PEACH 4 with ABINIT-MP: a general platform for classical and quantum simulations of biological molecules

A program package for molecular simulations of biological molecules was developed. The package, "PEACH version 4 with ABINIT-MP version 20021029," was constructed by incorporating ABINIT-MP, a program for the fragment molecular orbital (FMO) method [Chem.Phys. Lett. 313 (1999) 701], into PEACH, a program package for classical molecular dynamics simulations (MD). A few capabilities of the package were demonstrated. First, high parallel efficiency of FMO was demonstrated in a single point calculation of a protein. Second,FMO-MD simulations [Chem. Phys. Lett. 372 (2003) 342] of a peptide were performed with and without explicit solvent, and the simulations showed the influence of the solvent on the electronic state of the peptide.     

Application of fragment molecular orbital scheme to silicon-containing systems

The fragment molecular orbital (FMO) scheme has been successfully used for a variety of large-scale molecules such as proteins and nucleic acids so far. We have applied the FMO calculations to the silicon-containing systems like polysilanes. The error caused by the fragmentation was examined by the Hartree–Fock method and the second-order Møller-Plesset (MP2) perturbation method for the ground state energy. The dynamic polarizability as a linear response property was also evaluated with and without the fragmentation. A series of numerical comparisons showed that the FMO scheme is applicable to silicon-based molecules with reasonable accuracy. This implied a potential availability of FMO calculations for the issues relevant to nanoscience and nanotechnology.     

Application of resolution of identity approximation of second-order M?ller�CPlesset perturbation theory to three-body fragment molecular orbital method

The resolution of identity (RI) approximation of second-order M?ller?CPlesset perturbation (MP2) theory, termed as RI-MP2, is applied to three-body fragment molecular orbital (FMO3) method. New implementation of FMO3 RI-MP2 is developed based on an efficient parallel RI-MP2 code developed recently in our group. Using this new implementation, the accuracy and computational time of FMO3 RI-MP2 calculations are assessed for water clusters, polyalanines, and proteins. The errors arising from RI-MP2 are sufficiently small in the FMO3 MP2 calculations that they give quantitative accuracy for practical chemical applications. Considerable time savings are attained in the FMO3 MP2 calculations with the application of RI-MP2.     

Geometry optimizations of open-shell systems with the fragment molecular orbital method

The ability to perform geometry optimizations on large molecular systems is desirable for both closed- and open-shell species. In this work, the restricted open-shell Hartree-Fock (ROHF) gradients for the fragment molecular orbital (FMO) method are presented. The accuracy of the gradients is tested, and the ability of the method to reproduce adiabatic excitation energies is also investigated. Timing comparisons between the FMO method and full ab initio calculations are also performed, demonstrating the efficiency of the FMO method in modeling large open-shell systems.     

Ab initio quantum chemical calculation of electron density,electrostatic potential,and electric field of biomolecule based on fragment molecular orbital method

Efficient quantum chemical calculations of electrostatic properties, namely, the electron density (EDN), electrostatic potential (ESP), and electric field (EFL), were performed using the fragment molecular orbital (FMO) method. The numerical errors associated with the FMO scheme were examined at the HF, MP2, and RI‐MP2 levels of theory using 4 small peptides. As a result, the FMO errors in the EDN, ESP, and EFL were significantly smaller than the magnitude of the electron correlation effects, which indicated that the FMO method provides sufficiently accurate values of electrostatic properties. In addition, an attempt to reduce the computational effort was proposed by combining the FMO scheme and a point charge approximation. The error due to this approximation was examined using 2 proteins, prion protein and human immunodeficiency virus type 1 protease. As illustrative examples, the ESP values at the molecular surface of these proteins were calculated at the MP2 level of theory.     

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.     

Assessment and acceleration of binding energy calculations for protein–ligand complexes by the fragment molecular orbital method

In the field of drug discovery, it is important to accurately predict the binding affinities between target proteins and drug applicant molecules. Many of the computational methods available for evaluating binding affinities have adopted molecular mechanics‐based force fields, although they cannot fully describe protein–ligand interactions. A noteworthy computational method in development involves large‐scale electronic structure calculations. Fragment molecular orbital (FMO) method, which is one of such large‐scale calculation techniques, is applied in this study for calculating the binding energies between proteins and ligands. By testing the effects of specific FMO calculation conditions (including fragmentation size, basis sets, electron correlation, exchange‐correlation functionals, and solvation effects) on the binding energies of the FK506‐binding protein and 10 ligand complex molecule, we have found that the standard FMO calculation condition, FMO2‐MP2/6‐31G(d), is suitable for evaluating the protein–ligand interactions. The correlation coefficient between the binding energies calculated with this FMO calculation condition and experimental values is determined to be R = 0.77. Based on these results, we also propose a practical scheme for predicting binding affinities by combining the FMO method with the quantitative structure–activity relationship (QSAR) model. The results of this combined method can be directly compared with experimental binding affinities. The FMO and QSAR combined scheme shows a higher correlation with experimental data (R = 0.91). Furthermore, we propose an acceleration scheme for the binding energy calculations using a multilayer FMO method focusing on the protein–ligand interaction distance. Our acceleration scheme, which uses FMO2‐HF/STO‐3G:MP2/6‐31G(d) at R_int = 7.0 Å, reduces computational costs, while maintaining accuracy in the evaluation of binding energy. © 2015 Wiley Periodicals, Inc.     

A parallelized integral-direct second-order Møller–Plesset perturbation theory method with a fragment molecular orbital scheme

We propose a parallelized integral-direct algorithm of the second-order Møller–Plesset perturbation theory (MP2) as a size-consistent correlated method. The algorithm is a modification of the recipe by Mochizuki et al. [(1996) Theor Chim Acta 93:211]. There is no need to communicate the bulky data of integrals across worker processes, keeping the formal fifth-power dependence on the number of basis functions. A multiple integral screening procedure is incorporated to reduce the operation costs effectively. An approximate MP2 density matrix can also be directly calculated through the integral contraction with orbital energies. We implement the MP2 code by accepting Kitauras fragment molecular orbital (FMO) scheme as in the program ABINIT-MP developed by Nakano et al. [(2002) Chem Phys Lett 351:475]. The error in the FMO–MP2 energies is found to be within the order of the chemical accuracy. Timing and parallel acceleration results are shown for test molecules.     

Multilayer formulation of the fragment molecular orbital method (FMO)

The fragment molecular orbital method (FMO) has been generalized to allow for multilayer structure. Fragments are assigned to layers, and each layer can be described with a different basis set and/or level of electron correlation. Interlayer boundaries are treated in the general spirit of the FMO method since they also coincide with some interfragment boundaries. The question of the one- and two-layer FMO accuracy dependence upon the fragmentation scheme is also addressed. The new method has been applied to predict the reaction barrier and the reaction heat for the Diels-Alder reaction with a representative set of reactants based on dividing fragments in two layers. The 6-31G* basis set has been used for the active site and the 6-31G*, 6-31G, 3-21G, and STO-3G basis sets have been used for the substituents. Different levels of electron correlation (RHF, B3LYP, and MP2) have been applied to layers in systematic fashion. The one-layer FMO errors in the reaction barrier and the reaction heat were 2.0 kcal/mol or less for all levels applied (RHF, B3LYP, and MP2), relative to full ab initio methods. For the two-layer method the error was found to be several kcal/mol. Benchmark calculations of the activation barrier for the decarboxylation of phenylcyanoacetate by beta-cyclodextrin demonstrated that the two-layer calculations are efficient, being 36 times faster than the regular DFT, as well as accurate, with the error being 1.0 kcal/mol.     

Theoretical study of the prion protein based on the fragment molecular orbital method

We performed fragment molecular orbital (FMO) calculations to examine the molecular interactions between the prion protein (PrP) and GN8, which is a potential curative agent for prion diseases. This study has the following novel aspects: we introduced the counterpoise method into the FMO scheme to eliminate the basis set superposition error and examined the influence of geometrical fluctuation on the interaction energies, thereby enabling rigorous analysis of the molecular interaction between PrP and GN8. This analysis could provide information on key amino acid residues of PrP as well as key units of GN8 involved in the molecular interaction between the two molecules. The present FMO calculations were performed using an original program developed in our laboratory, called “Parallelized ab initio calculation system based on FMO (PAICS)”. © 2009 Wiley Periodicals, Inc. J Comput Chem 2009     

Exploring chemistry with the fragment molecular orbital method

The fragment molecular orbital (FMO) method makes possible nearly linear scaling calculations of large molecular systems, such as water clusters, proteins and DNA. In particular, FMO has been widely used in biochemical applications involving protein-ligand binding and drug design. The method has been efficiently parallelized suitable for petascale computing. Many commonly used wave functions and solvent models have been interfaced with FMO. We review the historical background of FMO, and summarize its method development and applications.     

Analytic gradient for second order Møller-Plesset perturbation theory with the polarizable continuum model based on the fragment molecular orbital method

A new energy expression is proposed for the fragment molecular orbital method interfaced with the polarizable continuum model (FMO/PCM). The solvation free energy is shown to be more accurate on a set of representative polypeptides with neutral and charged residues, in comparison to the original formulation at the same level of the many-body expansion of the electrostatic potential determining the apparent surface charges. The analytic first derivative of the energy with respect to nuclear coordinates is formulated at the second-order M?ller-Plesset (MP2) perturbation theory level combined with PCM, for which we derived coupled perturbed Hartree-Fock equations. The accuracy of the analytic gradient is demonstrated on test calculations in comparison to numeric gradient. Geometry optimization of the small Trp-cage protein (PDB: 1L2Y) is performed with FMO/PCM/6-31(+)G(d) at the MP2 and restricted Hartree-Fock with empirical dispersion (RHF/D). The root mean square deviations between the FMO optimized and NMR experimental structure are found to be 0.414 and 0.426 A? for RHF/D and MP2, respectively. The details of the hydrogen bond network in the Trp-cage protein are revealed.     

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.     

All electron quantum chemical calculation of the entire enzyme system confirms a collective catalytic device in the chorismate mutase reaction

To elucidate the catalytic power of enzymes, we analyzed the reaction profile of Claisen rearrangement of Bacillus subtilis chorismate mutase (BsCM) by all electron quantum chemical calculations using the fragment molecular orbital (FMO) method. To the best of our knowledge, this is the first report of ab initio-based quantum chemical calculations of the entire enzyme system, where we provide a detailed analysis of the catalytic factors that accomplish transition-state stabilization (TSS). FMO calculations deliver an ab initio-level estimate of the intermolecular interaction between the substrate and the amino acid residues of the enzyme. To clarify the catalytic role of Arg90, we calculated the reaction profile of the wild-type BsCM as well as Lys90 and Cit90 mutant BsCMs. Structural refinement and the reaction path determination were performed at the ab initio QM/MM level, and FMO calculations were applied to the QM/MM refined structures. Comparison between three types of reactions established two collective catalytic factors in the BsCM reaction: (1) the hydrogen bonds connecting the Glu78-Arg90-substrate cooperatively control the stability of TS relative to the ES complex and (2) the positive charge on Arg90 polarizes the substrate in the TS region to gain more electrostatic stabilization.     

Rapid and accurate assessment of GPCR–ligand interactions Using the fragment molecular orbital‐based density‐functional tight‐binding method

The reliable and precise evaluation of receptor–ligand interactions and pair‐interaction energy is an essential element of rational drug design. While quantum mechanical (QM) methods have been a promising means by which to achieve this, traditional QM is not applicable for large biological systems due to its high computational cost. Here, the fragment molecular orbital (FMO) method has been used to accelerate QM calculations, and by combining FMO with the density‐functional tight‐binding (DFTB) method we are able to decrease computational cost 1000 times, achieving results in seconds, instead of hours. We have applied FMO‐DFTB to three different GPCR–ligand systems. Our results correlate well with site directed mutagenesis data and findings presented in the published literature, demonstrating that FMO‐DFTB is a rapid and accurate means of GPCR–ligand interactions. © 2017 Authors. Journal of Computational Chemistry Published by Wiley Periodicals, Inc.     

Fragmentation at sp2 carbon atoms in fragment molecular orbital method

In the fragment molecular orbital (FMO) method, a given molecular system is usually fragmented at sp³ carbon atoms. However, fragmentation at different sites sometimes becomes necessary. Hence, we propose fragmentation at sp² carbon atoms in the FMO method. Projection operators are constructed using sp² local orbitals. To maintain practical accuracy, it is essential to consider the three-body effect. In order to suppress the corresponding increase of computational cost, we propose approximate models considering local trimers. Numerical verification shows that the present models are as accurate as or better than the standard FMO2 method in total energy with fragmentation at sp³ carbon atoms.     

Is large-scale ab initio Hartree-Fock calculation chemically accurate? Toward improved calculation of biological molecule properties

Numerical errors in total energy values in large-scale Hartree–Fock calculations are discussed. To obtain total energy values within chemical accuracy, 0.01 kcal/mol, stricter numerical accuracy is required as basis size increases. In molecules with 10,000 basis sizes, such as proteins, numerical accuracy for total energy values must be retained to at least 11 digits (i.e., to the order of 1.0D-10) to keep accumulation of numerical errors less than the chemical accuracy (0.01 kcal/mol). With this criterion, we examined the sensitivity analysis of numerical accuracy in Hartree–Fock calculation by uniformly replacing the last bit of the mantissa part of a double-precision real number by zero in the Fock matrix construction step, the total energy calculation step, and the Fock matrix diagonalization step. Using a partial summation technique in the Fock matrix generation step, the numerical error for total energy value of molecules with basis size greater than 10,000 was within chemical accuracy (0.01 kcal/mol), whereas with the conventional method the numerical error with several thousand basis sets was larger than chemical accuracy. Computation of one Fock matrix element with parallel machines can include the partial summation technique automatically, so that parallel calculation yields not only high-performance computing but also more precise numerical solutions than the conventional sequential algorithm. We also found that the numerical error of the Householder-QR diagonalization routine is equal to or less than chemical accuracy, even with a matrix size of 10,000. ©1999 John Wiley & Sons, Inc. J Comput Chem 20: 443–454, 1999     

Three‐body expansion of the fragment molecular orbital method combined with density‐functional tight‐binding

The three‐body fragment molecular orbital (FMO3) method is formulated for density‐functional tight‐binding (DFTB). The energy, analytic gradient, and Hessian are derived in the gas phase, and the energy and analytic gradient are also derived for polarizable continuum model. The accuracy of FMO3‐DFTB is evaluated for five proteins, sodium cation in explicit solvent, and three isomers of polyalanine. It is shown that FMO3‐DFTB is considerably more accurate than FMO2‐DFTB. Molecular dynamics simulations for sodium cation in water are performed for 100 ps, yielding radial distribution functions and coordination numbers. © 2017 Wiley Periodicals, Inc.     

Second order Møller-Plesset perturbation theory based upon the fragment molecular orbital method

The fragment molecular orbital (FMO) method was combined with the second order M?ller-Plesset (MP2) perturbation theory. The accuracy of the method using the 6-31G(*) basis set was tested on (H(2)O)(n), n=16,32,64; alpha-helices and beta-strands of alanine n-mers, n=10,20,40; as well as on (H(2)O)(n), n=16,32,64 using the 6-31 + + G(**) basis set. Relative to the regular MP2 results that could be afforded, the FMO2-MP2 error in the correlation energy did not exceed 0.003 a.u., the error in the correlation energy gradient did not exceed 0.000 05 a.u./bohr and the error in the correlation contribution to dipole moment did not exceed 0.03 debye. An approximation reducing computational load based on fragment separation was introduced and tested. The FMO2-MP2 method demonstrated nearly linear scaling and drastically reduced the memory requirements of the regular MP2, making possible calculations with several thousands basis functions using small Pentium clusters. As an example, (H(2)O)(64) with the 6-31 + + G(**) basis set (1920 basis functions) can be run in 1 Gbyte RAM and it took 136 s on a 40-node Pentium4 cluster.     

Parallel Fock matrix construction with distributed shared memory model for the FMO‐MO method

A parallel Fock matrix construction program for FMO‐MO method has been developed with the distributed shared memory model. To construct a large‐sized Fock matrix during FMO‐MO calculations, a distributed parallel algorithm was designed to make full use of local memory to reduce communication, and was implemented on the Global Array toolkit. A benchmark calculation for a small system indicates that the parallelization efficiency of the matrix construction portion is as high as 93% at 1,024 processors. A large FMO‐MO application on the epidermal growth factor receptor (EGFR) protein (17,246 atoms and 96,234 basis functions) was also carried out at the HF/6‐31G level of theory, with the frontier orbitals being extracted by a Sakurai‐Sugiura eigensolver. It takes 11.3 h for the FMO calculation, 49.1 h for the Fock matrix construction, and 10 min to extract 94 eigen‐components on a PC cluster system using 256 processors. © 2010 Wiley Periodicals, Inc. J Comput Chem, 2010