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.     

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.     

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.     

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.     

An efficient approach for ab initio energy calculation of biopolymers

We present a new method for efficient total-energy calculation of biopolymers using the density-matrix (DM) scheme based on the molecular fractionation with conjugate caps (MFCC) approach. In this MFCC-DM method, a biopolymer such as a protein is partitioned into properly capped fragments whose density matrices are calculated by conventional ab initio methods which are then assembled to construct the full system density matrix. The assembled full density matrix is then employed to calculate the total energy and dipole moment of the protein using Hartree-Fock or density-functional theory methods. Using this MFCC-DM method, the self-consistent-field procedure for solving the full Hamiltonian problem is avoided and an efficient approach for ab initio energy calculation of biopolymers is achieved. Two implementations of the approach are presented in this paper. Systematic numerical studies are carried out on a series of extended polyglycines CH3CO-(GLY)n-NHCH3(n = 3-25) and excellent results are obtained.     

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.     

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.     

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.     

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.     

An efficient fragment-based approach for predicting the ground-state energies and structures of large molecules

An efficient fragment-based approach for predicting the ground-state energies and structures of large molecules at the Hartree-Fock (HF) and post-HF levels is described. The physical foundation of this approach is attributed to the "quantum locality" of the electron correlation energy and the HF total energy, which is revealed by a new energy decomposition analysis of the HF total energy proposed in this work. This approach is based on the molecular fractionation with conjugated caps (MFCC) scheme (Zhang, D. W.; Zhang, J. Z. H. J. Chem. Phys. 2003, 119, 3599), by which a macromolecule is partitioned into various capped fragments and conjugated caps formed by two adjacent caps. We find that the MFCC scheme, if corrected by the interaction between non-neighboring fragments, can be used to predict the total energy of large molecules only from energy calculations on a series of small subsystems. The approach, named as energy-corrected MFCC (EC-MFCC), computationally achieves linear scaling with the molecular size. Our test calculations on a broad range of medium- and large molecules demonstrate that this approach is able to reproduce the conventional HF and second-order Moller-Plesset perturbation theory (MP2) energies within a few millihartree in most cases. With the EC-MFCC optimization algorithm described in this work, we have obtained the optimized structures of long oligomers of trans-polyacetylene and BN nanotubes with up to about 400 atoms, which are beyond the reach of traditional computational methods. In addition, the EC-MFCC approach is also applied to estimate the heats of formation for a series of organic compounds. This approach provides an appealing approach alternative to the traditional additivity rules based on either bond or group contributions for the estimation of thermochemical properties.     

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.     

A new fragment-based approach for calculating electronic excitation energies of large systems

We present a new fragment-based scheme to calculate the excited states of large systems without necessity of a Hartree-Fock (HF) solution of the whole system. This method is based on the implementation of the renormalized excitonic method [M. A. Hajj et al., Phys. Rev. B 72, 224412 (2005)] at ab initio level, which assumes that the excitation of the whole system can be expressed by a linear combination of various local excitations. We decomposed the whole system into several blocks and then constructed the effective Hamiltonians for the intra- and inter-block interactions with block canonical molecular orbitals instead of widely used localized molecular orbitals. Accordingly, we avoided the prerequisite HF solution and the localization procedure of the molecular orbitals in the popular local correlation methods. Test calculations were implemented for hydrogen molecule chains at the full configuration interaction, symmetry adapted cluster/symmetry adapted cluster configuration interaction, HF/configuration interaction singles (CIS) levels and more realistic polyene systems at the HF/CIS level. The calculated vertical excitation energies for lowest excited states are in reasonable accordance with those determined by the calculations of the whole systems with traditional methods, showing that our new fragment-based method can give good estimates for low-lying energy spectra of both weak and moderate interaction systems with economic computational costs.     

Computation of dipole, quadrupole, and octupole surfaces from the variational two-electron reduced density matrix method

Recent advances in the direct determination of the two-electron reduced density matrix (2-RDM) by imposing known N-representability conditions have mostly focused on the accuracy of molecular potential energy surfaces where multireference effects are significant. While the norm of the 2-RDM's deviation from full configuration interaction has been computed, few properties have been carefully investigated as a function of molecular geometry. Here the dipole, quadrupole, and octupole moments are computed for a range of molecular geometries. The addition of Erdahl's T2 condition [Int. J. Quantum Chem. 13, 697 (1978)] to the D, Q, and G conditions produces dipole and multipole moments that agree with full configuration interaction in a double-zeta basis set at all internuclear distances.     

Electron smearing in DFT calculations: A case study of doxorubicin interaction with single‐walled carbon nanotubes

To address the choice of an appropriate value of electron smearing to facilitate self‐consistent field (SCF) convergence, we studied the interaction of doxorubicin with short armchair and zigzag single‐walled carbon nanotube models with closed caps, at the PWC/DNP level of density functional theory. By gradually reducing the electron smearing value from a large and most commonly used one of 0.005 Ha to zero (Fermi occupation), we monitored the changes in close contacts between the interacting species, total energy of the molecular system, highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) energy and isosurfaces, HOMO‐LUMO gap energy, and plots of electrostatic potential. It became evident that the commonly used smearing values of ≥0.001 Ha can alter the results significantly (for example, by one order of magnitude for HOMO–LUMO gap energy). We suggest the setting of electron smearing value at 0.0001 Ha, which does not imply too high computation cost and can guarantee the results close to the ones obtained with Fermi occupation. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2011     

A new quantum method for electrostatic solvation energy of protein

A new method that incorporates the conductorlike polarizable continuum model (CPCM) with the recently developed molecular fractionation with conjugate caps (MFCC) approach is developed for ab initio calculation of electrostatic solvation energy of protein. The application of the MFCC method makes it practical to apply CPCM to calculate electrostatic solvation energy of protein or other macromolecules in solution. In this MFCC-CPCM method, calculation of protein solvation is divided into calculations of individual solvation energies of fragments (residues) embedded in a common cavity defined with respect to the entire protein. Besides computational efficiency, the current approach also provides additional information about contribution to protein solvation from specific fragments. Numerical studies are carried out to calculate solvation energies for a variety of peptides including alpha helices and beta sheets. Excellent agreement between the MFCC-CPCM result and those from the standard full system CPCM calculation is obtained. Finally, the MFCC-CPCM calculation is applied to several real proteins and the results are compared to classical molecular mechanics Poisson-Boltzmann (MM/PB) and quantum Divid-and-Conque Poisson-Boltzmann (D&C-PB) calculations. Large wave function distortion energy (solute polarization energy) is obtained from the quantum calculation which is missing in the classical calculation. The present study demonstrates that the MFCC-CPCM method is readily applicable to studying solvation of proteins.     

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.     

An alternative series solution to the isotropic quartic oscillator inN dimensions

The series solution of theN-dimensional isotropic quartic oscillator weighted by an appropriate function which exhibits the correct asymptotic behavior of the wave function is presented. The numerical performance of the solution in Hill's determinant picture is excellent, and yields the energy spectrum of the system to any desired accuracy for the full range of the coupling constant. Furthermore, it converges to the well-known exact solution of the unperturbed harmonic oscillator wave function, when the anharmonic interaction vanishes.     

A hybrid approach combining energy density analysis with the interaction energy decomposition method

We propose a new analysis technique for characterizing molecular interactions that combines an energy decomposition scheme, such as the Kitaura-Morokuma decomposition method, with energy density analysis, which partitions the total energy of the system into atomic contributions. The combined scheme, termed Interaction-EDA, enables us to estimate the local contribution of interaction energy components, such as electrostatic, exchange, polarization, and charge transfer. The evaluation of the local interaction energy is rather important in large systems. For a numerical assessment, the Interaction-EDA method is applied to the process of CO adsorption on Si(100) - (2 x 1) surface.     

B-spline method for energy minimization in grid-based molecular mechanics calculations

A method is described for molecular mechanics calculations based on a cubic B-spline approximation of the potential energy. This method is useful when parts of the system are allowed to remain fixed in position so that a potential energy grid can be precalculated and used to approximate the interaction energy between parts of a molecule or between molecules. We adapted and modified the conventional B-spline method to provide an approximation of the Empirical Conformational Energy Program for Peptides (ECEPP) potential energy function. The advantage of the B-spline method over simpler approximations is that the resulting B-spline function is C2 continuous, which allows minimization of the potential energy by any local minimization algorithm. The standard B-spline method provides a good approximation of the electrostatic energy; but in order to reproduce the Lennard–Jones and hydrogen-bonding functional forms accurately, it was necessary to modify the standard B-spline method. This modification of the B-spline method can also be used to improve the accuracy of trilinear interpolation for simulations that do not require continuous derivatives. As an example, we apply the B-spline method to rigid-body docking energy calculations using the ECEPP potential energy function. Energies are calculated for the complex of Phe-Pro-Arg with thrombin. For this system, we compare the performance of the B-spline method to that of the standard pairwise summation in terms of speed, accuracy, and overhead costs for a variety of grid spacings. In our rigid-body docking calculations, the B-spline method provided an accurate approximation of the total energy of the system, and it resulted in an 180-fold reduction in the time required for a single energy and gradient calculation for this system. © 1998 John Wiley & Sons, Inc. J Comput Chem 19: 71–85, 1998