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Enumerating and indexing many-body intramolecular interactions: a graph theoretic approach.

Penfold R, Wilde PJ - J Math Chem (2015)

Bottom Line: 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.

View Article: PubMed Central - PubMed

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

A plausible effective model of methylcyclopropane is represented by the graph . The iterated line graphs  are also shown for . Vertices  and edges  are labelled. The corresponding directed acyclic graph (DAG) generated by the line graph hierarchy is given where the vertices at each level  are denoted by the inherited sequence of atomic labels from  as indicated by the directed edges. Inspection of  confirms the -body interactions identified by the DAG sinks and comprise of: two proper torsions (denoted “p”) with distinct hinge atoms - and -, respectively; a single improper dihedral (denoted “i”) with common hinge ; and a single -cycle (denoted “c”) of atoms ,  and
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Fig2: A plausible effective model of methylcyclopropane is represented by the graph . The iterated line graphs are also shown for . Vertices and edges are labelled. The corresponding directed acyclic graph (DAG) generated by the line graph hierarchy is given where the vertices at each level are denoted by the inherited sequence of atomic labels from as indicated by the directed edges. Inspection of confirms the -body interactions identified by the DAG sinks and comprise of: two proper torsions (denoted “p”) with distinct hinge atoms - and -, respectively; a single improper dihedral (denoted “i”) with common hinge ; and a single -cycle (denoted “c”) of atoms , and

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


Enumerating and indexing many-body intramolecular interactions: a graph theoretic approach.

Penfold R, Wilde PJ - J Math Chem (2015)

A plausible effective model of methylcyclopropane is represented by the graph . The iterated line graphs  are also shown for . Vertices  and edges  are labelled. The corresponding directed acyclic graph (DAG) generated by the line graph hierarchy is given where the vertices at each level  are denoted by the inherited sequence of atomic labels from  as indicated by the directed edges. Inspection of  confirms the -body interactions identified by the DAG sinks and comprise of: two proper torsions (denoted “p”) with distinct hinge atoms - and -, respectively; a single improper dihedral (denoted “i”) with common hinge ; and a single -cycle (denoted “c”) of atoms ,  and
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Fig2: A plausible effective model of methylcyclopropane is represented by the graph . The iterated line graphs are also shown for . Vertices and edges are labelled. The corresponding directed acyclic graph (DAG) generated by the line graph hierarchy is given where the vertices at each level are denoted by the inherited sequence of atomic labels from as indicated by the directed edges. Inspection of confirms the -body interactions identified by the DAG sinks and comprise of: two proper torsions (denoted “p”) with distinct hinge atoms - and -, respectively; a single improper dihedral (denoted “i”) with common hinge ; and a single -cycle (denoted “c”) of atoms , and
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

Bottom Line: 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.

View Article: PubMed Central - PubMed

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