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Register a new userA central partition of molecular conformational space. II. Embedding 3D structures
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Added by
Jacques Gabarro-Arpa
Publication date
2004
fields
Physics
and research's language is
English
Authors
Jacques Gabarro-Arpa
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A combinatorial model of molecular conformational space that was previously developped (J. Gabarro-Arpa, Comp. Biol. and Chem. 27, (2003) 153-159), had the drawback that structures could not be properly embedded beacause it lacked explicit rotational symmetry. The problem can be circumvented by sorting the elementary 3D components of a molecular system into a finite set of classes that can be separately embedded. This also opens up the possibility of encoding the dynamical states into a graph structure.
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On the basis of empirical evidence from molecular dynamics simulations, molecular conformational space can be described by means of a partition of central conical regions characterized by the dominance relations between cartesian coordinates. This work presents a geometric and combinatorial description of this structure.
In a previous work a procedure was decribed for dividing the $3 times N$-dimensional conformational space of a molecular system into a number of discrete cells, this partition allowed the building of a combinatorial structure from data sampled in molecular dynamics trajectories: the graph of cells or G, encoding the set of cells in conformational space that are visited by the system in its thermal wandering. The information in G however, is encoded in a great number of fragments that must be aggregated. We describe here the algorithmic procedures 1) for aggregating the information from G into an hypergraph allowing to enumerate the relevant cells from conformational space, and 2) for puttting the data in a very compact format.
In previous works [physics/0204035, physics/0404052, physics/0509126] a procedure was described for dividing the $3 times N$-dimensional conformational space of a molecular system into a number of discrete cells, this partition allowed the building of a combinatorial structure from data sampled in molecular dynamics trajectories: the graph of cells, that encodes the set of cells in conformational space that are visited by the system in its thermal wandering. Here we outline a set of procedures for extracting useful information from this structure: 1st) interesting regions in the volume occupied by the system in conformational space can be bounded by a polyhedral cone whose faces are determined empirically from a set of relations between the coordinates of the molecule, 2nd) it is also shown that this cone can be decomposed into a hierarchical set of smaller cones, 3rd) the set of cells in a cone can be encoded by a simple combinatorial sequence.
In the first work of this series [physics/0204035] it was shown that the conformational space of a molecule could be described to a fair degree of accuracy by means of a central hyperplane arrangement. The hyperplanes divide the espace into a hierarchical set of cells that can be encoded by the face lattice poset of the arrangement. The model however, lacked explicit rotational symmetry which made impossible to distinguish rotated structures in conformational space. This problem was solved in a second work [physics/0404052] by sorting the elementary 3D components of the molecular system into a set of morphological classes that can be properly oriented in a standard 3D reference frame. This also made possible to find a solution to the problem that is being adressed in the present work: for a molecular system immersed in a heat bath we want to enumerate the subset of cells in conformational space that are visited by the molecule in its thermal wandering. If each visited cell is a vertex on a graph with edges to the adjacent cells, here it is explained how such graph can be built.
Realistic 3D-conformations of protein structures can be embedded in a cubic lattice using exclusively integer numbers, additions, subtractions and boolean operations.
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