ACCURACY AND APPLICABILITY OF METHODS - Accuracy and Applicability of Quantum Chemical Methods

Accuracy and Applicability of Quantum Chemical Methods in Computational

3. ACCURACY AND APPLICABILITY OF METHODS

So much for the theories. We now turn to their practicality and usefulness as regards computational medicinal chemistry. The current state of the ﬁeld can be thought of as a sliding scale of accuracy for each quantum chemical method, along with a con-comitant list of applicable system sizes due to limited computational power. (Con-sidering the vast dissimilarity among the techniques, it is not at all surprising that computational quantum chemists ﬁnd themselves continually arguing over what constitutes an‘‘accurate’’ calculation on a ‘‘large molecule.’’) In fact, ‘‘scaling’’ is an altogether appropriate word, for in computational science it denotes how quickly the time required for a calculation increases with an increase in size. Scalings for ab initio methods vary widely, so we will consider each in turn.

3.1. Ab Initio

The fastest scaling for an ab initio method is Hartree–Fock theory utilizing the self-consistent-ﬁeld procedure (HF/SCF), which for a given basis set scales as the number of electrons N⁴. In other words, double the size of the calculation and it will take

around 16 times as long. This scaling measures only the generation of the energy of a single point on the potential energy surface, yet geometry optimizations require multiple single points or analytic derivative computations to determine a minimum.

Thus, as a practical matter, a method can be used as a primary, day-to-day tool only when molecular geometries can be computed (although when necessary to determine energy barriers, a few higher-level energy points at the lower-level geometries may provide a decent substitute). Today’s computers make full geometry optimizations or energy points with HF/SCF realistic for systems of 150 or 500 heavy atoms, respectively. For organic molecules (and most drugs and receptor sites are in that category), such a calculation will undoubtedly provide structures within 0.1 A˚ of bond lengths and 5j of bond angles and relative energies often within 5–10 kcal/mol of experiment. Perturbation theory’s workhorse, MP2, scales as N⁵. So in 2002, it is constructive to use MP2 with full geometry optimization only on systems up to 50 heavy atoms, and energy points are useful up to 200 heavy atoms. The additional cost of electron correlation generally provides better results (0.05 A˚, 2j, 3 kcal/mol) and is especially helpful when determining the relative energetics of reactants, products, and transition states. When the desired accuracy involves distinguishing systems separated by only 1 kcal/mol, coupled cluster theory is the only practical solution. Its practicality is tempered by its N⁷scaling, however, and as such it is really only useful at present for systems of 20 heavy atoms or less (or up to 70 as an energy point). CCSD(T) can usually provide geometries to 0.01 A˚ and 0.5j with relative energies within 0.2 kcal/

mol of experiment [3,23].

Is there a place for the very highest-level ab initio techniques in computational medicinal chemistry? Sadly, not at this time. To acquire all of the electron correlation, theorists use the FCI method described above, but only for diatomic and triatomic systems that can justify the prohibitive N! cost. The explicit Hylleraas methods are so expensive that their full implementation may never be useful for systems with more than a few electrons, although the related R12 techniques (especially MP2-R12) might be [24]. It is worth mentioning that in certain situations involving excited states (i.e., electronic spectroscopy) or metal-containing systems, none of the practical ab initio techniques can be exhaustively accurate, as they are all based on a single-reference description of the wavefunction that fails when excited states are close in energy to the ground state. As traditional multiconfiguration SCF and CI are out of the question for large systems, the best choice is probably a linear response method [25] such as Configuration Interaction Singles (CIS) coupled with whatever reference is affordable.

Overall, perhaps the best argument for the use of ab initio quantum chemistry is its versatility. Ab initio methods are unmatched in the area of molecular property computation. Energy, being a zeroth-order property, is easily acquired. A myriad of other properties can be understood using analytic derivative theory; routines exist for infrared, Raman, NMR chemical shifts, circular dichroism, magnetic susceptibility, dipole moments, and spin-orbit coupling, among others [26]. Properties that relate to the essential thermodynamics or kinetics of a chemical reaction can be computed with mature, robust techniques such as Variational Transition State Theory (VTST) [27]

and emerging dynamical methods such as the Reaction Path Hamiltonian [28]. Even bulk eﬀects such as solvation can be treated, either explicitly through the addition of solvent molecules to the calculation, or in an averaged fashion utilizing the Polarizable Continuum Model (PCM) [29] or the Self-Consistent Reaction Field (SCRF) [30,31].

In the case of truly exotic physical phenomena such as the Mo¨ssbauer eﬀect, ab initio

methods are the only choice, and one must simply build a model compound to eﬀectively estimate the property on the largest systems of interest. Today, the use of ab initio quantum chemical methods in computational medicinal chemistry is wide-spread, albeit not so widespread as it could be if computers were faster. The consensus view is that such techniques are extremely valuable for smaller systems, but the ad-ditional accuracy is not worth the computational cost for larger systems. Hartree–

Fock theory, however, seems fast enough to warrant general use for many studies.

Indeed, HF/3-21G (Hartree–Fock with the split-valence Gaussian basis set 3-21G) has become a standard of sorts when a relatively large amount of accurate conformational information is desired, as in a recent conformational analysis of the glycoprotein model compounds N-formyl-L-asparaginamide and N-acetyl-L-asparagine N-methyl-amide [32]. Structural studies of the conventional variety can bear fruit also, especially if the chemical problem involves possible adverse reactions in DNA [energies determined using an impressive MP2/6-311G(2d,p) treatment to be positively sure]

[33]. In the end, when chemical intuition fails us, ab initio quantum chemistry is the time-tested method of last resort: CCSD(T) on acetone easily explains the dipole moment Stark eﬀect shifts in the photosynthetic reaction center of various Rhodo-bacter sphaeroidesmutants [34].

Berg and co-workers [35] plainly have a great deal of conﬁdence in ab initio methods for the advancement of medicine. They point out that it took only a few decades before it was possible to perform a full geometry optimization on a 126-atom, 372 degree-of-freedom chain of 12 alanines (see Fig. 1) [36], and they feel computa-tional power will continue to increase. In suggesting an ambitious computacomputa-tional eﬀort toward understanding peptide folding, they note that ab initio results, while expensive to obtain, will likely provide enough accuracy that they will not need to be recalculated for a long time. By contrast, they believe that the tremendous undertaking of computing millions of conformational parameters for all possible tripeptides requires rigor that semiempirical calculations cannot provide. Their initiative suggests that protein folding can be tackled three peptides at a time using methods param-eterized from their computations. The platform they have chosen: HF/3-21G for

Figure 1 The helical alanine 12-mer, a recent landmark for full ab initio geometry opti-mizations of biomolecules. See Ref. 36. (See color plate at end of chapter.)

geometry optimizations and the DFT functional B3LYP/6-31G* for energies. Why DFT and not MP2 or CCSD(T)? Let us consider it.

3.2. Density Functional Theory

Modern density functional theory in its current Kohn–Sham formulation is still very much a method in development. Its patchwork of varied and often peculiar-looking functionals does little to simplify matters for the nonspecialists, yet these workers are increasingly expected to use DFT in support of their research. A few functionals such as B3LYP [12,13] and BP86 [37,38] have been deemed useful in describing most chemical systems; no doubt newer functionals perform the same or better than these two, but they have been extensively tested and are therefore recommended for general use. It is important to realize that none of these so-called‘‘DFT functionals’’ can be shown to resemble the exact functional—the sooner that mythical creature is found, the better—but that their performance is roughly at the MP2 level with only an HF/

SCF level of computational cost. They have also proven to be unusually adept at modeling metal-containing systems, even when the usual all-electron basis set for the metal is replaced with a simpler eﬀective core potential (ECP) to model the inner electrons [39]. Many of these ECPs even account for important relativistic changes in the signiﬁcantly larger cores of heavy nuclei. Most of the properties available in a HF/

SCF calculation are available for DFT. Density functional theory is formally a ground state theory, but it has a linear-response formalism designated Time-Dependent DFT (TD-DFT) that can be used to produce an electronic spectrum and photochemical reaction data [40].

From a small molecule perspective, DFT functionals are still viewed with some suspicion due to their inconsistent record of accuracy for special cases such as an-nulenes [41]. However, for a large biomolecule, such problems are not likely to creep up at a significant rate, as unstrained organic chemistry is usually rather straightfor-ward theoretically. Density functional theory is not very sensitive to basis set effects, so a medium-size set is already approaching the limit of its accuracy; this makes it ideal for geometries of systems up to 150 heavy atoms (DFT typically runs within a factor of two of the HF/SCF required time). Energies of larger systems can be computed using Hartree–Fock geometries in order to verify reaction intermediates and transition states at the lower level of theory. Such an investigation was published recently by Rodriguez and co-workers [42], who showed using B3LYP/6-31G(d,p) energies and HF/6-31G(d,p) geometries that the keto-enol equilibrium is an important figure of merit for correlation with the antifungal activity of a-substituted acetophenones.

Density functional theory is often exploited along with HF/SCF for more energetic evidence when semiempirical methods are required to handle the geometries.

Sometimes the system is small enough to be studied with DFT alone, as in an article by Pan and McAllister. The tautomerization of steroids by D⁵-3 ketosteroid isomerase apparently proceeds by way of hydrogen bonding between the steroid and the Asp99 and/or Tyr14 residues. In this study, a model active site was constructed using formic acid and phenol as substitutes for Asp99 and Tyr14, respectively (see Fig. 2). The substrate was then placed inside the site and optimized using MP2/ and B3LYP/6-31+G(d,p) including solvation with the SCRF-SCIPCM [43] method. The results were unfortunately indeterminate, as the 1 kcal/mol diﬀerence between trial

structures was not enough to determine which H-bond-mediated mechanism was favored (if either) [44]; although uses abound, modern DFT functionals’ intermedi-ate-level accuracy proves insuﬃcient to describe bonding universally well.

3.3. Semiempirical Methods

Much of the important aspects of semiempirical methods has been previously discussed. Modern semiempirical methods include MNDO [45], AM1 [19], and PM3 [20], among others. MNDO is incorporated into the MOPAC software package [46], which is capable of computing many molecular properties including polar-izibilities, IR, NMR, Raman, and nuclear quadrupole resonance parameters. MNDO is not as often used, however, as it does not adequately reproduce hydrogen bonding or heats of formation to better than 14 kcal/mol [47]. Each semiempirical method is built around a diﬀerent eﬀective Hamiltonian, and thus some are more useful than others in various circumstances. It has been suggested that the PM3 Hamiltonian is superior for modeling hydrogen bonds for precisely this reason [48].

All current semiempirical methods suﬀer from their valence-only implementa-tion. They are also all parameterized for only the ground state for each nucleus of interest. As such, they are not very good choices for careful examinations of reactions.

On the whole, however, they can be expected to perform within 8–10 kcal/mol of experiment for heats of formation [47]. Solvation eﬀects may be easily included as well by way of the popular SMx technique [49]. Their most useful property is that they may be applied to systems of up to 500 atoms. Thus a full AM1 geometry optimization of the a-chymotrypsin (serine protease) active site is possible, along with its target N-acetyl-L-tryptophanamide [50]. In a far-looking viewpoint paper by Patel et al.,‘‘Will ab initioand DFT drug design be practical in the 21st century?’’, AM1 is used to probe

Figure 2 A model active site for D⁵-3 ketosteroid isomerase studied using density functional theory. See Ref. 44.

the optimal (unknown experimentally) arrangement among helices in the seven-helix h2-adrenergic G-protein coupled receptor [51]. The most likely use of semiempirical methods, though, is as the quantum mechanical part of a hybrid calculation.

3.4. Hybrid QM/MM

Hybrid Quantum Mechanics/Molecular Mechanics (QM/MM) methods have com-putational symbiosis as their goal. Quantum mechanics methods are readily appli-cable to 15–500 atom systems with good-to-excellent accuracy depending on the specific method. Molecular mechanical methods can generate decent results (close to HF/SCF accuracy, sometimes better for conventional systems) [52] for many thou-sands of atoms, provided nothing in the molecule requires accurate modeling of bond breaking, polarization effects, etc. In hybrid QM/MM methods, the unparalleled speed of molecular mechanics may be applied to the parts of the molecular system that have a negligible chemical impact, while some quantum mechanical theory of higher accuracy may attack the difficult-to-model catalytic active site [53]. Such cal-culations are, in principle, capable of handling systems with several thousand atoms, which is why such studies hold a significant share of research in computational me-dicinal chemistry.

As hybrid QM/MM uses each tool for maximum practicality in a pragmatic fashion, it is perhaps not too surprising that the first application of QM/MM was reported before all the theory had been constructed. Vibrational structures and electronic transitions in conjugated polyenes and retinal were the subjects of the very first hybrid QM/MM study by Warshel and Karplus [54]. The pioneering work of Warshel and Levitt laid down the ground rules for a consistent QM/MM algorithm [55]. Further research solidified the theory and made it robust enough to handle such historically difficult aspects as reactivity [56] and solvation [57]. The practitioners of the modern hybrid QM/MM procedures are legion because they recognize the efficacy of its compromise. It is not much of an exaggeration to say that hybrid QM/MM is capable of‘‘putting it all together’’ to achieve a thoughtful balance between accuracy and speed, making adjustments wherever necessary. Later chapters will deal with QM/

MM and all its allies in the world of accurate computations of large molecules, so a brief discussion of some quite recent applications should suﬃce.

Dıáz and co-workers recently published a mechanistic study that typifies the modern paradigm of computational medicinal chemistry. The purported mechanisms for benzyl penicillin acylation of class A TEM-1 h-lactamase (see Fig. 3) follow a number of pathways. To investigate these pathways, the relevant conformation of the reactive part was optimized using semiempirical QM/MM (PM3/AMBER, 66 atoms in the QM area). The Ser70 residue was considered essential to the proper catalytic activity. The target penicillin was optimized at the B3LYP/6-31+G* level, and tran-sition states for the reaction pathways were computed with MP2/6-31+G* as well as B3LYP. Energies for the structures were verified to be consistent by using the G2(MP2, SVP) scheme. The structures were believed to be connected to one another based on an Intrinsic Reaction Coordinate obtained at the HF/3-21G* level. Short-range solvent effects were treated with explicit solvent molecules when practical, while the rest of the solvent effects were explicitly included in the QM/MM treatment but approximated using SCRF in the ab initio treatments. The complexation energy had

to be carefully derived using a formula that took all the various treatments into consideration:

DE_compositecDE_B3LYP₌₆-31þG* ðactive siteÞ þ DDG^solvation ðprotein-penicillinÞ þ½DEPM3 ðprotein-penicillinÞ DEPM3 ðactive siteÞ ð6Þ The authors made a determination after examining their diverse and voluminous results that‘‘the acylation of class A h-lactamases by penicillin proceeds through a hydroxyl- and carboxylate-assisted mechanism’’ [58].

Another well-conceived research project was undertaken by Alhambra and co-workers which showcases the variety of problems that hybrid QM/MM can tackle with the help of conventional theory. The main interest of this work was the role of tunneling in the dynamics of the horse liver alcohol dehydrogenase (LADH) metalloenzyme. The specific kinetics had already been experimentally measured, so it was appropriate to compare those results to the best possible theory. Liver alco-hol dehydrogenase transforms benzyl alcoalco-holate into benzylaldehyde, and QM/MM (AM1/TIP3P, 9-31 atoms in the active site, depending on model) was used to explore the potential energy surface of that reaction to find stationary points. Those points were further refined with valence bond theory and the dynamics considered by VTST.

In order to properly model the tunneling behavior, a three-stage approach was devised which treated the outer part with MM and the inner part with QM, SEVB, and VTST in such a way as to allow for an‘‘equilibrium secondary zone’’ between the two parts.

This rather diﬃcult construction was no doubt complicated by the presence of a metal Figure 3 Class A TEM-1h-lactamase (PDB ID: 1BTL), the subject of a study employing QM/MM techniques. See Ref. 58.

in the active site, but in the end, it was possible to say that‘‘our computations conﬁrm . . . the experimental evidence for hydrogen tunneling in enzymatic reactions’’ [59].

That tunneling—something so fundamental to quantum mechanics—can be modeled in this way causes us to wonder what might be next.

3.5. The Future

In a 1996 review article on the future of quantum chemical methods, Head–Gordon paints a bleak picture for future conventional calculations. At the time, HF or DFT calculations on 100 atoms were feasible (it is closer to 150 today). He sets a goal of 10,000 atoms as the arrival point for the age of explicit calculations on entire proteins.

This is a 100-fold increase from 1996, but due to the scaling of conventional HF and DFT, such a task would require an unrealistic 600-fold increase in computational power. Furthermore, for high-level CCSD(T) calculations, even a 600-fold increase in processing speed would only improve applicability of the method by a factor of ﬁve.

Clearly, something must be done if this goal is to become a reality [23].

Fortunately, the solution may very well already exist. The rate-determining step in these calculations is the computation of two-electron repulsion integrals at N⁴, but superﬁcially it would seem that two-electron terms should really only scale as N². In fact, using the recently developed fast multipole methods [60,61], near-linear scaling can be achieved for Hartree–Fock and Density Functional theories. The conse-quences are staggering: the goal of 10,000 atoms could be reached with only a 100-fold increase in computer power! Additional work in the parallelization of quantum chemistry software [62] can further reduce this, because fully parallelized software can be run simultaneously on many machines and the work is additive. Linear scaling techniques utilizing sophisticated sparse matrix-multiply routines have demonstrated energy point HF and DFT calculations for, among other things, a 6304-atom nu-cleotide sequence and a nearly 20,000-atom polyglycine chain [63–65]. Even more impressive, semiempirical geometry optimizations of up to 3000 atoms including sol-vent eﬀects have already been computed over a 10-day period using similar processes

In document Computational Medicinal Chemistry for Drug Discovery (Page 154-161)