A central partition of molecular conformational space. II. embedding 3D structures

Abstract

Abstract-In the first work of this series [1] 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 [2] 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.

A Central Partition of Molecular Conformational Space. III. Combinatorial Determination of the Volume Spanned by a Molecular System

“…As it has been emphasized in [1][2] the −1 is because of the translation symmetry : the conformations outside the hyperplane U correspond to translated 3D structures.…”

“…This exemple leads to the general solution that was discussed in [2] : the 7 vectors (7) define a central partition dual to A 3 [9] that divides the 3D space in 32 cells. The sign vector of the cell that contains F α defines the sense of the shortest circular path that connects the projected vectors in the 7 ordered sets {e }.…”

A central partition of molecular conformational space. IV. Extracting information from the graph of cells

“…The aim of this series of papers [1][2][3][4] is to build a set of mathematical tools for studying the energy landscape of proteins [5,6,7], and the present paper is a step further towards this goal.…”

“…The aim of the preeceding papers [1][2][3][4] was to construct a graph whose nodes are the cells visited by the molecular system in its thermal wandering, two important properties of partition sequences make this construction possible :…”

“…The method for building the graph that was proposed in [2] consists in 1. Establishing a morphological classification of simplexes, where each class is defined by a set of geometrical constraints.…”