Fully quantum mechanical energy optimization for protein-ligand structure

Fragmentation-Based Decomposition of a Metalloenzyme-Substrate Interaction: A Case Study for a Lytic Polysaccharide Monooxygenase

We present a novel decomposition scheme for electronic interaction energies based on the flexible formulation of fragmentation schemes through fragment combination ranges (FCRs; J. Chem. Phys., 2021, 155, 164105). We devise a clear additive decomposition with contribution of nondisjoint fragments and correction terms for overlapping fragments and apply this scheme to the metalloenzyme-substrate complex of a lytic polysaccharide monooxygenase (LPMO) with an oligosaccharide. By this, we further illustrate the straightforward adaptability of the FCR-based schemes to novel systems. Our calculations suggest that the description of the electronic structure is a larger error source than the fragmentation scheme. In particular, we find a large impact of the basis set size on the interaction energies. Still, the introduction of three-body interaction terms in the fragmentation setup improves the agreement to the supermolecular reference. Yet, the qualitative results for the decomposition scheme with two-body terms only largely agree within the investigated electronic-structure approaches and basis sets, which are B97-3c, DFT (TPSS and B3LYP), and MP2 methods. The overlap contributions are found to be small, allowing analysis of the interaction energy into individual amino acid residues: We find a particularly strong interaction between the substrate and the LPMO copper active site.

Three-Layer Multiscale Approach Based on Extremely Localized Molecular Orbitals to Investigate Enzyme Reactions

Quantum mechanics/molecular mechanics (QM/MM) calculations are widely used embedding techniques to computationally investigate enzyme reactions. In most QM/MM computations, the quantum mechanical region is treated through density functional theory (DFT), which offers the best compromise between chemical accuracy and computational cost. Nevertheless, to obtain more accurate results, one should resort to wave function-based methods, which however lead to a much larger computational cost already for relatively small QM subsystems. To overcome this drawback, we propose the coupling of our QM/ELMO (quantum mechanics/extremely localized molecular orbital) approach with molecular mechanics, thus introducing the three-layer QM/ELMO/MM technique. The QM/ELMO strategy is an embedding method in which the chemically relevant part of the system is treated at the quantum mechanical level, while the rest is described through frozen ELMOs. Since the QM/ELMO method reproduces results of fully QM computations within chemical accuracy and with a much lower computational effort, it can be considered a suitable strategy to extend the range of applicability and accuracy of the QM/MM scheme. In this paper, other than briefly presenting the theoretical bases of the QM/ELMO/MM technique, we will also discuss its validation on the well-tested deprotonation of acetyl coenzyme A by aspartate in citrate synthase.

Libraries of Extremely Localized Molecular Orbitals. 3. Construction and Preliminary Assessment of the New Databanks

The fast and reliable determination of wave functions and electron densities of macromolecules has been one of the goals of theoretical chemistry for a long time, and in this context, several linear scaling techniques have been successfully devised over the years. Different approaches have been adopted to tackle this problem, and one of them exploits the fact that, according to the traditional chemical perception, molecules can be seen as constituted of recurring units (e.g., functional groups) with well-defined chemical features. This has led to the development of methods in which the global wave functions or electron densities of macromolecules are obtained by simply transferring density matrices or fuzzy electron densities associated with molecular fragments. In this context, we propose an alternative strategy that aims at quickly reconstructing wave functions and electron densities of proteins through the transfer of extremely localized molecular orbitals (ELMOs), which are orbitals strictly localized on small molecular units and, for this reason, easily transferable from molecule to molecule. To accomplish this task we have constructed original libraries of ELMOs that cover all the possible elementary fragments of the 20 natural amino acids in all their possible protonation states and forms. Our preliminary test calculations have shown that, compared to more traditional methods of quantum chemistry, the transfers from the novel ELMO databanks allow to obtain wave function and electron densities of large polypeptides and proteins at a significantly reduced computational cost. Furthermore, notwithstanding expected discrepancies, the obtained electron distributions and electrostatic potentials are in very good agreement with those obtained at Hartree-Fock and density functional theory (DFT) levels. Therefore, the results encourage to use the new libraries as alternatives to the popular pseudoatom-databases of crystallography in the refinement of crystallographic structures of macromolecules. In particular, in this context, we have already envisaged the coupling of the ELMO databanks with the promising Hirshfeld atom refinement technique to extend the applicability of the latter to very large systems.

Geometry optimizations with the incremental molecular fragmentation method

Nuclear energy gradients for the incremental molecular fragmentation (IMF) method presented in our previous work [Meitei OR, Heßelmann A, Molecular energies from an incremental fragmentation method, J Chem Phys 144(8):084109, 2016] have been derived. Using the second-order Møller–Plesset perturbation theory method to describe the bonded and nonbonded energy and gradient contributions and the uncorrelated Hartree–Fock method to describe the correction increment, it is shown that the IMF gradient can be easily computed by a sum of the underlying individual derivatives of the energy contributions. The performance of the method has been compared against the supermolecular method by optimizing the structures of a range of polyglycine molecules with up to 36 glycine residues in the chain. It is shown that with a sensible set of parameters used in the fragmentation the supermolecular structures can be fairly well reproduced. In a few cases the optimization with the IMF method leads to structures that differ from the supermolecular ones. It was found, however, that these are more stable geometries also on the supermolecular potential energy surface.

Atomic Spectral Methods for Ab Initio Molecular Electronic Energy Surfaces: Transitioning From Small-Molecule to Biomolecular-Suitable Approaches

Continuing attention has addressed incorportation of the electronically dynamical attributes of biomolecules in the largely static first-generation molecular-mechanical force fields commonly employed in molecular-dynamics simulations. We describe here a universal quantum-mechanical approach to calculations of the electronic energy surfaces of both small molecules and large aggregates on a common basis which can include such electronic attributes, and which also seems well-suited to adaptation in ab initio molecular-dynamics applications. In contrast to the more familiar orbital-product-based methodologies employed in traditional small-molecule computational quantum chemistry, the present approach is based on an "ex-post-facto" method in which Hamiltonian matrices are evaluated prior to wave function antisymmetrization, implemented here in the support of a Hilbert space of orthonormal products of many-electron atomic spectral eigenstates familiar from the van der Waals theory of long-range interactions. The general theory in its various forms incorporates the early semiempirical atoms- and diatomics-in-molecules approaches of Moffitt, Ellison, Tully, Kuntz, and others in a comprehensive mathematical setting, and generalizes the developments of Eisenschitz, London, Claverie, and others addressing electron permutation symmetry adaptation issues, completing these early attempts to treat van der Waals and chemical forces on a common basis. Exact expressions are obtained for molecular Hamiltonian matrices and for associated energy eigenvalues as sums of separate atomic and interaction-energy terms, similar in this respect to the forms of classical force fields. The latter representation is seen to also provide a long-missing general definition of the energies of individual atoms and of their interactions within molecules and matter free from subjective additional constraints. A computer code suite is described for calculations of the many-electron atomic eigenspectra and the pairwise-atomic Hamiltonian matrices required for practical applications. These matrices can be retained as functions of scalar atomic-pair separations and employed in assembling aggregate Hamiltonian matrices, with Wigner rotation matrices providing analytical representations of their angular degrees of freedom. In this way, ab initio potential energy surfaces are obtained in the complete absence of repeated evaluations and transformations of the one- and two-electron integrals at different molecular geometries required in most ab inito molecular calculations, with large Hamiltonian matrix assembly simplified and explicit diagonalizations avoided employing partitioning and Brillouin-Wigner or Rayleigh-Schrödinger perturbation theory. Illustrative applications of the important components of the formalism, selected aspects of the scaling of the approach, and aspects of "on-the-fly" interfaces with Monte Carlo and molecular-dynamics methods are described in anticipation of subsequent applications to biomolecules and other large aggregates.

Libraries of Extremely Localized Molecular Orbitals. 1. Model Molecules Approximation and Molecular Orbitals Transferability

Despite more and more remarkable computational ab initio results are nowadays continuously obtained for large macromolecular systems, the development of new linear-scaling techniques is still an open and stimulating field of research in theoretical chemistry. In this family of methods, an important role is occupied by those strategies based on the observation that molecules are generally constituted by recurrent functional units with well-defined intrinsic features. In this context, we propose to exploit the notion of extremely localized molecular orbitals (ELMOs) that, due to their strict localization on small molecular fragments (e.g., atoms, bonds, or functional groups), are in principle transferable from one molecule to another. Accordingly, the construction of orbital libraries to almost instantaneously build up approximate wave functions and electron densities of very large systems becomes conceivable. In this work, the ELMOs transferability is further investigated in detail and, furthermore, suitable rules to construct model molecules for the computation of ELMOs to be stored in future databanks are also defined. The obtained results confirm the reliable transferability of the ELMOs and show that electron densities obtained from the transfer of extremely localized molecular orbitals are very close to the corresponding Hartree-Fock ones. These observations prompt us to construct new ELMOs databases that could represent an alternative/complement to the already popular pseudoatoms databanks both for determining electron densities and for refining crystallographic structures of very large molecules.

Fragment quantum mechanical calculation of proteins and its applications

Conspectus The desire to study molecular systems that are much larger than what the current state-of-the-art ab initio or density functional theory methods could handle has naturally led to the development of novel approximate methods, including semiempirical approaches, reduced-scaling methods, and fragmentation methods. The major computational limitation of ab initio methods is the scaling problem, because the cost of ab initio calculation scales nth power or worse with system size. In the past decade, the fragmentation approach based on chemical locality has opened a new door for developing linear-scaling quantum mechanical (QM) methods for large systems and for applications to large molecular systems such as biomolecules. The fragmentation approach is highly attractive from a computational standpoint. First, the ab initio calculation of individual fragments can be conducted almost independently, which makes it suitable for massively parallel computations. Second, the electron properties, such as density and energy, are typically combined in a linear fashion to reproduce those for the entire molecular system, which makes the overall computation scale linearly with the size of the system. In this Account, two fragmentation methods and their applications to macromolecules are described. They are the electrostatically embedded generalized molecular fractionation with conjugate caps (EE-GMFCC) method and the automated fragmentation quantum mechanics/molecular mechanics (AF-QM/MM) approach. The EE-GMFCC method is developed from the MFCC approach, which was initially used to obtain accurate protein-ligand QM interaction energies. The main idea of the MFCC approach is that a pair of conjugate caps (concaps) is inserted at the location where the subsystem is divided by cutting the chemical bond. In addition, the pair of concaps is fused to form molecular species such that the overcounted effect from added concaps can be properly removed. By introducing the electrostatic embedding field in each fragment calculation and two-body interaction energy correction on top of the MFCC approach, the EE-GMFCC method is capable of accurately reproducing the QM molecular properties (such as the dipole moment, electron density, and electrostatic potential), the total energy, and the electrostatic solvation energy from full system calculations for proteins. On the other hand, the AF-QM/MM method was used for the efficient QM calculation of protein nuclear magnetic resonance (NMR) parameters, including the chemical shift, chemical shift anisotropy tensor, and spin-spin coupling constant. In the AF-QM/MM approach, each amino acid and all the residues in its vicinity are automatically assigned as the QM region through a distance cutoff for each residue-centric QM/MM calculation. Local chemical properties of the central residue can be obtained from individual QM/MM calculations. The AF-QM/MM approach precisely reproduces the NMR chemical shifts of proteins in the gas phase from full system QM calculations. Furthermore, via the incorporation of implicit and explicit solvent models, the protein NMR chemical shifts calculated by the AF-QM/MM method are in excellent agreement with experimental values. The applications of the AF-QM/MM method may also be extended to more general biological systems such as DNA/RNA and protein-ligand complexes.

The use of many-body expansions and geometry optimizations in fragment-based methods

Conspectus Chemists routinely work with complex molecular systems: solutions, biochemical molecules, and amorphous and composite materials provide some typical examples. The questions one often asks are what are the driving forces for a chemical phenomenon? How reasonable are our views of chemical systems in terms of subunits, such as functional groups and individual molecules? How can one quantify the difference in physicochemical properties of functional units found in a different chemical environment? Are various effects on functional units in molecular systems additive? Can they be represented by pairwise potentials? Are there effects that cannot be represented in a simple picture of pairwise interactions? How can we obtain quantitative values for these effects? Many of these questions can be formulated in the language of many-body effects. They quantify the properties of subunits (fragments), referred to as one-body properties, pairwise interactions (two-body properties), couplings of two-body interactions described by three-body properties, and so on. By introducing the notion of fragments in the framework of quantum chemistry, one obtains two immense benefits: (a) chemists can finally relate to quantum chemistry, which now speaks their language, by discussing chemically interesting subunits and their interactions and (b) calculations become much faster due to a reduced computational scaling. For instance, the somewhat academic sounding question of the importance of three-body effects in water clusters is actually another way of asking how two hydrogen bonds affect each other, when they involve three water molecules. One aspect of this is the many-body charge transfer (CT), because the charge transfers in the two hydrogen bonds are coupled to each other (not independent). In this work, we provide a generalized view on the use of many-body expansions in fragment-based methods, focusing on the general aspects of the property expansion and a contraction of a many-body expansion in a formally two-body series, as exemplified in the development of the fragment molecular orbital (FMO) method. Fragment-based methods have been very successful in delivering the properties of fragments, as well as the fragment interactions, providing insights into complex chemical processes in large molecular systems. We briefly review geometry optimizations performed with fragment-based methods and present an efficient geometry optimization method based on the combination of FMO with molecular mechanics (MM), applied to the complex of a subunit of protein kinase 2 (CK2) with a ligand. FMO results are discussed in comparison with experimental and MM-optimized structures.

Density functional calculations of extended, periodic systems using Coulomb corrected molecular fractionation with conjugated caps method (CC-MFCC)

A consistent DFT based formulation of the order-N molecular fractionation with conjugated caps method in which a molecular system is calculated considering a set of finite fragments, is proposed. Here we apply the method and test its performance on a periodic metal–organic framework system.

Thermodynamics of ideal proteinogenic homopolymer chains as a function of the energy spectrum E, helical propensity ω and enthalpic energy barrier

A reformulation and generalization of the Zwanzig model (ZW model) for ideal homopolymer chains poly-X, where X represents any of the twenty naturally occurring proteinogenic amino acid residues is presented. This reformulation and generalization provides a direct connection between coarse-grained parameters originally proposed in the ZW model with variables from the Lifson-Roig (LR) theory, such as the helical propensity per residue ω, and new variables introduced here, such as the energy gap Δ between unfolded and folded structures, as well as the ratio f of the energy scales involved. This enables us to discover the relevance of the energy spectrum E to the onset of configurational phase transitions. From the configurational partition function Q, thermodynamic properties such as the configurational entropy S, specific heat v and average energy <E> are calculated in terms of the number of residues K, temperature T, helical propensity ω and energy barrier ΔH for different poly-X chains in vacuo. Results obtained here provide substantial evidence that configurational phase transitions for ideal poly-X chains correspond to first-order phase transitions. An anomalous behavior of the thermodynamic functions <E>, Cv, S with respect to the number K of residues is also highlighted. On-going methods of solution are outlined.

Fragment-based quantum mechanical methods for periodic systems with Ewald summation and mean image charge convention for long-range electrostatic interactions

We describe an Ewald-summation method to incorporate long-range electrostatic interactions into fragment-based electronic structure methods for periodic systems. The present method is an extension of the particle-mesh Ewald technique for combined quantum mechanical and molecular mechanical (QM/MM) calculations, and it has been implemented into the explicit polarization (X-Pol) potential to illustrate the computational details. As in the QM/MM-Ewald method, the X-Pol-Ewald approach is a linear-scaling electrostatic method, in which the short-range electrostatic interactions are determined explicitly in real space and the long-range Ewald pair potential is incorporated into the Fock matrix as a correction. To avoid the time-consuming Fock matrix update during the self-consistent field procedure, a mean image charge (MIC) approximation is introduced, in which the running average with a user-chosen correlation time is used to represent the long-range electrostatic correction as an average effect. Test simulations on liquid water show that the present X-Pol-Ewald method takes about 25% more CPU time than the usual X-Pol method using spherical cutoff, whereas the use of the MIC approximation reduces the extra costs for long-range electrostatic interactions by 15%. The present X-Pol-Ewald method provides a general procedure for incorporating long-range electrostatic effects into fragment-based electronic structure methods for treating biomolecular and condensed-phase systems under periodic boundary conditions.

Fully ab initio protein-ligand interaction energies with dispersion corrected density functional theory

Dispersion corrected density functional theory (DFT-D3) is used for fully ab initio protein-ligand (PL) interaction energy calculation via molecular fractionation with conjugated caps (MFCC) and applied to PL complexes from the PDB comprising 3680, 1798, and 1060 atoms. Molecular fragments with n amino acids instead of one in the original MFCC approach are considered, thereby allowing for estimating the three-body and higher many-body terms. n > 1 is recommended both in terms of accuracy and efficiency of MFCC. For neutral protein side-chains, the computed PL interaction energy is visibly independent of the fragment length n. The MFCC fractionation error is determined by comparison to a full-system calculation for the 1060 atoms containing PL complex. For charged amino acid side-chains, the variation of the MFCC result with n is increased. For these systems, using a continuum solvation model with a dielectricity constant typical for protein environments (ϵ = 4) reduces both the variation with n and improves the stability of the DFT calculations considerably. The PL interaction energies for two typical complexes obtained ab initio for the first time are found to be rather large (-30 and -54 kcal/mol).

Multilevel X-Pol: a fragment-based method with mixed quantum mechanical representations of different fragments

The explicit polarization (X-Pol) method is a fragment-based quantum mechanical model, in which a macromolecular system or other large or complex system in solution is partitioned into monomeric fragments. The present study extends the original X-Pol method, where all fragments are treated using the same electronic structure theory, to multilevel representations, called multilevel X-Pol, in which different electronic structure methods are used to describe different fragments. The multilevel X-Pol method has been implemented into a locally modified version of Gaussian 09. A key ingredient that is used to couple interfragment electrostatic interactions at different levels of theory is the use of the response density for the post-self-consistent-field energy. (The response density is also called the generalized density.) The method is useful for treating fragments in a small region of the system such as a solute molecule or the substrate and amino acids in the active site of an enzyme with a high-level theory, and the fragments in the rest of the system by a lower-level and computationally more efficient method. Multilevel X-Pol is illustrated here by applications to hydrogen bonding complexes in which one fragment is treated with the hybrid M06 density functional, Møller-Plesset perturbation theory, or coupled cluster theory, and the other fragments are treated by Hartree-Fock theory or the B3LYP or M06 hybrid density functionals.

Optimization of the explicit polarization (X-Pol) potential using a hybrid density functional

The explicit polarization (X-Pol) method is a self-consistent fragment-based electronic structure theory in which molecular orbitals are block-localized within fragments of a cluster, macromolecule, or condensed-phase system. To account for short-range exchange repulsion and long-range dispersion interactions, we have incorporated a pairwise, empirical potential, in the form of Lennard-Jones terms, into the X-Pol effective Hamiltonian. In the present study, the X-Pol potential is constructed using the B3LYP hybrid density functional with the 6-31G(d) basis set to treat interacting fragments, and the Lennard-Jones parameters have been optimized on a dataset consisting of 105 bimolecular complexes. It is shown that the X-Pol potential can be optimized to provide a good description of hydrogen bonding interactions; the root mean square deviation of the computed binding energies from full (i.e., nonfragmental) CCSD(T)/aug-cc-pVDZ results is 0.8 kcal/mol, and the calculated hydrogen bond distances have an average deviation of about 0.1 Å from those obtained by full B3LYP/aug-cc-pVDZ optimizations.

SUPPLEMENTING THE PBSA APPROACH WITH QUANTUM MECHANICS TO STUDY THE BINDING BETWEEN CDK2 AND N2-SUBSTITUTED O6-CYCLOHEXYLMETHOXYGUANINE INHIBITORS

Because classical Poisson–Boltzman Surface Area (PBSA) model does not allow re-polarization of charges and does not account for charge transfer when a ligand binds to a protein, we have examined a hybrid approach in which we describe the protein–ligand interface by quantum mechanics and the rest of the system with the classical PBSA model. We found this approach to rank order the binding of five N2 -substituted O6 -cyclohexylmethoxyguanine inhibitors to CDK2 (cyclin-dependent kinase 2) properly. The calculated binding free energy correlated well with experimental Log(IC50) with a correlation coefficient of 0.94. A regression fit between experimental Log(IC50) and calculated binding free energy yielded a root-mean-square error of 0.48 when Log(IC50) spanned a range over three units. In addition, we observed charge transfer between the ligand and the protein at the interface — an effect not accounted for by the classical PBSA model. We also found that the direct interactions between the protein and the ligands provided the dominant factor to distinguish the binding affinity of the five ligands studied here. This hybrid approach can better prioritize derivatives of lead compounds for synthesis and biological evaluation.

DESIGN OF HYBRID INHIBITORS TO HIV-1 PROTEASE

A series of HIV-1 protease (PR) inhibitors are designed to increase the binding affinity with PR subsites based on the quantum analysis of the contributions of molecular fragments in six FDA-approved PR drugs to the total binding interaction. The binding free energies were estimated by modified linear interaction energy approach [Zoete H, Michielin O, Karplus M, J Comput Aided Mol Des17:861, 2003], in which the binding free energy is written as a linear combination of the electrostatic interaction energy between PR and the ligand, Eelec, the van der Waals interaction energy between PR and the ligand, E vdW , and the difference of the solvation free energies of the complex, the receptor and the isolated ligand, ΔG solv . The parameters of these energy terms were fitted for a training set of 14 HIV-1 protease–inhibitor complexes of known 3D structure with a correlation coefficient of 0.91 and an unsigned mean error of 0.83 kcal/mol.

QUANTUM CALCULATION OF PROTEIN SOLVATION AND PROTEIN–LIGAND BINDING FREE ENERGY FOR HIV-1 PROTEASE/WATER COMPLEX

HIV-1 protease (PR) is a primary target for anti-HIV therapeutics. A well conserved water molecule, denoted as W301, is found in almost all the crystallographic structures of PR/inhibitor complexes and it plays an important role in PR/inhibitor binding. As the PR/inhibitor interaction depends on the ionization state of the cleavage site which contains an aspartyl dyad (Asp25/Asp25′), the determination of the protonation states of aspartyl dyad in PR may be essential for drug design. In this study, a linear scaling quantum mechanical method, molecular fragmentation with conjugate caps (MFCC), is used for interaction study of PR/ABT-538 and W301 at four different monoprotonation states of the Asp25/Asp25′. Combined method of MFCC and conductor-like polarizable continuum model (CPCM) is applied in binding affinity calculation for four minimum energy structures which are extracted from four different molecular dynamics trajectories corresponding to four different monoprotonation states of Asp25/Asp25′. Our result is in good agreement with previous result obtained by FEP/TI method, showing that the conserved W301 contributes significantly to the binding free energy of PR/ABT-538 complex and different protonation states of Asp25/Asp25′ have significant impact on the binding free energy contribution from W301.

Systematic fragmentation method and the effective fragment potential: an efficient method for capturing molecular energies

The systematic fragmentation method fragments a large molecular system into smaller pieces, in such a way as to greatly reduce the computational cost while retaining nearly the accuracy of the parent ab initio electronic structure method. In order to attain the desired (sub-kcal/mol) accuracy, one must properly account for the nonbonded interactions between the separated fragments. Since, for a large molecular species, there can be a great many fragments and therefore a great many nonbonded interactions, computations of the nonbonded interactions can be very time-consuming. The present work explores the efficacy of employing the effective fragment potential (EFP) method to obtain the nonbonded interactions since the EFP method has been shown previously to capture nonbonded interactions with an accuracy that is often comparable to that of second-order perturbation theory. It is demonstrated that for nonbonded interactions that are not high on the repulsive wall (generally >2.7 A), the EFP method appears to be a viable approach for evaluating the nonbonded interactions. The efficacy of the EFP method for this purpose is illustrated by comparing the method to ab initio methods for small water clusters, the ZOVGAS molecule, retinal, and the alpha-helix. Using SFM with EFP for nonbonded interactions yields an error of 0.2 kcal/mol for the retinal cis-trans isomerization and a mean error of 1.0 kcal/mol for the isomerization energies of five small (120-170 atoms) alpha-helices.

Enabling ab initio Hessian and frequency calculations of large molecules

A linear scaling method, termed as cardinality guided molecular tailoring approach, is applied for the estimation of the Hessian matrix and frequency calculations of spatially extended molecules. The method is put to test on a number of molecular systems largely employing the Hartree-Fock and density functional theory for a variety of basis sets. To demonstrate its ability for correlated methods, we have also performed a few test calculations at the Moller-Plesset second order perturbation theory. A comparison of central processing unit and memory requirements for medium-sized systems with those for the corresponding full ab initio computation reveals substantial gains with negligible loss of accuracy. The technique is further employed for a set of larger molecules, Hessian and frequency calculations of which are not possible on commonly available personal-computer-type hardware.

The role of weakly polar and H-bonding interactions in the stabilization of the conformers of FGG, WGG, and YGG: an aqueous phase computational study

The energetics of intramolecular interactions on the conformational potential energy surface of the terminally protected N-Ac-Phe-Gly-Gly-NHMe (FGG), N-Ac-Trp-Gly-Gly-NHMe (WGG), and N-Ac-Tyr-Gly-Gly-NHMe (YGG) tripeptides was investigated. To identify the representative conformations, simulated annealing molecular dynamics (MD) and density functional theory (DFT) methods were used. The interaction energies were calculated at the BHandHLYP/aug-cc-pVTZ level of theory. In the global minima, 10%, 31%, and 10% of the stabilization energy come from weakly polar interactions, respectively, in FGG, WGG, and YGG. In the prominent cases 46%, 62%, and 46% of the stabilization energy is from the weakly polar interactions, respectively, in FGG, WGG, and YGG. On average, weakly polar interactions account for 15%, 34%, and 9% of the stabilization energies of the FGG, WGG, and YGG conformers, respectively. Thus, weakly polar interactions can make an important energetic contribution to protein structure and function.

Is quantum mechanics necessary for predicting binding free energy?

To take into account polarization effects, the linear interaction energy model with continuum electrostatic solvation (LIECE) is supplemented by the linear-scaling semiempirical quantum mechanical calculation of the intermolecular electrostatic energy (QMLIECE). QMLIECE and LIECE are compared on three enzymes belonging to different classes: the West Nile virus NS3 serine protease (WNV PR), the aspartic protease of the human immunodeficiency virus (HIV-1 PR), and the human cyclin-dependent kinase 2 (CDK2). QMLIECE is superior for 44 peptidic inhibitors of WNV PR because of the different amount of polarization due to the broad range of formal charges of the inhibitors (from 0 to 3). On the other hand, QMLIECE and LIECE show similar accuracy for 24 peptidic inhibitors of HIV-1 PR (20 neutral and 4 with one formal charge) and for 73 CDK2 inhibitors (all neutral). These results indicate that quantum mechanics is essential when the inhibitor/protein complexes have highly variable charge-charge interactions.

Evaluation of methods to cap molecular fragments in calculating energies of interaction in avian pancreatic polypeptide

The accuracy of the determination of the energy of interaction between Phe20 and the Pro5-Thr6-Tyr7-Pro8 complex inside the hydrophobic core of avian pancreatic polypeptide was investigated using three capping strategies for molecular fractionation with conjugated caps and DFT quantum chemical calculations at the BHandHLYP/cc-pVTZ level of theory. The most accurate determination resulted from acetylation of the alpha-amino group combined with methyl amidation of the alpha-carbonyl group with relative deviations less than 10%. Combinations of hydrogenation of the alpha-amino group with the replacement of the alpha-carbonyl group with a hydrogen and the hydrogenation of the alpha-amino group with methylation of the alpha-carbonyl group were less accurate, leading to relative deviations up to 35%. Choice of capping methods depends on the structural features of the polypeptide system, the desired accuracy and the available computational resources.

Extending the Power of Quantum Chemistry to Large Systems with the Fragment Molecular Orbital Method

Following the brief review of the modern fragment-based methods and other approaches to perform quantum-mechanical calculations of large systems, the theoretical development of the fragment molecular orbital method (FMO) is covered in detail, with the emphasis on the physical properties, which can be computed with FMO. The FMO-based polarizable continuum model (PCM) for treating the solvent effects in large systems and the pair interaction energy decomposition analysis (PIEDA) are described in some detail, and a range of applications of FMO to biological studies is introduced. The factors determining the relative stability of polypeptide conformers (alpha-helix, beta-turn, and extended form) are elucidated using FMO/PCM and PIEDA, and the interactions in the Trp-cage miniprotein construct (PDB: 1L2Y) are analyzed using PIEDA.

The generalized molecular fractionation with conjugate caps/molecular mechanics method for direct calculation of protein energy

A generalized molecular fractionation with conjugate caps/molecular mechanics (GMFCC/MM) scheme is developed for efficient linear-scaling quantum mechanical calculation of protein energy. In this GMFCC/MM scheme, the interaction energy between neighboring residues as well as between non-neighboring residues that are spatially in close contact are computed by quantum mechanics while the rest of the interaction energy is computed by molecular mechanics. Numerical studies are carried out to calculate torsional energies of six polypeptides using the GMFCC/MM approach and the energies are shown to be in general good agreement with the full system quantum calculation. Among those we tested is a polypeptide containing 396 atoms whose energies are computed at the MP26-31G* level. Our study shows that using GMFCC/MM, it is possible to perform high level ab initio calculation such as MP2 for applications such as structural optimization of protein complex and molecular dynamics simulation.

Molecular recognition mechanism of FK506 binding protein: An all-electron fragment molecular orbital study

The fragment molecular orbital (FMO) method has enabled electronic structure calculations and geometry optimizations of very large molecules with ab initio quality. We applied the method to four FK506 binding protein (FKBP) complexes (denoted by their PDB codes 1fkb, 1fkf, 1fkg, and 1fki) containing rapamycin, FK506, and two synthetic ligands. The geometries of reduced complex models were optimized at the restricted Hartree-Fock (FMO-RHF) level using the 3-21G basis set, and then for a better estimate of binding, the energetics were refined at a higher level of theory (2nd order Møller-Plesset perturbation theory FMO-MP2 with the 6-31G* basis set). Thus, obtained binding energies were -103.9 (-82.0), -102.2 (-69.2), -70.1 (-57.7), and -71.3 (-55.3) kcal/mol for 1fkb, 1fkf, 1fkg, and 1fki, respectively, where the correlation contribution is given in parentheses. The results show that the electron correlation contribution to binding is extremely important, and it accounts for 70-80% of the binding energy. The molecular recognition mechanism of FKBP was analyzed in detail based on the FMO-pair interactions between protein residues and the ligands. Solvation effects on the protein-ligand binding were estimated using the Poisson-Boltzmann/surface area model.

The Fragment Molecular Orbital Method for Geometry Optimizations of Polypeptides and Proteins

The fragment molecular orbital method (FMO) has been used with a large number of wave functions for single-point calculations, and its high accuracy in comparison to ab initio methods has been well established. We have developed the analytic derivative of the electrostatic interaction between far separated fragments and performed a number of restricted Hartree-Fock (RHF) geometry optimizations using FMO and ab initio methods. In particular, the alpha-helix, beta-turn, and extended conformers of a 10-residue polyalanine were studied and the good FMO accuracy was established (the rms deviations for the former two forms were about 0.2 A and for the latter structure about 0.001 A). Met-enkephalin dimer was used as a model for the polypeptide binding and computed at the 3-21G and 6-31G* levels with a similar accuracy achieved; the error in the binding energy predictions (FMO vs ab initio) was 1-3 kcal/mol. Chignolin (PDB: 1uao) and an agonist polypeptide of the erythropoietin receptor protein (emp1) were optimized at the 3-21(+)G level, with the rms deviation from ab initio of about 0.2 A, or 0.5 degrees in terms of bond angles. The effect of solvation on the structure optimization was studied in chignolin and the Trp-cage miniprotein construct (PDB:1l2y), by describing water with TIP3P. The computed structures in gas phase and solution are compared to each other and experiment.

Electrostatic field-adapted molecular fractionation with conjugated caps for energy calculations of charged biomolecules

An electrostatic field-adapted molecular fractionation with conjugated caps (EFA-MFCC) approach is implemented for treating macromolecules with several charge centers. The molecular fragmentation is performed in an "electrostatic field," which is described by putting point charges on charge centers, directly affecting the Hamiltonians of both fragments and conjugated caps. So the present method does not need truncation during the calculation of electrostatic interactions. Our test calculations on a series of charged model systems and biological macromolecules using the HF and B3LYP methods have demonstrated that this approach is capable of describing the electronic structure with accuracy comparable to other fragment-based methods. The EFA-MFCC approach is an alternative way for predicting the total energies of charged macromolecules with acyclic, loop, and intersectional loop structures and interaction energies between two molecules.

Quantum computational analysis for drug resistance of HIV-1 reverse transcriptase to nevirapine through point mutations

Quantum chemical calculation has been carried out to analyze binding interactions of nevirapine to HIV-1 reverse transcriptase (RT) and single point mutants Lys103 --> Asn (K103N) and Tyr181--> Cys (Y181C). In this study, the entire system of HIV-1 RT/nevirapine complex with over 15,000 atoms is explicitly treated by using a recently developed MFCC (molecular fractionation with conjugate caps) approach. Quantum calculation of protein-drug interaction energy is performed at Hartree-Fock and DFT levels. The RT-nevirapine interaction energies are computed at fixed geometries given by the crystal structures of the HIV-1 RT/nevirapine complexes from protein data bank (PDB). The present calculation provides a quantum mechanical interaction spectrum that explicitly shows interaction energies between nevirapine and individual amino-acid fragments of RT. Detailed interactions that are responsible for drug resistance of two major RT mutations are elucidated based on computational analysis in relation to the crystal structures of binding complexes. The present result provides a qualitative molecular understanding of HIV-1 RT drug resistance to nevirapine and gives useful guidance in designing improved inhibitors with better resistance to RT mutation.

Theoretical study of intramolecular interaction energies during dynamics simulations of oligopeptides by the fragment molecular orbital-Hamiltonian algorithm method

We analyzed the interaction energies between residues (fragments) in an oligopeptide occurring during dynamic simulations by using the fragment molecular orbital-Hamiltonian algorithm (FMO-HA) method, an ab initio MO-molecular dynamics technique. The FMO method enables not only calculation of large molecules based on ab initio MO but also easy evaluation of interfragment interaction energies. The glycine pentamer [(Gly)(5)] and decamer [(Gly)(10)] were divided into five and ten fragments, respectively. alpha-helix structures of (Gly)(5) and (Gly)(10) were stabilized by attractive interaction energies owing to intramolecular hydrogen bonds between fragments n and n+3 (and n+4), and beta-strand structures were characterized by repulsive interaction energies between fragments n and n+2. We analyzed interfragment interaction energies during dynamics simulations as the peptides' geometries changed from alpha helix to beta strand. Intramolecular hydrogen bonds between fragments 2-4 and 2-5 control the geometrical preference of (Gly)(5) for the beta-strand structure. The pitch of one turn of the alpha helix of (Gly)(10) elongated and thus weakened during dynamics due to a shifting of the intramolecular hydrogen bonds, and enabled the beta-strand structure to form. Changes in interaction energies due to the intramolecular hydrogen bonds controlled the tendency toward alpha-helix or beta-strand structure of (Gly)(5) and (Gly)(10). Evaluation of interfragment interaction energies during dynamics simulations thus enabled detailed analysis of the process of the geometrical changes occurring in oligopeptides.

A new method for direct calculation of total energy of protein

A new scheme is developed for efficient quantum mechanical calculation of total energy of protein based on a recently developed MFCC (molecular fractionation with conjugate caps) approach. In this scheme, the linear-scaling MFCC method is first applied to calculate total electron density of protein. The computed electron density is then employed for direct numerical integration in density functional theory (DFT) to yield total energy of protein, with the kinetic energy obtained by a proposed ansatz. Numerical studies are carried out to calculate torsional energies of two polypeptides using this approach and the energies are shown to be in good agreement with the corresponding full system DFT calculation.