High-performance transformation of protein structure representation from internal to Cartesian coordinates

We present a highly parallel algorithm to convert internal coordinates of a polymeric molecule into Cartesian coordinates. Traditionally, converting the structures of polymers (e.g., proteins) from internal to Cartesian coordinates has been performed serially, due to an inherent linear dependency along the polymer chain. We show this dependency can be removed using a tree-based concatenation of coordinate transforms between segments, and then parallelized efficiently on graphics processing units (GPUs). The conversion algorithm is applicable to protein engineering and fitting protein structures to experimental data, and we observe an order of magnitude speedup using parallel processing on a GPU compared to serial execution on a CPU.     

Replica exchange with guided annealing for accelerated sampling of disordered protein conformations

We critically examine a recently proposed convective replica exchange (cRE) method for enhanced sampling of protein conformation based on theoretical and numerical analysis. The results demonstrate that cRE and related replica exchange with guided annealing (RE‐GA) schemes lead to unbalanced exchange attempt probabilities and break detailed balance whenever the system undergoes slow conformational transitions (relative to the temperature diffusion timescale). Nonetheless, numerical simulations suggest that approximate canonical ensembles can be generated for systems with small conformational transition barriers. This suggests that RE‐GA maybe suitable for simulating intrinsically disordered proteins, an important class of newly recognized functional proteins. The efficacy of RE‐GA is demonstrated by calculating the conformational ensembles of intrinsically disordered kinase inducible domain protein. The results show that RE‐GA helps the protein to escape nonspecific compact states more efficiently and provides several fold speedups in generating converged and largely correct ensembles compared to the standard temperature RE. © 2014 Wiley Periodicals, Inc.     

Graphical approach for defining natural internal coordinates

A simple, customizable connectivity scheme is rigorously defined in which pairs of atoms are classified into three categories. The tools of graph theory are used to analyze the molecular graph and to efficiently find rings and ring assemblies through a combination of pruning and homeomorphic reduction. The definition of natural internal coordinates is extended in a nonredundant fashion for the various cases of weakly interacting components and for fused ring systems. The ring system coordinates were tested and found to be superior to Z-matrix coordinates. © 1998 John Wiley & Sons, Inc. J Comput Chem 19: 504–511, 1998     

Communication performance of d -meshes in molecular dynamics simulation

Communication algorithms, tailored for molecular dynamics simulation on d-meshes, are evaluated in terms of communication efficiency. It has been shown elsewhere that d-meshes are better than other regular topologies, e.g., hypercubes and standard toroidal 4-meshes, when compared in their diameter and average distance among nodes. Collective communication is needed in molecular dynamics simulation for the distribution of coordinates and calculation and distribution of new energies. We show that both collective communication patterns used in molecular dynamics can be efficiently solved with congestion-free algorithms for all-to-all communication based on store-and-forward routing and routing tables. Our results indicate that d-meshes compete with hypercubes in parallel computers. Therefore d-meshes can also be used as a communication upgrade of existing molecular dynamics simulation platforms and can be successfully applied to perform fast molecular dynamics simulation.     

Coordinate reduction for exploring chemical reaction paths

The potential energy surface for the reaction of a typical molecular system composed of N atoms is defined uniquely by 3N-6 coordinates. These coordinates can be defined by the Cartesian coordinates of the atomic centers (minus overall translation and rotation), or a set of internally defined coordinates such as bond stretches, angle bends, and torsions. By applying principal component analysis to the geometries along a reaction path, a reduced set of coordinates, d ≪ 3N-6, can be obtained. This reduced set of coordinates can reproduce the changes in geometry along the reaction path with chemical accuracy and may help improve the efficiency of reaction path optimization algorithms.     

Libcint: An efficient general integral library for Gaussian basis functions

An efficient integral library Libcint was designed to automatically implement general integrals for Gaussian‐type scalar and spinor basis functions. The library is able to evaluate arbitrary integral expressions on top of p, r and σ operators with one‐electron overlap and nuclear attraction, two‐electron Coulomb and Gaunt operators for segmented contracted and/or generated contracted basis in Cartesian, spherical or spinor form. Using a symbolic algebra tool, new integrals are derived and translated to C code programmatically. The generated integrals can be used in various types of molecular properties. To demonstrate the capability of the integral library, we computed the analytical gradients and NMR shielding constants at both nonrelativistic and 4‐component relativistic Hartree–Fock level in this work. Due to the use of kinetically balanced basis and gauge including atomic orbitals, the relativistic analytical gradients and shielding constants requires the integral library to handle the fifth‐order electron repulsion integral derivatives. The generality of the integral library is achieved without losing efficiency. On the modern multi‐CPU platform, Libcint can easily reach the overall throughput being many times of the I/O bandwidth. On a 20‐core node, we are able to achieve an average output 8.3 GB/s for C₆₀ molecule with cc‐pVTZ basis. © 2015 Wiley Periodicals, Inc.     

Peptide o‐Aminoanilides as Crypto‐Thioesters for Protein Chemical Synthesis

Fully unprotected peptide o‐aminoanilides can be efficiently activated by NaNO₂ in aqueous solution to furnish peptide thioesters for use in native chemical ligation. This finding enables the convergent synthesis of proteins from readily synthesizable peptide o‐aminoanilides as a new type of crypto‐thioesters. The practicality of this approach is shown by the synthesis of histone H2B from five peptide segments. Purification or solubilization tags, which are sometimes needed to improve the efficiency of protein chemical synthesis, can be incorporated into the o‐aminoanilide moiety, as demonstrated in the preparation of the cyclic protein lactocyclicin Q.     

Transformed Dirac equation for the hydrogen atom,comparison with previous approaches in momentum space,and the anomalous Zeeman effect in momentum representation

The solution of a unitarily transformed Dirac equation for the hydrogenic electron in zero magnetic field is investigated here. The momentum‐space representation is adopted as a natural recourse. The spinor part of the transformed wavefunction in momentum space can be easily prescribed for a central potential. Hence, for the Coulomb potential, a pair of equations is obtained for the radial components in momentum space. It is shown that starting from these radial equations, one can recover the equations previously derived by Rubinowicz, Lévy, and Lombardi for the problem of the Dirac hydrogen atom in momentum space. This establishes equivalence among different approaches based on the momentum representation, including the current treatment. The recovery of the equations due to Rubinowicz permits the exact eigenvalues to be written down and exact expressions to be derived for the radial components of the transformed wavefunction in momentum space. A new approach is adopted to carry out a reduction to the nonrelativistic regime and the nonrelativistic limit. At first the transformed momentum‐space equation for the hydrogen atom is rewritten in terms of the hyperspherical coordinates. The zeroth‐order solutions of the new equation are recovered in the limit c → ∞ where c is the speed of light. These are manifestly separable into positive‐ and negative‐energy forms. For positive energy, these solutions have nonvanishing upper components that are two‐component spinors. The latter exactly correspond to the single‐component, nonrelativistic, momentum‐space solutions derived by Fock. It is shown that when the upper component is corrected through first order in v²/c² but the separability is still maintained for the transformed wavefunction, one retrieves the Pauli equation in momentum space. It is also shown that for a hydrogen atom placed in a uniform magnetic field, the nonvanishing momentum‐space matrix elements representing the anomalous Zeeman effect have a simple form, namely, the product of a radial integral and an angular integral. These integrals are equal to the well‐known radial and angular integrals in coordinate representation. The matrix elements can be easily evaluated. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2003     

Techniques for geometry optimization: A comparison of cartesian and natural internal coordinates

A comparison is made between geometry optimization in Cartesian coordinates, using an appropriate initial Hessian, and natural internal coordinates. Results on 33 different molecules covering a wide range of symmetries and structural types demonstrate that both coordinate systems are of comparable efficiency. There is a marked tendency for natural internals to converge to global minima whereas Cartesian optimizations converge to the local minimum closest to the starting geometry. Because they can now be generated automatically from input Cartesians, natural internals are to be preferred over Z-matrix coordinates. General optimization strategies using internal coordinates and/or Cartesians are discussed for both unconstrained and constrained optimization. © John Wiley & Sons, Inc.     

A formulation of the quantum-mechanical many-body problem in terms of hyperspherical coordinates

Hyperspherical coordinates are used to construct a matrix representation of a general N-particle Hamiltonian in the case where the interaction is electrostatic. The Yukawa interaction can be treated similarly, as is shown in an appendix. The basis functions used to construct the matrix representation of H are mononomials inthe 3N coordinates of the particles, multiplied by functions of the hyperradius. Methods for transforming from this representation to a symmetry-adapted representation are also discussed.     

Multidimensional virtual-system coupled canonical molecular dynamics to compute free-energy landscapes of peptide multimer assembly

An enhanced-sampling method termed multidimensional virtual-system coupled canonical molecular dynamics (mD-VcMD) method is developed. In many cases, generalized-ensemble methods realizing enhanced sampling, for example, adaptive umbrella sampling, apply an effective potential, which is derived from temporarily assumed canonical distribution as a function of one or more arbitrarily defined reaction coordinates. However, it is not straightforward to estimate the appropriate canonical distribution, especially for cases applying multiple reaction coordinates. The current method, mD-VcMD, does not rely on the form of the canonical distribution. Therefore, it is practically useful to explore a high-dimensional reaction-coordinate space. In this article, formulation of mD-VcMD and its evaluation with the simple molecular models consisting of three or four alanine peptides are presented. We confirmed that mD-VcMD efficiently searched 2D and 3D reaction-coordinate spaces defined as interpeptide distances. Direct comparisons with results of long-term canonical MD simulations revealed that mD-VcMD produces correct canonical ensembles. © 2019 Wiley Periodicals, Inc.     

Unified treatment of complete orthonormal sets of nonrelativistic,quasirelativistic and relativistic sets of spinor wave functions,and Slater spinor orbitals in coordinate,momentum and four-dimensional spaces

Using the properties of tensor spherical harmonics introduced by the author in previous paper (Guseinov, Phys Lett A 372:44, 2007) and complete orthonormal scalar basis sets of nonrelativistic -exponential type orbitals ( -ETO), - momentum space orbitals ( -MSO) and z ^α-hyperspherical harmonics (z ^α-HSH) for particles with spin s = 0 the new analytical relations for the quasirelativistic and relativistic spinor wave functions and Slater spinor orbitals in coordinate, momentum and four-dimensional spaces are derived, where α = 1, 0, −1, −2,.... The 2-component quasirelativistic and 4-component relativistic spinor wave functions obtained are complete without the inclusion of the continuum. The relativistic spinor wave function sets and Slater spinor orbitals are expressed through the corresponding quasirelativistic spinor wave functions and Slater spinor orbitals, respectively. The analytical formulas for overlap integrals over quasirelativistic and relativistic Slater spinor orbitals with the same screening constants in coordinate space are also derived.     

Multiobjective evolutionary algorithm with many tables for purely ab initio protein structure prediction

This article focuses on the development of an approach for ab initio protein structure prediction (PSP) without using any earlier knowledge from similar protein structures, as fragment‐based statistics or inference of secondary structures. Such an approach is called purely ab initio prediction. The article shows that well‐designed multiobjective evolutionary algorithms can predict relevant protein structures in a purely ab initio way. One challenge for purely ab initio PSP is the prediction of structures with β‐sheets. To work with such proteins, this research has also developed procedures to efficiently estimate hydrogen bond and solvation contribution energies. Considering van der Waals, electrostatic, hydrogen bond, and solvation contribution energies, the PSP is a problem with four energetic terms to be minimized. Each interaction energy term can be considered an objective of an optimization method. Combinatorial problems with four objectives have been considered too complex for the available multiobjective optimization (MOO) methods. The proposed approach, called “Multiobjective evolutionary algorithms with many tables” (MEAMT), can efficiently deal with four objectives through the combination thereof, performing a more adequate sampling of the objective space. Therefore, this method can better map the promising regions in this space, predicting structures in a purely ab initio way. In other words, MEAMT is an efficient optimization method for MOO, which explores simultaneously the search space as well as the objective space. MEAMT can predict structures with one or two domains with RMSDs comparable to values obtained by recently developed ab initio methods (GAPF_CG, I‐PAES, and Quark) that use different levels of earlier knowledge. © 2013 Wiley Periodicals, Inc.     

Unique Euler angles and self‐consistent multiplication tables for double point groups

The standard Euler angle parameterization of rotations is not unique. This is a particular problem when considering spinor representations. We enlarge the domain of the Euler angles from an SO₃ covering to an SU₂ covering, 0≤α<2π, 0≤β≤π, 0≤γ<4π. With this modification we can find unique Euler angles for operations of the double groups and thus construct self‐consistent group tables for those groups. Factor systems can then be described for the projective representations. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 75: 1–9, 1999     

T-Analyst: a program for efficient analysis of protein conformational changes by torsion angles

T-Analyst is a user-friendly computer program for analyzing trajectories from molecular modeling. Instead of using Cartesian coordinates for protein conformational analysis, T-Analyst is based on internal bond-angle-torsion coordinates in which internal torsion angle movements, such as side-chain rotations, can be easily detected. The program computes entropy and automatically detects and corrects angle periodicity to produce accurate rotameric states of dihedrals. It also clusters multiple conformations and detects dihedral rotations that contribute hinge-like motions. Correlated motions between selected dihedrals can also be observed from the correlation map. T-Analyst focuses on showing changes in protein flexibility between different states and selecting representative protein conformations for molecular docking studies. The program is provided with instructions and full source code in Perl.     

Calculation of molecular vibrations: Selective scaling factors for semiempirical force constants

The complete force constant matrices of a set of 50 aliphatic and aromatic hydrocarbons are calculated at the density functional theory B3LYP/6–31+G(d, p) and semiempirical PM3 levels of theory. After transformation from Cartesian to nonredundant internal coordinates, the errors in the semiempirical force constants are systematically analyzed. The force constants of the C(SINGLE BOND)C stretching coordinates can be easily corrected by a second-order fit. Thus, only two parameters are needed to reduce the mean error from 21.2 to 1.23%. The errors of other internal coordinates, particulary those including torsional modes, exhibit a larger diversity. The performance of the correction scheme in predicting vibrational spectra is shown for several examples including buckminsterfullerene (C₆₀). © 1997 John Wiley & Sons, Inc. J Comput Chem 18 : 2050–2059, 1997     

Development of spin‐dependent relativistic open‐shell Hartree–Fock theory with time‐reversal symmetry (II): The restricted open‐shell approach

An open‐shell Hartree–Fock (HF) theory for spin‐dependent two‐component relativistic calculations, termed the Kramers‐restricted open‐shell HF (KROHF) method, is developed. The present KROHF method is defined as a relativistic analogue of ROHF using time‐reversal symmetry and quaternion algebra, based on the Kramers‐unrestricted HF (KUHF) theory reported in our previous study (Int. J. Quantum Chem., doi: 10.1002/qua.25356 ). As seen in the nonrelativistic ROHF theory, the ambiguity of the KROHF Fock operator gives physically meaningless spinor energies. To avoid this problem, the canonical parametrization of KROHF to satisfy Koopmans' theorem is also discussed based on the procedure proposed by Plakhutin et al. (J. Chem. Phys. 2006 , 125, 204110). Numerical assessments confirmed that KROHF using Plakhutin's canonicalization procedure correctly gives physical spinor energies within the frozen‐orbital approximation under spin–orbit interactions.     

Determination of galacturonic acid content in pectin from fruit juices by liquid chromatographydiode array detection-electrospray ionization tandem mass spectrometry

A reversed-phase high-performance liquid chromatographic (HPLC) method is applied for the determination of galacturonic acid (GA) of pectins in different commercial fruit juices. The separation was carried out on a C₁₈ column using precolumn derivatization with p-aminobenzoic acid (p-ABA) and UV detection at 304 nm. The identification of GA was confirmed by high-performance liquid chromatography/electrospray ionization tandem mass spectrometry (HPLC-ESI-MS/MS) in positive ion mode. The concentration of GA in the samples analyzed ranged from 12.9 ± 0.5 to 49.4 ± 0.5 mg_GA L⁻¹. Amongst the samples analyzed, mango juice was found to be richest in GA content, and therefore a good source of pectins. Detection and quantification limits of the described methodology were 1.2 and 3.9 mg L⁻¹, respectively. Quantitative GA recoveries in the beverages had a range between 90 and 98%. The results showed that the HPLC method proposed was precise and suitable for the identification and quantification of GA in commercial fruit juices.     

Spot overlapping in two-dimensional polyacrylamide gel electrophoresis separations: a statistical study of complex protein maps

A statistical approach able to extract the information contained in a two-dimenisional polyacrylamide gel electrophoresis (2-D PAGE) separation is here reported. The method is based on the quantitative theory of peak overlapping, a procedure previously developed by the authors and here extended to 2-D separations. The whole map is divided into many strips in order to obtain 1-D separations on which the statistic procedure is applied: the developed algorithms, on the basis of spot experimental data (intensity and spatial coordinates) permit to estimate the intrinsic number of components and to single out the specific order present in spot positions. The procedure was validated on computer-simulated maps. Its applicability to real samples was tested on maps obtained from literature sources. The following important information on protein mixtures can be extracted: (i) the number of proteins can be accurately estimated, on the basis of the spatial coordinates and intensities of spots detected in the 2-D PAGE map; (ii) the model describing distribution of interdistance between adjacent spots can be identified in both the separation dimensions; (iii) the presence of repeated interdistances in spot positions in the maps can be easily singled out: these regularities suggest specific protein modifications.     

Constrained optimization in delocalized internal coordinates

Using the recently introduced delocalized internal coordinates, in conjunction with the classical method of Lagrange multipliers, an algorithm for constrained optimization is presented in which the desired constraints do not have to be satisfied in the starting geometry. The method used is related to a previous algorithm by the same author for constrained optimization in Cartesian coordinates [J. Comput. Chem., 13 , 240 (1992)], but is simpler and far more efficient. Any internal (distance or angle/torsion) constraint can be imposed between any atoms in the system whether or not the atoms involved are formally bonded. Imposed constraints can be satisfied exactly. © 1997 John Wiley & Sons, Inc. J Comput Chem 18 :1079–1095, 1997