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

mutation. In principle, such an equilibrium can be obtained with sufficiently small changes in λ at each step, but in practice, numerical limitations in computing energy differences as a function of λ place a lower limit on that increment (as, of course, does the requirement that the total number of steps required not exceed available computational resources). Steps that are too large, on the other hand, lead to chaos, since high-energy interactions run the simulation into unrepresentative regions of phase space from which the system is unlikely ever to return.

In formalism, many aspects of free-energy simulations lend themselves more to implementation within a Monte Carlo sampling scheme than within a molecular dynamics scheme. Unfortunately, MC schemes applied to large flexible molecules (e.g., proteins) tend to be very inefficient, since most proposed moves of the large molecule are rejected as being too energetically unreasonable, so MD simulations remain the standard. Innovative attempts to combine some of the best features of both have been described, as already noted in Chapter 3.

Possibly the most vexing aspect of free energy calculations is that, as with most simulations of sufficiently complex systems, meaningful error analysis is almost impossible. The difficulty of demonstrating a properly converged sampling of phase space for any simulation has already been amply discussed in Section 3.6.4. Given that a free-energy simulation typically comprises some 10 or more individual simulations, whose errors may be expected to be highly correlated and the results from which are pieced together, uncertainty only grows. The cost of the simulations is such that they are rarely carried out more than once (using, for example, different starting conditions) to assess the statistical reliability of the free-energy change.

As a result, many free-energy simulations are carried out not necessarily to predict a specific value but rather to demonstrate agreement with experiment, after which interpretation of the simulation results can be carried out with enhanced confidence to understand why the free-energy change is what it is. This process in itself can be quite ambiguous, however. A typical analysis involves decomposing the free-energy change into constituent components, i.e., changes in electrostatic interactions, van der Waals interactions, bond torsion contributions, etc. However, while the total free-energy change is path independent, the changes in the components are not, so such a decomposition must be interpreted with caution.

12.3 Other Thermodynamic Properties

Properties other than free-energy changes are usually considerably more difficult to evaluate to an equivalent level of accuracy. One approach is simply to attempt a brute force calculation for different systems analogous to that outlined for U in Eqs. (12.9) and (12.10). However, this approach has little value in any but the simplest of systems owing to the large uncertainties in the absolute values of the thermodynamic quantities.

Another approach is to carry out free-energy simulations at several different temperatures, and then construct the equivalent of a van’t Hoff plot to separate, say, the enthalpic and entropic contributions to the free energy. This approach is obviously extraordinarily demanding of resources, since every temperature point requires a new free-energy simulation, and unless there are many points, the error in the temperature dependence of the free energy determined by linear regression of the latter on the former may be rather large.

In some cases, it is possible to take advantage of various thermodynamic relationships to write some property as a fluctuation-dependent quantity. Thus, for example, the entropy change may be computed from

evaluating the integral numerically, as is done for standard TI. Peter et al. (2004) have carried out a more detailed analysis of the accuracy and convergence of various approaches for computing entropy.

Similarly, the absolute constant-volume heat capacity may be computed as

As a rule, fluctuations are much slower to converge statistically than are the properties that are fluctuating, so analyses of Eqs. (12.28) and (12.29) require very long simulation times. Of course, Eq. (12.29) is simpler to evaluate than Eq. (12.28) since the latter does not involve a perturbation of one system into another.

12.4 Solvent Models

If a solvent is to be considered as ‘explicitly’ present in a simulation, obviously there must be some atomistic manner in which it is represented in the energy expression – this being the fundamental distinction from a continuum solvation model. However, since the solvent molecules greatly outnumber the solute molecule(s), there are advantages of efficiency that accrue from adopting as simple a representation as possible, and that is reflected in many of the solvent models in common use.

12.4.1Classical Models

The simplest model for a solvent molecule is clearly one that is molecular-mechanics-like. That being said, various levels of complexity remain even within the choice of a classical representation. Of all possible solvents of interest to chemists, water is arguably the most important, and not surprisingly it has spawned the largest number of models. Besides differing in parameter values, the various classical models differ in the total number of interacting sites. Probably the simplest possible model for water is to treat it as a Lennard – Jones sphere, inside which two charges are embedded of equal magnitude and opposite sign to mimic water’s dipole moment. A solute molecule thus sees three interaction sites: the center of the sphere characterized by characteristic ε and σ values (see Eqs. (2.16), (2.30), and (2.31)), and the two point charges. (An alternative model would be to put a point dipole at the center of the sphere, but the evaluation of dipole – dipole interactions is sufficiently more time-consuming than that of charge – charge interactions that there is no real simplification inherent in this approach.)

A slightly more complex representation is to put equal positive atomic charges on the hydrogen atoms and a negative charge on the symmetry axis, or equal negative charges in the lone pair regions, again to mimic water’s dipole moment, but also to better represent its overall charge distribution. Such very simple models, with careful parameterization, do remarkably well in reproducing many properties of liquid water, e.g., bulk density, heat of vaporization, compressibility, heat capacity, etc. The most successful models along these lines, that are widely used in modern simulations, are the transferable-intermolecular- potentials-3- and -4-point-charge water models (TIP3P and TIP4P; Jorgensen et al. 1983). The similarly designed SPC (simple point charge) water model also continues to see modern use (Berendsen et al. 1981) including forms recently modified to improve its dielectric and diffusive properties (Glattli, Daura, and van Gunsteren 2003).

For non-aqueous solvents, the approximation of the solvent as a LJ sphere is usually less practical. However, substantial time savings can be realized by employing a united-atom approach for carbon atoms and their attached hydrogens. The most complete parameterization of organic solvents has been accomplished as part of the OPLS force field, including inter alia alkanes, aromatics, carbon tetrachloride, chloroform, furan, n-octanol, and pyrrole, many in both UA and AA representations (see Table 2.1 and also Jorgensen, Briggs, and Contreras 1990; Kaminski et al. 1994; McDonald and Jorgensen 1998). In these cases, as for water, solvent parameters were optimized based on comparison of bulk solvent properties to experimental measurements.

At the next level of complexity, the polarity of solvent models, as made manifest by their atomic partial charges, can be augmented with a polarizability. This allows the solvent molecule to respond to its surroundings in a fashion conceptually similar to the electronic component of the solvent polarization described in Section 11.1.1. Typically a polarizability tensor α is assigned either to the solvent molecule as a whole or to individual atoms. Then, the induced dipole moment at each polarizable position can be determined from

where E is the total electric field arising from all of the atomic point charges and all of the induced dipoles. Thus, µind must be determined iteratively, with convergence potentially being problematic. Once converged, the additional contribution to the total electrostatic energy from the charge-induced dipole interactions can be computed according to

where i runs over charge sites and j runs over polarizability sites and r is the intersite distance. In addition, induced-dipole –induced-dipole interactions contribute according to Eq. (2.23).

Owing to its particular importance, polarizable solvent models have largely been restricted to water, for which a sizable number have been developed (see, for example, Dang 1992; Rick, Stuart, and Berne 1994; Bernardo et al. 1994; Zhu and Wong 1994; Lefohn, Ovchinnikov, and Voth 2001). Because evaluating the terms deriving from solvent polarizability

increases the amount of time required for a simulation by roughly an order of magnitude, the use of polarizable solvents has been primarily either for technical comparisons in model development, or for the simulation of particularly simple systems, where convergence for a given property of interest may be expected to occur quickly. Developers tend to focus on properties for which non-polarizable water models do poorly, e.g., the density anomaly in water where below 4 ◦C the liquid density begins to decrease with decreasing temperature. However, the failures of prior models to function well for such properties is not necessarily intrinsic, but may simply reflect a failure to have considered the property in the development of the non-polarizable model (Mahoney and Jorgensen 2000). More recent work with polarizable acetonitrile and acetone solvent models has indicated, not surprisingly, that polarizability critically improves the description of solvation structures and interaction energies associated with the solvation of monatomic ions in these solvents (Fischer et al. 2002).

A yet more complete but still formally classical solvent model has been developed for use when the solute is represented quantum mechanically. The electrostatic interactions between a classical solvent and a quantum mechanical solute are relatively simple to represent, and are discussed in detail in the next chapter on mixed QM/MM methods. The non-bonded interactions are somewhat more challenging. Gordon et al. (2001) have described an approach that they call the effective fragment potential (EFP) method that, by analogy to ECPs, replaces the direct computation of dispersion and exchange-repulsion interactions between solute and solvent electrons by an interaction between solute electrons and a solvent pseudopotential. The solvent pseudopotentials (and the representation of its electrostatic distribution and polarizability) are determined parametrically in order to create a transferable solvent model especially suitable for use in QM/MM calculations using HF theory as the QM component. The EFP model has since been extended to DFT as the QM component as well (Adamovic, Freitag, and Gordon 2003).

12.4.2Quantal Models

When one refers to a quantum mechanical solvent model, the word ‘model’ reverts to its usual sense in the context of QM methods: it is the level of electronic structure theory used to describe the solvent. Thus, there is no real distinction between the solvent and the solute in terms of computational technology – the wave function for the complete supersystem (or the DFT equivalent) is computed without resort to methodological approximations beyond those inherent to the level of electronic structure theory. To avoid problems with basis-set imbalances, one might expect calculations representing the solvent in a fully QM fashion to employ a common level of theory for all particles, but this does not have to be the case.

At several points in this book, it has been emphasized that the prevalence of classical MC and MD simulations derives from the impracticality of carrying out fully QM dynamics. While this is largely true, for systems of only modest size where short trajectories may be profitably analyzed, fully QM MD simulations using the so-called Car – Parrinello technique are a viable option (Car and Parrinello 1985). In its most widely used formulation, the Car –Parrinello method employs DFT as the electronic-structure method of choice. In

principle, every MD step should involve taking a phase point, computing the energy and the gradients for that point given the nuclear positions, and propagating a short time step prior to repeating this process. This formalism is extremely time-consuming. Car and Parrinello showed, however, that one does not need to fully converge the KS wave function at every step. Instead, the KS MO coefficients are treated as dynamical variables. That is, they are assigned a fictitious mass and have their ‘coordinates’ added to the usual 3N positional dimensions of phase space. By careful choice of the masses for the electronic degrees of freedom, and the time steps for the electronic and nuclear movements, it is possible to obtain a relevant MD sampling of phase space in favorable systems. To further increase speed, the method usually uses a plane-wave basis set, which is ideally suited to the periodic boundary conditions usually imposed in a condensed-phase simulation and allows fast Fourier transform methods to facilitate solution of the SCF equations.

The obvious advantage of a fully QM solvent representation is that intimate solvent participation in reactions, say as a proton donor or acceptor, or simply a charge-transfer partner with the solute, is handled entirely naturally. With improved DFT functionals and everincreasing computer speeds, this method holds great promise for the future, although it is still sufficiently time-consuming that present day applications remain somewhat limited.

12.5 Relative Merits of Explicit and Implicit Solvent Models

The fundamental difference between the explicit and implicit solvent models is not that one has solvent and the other does not. Rather, the difference is that the implicit model employs a homogeneous medium to represent the solvent where the explicit model uses atomistically represented molecules. While the latter choice is clearly the more physically realistic, the practical limitations imposed by explicit representation dictate that it is not necessarily the best choice for a given problem of interest. This section compares and contrasts the relative strengths and weaknesses of the two models, including some illustrative applications.

12.5.1Analysis of Solvation Shell Structure and Energetics

A reaction that has received a substantial amount of study using a variety of alternative solvent (and solute) models is the Claisen rearrangement, a [3,3] sigmatropic shift that converts an allyl vinyl ether into a γ ,δ-unsaturated aldehyde (Figure 11.5). The motivation for its study has been two-fold. First, the conversion of chorismate to prephenate, which is the first committed step in the biosynthesis of aromatic amino acids in plants, involves an enzyme-catalyzed Claisen rearrangement. Secondly, although pericyclic reactions are conventionally thought of as being relatively insensitive to solvent effects, the rate acceleration for rearrangement of the parent allyl vinyl ether comparing the gas phase to aqueous solution has been estimated to be on the order of 1000-fold. How does aqueous solvation effect this large rate acceleration?

Storer et al. (1994) employed a SMx GB continuum solvent model to investigate this question. Because of the efficiency of the continuum model, they were able to examine various levels of electronic-structure theory in assessing the influence of aqueous solvation

on the reaction coordinate. Their key findings were that (i) the TS structure was significantly better solvated than the reactant, primarily because of increased polarization free energy associated with increased polarity in the rearranging fragments and (ii) the solvation effect favored a change in the TS structure so that the rearranging fragments were separated by a larger distance, thus enhancing its polarity (Figure 11.5). These effects, combined with a very small hydrophobic acceleration associated with the two hydrocarbon termini coming together in the forming C – C bond, were sufficient to account for the full range of aqueous acceleration inferred experimentally.

By way of contrast, Severance and Jorgensen (1992) addressed the same problem using an explicit solvent model. In particular, they first generated the intrinsic reaction coordinate (a concept explained more fully in Chapter 15) for the Claisen rearrangement in the gas phase. They then selected specific structures along the reaction coordinate to serve as intermediates in a free-energy simulation using FEP methods. The rigid solute structures were treated classically using OPLS non-bonded parameters and ESP charges determined from gas-phase HF/6-31G(d) calculations, and the aqueous solvent was modeled with the TIP4P water model. By analyzing free energy changes as λ perturbed from one structure to the next along the reaction coordinate using MC simulations, they determined a rate acceleration in good agreement with that inferred experimentally. An investigation of the factors causing acceleration included an analysis of the radial distribution functions of water about the solute oxygen atom. The simulations indicated that the average number of hydrogen bonds to the reactant’s ether-like oxygen atom was slightly in excess of one, while the number to the TS oxygen atom was closer to two. Moreover, the strengths of the solute –water interactions for hydrogen bonded waters were greater in the TS structure than in the reactant.

Thus, both studies came to similar conclusions with respect to the source of acceleration: greater polarity of the TS structure contributing to stronger aqueous solvation. However, the ‘language’ of the continuum model restricts the expression of that result to broader electrostatic terms while the explicit nature of the simulation permits a more fine-grained analysis that illustrates how improved hydrogen bonding is a part of the electrostatic component. In terms of describing the reaction path, the explicit model restricted itself to an analysis of the solvation of the gas-phase reaction coordinate. The continuum model, on the other hand, considered movement off the reaction coordinate and found that to be important in stabilizing the TS structure. Furthermore, the SCRF nature of the continuum model allowed for an analysis of the relative polarizability of the TS compared to the reactant, and the TS was found to be considerably more polarizable, again contributing to its improved solvation. (The use of HF/6-31G(d) ESP charges in the explicit simulation was motivated in part because the known systematic tendency for these charges to be too large in the gas phase may be taken as a compensating error for not including the effect of solvation on the electronic structure.)

One key issue, then, in deciding upon what type of solvation model to employ is the level of detail in solvent structure that is of interest to the researcher. An important point to make in this regard is that some solvent molecules really should not be thought of as solvent per se. For instance, various inhibitors of HIV-1 protease are known to bind strongly to the enzyme’s active site only because there is an accompanying water molecule also bound in the site. Such a water cannot be considered simply to be a bulk region having a high dielectric

constant; rather, it is more a component of a supermolecular solute. Such solvent molecules are something of a technical challenge for both kinds of solvent models. A continuum model may function perfectly well if the special solvent molecule(s) are included explicitly, but one needs to know ahead of time to include them. Similarly, in an explicit simulation, if they are not placed in the appropriate location, the timescale for entry from the bulk solution may be such that no solvent molecule ever occupies the correct position throughout the length of the simulation, in which case the results simply represent an unbalanced sampling of phase space and are not useful.

Interestingly, Morreale et al. (2003) have shown that when solvation free energies are decomposed into atom/group-specific contributions, there is in general good agreement between results obtained from continuum and explicit-solvent calculations. This observation suggests that future analyses of such fragment solvation free energies may assist in comparison of results from disparate solvation protocols.

12.5.2Speed/Efficiency

For equivalent levels of theory used to represent the solute, continuum solvation models are inevitably several orders of magnitude faster for the analysis of solvation free energies than are explicit solvent models. As a rule, at the QM level the SCRF portion of a continuum solvation calculation usually adds no more than 15 percent or so to the total time of an SCF calculation; at the MM level a factor of 2 or so is involved. Obviously, however, there is no particular virtue in the speed with which a wrong answer (or one that fails to answer the question of interest) may be obtained. Thus, in those instances where an explicit model is called for, time must be invested in the calculation. The variety of system sizes that may be envisioned precludes generalizations about time requirements, but the current state of the art for simulations of solvated biomolecules having molecular weights in the range 10 to 100 kDa is typically a few nanoseconds for roughly one cpu-week of time on a modern processor. Advances in algorithms and hardware speeds are constantly improving on this.

12.5.3Non-equilibrium Solvation

In Section 11.4.6, the limitations of continuum models in their ability to treat non-equilibrium solvation, at least in their simplest incarnations, were noted and discussed. In principle, explicit solvent models might be expected to be more appropriate for the study of chemical processes characterized by non-equilibrium solvation. In practice, however, the situation is not much better for the explicit models than for the implicit.

Consider a typical event that would be expected to exhibit non-equilibrium solvation, e.g., a chemical reaction with significant rearrangement of charge density in the region of the transition-state structure. The short lifetime of the TS and the corresponding ‘sudden’ change in charge distribution would be expected to limit the solvent’s ability to solvate the reaction coordinate in a fully equilibrated fashion. In the abstract, it might seem that a molecular dynamics trajectory of the reaction would offer a useful tool for studying the nonequilibrium effects, since the time course of the reaction is realistically mimicked to within

the accuracy of the employed force field. Unfortunately, the likelihood of any trajectory spontaneously following a reactive path is unacceptably small unless the barrier is quite low (and, if there is enough charge reorganization to give rise to significant non-equilibrium solvation effects, a low barrier is not expected).

In the absence of being able to observe spontaneous reactive events, one can attempt to take advantage of sampling methods like those outlined in Section 12.2.5 to force trajectories along the reaction coordinate. However, by changing the PES through addition of a biasing potential, or sampling over an extended period at constrained reaction coordinate values, one changes the length of time the trajectory spends in given regions of phase space, and the solvation is likely to become more equilibrium-like in character.

With Monte Carlo methods, the adoption of the Metropolis sampling scheme intrinsically assumes equilibrium Boltzmann statistics, so special modifications are required to extend MC methods to non-equilibrium solvation as well. Fortunately, for a wide variety of processes, ignoring non-equilibrium solvation effects seems to introduce errors no larger than those already inherent from other approximations in the model, and thus both implicit and explicit models remain useful tools for studying chemical reactivity.

12.5.4Mixed Explicit/Implicit Models

Having identified the strongest points of the explicit and implicit solvent models, it seems an obvious step to try to combine them in a way that takes advantage of the strengths of each. For instance, to the extent first-solvation-shell effects are qualitatively different from those deriving from the bulk, one might choose to include the first solvation shell explicitly and model the remainder of the system with a continuum (see, for instance, Chalmet, Rinaldi, and Ruiz-Lopez,´ 2001).

There are certain instances where this approach may be regarded as an attractive option. For example, Cossi and Crescenzi (2003) found that accurate computation of 17O NMR chemical shifts for alcohols, ethers, and carbonyls in aqueous solution required at least one explicit solvent shell, but that beyond that shell a continuum could be used to replace what would otherwise be a need for a much larger cluster. However, just as the strengths of the two models are combined, so are the weaknesses. A typical first shell of solvent for a small molecule may be expected to be composed of a dozen or so solvent molecules. The resulting supermolecular cluster will inevitably be characterized by a large number of accessible structures that are local minima on the cluster PES, so that statistical sampling will have to be undertaken to obtain a proper equilibrium distribution. Thus, QM methods require a substantial investment of computational resources. In addition, certain technical points require attention, e.g., how does one keep the first solvent shell from ‘exchanging’ with the continuum since both, in principle, foster identical solvation interactions?

So, while there is growing interest in hybrid models of all sorts (as discussed in more detail in the next chapter), the choice of a mixed solvent model is not necessarily intrinsically better than a pure explicit or pure implicit model. In general, unless there is a strong suspicion that first-solvation-shell effects are drastically different from those more typically encountered, there is no particularly compelling reason to pursue a mixed modeling strategy. An example

of such a situation might be the aqueous coordination sphere surrounding a highly charged metal cation. In that case, the electrostriction of the first shell makes the water molecules more ligand-like than solvent-like, and their explicit inclusion in the solute complex is entirely warranted.

12.6 Case Study: Binding of Biotin Analogs to Avidin

Synopsis of Kuhn and Kollman (2000) ‘A Ligand That Is Predicted to Bind Better to Avidin than Biotin: Insights from Computational Fluorine Scanning’.

One of the strongest known interactions between a biopolymer and a small-molecule substrate is that between the protein avidin and D-(+)-biotin, the structure of which is shown in Figure 12.6. The binding energy for this complex has been measured to be −20.8 kcal mol−1. While this represents an extraordinarily strong interaction, Kuhn and Kollman suggested that it might be possible to make it still stronger by replacing one or more hydrogen atoms on the biotin framework with fluorine atoms. Fluorine is roughly isosteric with hydrogen (i.e., the C – F and C – H bond lengths have roughly similar lengths and F and H have similar covalent radii), but is considerably more hydrophobic. Thus, if a region of the binding pocket interacts with biotin via non-polar interactions, and is adequately shaped to accommodate the very slightly larger fluorine atom, decreased aqueous solvation of the fluorinated analog would be expected to increase the binding free energy (note that the lower polarizability of fluorine compared to alkyl hydrogen also suggests the favorable dispersion interactions between the biotin analog and the protein will be reduced, but this is generally a smaller effect than enhanced hydrophobicity in the absence of steric constraints).

Kuhn and Kollman pursue several different algorithmic approaches to estimating the binding free energies of different fluorobiotins. The fastest approach, which they refer to as fluorine scanning, involves a combination of explicit and implicit solvation models to compute the horizontal legs of the free-energy cycle in Figure 12.6. First, an MD trajectory of the avidin – biotin complex is obtained under standard MD conditions, including explicit solvent and using periodic boundary conditions.

The trajectory is then ‘post-processed’ to determine absolute free energies in solution for biotin, avidin, and the avidin – biotin complex. This process begins by stripping the water from the trajectory, and then computing absolute free energy as

where EMM is the force-field energy, Gsolv is computed from a continuum solvation model (in this case a finite difference Poisson – Boltzmann (FDPB) model with hydrophobic atomic surface tensions), and the expectation value is taken over the snapshots of the MD trajectory. Evaluations of Eq. (12.32) for isolated biotin and avidin are carried out using the same snapshots as those for the complex, i.e., using those geometries found in the complex, but only the atoms of the individual component are retained. The solute entropies S are determined from the usual statistical mechanical formulae (Section 10.3) with the vibrational frequencies being determined from normal mode analysis of each solute optimized separately using a distance-dependent dielectric constant to mimic the effects of solvation.

S

Figure 12.6 Free-energy cycle associated with the binding of biotin and fluorobiotin analogs to avidin. What issues arise in choosing a force field for explicit simulation of these systems? What methods are better suited to computing the vertical legs of the cycle and what methods the horizontal ones?

Note that since the free energies of the isolated components are computed using the same geometries as are employed in computing the free energy of the complex, the internal force-field energies cancel in computing a free energy of binding as

Only the force-field energy term associated with interactions between the biotin and avidin fragments remains. This is added to the differential solvation free energies and differential thermal terms to determine the full binding free energy.

To avoid the cost of multiple MD simulations, Eqs. (12.32) and (12.33) for fluorosubstituted biotin analogs are also evaluated using geometries from the original MD trajectory. The relevant hydrogen is simply replaced by a fluorine atom having the appropriate bond length oriented along the original C – H bond axis. Thus, there is no relaxation of the complex to relieve steric clashes if they are introduced. Again, the only forcefield energy terms that survive in computing the free energy of binding are the interaction energies between the biotin analog and the avidin. It is further assumed that the entropy change computed for complexation with biotin remains the same for a fluorinated biotin.

The results of this rapid fluorine scanning are that substitution at positions pro-R 6 and 9 and pro-S 7, 8, and 9 are all predicted to decrease binding by more than 4 kcal mol−1, substitution at position pro-S 6 is predicted to decrease binding by about 2 kcal mol−1, substitution at position pro-R 7 is predicted to have only a small unfavorable effect, and substitution at position pro-R 8 is predicted to increase binding by a little less than 1 kcal mol−1. The absolute free energy of binding for biotin itself with this method is computed to be −18.8 kcal mol−1, although the fluctuations in the ensemble average and