Topic

# Coordinate space

About: Coordinate space is a research topic. Over the lifetime, 2166 publications have been published within this topic receiving 59763 citations.

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TL;DR: In this article, the authors compared the canonical distribution in both momentum and coordinate space with three recently proposed constant temperature molecular dynamics methods by: (i) Nose (Mol. Phys., to be published); (ii) Hoover et al. [Phys. Rev. Lett. 77, 63 (1983); and (iii) Haile and Gupta [J. Chem. Phys. 79, 3067 (1983).

Abstract: Three recently proposed constant temperature molecular dynamics methods by: (i) Nose (Mol. Phys., to be published); (ii) Hoover et al. [Phys. Rev. Lett. 48, 1818 (1982)], and Evans and Morriss [Chem. Phys. 77, 63 (1983)]; and (iii) Haile and Gupta [J. Chem. Phys. 79, 3067 (1983)] are examined analytically via calculating the equilibrium distribution functions and comparing them with that of the canonical ensemble. Except for effects due to momentum and angular momentum conservation, method (1) yields the rigorous canonical distribution in both momentum and coordinate space. Method (2) can be made rigorous in coordinate space, and can be derived from method (1) by imposing a specific constraint. Method (3) is not rigorous and gives a deviation of order N−1/2 from the canonical distribution (N the number of particles). The results for the constant temperature–constant pressure ensemble are similar to the canonical ensemble case.

11,619 citations

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TL;DR: It is concluded that the most promising detailed approach to the protein‐folding problem would consist of some coarse global sampling strategy combined with the local energy minimization in the torsion coordinate space.

Abstract: An efficient methodology, further referred to as ICM, for versatile modeling operations and global energy optimization on arbitrarily fixed multimolecular systems is described. It is aimed at protein structure prediction, homology modeling, molecular docking, nuclear magnetic resonance (NMR) structure determination, and protein design. The method uses and further develops a previously introduced approach to model biomolecular structures in which bond lengths, bond angles, and torsion angles are considered as independent variables, any subset of them being fixed. Here we simplify and generalize the basic description of the system, introduce the variable dihedral phase angle, and allow arbitrary connections of the molecules and conventional definition of the torsion angles. Algorithms for calculation of energy derivatives with respect to internal variables in the topological tree of the system and for rapid evaluation of accessible surface are presented. Multidimensional variable restraints are proposed to represent the statistical information about the torsion angle distributions in proteins. To incorporate complex energy terms as solvation energy and electrostatics into a structure prediction procedure, a “double-energy” Monte Carlo minimization procedure in which these terms are omitted during the minimization stage of the random step and included for the comparison with the previous conformation in a Markov chain is proposed and justified. The ICM method is applied successfully to a molecular docking problem. The procedure finds the correct parallel arrangement of two rigid helixes from a leucine zipper domain as the lowest-energy conformation (0.5 A root mean square, rms, deviation from the native structure) starting from completely random configuration. Structures with antiparallel helixes or helixes staggered by one helix turn had energies higher by about 7 or 9 kcal/mol, respectively. Soft docking was also attempted. A docking procedure allowing side-chain flexibility also converged to the parallel configuration starting from the helixes optimized individually. To justdy an internal coordinate approach to the structure prediction as opposed to a Cartesian one, energy hypersurfaces around the native structure of the squash seeds trypsin inhibitor were studied. Torsion angle minimization from the optimal conformation randomly distorted up to the rms deviation of 2.2 A or angular rms deviation of l0° restored the native conformation in most cases. In contrast, Cartesian coordinate minimization did not reach the minimum from deviations as small as 0.3 A or 2°. We conclude that the most promising detailed approach to the protein-folding problem would consist of some coarse global sampling strategy combined with the local energy minimization in the torsion coordinate space. © 1994 by John Wiley & Sons, Inc.

1,458 citations

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TL;DR: In this article, a general coefficient of monotonicity, whose maximization is equivalent to optimal satisfaction of the Monotonicity condition, is defined, and which allows various options both for treatment of ties and for weighting error-of-fit.

Abstract: LetA
1,A
2, ...,A
n be anyn objects, such as variables, categories, people, social groups, ideas, physical objects, or any other. The empirical data to be analyzed are coefficients of similarity or distance within pairs (A
i,A
i
), such as correlation coefficients, conditional probabilities or likelihoods, psychological choice or confusion, etc. It is desired to represent these data parsimoniously in a coordinate space, by calculatingm coordinates {x
ia
} for eachA
i
for a semi-metricd of preassigned formd
ij
=d(|x
i1 -x
j1
|, |x
i2 -x
j2|, ..., |x
im
-x
jm
|). The dimensionalitym is sought to be as small as possible, yet satisfy the monotonicity condition thatd
ij

1,288 citations

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TL;DR: In this article, the concept of an off-diagonai long-range order of the reduced density matrices in the coordinate space representation is introduced and the onset of such an order is assumed to lead to a new thermodynamic phase of the system.

Abstract: This review is concerned with the concept that in a many-body system of bosons or fermions, it is possible to have an off-diagonai long-range order of the reduced density matrices in the coordinate space representation. The onset of such an order ieads to a new thermodynamic phase of the system. It is reasonabie to assume that superfluid He II and the superconductors are phases characterized by the existence of such an order. (W.D.M.)

1,003 citations

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TL;DR: The form of the wavefunction psi for a semiclassical regular quantum state (associated with classical motion on an N-dimensional torus in the 2N-dimensional phase space) is very different from the form of psi for an irregular state associated with stochastic classical motion in all or part of the (2N-1) energy surface in phase space as discussed by the authors.

Abstract: The form of the wavefunction psi for a semiclassical regular quantum state (associated with classical motion on an N-dimensional torus in the 2N-dimensional phase space) is very different from the form of psi for an irregular state (associated with stochastic classical motion on all or part of the (2N-1)-dimensional energy surface in phase space). For regular states the local average probability density Pi rises to large values on caustics at the boundaries of the classically allowed region in coordinate space, and psi exhibits strong anisotropic interference oscillations. For irregular states Pi falls to zero (or in two dimensions stays constant) on 'anticaustics' at the boundary of the classically allowed region, and psi appears to be a Gaussian random function exhibiting more moderate interference oscillations which for ergodic classical motion are statistically isotropic with the autocorrelation of psi given by a Bessel function.

1,002 citations