Internal Coordinate Space Research Articles

Sampling small-scale and large-scale conformational changes in proteins and molecular complexes

Sampling of small-scale and large-scale motions is important in various computational tasks, such as protein-protein docking and ligand binding. Here, we report further development and applications of the activation-relaxation technique for internal coordinate space trajectories (ARTIST). This method generates conformational moves of any complexity and size by identifying and crossing well-defined saddle points connecting energy minima. Simulations on two all-atom proteins and three protein complexes containing between 70 and 300 amino acids indicate that ARTIST opens the door to the full treatment of all degrees of freedom in dense systems such as protein-protein complexes.

ARTIST: An activated method in internal coordinate space for sampling protein energy landscapes

We present the first applications of an activated method in internal coordinate space for sampling all-atom protein conformations, the activation-relaxation technique for internal coordinate space trajectories (ARTIST). This method differs from all previous internal coordinate-based studies aimed at folding or refining protein structures in that conformational changes result from identifying and crossing well-defined saddle points connecting energy minima. Our simulations of four model proteins containing between 4 and 47 amino acids indicate that this method is efficient for exploring conformational space in both sparsely and densely packed environments, and offers new perspectives for applications ranging from computer-aided drug design to supramolecular assembly.

Molecular Structure of Proline

The molecular structures of the two lowest-energy conformers of proline, Pro-I and Pro-II, have been characterized by ab initio electronic structure computations. An extensive MP2/6-31G* quartic force field for Pro-I, containing 62,835 unique elements in the internal coordinate space, was computed to account for anharmonic vibrational effects, including total zero-point contributions to isotopomeric rotational constants. New re and improved r0 least-squares structural refinements were performed to determine the heavy-atom framework of Pro-I, based on experimentally measured (A. Lesarri, S. Mata, E. J. Cocinero, S. Blanco, J. C. Lopez, J. L. Alonso, Angew. Chem. 2002, 114, 4867; Angew. Chem. Int. Ed. 2002, 41, 4673) rotational constant sets of nine isotopomers and our ab initio data for structural constraints and zero-point vibrational (ZPV) shifts. Without the ab initio constraints, even the extensive set of empirical rotational constants cannot satisfactorily fix the molecular structure of the most stable conformer of proline, a 17-atom molecule with no symmetry. After imposing the ab initio constraints, excellent agreement between theory and experiment is found for the heavy-atom geometric framework, the root-mean-square (rms) residual of the empirical rotational constant fit being cut in half by adding ZPV corrections. The most significant disparity, about 0.07 A, between the empirical and the best ab initio structures, concerns the r(N...H) distance of the intramolecular hydrogen bond. Some of the experimental quartic centrifugal distortion constants assigned to Pro-II have been corrected based on data obtained from a theoretical force field.

Analytical treatment of fragmentation‐diffusion population balance

AbstractThe quest for advanced models to realistically simulate various processes involving populations of “particles” (aerosols, dispersions of all kinds, minerals, cultures of microorganisms) necessitates the extension of the classical population balance formulation, which is of zero dimension (lumped) in external and one‐dimensional (1‐D) (single particle‐characterization variable) in internal coordinate space, to more dimensions in both spaces. In order to contribute in this direction, this work is focused on a spatially distributed fragmentation equation, the spatial nonuniformity being governed by a diffusion mechanism and by deposition of particles on solid boundaries. The combined effect of diffusion and fragmentation on the particle‐size distribution is of interest here. The linear form of the fragmentation‐diffusion equation allows a convenient analytical manipulation, based on decomposition of solutions in spatial and temporal modes. Generalized solutions are derived for the above equation for arbitrary spatial domains, fragmentation functions, deposition rates and particle sources. The above approach can be combined with the well‐known approaches for the numerical solution of the lumped fragmentation equation to derive solutions for the fragmentation‐diffusion equation under arbitrary conditions. This study, as a first step, is restricted to obtaining explicit results for some rather simple cases, for example, power law fragmentation and deposition rates, fragmentation in equal sizes and simple shapes of the domain under consideration. © 2004 American Institute of Chemical Engineers AIChE J, 50: 1746–1759, 2004

Role of isomerization channel in unimolecular dissociation reaction H2CO→H2+CO: Ab initio global potential energy surface and classical trajectory analysis

We constructed a full dimensional potential energy function of H2CO that can describe both the dissociation and isomerization channels by the modified Shepard interpolation method. Ab initio calculations at the MP2/cc-pVTZ level were carried out to obtain the local potential functions at about 4700 points. The interpolant points were sampled by classical trajectory calculations and by the grid searches in the internal coordinate space. Classical trajectory calculations were performed to examine the intramolecular dynamics associated with the dissociation as well as the product state distributions. The time scale of intramolecular vibrational energy randomization was much faster than that of the dissociation reaction. The dissociation rate was obtained from the classical trajectory results and the effect of the isomerization channel on the dissociation was estimated. The calculated rate constants were compared with those by Rice–Ramsperger–Kassel–Marcus theory.

Visualizing protein conformational changes on a personal computer – alpha carbon pseudo bonding as a constraint for interpolation in internal coordinate space

A Java program has been developed to produce animations and movies of protein conformation changes. The animations, primarily intended for teaching purposes, are produced by visualizing a series of aligned structures interpolated between two forms of the same protein. To produce plausible intermediate structures, the interpolation is performed in internal coordinate space and uses a simple constraint to avoid the production of artifactual movements of the protein backbone. This constraint involves the introduction of ‘pseudo’ bonds linking the protein alpha carbon atoms. All of the steps from alignment of initial structures to the production of AVI movies can be performed on readily available personal computers.

Efficient calculation of Franck–Condon factors and vibronic couplings in polyatomics

AbstractWe present a technique for the calculation of Franck–Condon factors and other integrals between vibronic wave functions belonging to different electronic states. The technique is well suited for the determination of the nonadiabatic or spin‐orbit couplings related to radiationless decays in polyatomics. Rigorous or approximate partitions of the internal coordinate space are exploited to achieve better efficiency and/or to go beyond the harmonic approximation. The technique is tested by computing the Internal Conversion and InterSystem Crossing rates of (CH3)3CNO in its 1(n→π*) state. © 2001 John Wiley & Sons, Inc. J Comput Chem 22: 968–975, 2001

Statistical thermodynamics of hindered rotation from computer simulations

Due to the nonlinearity of internal molecular coordinate space, comprehensive statistical mechanical treatment of polyatomic molecules presents a formidable theoretical problem where traditional statistics does not hold. Here we introduce a new theoretical approach to the hindered internal rotor problem, which relies on simulated statistics of microstates, rather than on first-principles calculations of the partition function. The concept links traditional statistical-mechanical thermodynamics to an authentic treatment of curvilinear descriptive statistics. The results are illustrated by classical molecular dynamics simulations.

Internal coordinate phase space analysis of macromolecular systems

Abstract In recent studies, as well as the results presented herein, it has been shown that the classical dynamics of macromolecular systems exhibit chaos far below the zero-point energy consequently resulting in the loss of a qualitative correspondence to quantum behavior. Mechanisms responsible for this undesirable and unrealistic dynamics have been shown to be due to the flow of energy out of the high frequency modes into the low frequency, large amplitude modes. A very powerful tool for eliminating the high frequency modes in macromolecular systems has been the development of internal coordinate molecular dynamics. This method only integrates the chosen degrees of freedom that determine the overall structure of the molecular system (the torsion). In this paper, we have used this technique with the appropriate analysis from semi-classical theory and nonlinear dynamics to study the trajectories generated for some simple polymer fragments. The method does effectively eliminate most of the problems associated with zero-point energy flow and the resulting phase space structure exhibits a high degree of stable quasiperiodic motion. However, the semi-classical relationships to quantum mechanics in this quasiperiodic regime require that initial conditions be chosen carefully in order for the resulting trajectory to have any quantum relevance.

A density functional derived vibrational force field for β-ionone : Use of the ultraviolet resonance Raman intensities to check the vibrational analysis accuracy

In the present article we aim to determine an accurate and transferable molecular force field for β-ionone in order to perform molecular dynamics calculations on the retinal isomers. For β-ionone, a force field is derived from calculations using the Density Functional Theory (DFT). The force constants expressed in the internal coordinate space were scaled fitting theoretical to experimental vibrational wavenumbers. The validity of the force field was checked using a comparison between calculated and observed resonance Raman intensities obtained from the A-term part (Franck–Condon) of the scattering tensor.

Spinning strings of the Neveu-Schwarz-Ramond type in 3, 4, 6, and 10 space-time dimensions

A model of a spinning string with an internal coordinate index is proposed and studied. When the action for this model is taken to be diagonal in this internal coordinate space and quantized in the light-cone gauge it is found to be Lorentz covariant in four-dimensional space-time provided that the internal coordinate space is four dimensional.This combination of space-time dimension, D, and internal coordinate space dimension, N, is just one of four possible sets, the other three corresponding to D = 3, 6, and 10, precisely the same values for which it is possible to formulate Yang-Mills theories with simple supersymmetry. By comparing the number of propagating degrees of freedom at the zero-mass level in the open string bosonic and fermionic sectors it is found that a supersymmetric interpretation of this model is possible provided that all physical states in the bosonic sector have even G-parity and the ground-state spin or in the fermionic sector have positive chirality. A possible interpretation of the connection betweenthe N components of each of the D space-time coordinates is presentedon the basis that the space-time coordinates are scalars in the internal coordinate space. This interpretation would appear to be reasonable given the fact that the field variables in the Lagrangian density do not necessarily have to represent physically measurable quantities but can, instead, only represent physically measurable quantities when combined in some manner, the simplest of which being a linear combination. The Lagrangian density simply produces the equations of motion and the constraint equations for the independent variables, only linear combinations of which represent the four dimensions of physical space-time.PACS Nos.: 11.17.+y, 11.10.Qr, 1.30.Cp, 11.30.Pb

Ab InitioFolding of Peptides by the Optimal-Bias Monte Carlo Minimization Procedure

Prediction of three-dimensional structures of proteins and peptides by global optimization of the free energy estimate has been attempted without much success for over thirty years. The key problems were the insufficient accuracy of the free energy estimate and the giant size of the conformational space. Global optimization of the free energy function of a peptide in internal coordinate space is a powerful method of structure prediction that outperforms both Molecular Dynamics, bound by the continuity requirement, and Monte Carlo, bound by the Boltzmann ensemble generation requirement. We demonstrate that stochastic global optimization algorithms of the first order, i.e., with local minimization after each iteration (e.g., Monte Carlo-Minimization), have a greater chance of finding the global minimum after a fixed number of function evaluations. Recently, the principle of optimal bias was mathematically justified and the Optimal-Bias Monte Carlo-Minimization algorithm (a.k.a. Biased Probability Monte Carlo-minimization) was successfully applied to theoreticalab initiofolding of several peptides, resulting in more than a 10-fold increase in efficiency compared to the Monte Carlo-Minimization method. The square-root bias is shown to be comparable in performance with the previously derived linear bias. A 23-residue peptide of beta-beta-alpha structure can be predicted from any random starting conformation.

Force-constant weighted redundant coordinates in molecular geometry optimizations

A novel procedure to select internal coordinates for molecular geometry optimizations is presented. The procedure has features in common with other so-called redundant internal coordinates schemes. It is a black-box method which automatically selects an appropriate set of internal coordinates in which the geometry optimization is performed. The method is explicitly expressed in the non-redundant parameter space, thus avoiding the need for projections from the redundant internal coordinate space. The new procedure introduces a weighting in which the redundancy is modified prior to the generation of the non-redundant internal coordinates. The new method favors those redundant internal coordinates which are the most significant. It has favorable properties for the automatic generation of molecular coordinates in van der Waals complexes and transition state optimizations.

Fluctuations and non-equilibrium thermodynamics

A closed macroscopic system with fluctuating local properties is treated as a thermodynamic system with internal degrees of freedom. Gibb’s entropy postulate is used to define the systems’ entropy as a functional of the probability density in internal coordinate space. It is shown that application of the scheme of thermodynamics of irreversible processes then leads directly to the theory of fluctuations as Markov processes described by a multivariate Fokker–Planck equation. In this perspective fluctuation theory may be said to have become integrated into non-equilibrium thermodynamics.

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

Conventional molecular dynamics simulations macromolecules require long computational times because the most interesting motions are very slow compared with 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 to eliminate all uninteresting fast degrees of freedom from the system. This paper presents a new method for internal coordinate molecular dynamics simulations of macromolecules. Equations of motion are derived which 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 a 56 residue globular protein. 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 fsec, that is two times higher than with fixed bond lengths only. The step size can be increased up to 11 fsec when rotation of hydrogen atoms is suppressed.

Estimation of an Urey–Bradley force field from ab initio calculations

Urey–Bradley (UB) type force constants have been derived from the Cartesian force constant as obtained through ab initio calculations. The matrix of the force constants is expressed in the internal coordinate space and its elements are scaled so that the theoretical vibrational frequencies fit the experimental ones. Predicted UB type force constants are then extracted from this scaled internal force constant matrix using a Z matrix representing internal coordinate interactions in the UB space. We also include a Jacobian matrix determination into the SPASIBA program to help in refining the predicted force constants. These force constants must lead to a good vibrational force field together with an accurate reproduction of the equilibrium geometries. This process was applied to H 2O 2, H 2S 2, NH 2OH, N 2H 4 and CH 3OH. © 1997 Elsevier Science B.V.

Flexible protein-ligand docking by global energy optimization in internal coordinates.

Eight protein-ligand complexes were simulated by using global optimization of a complex energy function, including solvation, surface tension, and side-chain entropy in the internal coordinate space of the flexible ligand and the receptor side chains [Abagyan, R.A., Totrov, M.M. J. Mol. Biol. 235: 983-1002, 1994]. The procedure uses two types of efficient random moves, a pseudobrownian positional move [Abagyan, R.A., Totrov, M.M., Kuznetsov, D.A. J. Comp. Chem. 15:488-506, 1994] and a Biased-Probability multitorsion move [Abagyan, R.A., Totrov, M.M. J. Mol. Biol. 235: 983-1002, 1994], each accompanied by full local energy minimization. The best docking solutions were further ranked according to the interaction energy, which included intramolecular deformation energies of both receptor and ligand, the interaction energy, surface tension, side-chain entropic contribution, and an electrostatic term evaluated as a boundary element solution of the Poisson equation with the molecular surface as a dielectric boundary. The geometrical accuracy of the docking solutions ranged from 30% to 70% according to the relative displacement error measure at a 1.5 A scale. Similar results were obtained when the explicit receptor atoms were replaced with a grid potential.

The simulation of substantial conformational transitions of proteins: progress in the application of path energy minimization in internal coordinate space.

The simulation of substantial conformational transitions of proteins: progress in the application of path energy minimization in internal coordinate space.

An object-oriented approach to the calculation of the inverse kinetic energy matrix

A novel approach to the calculation of the elements of the kinetic energy matrix G is presented. In the proposed method the matrix elements are calculated directly in the internal coordinate space, without determination of the intermediate B matrix, using the analytical expressions given by Decius. The C++ program exploits a new method for the representation the geometry and constitution of chemical structures, which enables both molecular topology manipulation and geometry calculation. Starting with the set of internal coordinates, all possible types of the G matrix elements are found for the molecule in an automatic search and subsequently evaluated. The method is exemplified by calculation on the water and cyclopentanone molecules.

Small‐amplitude protein conformational dynamics: Second‐order analytic relation between cartesian coordinates and dihedral angles

AbstractAn analytic expression for protein atomic displacements in Cartesian coordinate space (CCS) against small changes in dihedral angles is derived. To study time‐dependent dynamics of a native protein molecule in CCS from dynamics in the internal coordinate space (ICS), it is necessary to convert small changes of internal coordinate variables to Cartesian coordinate variables. When we are interested in molecular motion, six degrees of freedom for translational and rotational motion of the molecule must be eliminated in this conversion, and this conversion is achieved by requiring the Eckart condition to hold. In this article, only dihedral angles are treated as independent internal variables (i.e., bond angles and bond lengths are fixed), and Cartesian coordinates of atoms are given analytically by a second‐order Taylor expansion in terms of small deviations of variable dihedral angles. Coefficients of the first‐order terms are collected in the K matrix obtained previously by Noguti and Go (1983) (see ref. 2). Coefficients of the second‐order terms, which are for the first time derived here, are associated with the (newly termed) L matrix. The effect of including the resulting quadratic terms is compared against the precise numerical treatment using the Eckart condition. A normal mode analysis (NMA) in the dihedral angle space (DAS) of the protein bovine pancreatic trypsin inhibitor (BPTI) has been performed to calculate shift of mean atomic positions and mean square fluctuations around the mean positions. The analysis shows that the second‐order terms involving the L matrix have significant contributions to atomic fluctuations at room temperature. This indicates that NMA in CCS involves significant errors when applied for such large molecules as proteins. These errors can be avoided by carrying out NMA in DAS and by considering terms up to second order in the conversion of atomic motion from DAS to CCS. © 1995 by John Wiley & Sons, Inc.