Cramer C.J. Essentials of Computational Chemistry Theories and Models

the possible error in the entropy calculation are sufficiently large to make the good agreement with the experimental value quoted above seem potentially slightly misleading.

Irrespective of the accuracy of the absolute binding free energies, the major goal of the scanning is to identify possible substitutions meriting further study by a more accurate methodology. First, as a check on the assumptions of the model, binding free energies for two substituted cases were computed from Eqs. (12.32) and (12.33) but using MD trajectories generated for the proper complexes. The results were sufficiently close to those obtained using the unsubstituted trajectory that no concerns were generated. Then, full FEP calculations using TI and explicit solvent were carried out mutating biotin into 8R- fluoroavidin and 8S-fluoroavidin, i.e., computing the vertical legs in Figure 12.6 (mutations were run in both the backward and forward directions). For the 8R-fluoro analog, the binding free energy was computed to be 1.5 kcal mol−1 stronger than biotin, in reasonable agreement with the fluorine scanning value of 0.9 kcal mol−1.

There are a few technical details in this paper that are rather more ill-defined than ideal for a ‘canned’ strategy – the description above of the fluorine scanning procedure glosses over some of the finer details associated with evaluating binding free energies for the substituted analogs. Nevertheless, this paper presents an interesting comparison of more and less time-consuming models for estimating differential binding free energies from explicit simulation. The joint application of explicit solvent and continuum solvent methodologies for biomolecular studies seems destined to increase in frequency.

Extensions of this case study are available for the interested reader. First, Dixon et al. (2002) have expanded the analysis presented above to include consideration of methylated biotin analogs and in the process developed a graphical approach for visualizing free energy changes. In addition, Lazaridis, Masunov, and Gandolfo (2002) have also considered the binding free energies of various ligands, including biotin and biotin analogs, to avidin and streptavidin. These authors decompose results from MD simulations with implicit solvation into ligand/enzyme interaction energies, reorganization energies, and entropy changes, and they conclude that the most difficult component to predict with acceptable precision is the reorganization energy of the macromolecule. The results from all three of these studies have important implications for docking models in general, and in particular for models that employ static ligand and/or receptor structures to improve efficiency, and thereby ignore relaxation energetics.

Bibliography and Suggested Additional Reading

Blondel, A. 2004. ‘Ensemble Variance in Free Energy Calculations by Thermodynamic Integration: Theory, Optimal “Alchemical” Path, and Practical Solution’, J. Comput. Chem., 25, 985.

Brooks, C. L., III and Case, D. A. 1993. ‘Simulations of Peptide Conformational Dynamics and Thermodynamics’ Chem. Rev. 93, 2487.

Boresch, S. 2002. ‘The Role of Bonded Energy Terms in Free Energy Simulations – Insights from Analytical Results’, Mol. Sim., 28, 13.

Frenkel, D. and Smit, B. 1996. Understanding Molecular Simulation, Academic Press: New York. Jensen, F. 1999. Introduction to Computational Chemistry , Wiley: Chichester.

Kollman, P. 1993. ‘Free Energy Calculations: Applications to Chemical and Biochemical Phenomena’,

Chem. Rev., 93, 2395.

Levy, R. M. and Gallicchio, E. 1998. ‘Computer Simulations with Explicit Solvent – Recent Progress in the Thermodynamic Decomposition of Free Energies and in Modeling Electrostatic Effects’,

Annu. Rev. Phys. Chem., 49, 531.

Lybrand, T. P. 1990. ‘Computer Simulation of Biomolecular Systems Using Molecular Dynamics and Free Energy Perturbation Methods’, in Reviews in Computational Chemistry , Vol. 1, Lipkowitz, K. B. and Boyd, D. B. Eds., VCH: New York, 295.

Orozco, M. and Luque, F. J. 2000. ‘Theoretical Methods for the Description of the Solvent Effect on Biomolecular Systems’, Chem. Rev., 100, 4187.

Sen, S. and Nilsson, L. 1999. ‘Some Practical Aspects of Free Energy Calculations from Molecular Dynamics Simulation’, J. Comput. Chem., 20, 877.

Straatsma, T. P. 1996. ‘Free Energy by Molecular Simulation’, in Reviews in Computational Chemistry , Vol. 9, Lipkowitz, K. B. and Boyd, D. B. Eds., VCH: New York, 81.

van Gunsteren, W. F., Luque, F. J., Timms, D., and Torda, A. E. 1994. ‘Molecular Mechanics in Biology: From Structure to Function, Taking Account of Solvation’, Annu. Rev. Biophys. Biomol. Struct., 23, 847.

References

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Berendsen, H. J. C., Postma, J. P. M., van Gunsteren, W. F., and Hermans, J. 1981. In: Intermolecular Forces , Pullman, B., Ed., Reidel: Dordrecht.

Bernardo, D. N., Ding, Y., Krogh-Jespersen, K., and Levy, R. M. 1994. J. Phys. Chem., 98, 4180. Car, R. and Parrinello, M. 1985. Phys. Rev. Lett., 55, 2471.

Chalmet, S., Rinaldi, D., and Ruiz-Lopez,´ M. F. 2001. Int. J. Quantum Chem., 84, 559. Cossi, M. and Crescenzi, O. 2003. J. Chem. Phys., 118, 8863.

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Dixon, R. W., Radmer, R. J., Kuhn, B., Kollman, P. A., Yang, J., Raposo, C., Wilcox, C. S., Klumb, L. A. Stayton, P. S., Behnke, C., Le Trong, I., Stenkamp, R. 2002. J. Org. Chem., 67, 1827.

Fischer, R., Richardi, J., Fries, P. H., and Krienke, H. 2002. J. Chem. Phys., 117, 8467. Glattli, A, Daura, X., and van Gunsteren, W. F. 2003. J. Comput. Chem., 24, 1087.

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Jorgensen, W. and Ravimohan, C. 1985. J. Chem. Phys., 83, 3050. Jorgensen, W. L. and Nguyen, T. B. 1993. J. Comput. Chem., 13, 195.

Jorgensen, W. L., Briggs, J. M., and Contreras, M. L. 1990. J. Phys. Chem., 94, 1683.

Jorgensen, W. L., Chandrasekhar, J., Madura, J. D., Impey, R. W., and Klein, M. L. 1983. J. Chem. Phys., 79, 926.

Kaminski, G., Duffy, E. M., Matsui, T., and Jorgensen, W. L. 1994. J. Phys. Chem., 98, 13 077. Kuhn, B. and Kollman, P. A. 2000. J. Am. Chem. Soc., 122, 3909.

Kumar, S., Bouzida, D., Swendsen, R. H., Kollman, P. A., and Rosenberg, J. M. 1992. J. Comput. Chem., 13, 1011.

Lazaridis, T., Masunov, A., Gandolfo, F. 2002. Proteins, 47, 194.

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Mahoney, M. W. and Jorgensen, W. L. 2000. J. Chem. Phys., 112, 8910. McDonald, N. A. and Jorgensen, W. L. 1998 J. Phys. Chem. B , 102, 8049.

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Peter, C., Oostenbrink, C., van Dorp, A., and van Gunsteren, W. F. 2004. J. Chem. Phys., 120, 2652. Rick, S. W., Stuart, S. J., and Berne, B. J. 1994. J. Chem. Phys., 101, 6141.

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Liotard, D. A. 1994. In: Structure and Reactivity in Aqueous Solution, ACS Symposium Series, Vol. 568, Cramer, C. J. and Truhlar, D. G., Eds., American Chemical Society: Washington, DC, 24.

Zhu, S.-B. and Wong C. F. 1994. J. Phys. Chem., 98, 4695.

13

Hybrid Quantal/Classical Models

13.1 Motivation

An interest in understanding solvent structure represents one example of a situation that requires the explicit representation of a large system, as described in detail in the preceding chapter. For reasons of efficiency, such representation is most typically carried out at the molecular mechanics level. The chief drawback of the MM level of theory, however, is that it is almost never appropriate for the description of processes involving bond-making or bond-breaking, i.e., chemical reactions. To adequately model such processes, QM methods are required. However, the region of space within which significant changes in electronic structure occur along the course of a reaction coordinate is often relatively small compared to the size of the reacting system as a whole. For instance, a very large enzyme may catalyze the conversion of its substrate from one molecule to another, but the volume of space within which bonds are being made and broken is usually limited to the relatively small active site. The remainder of the enzyme may be important for maintaining its structure, recognizing other enzymes with which it works, folding, etc., but fails to exert any quantum mechanical influence on the catalytic active site.

Thus, from a modeling perspective, we may regard the situation in the abstract as described by Figure 13.1. Within a limited region, we wish to make use of the tools of quantum mechanics to accurately model an electronic-structure problem, while in the surrounding region the explicit representation of the supersystem is important, but the level of model applied can be reduced in complexity owing to the more simply understood influence of the outer region on the process as a whole. When the level applied to the outer system is MM, the complete Hamiltonian for the system must be some kind of hybrid of QM and MM methodologies, defining a so-called QM/MM technique. Put in a disarmingly simple fashion

where HQM accounts for the full interaction energy of all quantum mechanical particles with one other, HMM accounts for the full interaction energy of all classical particles with one other, and HQM/MM accounts for the energy of all interactions between one quantum mechanical particle and one classical particle. Methods for the evaluation of the first two

Essentials of Computational Chemistry, 2nd Edition Christopher J. Cramer

2004 John Wiley & Sons, Ltd ISBNs: 0-470-09181-9 (cased); 0-470-09182-7 (pbk)

MM

QM

Figure 13.1 In large systems that require explicit representation, understanding bond-making/- bond-breaking processes can often be accomplished using a quantum mechanical representation of only a portion of the full system, with a molecular mechanics representation of the rest

terms on the r.h.s. of Eq. (13.1) have already been the subject of much discussion in preceding chapters – the devil for a hybrid method is in the details of the final term, and those are the subject of this chapter.

Many QM/MM modeling schemes have been described with varying levels of formalism. In terms of classification, perhaps the most fundamental distinction is whether or not the boundary separating the QM region from the MM region in Figure 13.1 cuts across any chemical bonds. If it does not, the coupling of the QM and MM regions can be represented with a reasonable degree of simplicity. If so clean a separation is not practical, however, e.g., the QM region consists of the substrate for a large enzyme and at least one atom from a side chain residue in the active site (that serves to accept a proton from the substrate, for example), then more complicated coupling schemes must be employed to stitch together the distinct subspaces.

13.2 Boundaries Through Space

In some sense, the simplest example of what might be called a QM/MM approach with a through-space boundary has already been alluded to in Section 12.2.5 and illustrated with the specific example of the Claisen rearrangement in Section 12.5.1. To evaluate the PMF for a reaction in solution, one useful approach is to compute the reaction coordinate using a QM method in the gas phase, and then determine changes in solvation free energy as the system is driven from one end of the coordinate to the other by the coupling parameter λ. For the FEP calculations themselves, the reacting system is represented classically (e.g., using fixed geometries, partial atomic point charges, and van der Waals parameters), but the gas-phase energies to which the solvation free energies are added, and also often the atomic partial charges, are taken from the antecedent QM calculations. As has already been emphasized, this

approach ignores the effect solvent has on coordinates other than the reaction coordinate, and on the solute wave function, but it nevertheless may legitimately be referred to as a ‘weakly coupled’ QM/MM calculation. We now proceed to consider increasingly tightly coupled protocols for joining the two regions.

13.2.1Unpolarized Interactions

A significant issue with modern force fields is that it can be difficult to simultaneously address both generality and suitability for use in condensed-phase simulations. For example, the MMFF94 force field is reasonably robust for gas-phase conformational analysis over a broad range of chemical functional groups, but erroneously fails to predict a periodic box of n-butane to be a liquid at −0.5 ◦C (Kaminski and Jorgensen 1996). The OPLS force field, on the other hand, is very accurate for condensed-phase simulations of molecules over which it is defined, but it is an example of a force field whose parameterization is limited primarily to functionality of particular relevance to biomolecules, so it is not obvious how to include arbitrary solutes in the modeling endeavor.

Kaminski and Jorgensen (1998) have proposed one particularly simple QM/MM approach to address this problem, which they refer to as AM1/OPLS/CM1 (AOC). In AOC, Monte Carlo calculations are carried out for solute molecules represented by the AM1 Hamiltonian embedded in periodic boxes of solvent molecules represented by the OPLS force field. Thus, HQM in Eq. (13.1) is simply the AM1 energy for the solute, and HMM is evaluated for all solvent– solvent interactions using the OPLS force field. The QM/MM interaction energy is computed in a fashion closely resembling the standard approach for MM non-bonded interactions

j

where the Lennard–Jones parameters are determined from the usual combining rules (Eqs. (2.30) and (2.31)) assuming the solute atoms have ε and σ values characteristic for their atomic type in the OPLS force field. The single feature that is quantum mechanical is that the solute charges are determined from the CM1 charge model applied to the AM1 wave function (see Section 9.1.3.4). For charged molecules, the constant α is 1.0, while for neutral molecules, it is 1.2 to approximate the effect of solvent polarization on the gas-phase charge distribution.

The choice of AM1 as a particularly efficient level of electronic-structure theory is motivated by the large number of QM calculations potentially required in the MC sampling. In the standard AOC MC protocol, moves of solute internal coordinates are attempted every 50 MC steps, and accepted or rejected according to the standard Metropolis protocol as described in Section 3.4.2. If the move is accepted, the QM energy and CM1 charges are updated and used in Eq. (13.2) until the next accepted change of solute geometry. Note that QM calculations are not required unless a solute move is being attempted.

The AOC method successfully predicts the effects of polar solvents on rotameric equilibria for 1,2-dichloroethane and 2-furfural, as illustrated in Table 13.1. However, it is not very

GoS (gauche) − GoS(trans)

a Kaminski and Jorgensen (1998). b Chambers et al. (1999).

c Wiberg et al. (1995).

d Depaepe and Ryckaert (1995). e Abraham and Siverns (1972).

successful at predicting solvation effects on these equilibria in non-polar solvents, since the OPLS solvent molecules are not electronically polarizable. Such effects are included in continuum solvation models like SM5.4/AM1, being implicit in solvent dielectric constants on the order of 2, and data from that model may be compared for the 1,2-dichloroethane equilibrium in Table 13.1.

The AOC model has also proven efficient for modeling solvation free energy differences along a reaction coordinate generated from gas-phase calculations (as previously described in Section 12.5.1). Chandrasekhar, Shariffskul, and Jorgensen (2002) used this technique to obtain excellent agreement with experiment in predicting the aqueous acceleration of Diels– Alder cycloadditions of cyclopentadiene, supporting the conclusion from prior purely MM simulations that enhanced hydrogen bonding to hydrophilic functionality in the TS structures is responsible for the acceleration.

With respect to further developments of the AOC protocol, Udier-Blagovic et al. (2004) recently assessed the relative utility of scaled CM1 and CM3 charges from AM1 and PM3 calculations for use in computation of absolute solvation free energies via AOC. On an

initial test set of 13 organic molecules, they found neither charge model based on PM3 to provide acceptable accuracy. However, scaling CM1 and CM3 charges derived from AM1 by factors of 1.14 and 1.15, respectively, gave average errors of only 1.0 and 1.1 kcal mol−1, respectively, over a diverse test set of 25 organic molecules.

13.2.2Polarized QM/Unpolarized MM

The next level of complexity involves accounting for environmentally induced relaxation of the QM wave function explicitly (as compared to, say, the implicit scaling factor α in Eq. (13.2)). The coupling Hamiltonian HQM/MM remains similar in spirit to that described by Eq. (13.2), in the sense that the interaction must be represented as a sum of electrostatic and other non-bonded interactions, but the next step is to determine the Fock operator (or analogous DFT operator) that is used to obtain orbitals minimizing the complete Hamiltonian. This is quite straightforward in practice given that we may write the coupling term as

Thus, the electrostatic interaction term of Eq. (13.2) has been separated into an operator acting on the QM electrons (the first term on the r.h.s. of Eq. (13.3)) and the classical term for the interaction of the MM atoms with the solute nuclei. The Lennard–Jones term is the same in Eqs. (13.2) and (13.3) (although the parameters may certainly be different from one model to another).

The next step is to find orbitals that minimize the expectation value of Hcomplete in Eq. (13.1), given Eq. (13.3) for HQM/MM. If we take as our wave function a standard normalized Slater determinant, we have

where i and j run over N QM electrons, k and l run over the K nuclei in the QM fragment, and m runs over the M molecular mechanics atoms. The second equality in Eq. (13.4) simply expands the QM Hamiltonian into its usual individual terms and uses Eq. (13.3) to expand the QM/MM component of the Hamiltonian. The terms having no dependence on the electronic coordinates – HMM, the QM-nuclei/MM-atom electrostatic interactions, and the LJ interactions – may be taken outside of expectation value integrals, which are then simply one by normalization of the wave function. The third equality simply collects terms together in a convenient fashion.

Note that the only operator acting on the electronic wave function for the QM/MM system that would not be present in the isolated QM system is that involving the charges of the MM atoms. In operator formalism, these atoms behave exactly like QM nuclei, except that they bear partial atomic charges instead of atomic-number-based charges. As such, they enter into the standard Fock operator just as nuclear charges do, i.e., as part of the one-electron operator. Elements of the QM/MM Fock matrix that minimize the energy computed from Eq. (13.4) are thus calculated from the generalization of Eq. (4.54) as

where only the third term on the r.h.s. is different from the usual QM expression. The third term involves the computation of M one-electron integrals. Insofar as the bottlenecks in HF theory tend to be assembly of the two-electron integrals or diagonalization of the Fock matrix, the actual increase in computational time required for a QM/MM calculation compared to a purely QM calculation on the same fragment can be quite small.

DFT equations analogous to Eqs. (13.4) and (13.5) can be derived in a similarly straightforward way. Again, the ultimate influence of the MM system on the KS orbitals is made manifest only by the appearance of additional one-electron integrals associated with the MM atoms in the KS operator.

Of course, the simplicity of the QM/MM operator does not imply that it has only a small effect. Large atomic partial charges placed near the QM fragment would be expected to polarize the system strongly. Table 13.2 compares the dipole moments of the standard nucleic acid bases at the AM1 level evaluated in the gas phase and in a QM/MM calculation carried out modeling aqueous solvation with a periodic box of TIP3P water molecules. For comparison, results from the AM1-SM2 aqueous continuum solvation model are also provided.

It is important to recognize how a QM/MM calculation like that for the nucleic acid base solvated dipole moments is accomplished. We outline here a typical series of steps

1.Choose the particular QM and MM levels to be used.

2.Given those QM and MM levels, select a set of LJ parameters for the QM fragment. One option is to use the same parameters for atoms in the QM fragment as those that would

Table 13.2 Computed dipole moments (D) of the nucleic acid bases in the gas phase and in aqueous solution

a Gao 1994.

b Cramer and Truhlar 1992, 1993.

be applied to those atoms were they to be in the MM region (like the AOC model). Another option is to develop separate transferable LJ parameters to be used for the QM fragment whenever the particular QM/MM choice has been made (see, for example, Martin et al. 2002). The data in Table 13.2 were determined using such a procedure, with the parameters thus being part of the definition of the AM1/TIP3P model (Gao and Xia 1992).

3.Where necessary, determine system properties as ensemble expectation values (for the data in Table 13.2, a MC sampling scheme was employed, but MD methods are equally applicable). Every time the coordinates of any atom in the system change, i.e., at each time step in a MD trajectory or following an accepted MC move of either a QM or MM atom, recompute the QM wave function (since at least one term in the operator involving the relative positions of the QM electrons and the MM atoms must be different). Note the contrast with the AOC method, where only internal moves in the QM system’s coordinates necessitate a recomputation of the wave function.

4.At each simulation step, the property or properties of interest are included in the ensemble average. For Table 13.2, the property is the evaluation of the dipole moment operator as an electronic expectation value over the QM subsystem. Thus, the QM/MM result for this case is an MC ensemble expectation value of a quantum mechanical operator expectation value.

A different application of the AM1/TIP3P model nicely illustrates the ability of QM/MM models to permit the analysis of quantities not typically simultaneously available to either pure QM or MM models. Gao (1994) employed the AM1/TIP3P model to determine the PMF for the Claisen rearrangement in water, a reaction already discussed in some detail in the context of pure continuum or explicit solvation models in Section 12.5.1 (see also Section 11.1.2). Similarly to the pure MM simulation, the computational protocol involved FEP along the gas-phase reaction coordinate using λ to drive the structure of the initial allyl vinyl ether through the TS to the unsaturated aldehyde product. At the AM1/TIP3P level, the same increase in hydrogen bonding to the ether oxygen noted in the pure MM study was observed. In the QM/MM model, however, the effect of this increased hydrogen bonding on