Paul D. Lyne

tional quantum mechanical methods [5]. A typical simulation of a solvated enzyme could include approximately 10,000 atoms, which makes the simulation using quantum mechanics intractable even with current computing facilities. A similification of the problem can be achieved by considering only the substrate and selected active site residues while neglecting the remainder of the enzyme and solvent. However, the remainder of the enzyme and solvent, although not having a covalent effect on the reaction, can significantly alter the energetics of the system and thus need to be included in the simulation.

The emergence of hybrid quantum mechanical–molecular mechanical (QM–MM) methods in recent years addresses this problem. Pioneering studies of this type were made by Warshel and Levitt [6]. The method entails the division of the system of interest into a small region that is treated quantum mechanically, with the remainder of the system treated with computationally less expensive classical methods. The quantum region includes all the atoms that are directly involved in the chemical reaction being studied, and the remainder of the system, which is believed to change little during the reaction, is treated with a molecular mechanics force field [7]. The atoms in each system influence the other system through a coupled potential that involves electrostatic and van der Waals interactions [8–18]. Several molecular mechanics programs have been adapted to perform hybrid QM–MM simulations. In the majority of the implementations the quantum region has been treated either by empirical valence bond methods [19] or with a semiempirical method (usually AM1 [20]). These implementations have been applied, for example, to study solvation [12,21], condensed phase spectroscopy [22], conformational flexibility [23], and chemical reactivity in solution [24,25], in enzymes [18,26–36], and in DNA [37].

Although semiempirical methods have the advantage of being computationally inexpensive, they have a number of limitations [38–40]. The major limitations concern their accuracy and reliability. In general, they are less accurate than high level methods, and since they have been parametrized to reproduce the ground-state properties of molecules, they are often not well suited to the study of chemical reactions. A further disadvantage of the semiempirical methods is the limited range of elements for which parameters have been determined.

To overcome these limitations, the hybrid QM–MM potential can employ ad initio or density function methods in the quantum region. Both of these methods can ensure a higher quantitative accuracy, and the density function methods offer a computaitonally less expensive procedure for including electron correlation [5]. Several groups have reported the development of QM–MM programs that employ ab initio [8,10,13,16] or density functional methods [10,41–43].

This chapter presents the implementaiton and applicable of a QM–MM method for studying enzyme-catalyzed reactions. The application of QM–MM methods to study solution-phase reactions has been reviewed elsewhere [44]. Similiarly, empirical valence bond methods, which have been successfully applied to studying enzymatic reactions by Warshel and coworkers [19,45], are not covered in this chapter.

II. BACKGROUND

A. QM–MM Methodology

In the combined QM–MM methodology the system being studies is partitioned into a quantum mechanical region and a molecular mechanical region (Fig. 1). The quantum

Figure 1 Schematic diagram depicting the partitioning of an enzymatic system into quantum and classical regions. The side chains of a tyrosine and valine are treated quantum mechanically, whereas the remainder of the enzyme and added solvent are treated with a classical force field.

region will normally include the substrate, side chains of residues believed to be involved in the reaction, and any cofactors. The remainder of the protein and solvent is included in the molecular mechanics region. For the QM region, the wave function of the system, Ψ, is a Slater determinant of one-electron molecular orbitals, ψi (or Kohn–Sham orbitals in the case of density functioned theory [46]),

where α and β refer to spin eigenfunctions, r the coordinates of the electrons, Rq the coordinates of the QM nuclei, and Rc the coordinates of the atoms in the MM region. The total energy of the system is evaluated by solving the Schro¨dinger equation with an effective Hamiltonian for the system:

For QM–MM methods it is assumed that the effective Hamiltonian can be partitioned into quantum and classical components by writing [9]

To date the majority of QM–MM applications have employed density functional methods ab initio or semiempirical methods in the quantum region. The energy terms evaluated in these methods are generally similar, but there are specific differences. The relevant equations for the density functional based methods are described first, and this is followed by a description of the specific differences associated with the other methods.

For density functional based QM–MM methods the electronic energy terms depend explicitly on the electron density, ρ(r), of the atoms in the quantum region [46]. The

electron density is determined by solving the one-electron Kohn–Sham equations [46, 47]

where ei are the eigenfunctions associated with the Kohn–Sham orbitals. The quantum

ˆ

Hamiltonian HDF is given by

In this equation EXC is the exchange correlation functional [46], qc is the partial charge of an atom in the classical region, Zq is the nuclear charge of an atom in the quantum region, riq is the distance between an electron and quantum atom q, ric is the distance between an electron and a classical atom c; Rqq′ is the distance between two quantum nuclei, and r′ is the coordinate of a second electron. Once the Kohn–Sham equations have been solved, the various energy terms of the DF–MM method are evaluated as

and

where Vqc is the van der Waals interaction energy between the quantum and classical regions. This Lennard-Jones term is given as

This term is essential to obtain the correct geometry, because there is no Pauli repulsion between quantum and classical atoms. The molecular mechanics energy term, EMM, is calculated with the standard potential energy term from CHARMM [48], AMBER [49], or GROMOS [50], for example.

The implementation of the method using ab initio methods for the quantum region is straightforward. The analogous equations for the electronic Hamiltonian and the corresponding energies in this case are [51]

The indices µ and v refer to the basis set orbitals, φ, and Pµv and Fµv are elements of the density and Fock matrices, respectively.

For QM–MM methods employing semiempirical methods in the quantum region, the implementaiton is not as straightforward. In semiempirical methods such s MNDO, AM1, or PM3 [38], a minimum basis set representation of the valence levels of each atom is employed. Thus, for example, for first row main group elements, the basis set comprises just four atomic orbitals, namely one s and three p orbitals. The remainder of the electrons in the atoms are grouped into core terms. Therefore, the added complication for QM– MM methods involving semiempirical Hamiltonians lies in how to treat the interactions between the core terms on the quantum atoms and the partial charges on atoms in the classical region. The treatment that is currently used follows that suggested by Field et al. [9], where the interaction between the semiempirical cores and the classical atoms are treated in the same manner as pure quantum interactions, with the atoms in the classical regions carrying an s orbital for this purpose. For Am1 or PM3 the interaction between the classical atoms and the cores of the quantum atoms is given as [20]

EcoreQM–MM qcZq(SqSq |ScSc)[1 exp ( αcRqc) exp ( αqRqc)]

From this equation it is seen that parameters have been introduced into the QM–MM method, with Kci , Lic, Mic, and αc corresponding to the pseudo s orbital on the classical atom. These parameters can be optimized to reproduce experimental or high level theoretical data. Field et al. [9] performed extensive investigations of the values of these extra parameters and suggested that the parameters Kci , Lci , and Mic (i 1, . . . , 4) can be set to zero and that αc should take a value of 5.0. These are generally the values used in most current QM–MM implementations that employ semiempirical methods in the quantum region.

Finally, the parametrization of the van der Waals part of the QM–MM interaction must be considered. This applies to all QM–MM implementations irrespective of the quantum method being employed. From Eq. (9) it can be seen that each quantum atom needs to have two Lennard-Jones parameters associated with it in order to have a van der Walls interaction with classical atoms. Generally, there are two approaches to this problem. The first is to derive a set of parameters, εq and σq, for each common atom type and then to use this standard set for any study that requires a QM–MM study. This is the most common aproach, and the ‘‘derived’’ Lennard-Jones parameters for the quantum atoms are simply the parameters found in the MM force field for the analogous atom types. For example, a study that employed a QM–MM method implemented in the program CHARMM [48] would use the appropriate Lennard-Jones parameters of the CHARMM force field [52] for the atoms in the quantum region.