Enumerating and indexing many-body intramolecular interactions: a graph theoretic approach.
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The hierarchical structure of atomic index lists for each interaction order [Formula: see text] is compactly expressed as a directed acyclic graph.With suitable data structures (e.g., edge lists or adjacency matrices), automatic enumeration and indexing of [Formula: see text]-body interactions can be implemented straightforwardly to handle large bio-molecular systems.Explicit examples are discussed, including a chemically relevant effective potential model of taurocholate bile salt.
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Affiliation: Institute of Food Research, Norwich Research Park, Colney, Norwich, NR4 7UA UK.
ABSTRACT
The central idea observes a recursive mapping of [Formula: see text]-body intramolecular interactions to [Formula: see text]-body terms that is consistent with the molecular topology. Iterative application of the line graph transformation is identified as a natural and elegant tool to accomplish the recursion. The procedure readily generalizes to arbitrary [Formula: see text]-body potentials. In particular, the method yields a complete characterization of [Formula: see text]-body interactions. The hierarchical structure of atomic index lists for each interaction order [Formula: see text] is compactly expressed as a directed acyclic graph. A pseudo-code description of the generating algorithm is given. With suitable data structures (e.g., edge lists or adjacency matrices), automatic enumeration and indexing of [Formula: see text]-body interactions can be implemented straightforwardly to handle large bio-molecular systems. Explicit examples are discussed, including a chemically relevant effective potential model of taurocholate bile salt. No MeSH data available. Related in: MedlinePlus |
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Mentions: The molecular graph indicated in Fig. 2 presents amongst other possibilities a plausible lumped model for methylcyclopropane where hydrogen atoms are absorbed onto the carbon backbone in the usual way. Direct inspection of immediately establishes four -body interactions in the presence of an external field (that is, the number of “atoms” ) and also four -body bonds . Atom is the hinge for three distinct -body bends with a further two hinged at atoms and respectively, to give . Among the -body interactions there are two proper torsions with distinct hinge atoms - and -, respectively, as well as a single improper dihedral with common hinge atom . A single -cycle is present so that . Adjacency matrices for the iterated line graphs necessary for describing interactions up to the -body level are given by\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\mathbf{A}(G) = \begin{pmatrix} 0 &{}\quad 1^{1} &{}\quad 0 &{}\quad 0 \\ 1^{1} &{}\quad 0 &{}\quad 1^{2} &{}\quad 1^{3} \\ 0 &{}\quad 1^{2} &{}\quad 0 &{}\quad 1^{4} \\ 0 &{}\quad 1^{3} &{}\quad 1^{4} &{}\quad 0 \end{pmatrix} , \quad \mathbf{A}\bigl (L(G)\bigr ) = \begin{pmatrix} 0 &{}\quad 1^{1} &{}\quad 1^{2} &{}\quad 0 \\ 1^{1} &{}\quad 0 &{}\quad 1^{3} &{}\quad 1^{4} \\ 1^{2} &{}\quad 1^{3} &{}\quad 0 &{}\quad 1^{5} \\ 0 &{}\quad 1^{4} &{}\quad 1^{5} &{}\quad 0 \end{pmatrix} ,\\&\mathbf{A}\bigl (L^{2}(G)\bigr ) = \begin{pmatrix} 0 &{}\quad 1^{1} &{}\quad 1^{2} &{}\quad 1^{4} &{}\quad 0 \\ 1^{1} &{}\quad 0 &{}\quad 1^{3} &{}\quad 0 &{}\quad 1^{6} \\ 1^{2} &{}\quad 1^{3} &{}\quad 0 &{}\quad 1^{5} &{}\quad 1^{7} \\ 1^{4} &{}\quad 0 &{}\quad 1^{5} &{}\quad 0 &{}\quad 1^{8} \\ 0 &{}\quad 1^{6} &{}\quad 1^{7} &{}\quad 1^{8} &{}\quad 0 \end{pmatrix}.\\ \end{aligned}$$\end{document}A(G)=011001101213012014013140,A(L(G))=01112011013141213015014150,A(L2(G))=01112140110130161213015171401501801617180.Fig. 2 |
View Article: PubMed Central - PubMed
Affiliation: Institute of Food Research, Norwich Research Park, Colney, Norwich, NR4 7UA UK.
The central idea observes a recursive mapping of [Formula: see text]-body intramolecular interactions to [Formula: see text]-body terms that is consistent with the molecular topology. Iterative application of the line graph transformation is identified as a natural and elegant tool to accomplish the recursion. The procedure readily generalizes to arbitrary [Formula: see text]-body potentials. In particular, the method yields a complete characterization of [Formula: see text]-body interactions. The hierarchical structure of atomic index lists for each interaction order [Formula: see text] is compactly expressed as a directed acyclic graph. A pseudo-code description of the generating algorithm is given. With suitable data structures (e.g., edge lists or adjacency matrices), automatic enumeration and indexing of [Formula: see text]-body interactions can be implemented straightforwardly to handle large bio-molecular systems. Explicit examples are discussed, including a chemically relevant effective potential model of taurocholate bile salt.
No MeSH data available.