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.     

Quantum mechanical map for protein-ligand binding with application to beta-trypsin/benzamidine complex

We report full ab initio Hartree-Fock calculation to compute quantum mechanical interaction energies for beta-trypsin/benzamidine binding complex. In this study, the full quantum mechanical ab initio energy calculation for the entire protein complex with 3238 atoms is made possible by using a recently developed MFCC (molecular fractionation with conjugate caps) approach in which the protein molecule is decomposed into amino acid-based fragments that are properly capped. The present MFCC ab initio calculation enables us to obtain an "interaction spectrum" that provides detailed quantitative information on protein-ligand binding at the amino acid levels. These detailed information on individual residue-ligand interaction gives a quantitative molecular insight into our understanding of protein-ligand binding and provides a guidance to rational design of potential inhibitors of protein targets.     

QM/MM-PBSA method to estimate free energies for reactions in proteins

We have developed a method to estimate free energies of reactions in proteins, called QM/MM-PBSA. It estimates the internal energy of the reactive site by quantum mechanical (QM) calculations, whereas bonded, electrostatic, and van der Waals interactions with the surrounding protein are calculated at the molecular mechanics (MM) level. The electrostatic part of the solvation energy of the reactant and the product is estimated by solving the Poisson-Boltzmann (PB) equation, and the nonpolar part of the solvation energy is estimated from the change in solvent-accessible surface area (SA). Finally, the change in entropy is estimated from the vibrational frequencies. We test this method for five proton-transfer reactions in the active sites of [Ni,Fe] hydrogenase and copper nitrite reductase. We show that QM/MM-PBSA reproduces the results of a strict QM/MM free-energy perturbation method with a mean absolute deviation (MAD) of 8-10 kJ/mol if snapshots from molecular dynamics simulations are used and 4-14 kJ/mol if a single QM/MM structure is used. This is appreciably better than the original QM/MM results or if the QM energies are supplemented with a point-charge model, a self-consistent reaction field, or a PB model of the protein and the solvent, which give MADs of 22-36 kJ/mol for the same test set.     

Generalized born model with a simple smoothing function

Based on recent developments in generalized Born (GB) theory that employ rapid volume integration schemes (M. S. Lee, F. R. Salabury, Jr., and C. L. Brooks III, J Chem Phys 2002, 116, 10606) we have recast the calculation of the self-electrostatic solvation energy to utilize a simple smoothing function at the dielectric boundary. The present GB model is formulated in this manner to provide consistency with the Poisson-Boltzmann (PB) theory previously developed to yield numerically stable electrostatic solvation forces based on finite-difference methods (W. Im, D. Beglov, and B. Roux, Comp Phys Commun 1998, 111, 59). Our comparisons show that the present GB model is indeed an efficient and accurate approach to reproduce corresponding PB solvation energies and forces. With only two adjustable parameters--a(0) to modulate the Coulomb field term, and a(1) to include a correction term beyond Coulomb field--the PB solvation energies are reproduced within 1% error on average for a variety of proteins. Detailed analysis shows that the PB energy can be reproduced within 2% absolute error with a confidence of about 95%. In addition, the solvent-exposed surface area of a biomolecule, as commonly used in calculations of the nonpolar solvation energy, can be calculated accurately and efficiently using the simple smoothing function and the volume integration method. Our implicit solvent GB calculations are about 4.5 times slower than the corresponding vacuum calculations. Using the simple smoothing function makes the present GB model roughly three times faster than GB models, which attempt to mimic the Lee-Richards molecular volume.     

Accurate pK(a) calculations for carboxylic acids using complete basis set and Gaussian-n models combined with CPCM continuum solvation methods

Complete Basis Set and Gaussian-n methods were combined with CPCM continuum solvation methods to calculate pK(a) values for six carboxylic acids. An experimental value of -264.61 kcal/mol for the free energy of solvation of H(+), DeltaG(s)(H(+)), was combined with a value for G(gas)(H(+)) of -6.28 kcal/mol to calculate pK(a) values with Cycle 1. The Complete Basis Set gas-phase methods used to calculate gas-phase free energies are very accurate, with mean unsigned errors of 0.3 kcal/mol and standard deviations of 0.4 kcal/mol. The CPCM solvation calculations used to calculate condensed-phase free energies are slightly less accurate than the gas-phase models, and the best method has a mean unsigned error and standard deviation of 0.4 and 0.5 kcal/mol, respectively. The use of Cycle 1 and the Complete Basis Set models combined with the CPCM solvation methods yielded pK(a) values accurate to less than half a pK(a) unit.     

Fractionation of peptide with disulfide bond for quantum mechanical calculation of interaction energy with molecules

We present a computational study of a recently developed molecular fractionation with conjugated caps (MFCC) method for application to peptide/protein that has disulfide bonds. Specifically, we employ the MFCC approach to generate peptide fragments in which a disulfide bond is cut and a pair of conjugated caps are inserted. The method is tested on two peptides interacting with a water molecule. The first is a dipeptide consisting of two cysteines (Cys-Cys) connected by a disulfide bond and the second is a seven amino acid peptide consisting of Gly-Cys-Gly-Gly-Gly-Cys-Gly with a disulfide cross link. One-dimensional peptide-water potential curves are computed using the MFCC method at various ab initio levels for a number of interaction geometries. The calculated interaction energies are found to be in excellent agreement with the results obtained from the corresponding full system ab initio calculations for both peptide/water systems. The current study provides further numerical support for the accuracy of the MFCC method in full quantum mechanical calculation of protein/peptide that contains disulfide bonds.     

Molecular fractionation with conjugated caps density matrix with pairwise interaction correction for protein energy calculation

Pairwise interaction correction (PIC) is introduced to account for electron density polarization due to short-range interactions such as hydrogen bonding and close contact between molecular fragments in the molecular fractionation with conjugated caps density matrix (MFCC-DM) approach for energy calculation of protein and other polymers [Chen et al., J. Chem. Phys. 122, 184105 (2005)]. With this PIC, the accuracy of the calculated protein energy and other electronic properties are improved, and the MFCC approach can be applied to study real proteins with short-range structural complexity. In the present MFCC-DM-PIC approach, the short-range interresidual interactions are represented by a pair of small molecules (interacting units) which are made from the two residues that fall within a certain distance criterion. The density matrices of fragments, concaps, interacting units and pairs are calculated by conventional Hartree-Fock or density functional theory methods and are combined to construct the full density matrix which is finally employed to calculate the total energy, electron density, electrostatic potential, dipole moment, etc., of the protein. Numerical tests on seven conformationally varied peptides are presented to demonstrate the accuracy of the MFCC-DM-PIC method.     

Fully quantum mechanical energy optimization for protein-ligand structure

We present a quantum mechanical approach to study protein-ligand binding structure with application to a Adipocyte lipid-binding protein complexed with Propanoic Acid. The present approach employs a recently develop molecular fractionation with a conjugate caps (MFCC) method to compute protein-ligand interaction energy and performs energy optimization using the quasi-Newton method. The MFCC method enables us to compute fully quantum mechanical ab initio protein-ligand interaction energy and its gradients that are used in energy minimization. This quantum optimization approach is applied to study the Adipocyte lipid-binding protein complexed with Propanoic Acid system, a complex system consisting of a 2057-atom protein and a 10-atom ligand. The MFCC calculation is carried out at the Hartree-Fock level with a 3-21G basis set. The quantum optimized structure of this complex is in good agreement with the experimental crystal structure. The quantum energy calculation is implemented in a parallel program that dramatically speeds up the MFCC calculation for the protein-ligand system. Similarly good agreement between MFCC optimized structure and the experimental structure is also obtained for the streptavidin-biotin complex. Due to heavy computational cost, the quantum energy minimization is carried out in a six-dimensional space that corresponds to the rigid-body protein-ligand interaction.     

Calculating solvation energies by means of a fluctuating charge model combined with continuum solvent model

Continuum solvent models have shown to be very efficient for calculating solvation energy of biomolecules in solution. However, in order to produce accurate results, besides atomic radii or volumes, an appropriate set of partial charges of the molecule is needed. Here, a set of partial charges produced by a fluctuating charge model-the atom-bond electronegativity equalization method model (ABEEMσπ) fused into molecular mechanics is used to fit for the analytical continuum electrostatics model of generalized-Born calculations. Because the partial atomic charges provided by the ABEEMσπ model can well reflect the polarization effect of the solute induced by the continuum solvent in solution, accurate and rapid calculations of the solvation energies have been performed for series of compounds involving 105 small neutral molecules, twenty kinds of dipeptides and several protein fragments. The solvation energies of small neutral molecules computed with the combination of the GB model with the fluctuating charge protocol (ABEEMσπ∕GB) show remarkable agreement with the experimental results, with a correlation coefficient of 0.97, a slope of 0.95, and a bias of 0.34 kcal∕mol. Furthermore, for twenty kinds of dipeptides and several protein fragments, the results obtained from the analytical ABEEMσπ∕GB model calculations correlate well with those from ab initio and Poisson-Boltzmann calculations. The remarkable agreement between the solvation energies computed with the ABEEMσπ∕GB model and PB model provides strong motivation for the use of ABEEMσπ∕GB solvent model in the simulation of biochemical systems.     

FACTS: Fast analytical continuum treatment of solvation

An efficient method for calculating the free energy of solvation of a (macro)molecule embedded in a continuum solvent is presented. It is based on the fully analytical evaluation of the volume and spatial symmetry of the solvent that is displaced from around a solute atom by its neighboring atoms. The two measures of solvent displacement are combined in empirical equations to approximate the atomic (or self) electrostatic solvation energy and the solvent accessible surface area. The former directly yields the effective Born radius, which is used in the generalized Born (GB) formula to calculate the solvent-screened electrostatic interaction energy. A comparison with finite-difference Poisson data shows that atomic solvation energies, pair interaction energies, and their sums are evaluated with a precision comparable to the most accurate GB implementations. Furthermore, solvation energies of a large set of protein conformations have an error of only 1.5%. The solvent accessible surface area is used to approximate the nonpolar contribution to solvation. The empirical approach, called FACTS (Fast Analytical Continuum Treatment of Solvation), is only four times slower than using the vacuum energy in molecular dynamics simulations of proteins. Notably, the folded state of structured peptides and proteins is stable at room temperature in 100-ns molecular dynamics simulations using FACTS and the CHARMM force field.     

Dispersion terms and analysis of size- and charge dependence in an enhanced Poisson-Boltzmann approach

We implement a well-established concept to consider dispersion effects within a Poisson-Boltzmann approach of continuum solvation of proteins. The theoretical framework is particularly suited for boundary element methods. Free parameters are determined by comparison to experimental data as well as high-level quantum mechanical reference calculations. The method is general and can be easily extended in several directions. The model is tested on various chemical substances and found to yield good-quality estimates of the solvation free energy without obvious indication of any introduced bias. Once optimized, the model is applied to a series of proteins, and factors such as protein size or partial charge assignments are studied.     

New method for direct linear-scaling calculation of electron density of proteins

A new scheme for direct linear-scaling quantum mechanical calculation of electron density of protein systems is developed. The new scheme gives much improved accuracy of electron density for proteins than the original MFCC (molecular fractionation with conjugate caps) approach in efficient linear-scaling calculation for protein systems. In this new approach, the error associated with each cut in the MFCC approach is estimated by computing the two neighboring amino acids in both cut and uncut calculations and is corrected. Numerical tests are performed on six oligopeptide taken from PDB (protein data bank), and the results show that the new scheme is efficient and accurate.     

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 (? = 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). © 2012 Wiley Periodicals, Inc.     

New analytic approximation to the standard molecular volume definition and its application to generalized Born calculations

In a recent article (Lee, M. S.; Salsbury, F. R. Jr.; Brooks, C. L., III. J Chem Phys 2002, 116, 10606), we demonstrated that generalized Born (GB) theory provides a good approximation to Poisson electrostatic solvation energy calculations if one uses the same definitions of molecular volume for each. In this work, we present a new and improved analytic method for reproducing the Lee-Richards molecular volume, which is the most common volume definition for Poisson calculations. Overall, 1% errors are achieved for absolute solvation energies of a large set of proteins and relative solvation energies of protein conformations. We also introduce an accurate SASA approximation that uses the same machinery employed by our GB method and requires a small addition of computational cost. The combined methodology is shown to yield an efficient and accurate implicit solvent representation for simulations of biopolymers.     

Generalized energy-based fragmentation approach for computing the ground-state energies and properties of large molecules

We present a generalized energy-based fragmentation (GEBF) approach for approximately predicting the ground-state energies and molecular properties of large molecules, especially those charged and polar molecules. In this approach, the total energy (or properties) of a large molecule can be approximately obtained from energy (or properties) calculations on various small subsystems, each of which is constructed to contain a certain fragment and its local surroundings within a given distance. In the quantum chemistry calculation of a given subsystem, those distant atoms (outside this subsystem) are modeled as background point charges at the corresponding nuclear centers. This treatment allows long-range electrostatic interaction and polarization effects between distant fragments to be taken into account approximately, which are very important for polar and charged molecules. We also propose a new fragmentation scheme for constructing subsystems. Our test calculations at the Hartree-Fock and second-order M?ller-Plesser perturbation theory levels demonstrate that the approach could yield satisfactory ground-state energies, the dipole moments, and static polarizabilities for polar and charged molecules such as water clusters and proteins.     

Absolute pK(a) determinations for substituted phenols

The CBS-QB3 method was used to calculate the gas-phase free energy difference between 20 phenols and their respective anions, and the CPCM continuum solvation method was applied to calculate the free energy differences of solvation for the phenols and their anions. The CPCM solvation calculations were performed on both gas-phase and solvent-phase optimized structures. Absolute pK(a) calculations with solvated phase optimized structures for the CPCM calculations yielded standard deviations and root-mean-square errors of less than 0.4 pK(a) unit. This study is the most accurate absolute determination of the pK(a) values of phenols, and is among the most accurate of any such calculations for any group of compounds. The ability to make accurate predictions of pK(a) values using a coherent, well-defined approach, without external approximations or fitting to experimental data, is of general importance to the chemical community. The solvated phase optimized structures of the anions are absolutely critical to obtain this level of accuracy, and yield a more realistic charge separation between the negatively charged oxygen and the ring system of the phenoxide anions.     

Theoretical method for full ab initio calculation of DNA/RNA-ligand interaction energy

In this paper, we further develop the molecular fractionation with conjugate caps (MFCC) scheme for quantum mechanical computation of DNA-ligand interaction energy. We study three oligonuclear acid interaction systems: dinucleotide dCG/water, trinucleotide dCGT/water, and a Watson-Crick paired DNA segment, dCGT/dGCA. Using the basic MFCC approach, the nucleotide chains are cut at each phosphate group and a pair of conjugate caps (concaps) are inserted. Five cap molecules have been tested among which the dimethyl phosphate anion is proposed to be the standard concap for application. For each system, one-dimensional interaction potential curves are computed using the MFCC method and the calculated interaction energies are found to be in excellent agreement with corresponding results obtained from the full system ab initio calculations. The current study extends the application of the MFCC method to ab initio calculations for DNA- or RNA-ligand interaction energies.     

Absolute calculations of acidity of C-substituted tetrazoles in solution

The CBS-QB3 method was used to calculate the gas-phase free energy difference between nine tetrazole derivatives and their anions, and the DPCM and CPCM continuum solvation methods were applied to calculate the free energy differences of solvation. The calculations were performed on both gas-phase and solvent-phase optimized structures. Absolute pKa calculations using the CPCM method and the gas-phase optimized structures yielded mean unsigned error of 0.4 pKa unit. The calculations were made with the routine settings implemented in Gaussian 98. The study is as accurate as the best reported so far for six carboxylic acids and phenols and, to our knowledge, the best reported for the acidities of heterocyclic compounds in solution.     

Conformational dependence of charges in protein simulations

We have studied the conformational dependence of molecular mechanics atomic charges for proteins by calculating the charges fitted to the quantum mechanical (QM) electrostatic potential (ESP) for all atoms in complexes between avidin and seven biotin analogues for 20 snapshots from molecular dynamics simulations. We have studied how various other charge sets reproduce those charges. The QM charges, even if averaged over all snapshots or all residues, in general have a larger magnitude than standard Amber charges, indicating that the restraint toward zero in the restrained ESP method is too strong. This has a significant influence on the electrostatic conformational energies and the interaction energy between the biotin ligand and the protein, giving a difference between the QM and Amber charges of 43 and 8 kJ/mol for the negatively charged and neutral biotin analogues, respectively (3-4%). However, this energy difference is strongly reduced if the solvation energy (calculated by the Poisson-Boltzmann or Generalized Born methods) is added, viz., to 7 kJ/mol for charged and 3 kJ/mol for uncharged ligand. In fact, charges need to be recalculated with a QM method only for residues within 7 or 4 A of the ligand, if the error should be less than 4 kJ/mol. Unfortunately, the QM charges do not give significantly better MM/PBSA estimates of ligand-binding affinities than standard Amber charges.