is to determine the LennardJones parameters for the QM atoms of the hybrid potential that best fit the results of high-level ab initio QM calculations (energies and geometries) on complexes between a solute and a small number of solvent molecules. He has then shown that these parameters can give excellent results for the solvation free energies of a wide range of organic compounds and he has discussed their significance in terms of the solvent polarization effect, i.e., the energetic effect the solvent has on the distortion of the solute’s electron density.21 Although his initial work in this area involved the AM1 method, he and Freindorf have recently performed the same parametrization using solute/solvent complex data for a hybrid potential with an ab initio molecular orbital method and have shown that significantly better results can be obtained than with the previous potential.22 In addition to solvation free energies, Gao has studied the effects of solvent on the conformational equilibria of different molecules and on the free energy profiles for various reactions, including nucleophilic substitution and pericyclic reactions. Another area he has studied is the effect of solvent on solute molecules in their excited states and, in particular, the shifts that are observed in their spectra in different solvents. The calculations showed that, whereas the blue shifts of acetone arising from stabilization of the ground state in polar solvents could be successfully reproduced, the hybrid potentials could not account for the red shifts which arise in some nonpolar solvents. This is a limitation of the model as the red shifts are caused by changes in the dispersion interactions between the two states and the solvent.23 Thompson has also studied the effects of solvation on spectra. He has calculated the spectral shifts for the ! nŁ CCA26- COMBINED QUANTUM MECHANICAL AND MOLECULAR MECHANICAL POTENTIALS transition for some simple carbonyl compounds.24 This work uses an AM1 semiempirical method for the atoms in the QM region but is unusual in that it includes polarization on the atoms of the MM region and develops a consistent way for calculating the interactions between the induced dipoles on the MM atoms and the QM atom charges. Thompson and Schenter have applied a similar model with polarizable MM atoms to the calculation of spectra of the excited states of the bacteriochlorophyll b dimer from the photosynthetic reaction center of Rhodopseudomonas viridis.25 Thompson has also studied processes without MM atom polarization, including the association of the potassium cation, KC , with dimethyl ether.26 Other workers who have developed or used QM/MM semiempirical potentials include Bakowies and Thiel, who have discussed the integration of MM polarization into hybrid potentials and have applied their method to study a number of reactions in the gas phase.27 Bash and co-workers have calculated the free energy profile for the proton transfer between methanol and imidazole in solution.28,29 The different aspect of this work was that they reparametrized the semiempirical method to get accurate interactions between the QM and MM portions of the system using a method originated by Rossi and Truhlar.30 The majority of the studies in this section have been performed with hybrid potentials that employ a semiempirical QM method because these methods are computationally less demanding. However, there has been considerable interest in developing combined potentials which use a DFT model for the QM region.31 A number of workers have developed methods of this sort, although, as yet, their application has been limited to the study of fairly simple systems, such as a single QM water molecule in a bath of MM waters or the solvation of small ions.32 34 3.2 Enzymatic Reactions Hybrid potentials are an ideal way to study reaction processes in enzymes. Such studies are, however, more demanding than studies of the equivalent processes in solution. Figure 3 shows a schematic of the partitioning of an enzymatic system for simulation with a hybrid potential. Atoms constituting the active center are modeled with QM methods while the rest of the enzyme and the solvent are represented by an MM force field. A major difference to the simulation of reaction processes in solution is that the way in which the system has to be partitioned into different regions may no longer be clear and it will be necessary to investigate how many atoms within the active site need to be treated quantum mechanically. Another difference is that the protein will be divided between QM and MM regions and so some sort of ‘link atom’ approximation will need to be employed. It is probably true to say that the driving force for the development of hybrid potentials was the desire to investigate reactions in enzymes. Warshel and Levitt in their pioneering paper4 studied the stability of the carbonium ion intermediate formed in the cleavage of a glycosidic bond by lysozyme and concluded that electrostatic stabilization by the protein of the intermediate was an important factor in determining the protein’s catalytic activity. Since the work on lysozyme, Warshel has studied a large number of enzyme reactions with 5 Figure 3 The QM/MM hybrid potential approach applied to an enzymatic reaction. The atoms in red are QM while the rest of the atoms are MM; light blue atoms are mobile while the purple ones are fixed and only ‘feel’ the rest of the system through nonbonded interactions. The purple ring is the boundary region his EVB methods, including some serine proteases and various metalloenzymes.5 There have been a number of QM/MM studies of enzyme reactions by other workers. They all follow basically the same pattern outlined in the original paper by Warshel and Levitt, but they have used the QM(ab initio)/MM or QM(semiempirical)/MM types of hybrid potentials described in Refs. 6 and 10. It should be noted that, up to the present, calculations with these types of hybrid potentials on enzymes have been limited to minimization or reaction path studies. In contrast to the case with the EVB models, no free energy calculations have been performed due to their expense. Studies with ab initio types of hybrid potential include the early work of Weiner et al. on the nature of catalysis in trypsin8,9 and the studies of the catalytic activity of phospholipase A2 by Hillier et al.35 Investigations with semiempirical hybrid potentials are more extensive and include calculations of the reactions in triosephosphate isomerase by Bash et al.36 and in chorismate mutase by Lyne et al.37 and a study of the proton jump in the catalytic triad of human neutrophil elastase.14 The study of the chorismate mutase reaction was especially interesting because the enzyme is the only known one that catalyzes a pericyclic reaction that also occurs readily in solution. The results of the hybrid study were particularly lucid in this case because the enzyme works, not by chemically catalyzing the reaction, but by preferentially binding a distorted form of the substrate and stabilizing the transition state. 4 CONCLUSION It is apparent that hybrid QM/MM potentials complement existing methods of simulation. They are very useful tools CCA26- 6 COMBINED QUANTUM MECHANICAL AND MOLECULAR MECHANICAL POTENTIALS for studying the properties of condensed phase systems, such as solvation effects and reaction processes, that cannot be studied using alternative potentials, either because they are too expensive (purely QM potentials) or because they are not appropriate (traditional MM force fields). Certain trends in the development and application of hybrid potentials are evident. Up to now most studies have been done with potentials that use semiempirical methods as the approximation. It is likely that these potentials will remain the most widely used in the future as they are relatively inexpensive to apply, although they will be improved, either by reparametrizing the semiempirical method for each new problem of interest or by using newer semiempirical approximations. Wider application of potentials with ab initio methods is also likely. Particularly promising are hybrid DFT/MM potentials as DFT methods are one of the most inexpensive types of ab initio methods yet they can provide results of an accuracy equivalent to the more expensive correlated molecular orbital methods. The other area of improvement in the potentials will be in the description of the interactions between the QM and MM regions. Increasing numbers of studies, such as those investigating electronic spectra, have shown that the inclusion of polarization effects on the atoms in the MM region is important if accurate results are to be obtained. Equally, it is possible to imagine that a more accurate and consistent representation of the exchange repulsion and dispersion interactions between the QM and MM atoms than that obtained by using LennardJones terms could be developed. Such methods could involve, for example, putting pseudopotentials on the MM atoms or using frozen densities on the MM atoms in conjunction with a DFT method for the QM region. The other obvious area in need of investigation is the ‘link atom’ problem, i.e., how to represent accurately covalent interactions between QM and MM atoms. In addition to improvements in the potentials themselves, it is also clear that the rapid advance in computer technologies, such as the increasing speed of hardware components, and in techniques such as massively parallel computing, will make possible the application of hybrid potentials to larger numbers of exciting chemical problems. 5 RELATED ARTICLES AM1; Hybrid Quantum Mechanical/Molecular Mechanical (QM/MM) Methods; Combined Quantum Mechanics and Molecular Mechanics Approaches to Chemical and Biochemical Reactivity; Density Functional Theory, HF and the Self-consistent Field; Divide and Conquer for Semiempirical MO Methods; Electrostatic Catalysis; Force Fields: AMBER; Force Fields: CFF; CHARMM: The Energy Function and Its Parametrization; Force Fields: General Discussion; The GROMOS Force Field; Hybrid Methods; Mixed Quantum-classical Methods; MNDO; MNDO/d; Molecular Dynamics: Techniques and Applications to Proteins; OPLS Force Fields; Parameterization of Semiempirical MO Methods; PM3; Protein Force Fields; QM/MM; Quantum Mechanical/Molecular Mechanical Coupled Potentials Applied to Biological Systems; SINDO1: Parametrization and Application. 6 REFERENCES 1. M. P. Allen and D. J. Tildesley, ‘Computer Simulation of Liquids’, Oxford University Press, Oxford, 1987. 2. U. Burkett and N. L. Allinger, ‘Molecular Mechanics’, ACS Monographs 177, American Chemical Society, Washington, DC, 1972. 3. T.-S. Lee, D. M. York, and W. Yang, J. Chem. Phys., 1996, 105, 2744 2750. 4. A. Warshel and M. Levitt, J. Mol. Biol., 1976, 103, 227 249. 5. A. Warshel, ‘Computer Modeling of Chemical Reactions in Enzymes and Solutions’, Wiley, New York, 1992. 6. U. C. Singh and P. A. Kollman, J. Comput. Chem., 1986, 7, 718 730. 7. S. J. Weiner, P. A. Kollman, D. A. Case, U. C. Singh, C. Ghio, G. Alagona, S. Profeta, Jr., and P. Weiner, J. Am. Chem. Soc., 1984, 106, 765 784. 8. S. J. Weiner, U. C. Singh, and P. A. Kollman, J. Am. Chem. Soc., 1985, 107, 2219 2229. 9. S. J. Weiner, G. L. Seibel, and P. A. Kollman, Proc. Natl. Acad. Sci. USA, 1986, 83, 649 653. 10. M. J. Field, P. A. Bash, and M. Karplus, J. Comput. Chem., 1990, 11, 700 733. 11. M. J. S. Dewar, E. G. Zoebisch, E. F. Healy, and J. J. P. Stewart, J. Am. Chem. Soc., 1985, 107, 3902 3909. M. J. S. Dewar and W. Thiel, J. Am. Chem. Soc., 1977, 99, 4899 4907. 12. B. R. Brooks, R. E. Bruccoleri, B. D. Olafson, D. J. States, S. Swaminathan, and M. Karplus, J. Comput. Chem., 1983, 4, 187 219. 13. M. Svensson, S. Humbel, and K. Morokuma, J. Chem. Phys., 1996, 105, 3654 3661. 14. G. Monard, M. Loos, V. Théry, K. Baka, and J. L. Rivail, Int. J. Quantum Chem., 1996, 58, 153 159. 15. T. A. Wesolowski and A. Warshel, J. Phys. Chem., 1993, 97, 8050 8053. 16. Y.-T. Chang and W. H. Miller, J. Phys. Chem., 1990, 9, 5884 5888. 17. P. A. Bash, M. J. Field, and M. Karplus, J. Am. Chem. Soc., 1987, 109, 8092 8094. 18. W. L. Jorgensen, D. S. Maxwell, and J. Tirado-Rives, J. Am. Chem. Soc., 1996, 118, 11225 11236. 19. J. Gao, in ‘Reviews in Computational Chemistry’, eds. K. B. Lipkowitz and D. B. Boyd, Vol. 7, VCH, New York, 1995. 20. J. Gao, Acc. Chem. Res., 1996, 29, 298 307. 21. J. Gao and X. Xia, Science, 1992, 258, 631 635. 22. M. Freindorf and J. Gao, J. Comput. Chem., 1996, 17, 386 395. 23. J. Gao, J. Am. Chem. Soc., 1994, 116, 9324 9328. 24. M. A. Thompson, J. Phys. Chem., 1996, 100, 14492 14507. 25. M. A. Thompson and G. K. Schenter, J. Phys. Chem., 1995, 99, 6374 6386. 26. M. A. Thompson, J. Am. Chem. Soc., 1995, 117, 11341 11344. 27. D. Bakowies and W. Thiel, J. Phys. Chem., 1996, 100, 10580 10594. 28. L. L. Ho, A. D. MacKerell, Jr., and P. A. Bash, J. Phys. Chem., 1996, 100, 4466 4475. 29. P. A. Bash, L. L. Ho, A. D. MacKerell, Jr., D. Levine, and P. Hallstrom, Proc. Natl. Acad. Sci. USA, 1996, 93, 3698 3703. 30. I. Rossi and D. G. Truhlar, Chem. Phys. Lett., 1995, 233, 231 236. 31. R. G. Parr and W. Yang, ‘Density Functional Theory of Atoms, and Molecules’, Oxford University Press, London, 1989. 32. R. V. Stanton, D. S. Hartsough, and K. M. Merz, Jr., J. Phys Chem., 1993, 97, 11868 11870. 33. D. Wei and D. R. Salahub, Chem. Phys. Lett., 1994, 224, 291 296. CCA26- COMBINED QUANTUM MECHANICAL AND MOLECULAR MECHANICAL POTENTIALS 34. I. Tuñón, M. T. C. Martins-Costa, C. Millot, M. F. Ruiz-López, and J. L. Rivail, J. Comput. Chem., 1996, 17, 19 29. 35. B. Waszkowycz, I. H. Hillier, N. Gensmantel, and D. W. Payling, J. Chem. Soc., Perkin Trans. 2, 1991, 1819 1832. 7 36. P. A. Bash, M. J. Field, R. C. Davenport, G. A. Petsko, D. Ringe, and M. Karplus, Biochemistry, 1991, 30, 5826 5832. 37. P. D. Lyne, A. J. Mulholland, and W. G. Richards, J. Am. Chem. Soc., 1995, 117, 11345 11350.