Parallelized Natural Extension Reference Frame: Parallelized Conversion from Internal to Cartesian Coordinates

The conversion of polymer parameterization from internal coordinates (bond lengths, angles, and torsions) to Cartesian coordinates is a fundamental task in molecular modeling, often performed using the natural extension reference frame (NeRF) algorithm. NeRF can be parallelized to process multiple polymers simultaneously, but is not parallelizable along the length of a single polymer. A mathematically equivalent algorithm, pNeRF, has been derived that is parallelizable along a polymer's length. Empirical analysis demonstrates an order-of-magnitude speed up using modern GPUs and CPUs. In machine learning-based workflows, in which partial derivatives are backpropagated through NeRF equations and neural network primitives, switching to pNeRF can reduce the fractional computational cost of coordinate conversion from over two-thirds to around 10%. An optimized TensorFlow-based implementation of pNeRF is available on GitHub at https://github.com/aqlaboratory/pnerf © 2018 Wiley Periodicals, Inc.     

Quasi-Hamiltonian equations of motion for internal coordinate molecular dynamics of polymers

Conventional molecular dynamics simulations of macromolecules require long computational times because the most interesting motions are very slow compared to the fast oscillations of bond lengths and bond angles that limit the integration time step. Simulation of dynamics in the space of internal coordinates, that is, with bond lengths, bond angles, and torsions as independent variables, gives a theoretical possibility of eliminating all uninteresting fast degrees of freedom from the system. This article presents a new method for internal coordinate molecular dynamics simulations of macromolecules. Equations of motion are derived that are applicable to branched chain molecules with any number of internal degrees of freedom. Equations use the canonical variables and they are much simpler than existing analogs. In the numerical tests the internal coordinate dynamics are compared with the traditional Cartesian coordinate molecular dynamics in simulations of a 56 residue globular protein. For the first time it was possible to compare the two alternative methods on identical molecular models in conventional quality tests. It is shown that the traditional and internal coordinate dynamics require the same time step size for the same accuracy and that in the standard geometry approximation of amino acids, that is, with fixed bond lengths, bond angles, and rigid aromatic groups, the characteristic step size is 4 fs, which is 2 times higher than with fixed bond lengths only. The step size can be increased up to 11 fs when rotation of hydrogen atoms is suppressed. © 1997 by John Wiley & Sons, Inc. J Comput Chem 18 : 1354–1364, 1997     

Molecular chains with fixed bond lengths and angles: a study based on quantum statistical mechanics

This work studies large three-dimensional open molecular chains at thermal equilibrium in which bond lengths and angles are fixed (hard variables), based upon quantum statistics. A model for a chain formed by N particles interacting through harmonic-like vibrational potentials is treated in the high-frequency limit in which all bond lengths and angles become constrained, while other N angles (soft variables) remain unconstrained. The associated quantum partition function is bounded rigorously, using a variational inequality (related to the Born-Oppenheimer approximation), by another quantum partition function, Z. The total vibrational zero-point energy is shown to be independent of the soft variables thereby solving for this model a generic difficulty in the elimination of hard variables. Z depends only on soft variables and, under certain conditions, it can be approximated by a classical partition function Z_c. The latter satisfies the equipartition principle and it differs from other classical partition functions for related molecular chains. The extension of the model when only part of the bond angles become fixed in the high-frequency limit is outlined. As another generalization, a systematic study of macromolecules, as composed of electrons and heavy particles with Coulomb interactions, is also presented. Its exact quantum partition function is bounded, supposing that the effective molecular potential also tends to constrain all bond lengths and angles, and under suitable assumptions, by another quantum partition function. The latter depends only on the remaining soft variables and it generalizes the one obtained for the first model.     

Polymer dynamics in torsion space

The large scale motion of proteins, or covalently bonded polymers in general, is governed by the dynamics of the torsion angles, with bond lengths and bond angles kept approximately constant. In the present work, the Lagrangian equations of torsion motion are derived for a general macromolecule. The dynamics is implemented numerically for a test protein, using the velocity Verlet method as the integrator. The results indicate time steps of up to about 30 fs can be used for short time (up to at least 20 ps) simulations, before the dynamics and energy start to differ significantly from results obtained with smaller time steps. For longer time simulations, up to 1000 ps, a time step of 10 fs is relatively safe. © 2010 Wiley Periodicals, Inc. J Comput Chem, 2010     

Conformational analysis of nucleic acid molecules with flexible furanose rings in dihedral angle space

The computational algorithm that works in the coordinate space of dihedral angles (i.e., bond lengths and bond angles are kept fixed and only rotatable dihedral angles are treated as independent variables) is extended to deal with the pseudorotational m otion of furanose rings by introducing a variable of pseudorotation. Then, this algorithm is applied to a distance geometry calculation that generates three-dimensional (3D) structures that are consistent with given constraints of interatomic distances. This method efficiently generates 3D structures of an RNA hairpin loop which satisfy a set of experimental NMR data. © 1996 by John Wiley & Sons, Inc.     

A new method for fast and accurate derivation of molecular conformations

During molecular simulations, three-dimensional conformations of biomolecules are calculated from the values of their bond angles, bond lengths, and torsional angles. In this paper we study how to efficiently derive three-dimensional molecular conformations from the values of torsional angles. This case is of broad interest as torsional angles greatly affect molecular shape and are always taken into account during simulations. We first review two widely used methods for deriving molecular conformations, the simple rotations scheme and the Denavit-Hartenberg local frames method. We discuss their disadvantages which include extensive bookkeeping, accumulation of numerical errors, and redundancies in the local frames used. Then we introduce a new, fast, and accurate method called the atomgroup local frames method. This new method not only eliminates the disadvantages of earlier approaches but also provides lazy evaluation of atom positions and reduces the computational cost. Our method is especially useful in applications where many conformations are generated or updated such as in energy minimization and conformational search.     

A multiple chain Monte Carlo method for atomistic simulation of high molecular weight polymer melts

A new Monte Carlo method is proposed for the simulation of bulk systems of atomistically detailed polymers. Each move consists of a configurational rearrangement of the atoms in a specified region of the material, rather than a specified molecule. Thus atoms within different chains may be displaced cooperatively in each Monte Carlo move. Here, the method is implemented for the case of melts of linear chains, where the bond lengths and bond angles are held constant during the move. The performance of the algorithm is examined for linear polyethylene systems with chain lengths of 100 and 1000 backbone atoms, under a range of conditions. The method shows a considerable potential as a very general and flexible tool for simulating realistic polymer materials, subject to a number of performances limiting factors which are described in detail.     

MP-NeRF: A massively parallel method for accelerating protein structure reconstruction from internal coordinates

The conversion of proteins between internal and cartesian coordinates is a limiting step in many pipelines, such as molecular dynamics simulations and machine learning models. This conversion is typically carried out by sequential or parallel applications of the Natural extension of Reference Frame (NeRF) algorithm. This work proposes a massively parallel NeRF implementation which, depending on the polymer length, achieves speedups between 400 and 1200× over the previous state-of-the-art. It accomplishes this by dividing the conversion into three main phases: parallel composition of the monomer backbone, assembly of backbone subunits, and parallel elongation of sidechains; and by batching these computations into a minimal number of efficient matrix operations. Special emphasis is placed on reusability and ease of use. We open source the code (available at https://github.com/EleutherAI/mp_nerf ) and provide a corresponding python package.     

Extension of miyazawa's equations to the case of an arbitrary number of junction atoms in a molecular helix

An algorithm is proposed for the construction of Miyazawa's equations with an arbitrary number of junction atoms in the repeat unit of the macromolecule. In these equations the helical parameters of an infinite polymer chain and the internal parameters of the chain, bond lengths, bond angles, and internal rotation angles, are interrelated. © 1993 John Wiley & Sons, Inc.     

Molecular mechanics calculations of β‐diketonate,aqua, and aqua‐β‐diketonate complexes of lanthanide ions using Gillespie–Kepert model

A version of molecular mechanics based on the Gillespie–Kepert model of coordination bonds “repulsion” is applied to lanthanide complexes. The force field parameters are developed that describe the structure of β‐diketonate‐, aqua‐, and mixed aqua‐β‐diketonate complexes with good accuracy; the same parameters are applicable to various coordination numbers/polyhedra. For the aqua complexes, typical root‐mean‐square deviation (calculated vs. X‐ray experimental values) is 0.02 Å in Ln–O bond lengths and 2.0° in O–Ln–O valence angles. For most of the other compounds, the same precision is achieved in coordination bond lengths, while 3.5° is a typical precision for coordination bond angles. Calculations successfully reproduce the puckering of the β‐diketonate chelate rings, as well as the relative stability of isomers for a representative example. © 2000 John Wiley & Sons, Inc. J Comput Chem 22: 38–50, 2001     

CNDO/2 (complete neglect of differential overlap)-method for third-row molecules

An extension of the CNDO/2 method to compounds containing third-row elements (Germanium, Arsenic, Selenium and Bromine) is presented. Bond lengths, bond angles, dipole moments, and ionization potentials are considered.     

Molecular parameters for the prediction of polyurethane structures

Recent efforts to determine the structures of poly(MDI/diol) hard segments in polyurethane elastomers have relied on the structures determined by single-crystal x-ray methods for diphenylmethane urethane model compounds. We have surveyed the structure of six model compounds, and have derived average values for the bond lengths, bond angles, and bond torsion angles for use in future analyses. The applicability of these averages to polymer structures is discussed, and the data are used to derive models for the poly(MDI-butanediol) chain which are found to be consistent with the fiber repeat determine by x-ray methods.     

Hybridization in highly strained small ring hydrocarbons. V. Methylene and isopropylene substituted spiropentanes

Using the method of maximum overlap, the hybrids, the bond overlaps and the deviation angles as well as the interhybrid angles have been calculated for several methylene and isopropylene substituted spiropentanes. Considerably different hybrids are found for the spiro carbon when the two C₃ rings have a different number of substituants. They deviate appreciably from the usually assumed sp³ type, being as different as sp^2.70 and sp^3.34. A brief discussion and comparison of some experimental molecular properties and the corresponding calculated parameters is presented. A discussion of factors which influence the hybridization in small rings has been given with some critical remarks on the reliability of predictions of molecular geometry (bond lengths) and molecular shapes (bond angles) by the maximum overlap method.     

Crystal structure of the complex of trichosanthin with NADPH at 0.172 nm resolution

The orthorhombic crystal structure of the complex of trichosanthin with nicotinamide adenine dinucleotide phosphate has been determined by molecular replacement method using one of the molecules of the monoclinic crystal structure of trichosanthin at 0.27 nm resolution as the search model. The crystallographic refinement at 0.172 nm resolution led to a final R-factor of 17.4% with root-mean-square deviations of 0.0013 nm and 3.8 from the ideal bond lengths and bond angles, respectively. The quality of the structure, the polypeptide chain fold and the comparison of it with that of the monoclinic trichosanthin structure, the location of nicotinamide adenine dinucleotide phosphate, the active site structure as well as the solvent structure are described.     

Silicon nanotubes with distinct bond lengths

In this paper, we extend both the rolled-up and the polyhedral models for single-walled silicon nanotubes with equal bond lengths to models having distinct bond lengths. The silicon nanotubes considered here are assumed to be formed by sp³ hybridization with different bond lengths so that the nanotube lattice is assumed to comprise only skew rhombi. Beginning with the three postulates that all bonds lying on the same helix are equal, all adjacent bond angles are equal, and all atoms are equidistant from a common axis of symmetry, we derive exact formulae for the polyhedral geometric parameters such as chiral angles, bond angles, radius and unit cell length. The polyhedral model presented here with distinct bond lengths includes both the rolled-up model with distinct bond lengths which arises from the first term of an asymptotic expansion, and an existing polyhedral model of the authors which assumes equal bond lengths. Finally, some molecular dynamics simulations are undertaken for comparison with the geometric model. These simulations start with equal bond lengths and then stabilize in such a way that two distinct bond lengths emerge.     

Geometry optimization of periodic systems using internal coordinates

An algorithm is proposed for the structural optimization of periodic systems in internal (chemical) coordinates. Internal coordinates may include in addition to the usual bond lengths, bond angles, out-of-plane and dihedral angles, various "lattice internal coordinates" such as cell edge lengths, cell angles, cell volume, etc. The coordinate transformations between Cartesian (or fractional) and internal coordinates are performed by a generalized Wilson B-matrix, which in contrast to the previous formulation by Kudin et al. [J. Chem. Phys. 114, 2919 (2001)] includes the explicit dependence of the lattice parameters on the positions of all unit cell atoms. The performance of the method, including constrained optimizations, is demonstrated on several examples, such as layered and microporous materials (gibbsite and chabazite) as well as the urea molecular crystal. The calculations used energies and forces from the ab initio density functional theory plane wave method in the projector-augmented wave formalism.     

Quantum chemical study of electronic and structural properties of retinal and some aromatic analogs

The electronic and structural properties of retinal and four analogs were studied using semiempirical, ab initio Hartree-Fock, and density functional theory methods with the aim to evaluate the effects caused by some structural modifications in the ring bound to the polyenic chain and compared with the all-E-trans-retinal molecule. Therefore, some properties such as bond lengths, bond angles, atomic charges derived from electrostatic potential charges from electrostatic potential using grid based method as well as frontier orbitals of the polyenic chain were analyzed. Furthermore, the transition energies of the molecules were also calculated using the Zerner's intermediate neglect of differential overlap-spectroscopic, time-dependent Hartree-Fock, and time-dependent density functional theory methods. The results indicate that in spite of the structural modifications of retinal derivatives in comparison with all-E-trans-retinal, their properties seem similar. Thus, these molecules may behave similarly to all-E-trans-retinal and possibly be attempted in the search of novel molecular devices.     

Ab initio study of the Trinem antibiotic Sanfetrinem GV104326

We present a theoretical calculation of trinem antibiotic sanfetrinem GV104326 and its cis-isomer of proton in β-lactam ring and methoxy inversion isomer. Ab initio Hartree Fock method is employed to calculate the geometry optimization; optimized parameters, bond lengths, bond angles and dihedral angles of isomers. We also calculated the vibrational frequencies, the IR intensities, the Raman activities and the thermochemical parameters. Mulliken population analysis is used to compare bond strength of an active site. These results are compared with available experimental data and other related theoretical calculations. The calculated IR spectra are in good agreement with experiments. Calculated geometrical and conformational structures of sanfetrinem and its isomers support the previous study of the geometrical requirements for the antibacterial activities. The significance of the rotational conformation of the carboxyl group to chemical reactivities is discussed.