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Polyspherical coordinate systems on orbit spaces with applications to biomolecular conformation


A 2002 Preprint by D. Dix

  • 2002:04
  • The group $G _ a$ of rigid motions, i.e. the semidirect product of $R^n$ (translations) and $SO(n)$ (proper rotations), acts on $R^n$. Let N be a set  with N elements. Let $(R^n)^N$ denote the set of all mappings $i\mapsto{R _ i}$ from N to $R^n$. $G _ a$ also acts on $(R^n)^N$ by the diagonal action $(g\cdot{R}) _ i=g\cdot{R _ i}$. Let B consist of those $R\in(R^n)^N$ whose isotropy subgroup consists only of the identity in $G _ a$. When $N\geq{n}\geq2$ we study a particular atlas of polyspherical coordinate charts in the orbit manifold $G _ a\!\setminus\!B$ and a related atlas of local trivializations of the principle bundle $B\to{G _ a\!\setminus\!B}$ with structure group $G _ a$. The coordinate charts and indexed by new combinatorial structus we call Z-systems $\Gamma=(\Gamma^1,\ldots,$ $\Gamma^n _ * ),$ defined as follows. Let $(^{\ \ \!\!N} _ {k+1})$ denote the set of all abstract k-simplices, i.e. subsets of N with exactly $k+1$ elements. Define $\Gamma^0=(^N _ {\ \!1})$. For each $k=1,\ldots,n$ we assume that $\Gamma^k\subset(^{\ \ \!\!N} _ {k+1})$ such that the pair $(\Gamma^{k-1},\Gamma^k)$ is a tree hypergraph, where $\Gamma^{k-1}$ is the set of vertices and $\Gamma^{k}$ is the set of edges, and where a $k-1$ simplex (vertex) v and a k simplex (edge) e are incident if $v\subset{e}$. Furthermore if $v _ 1,v _ 2\in\Gamma^{k-1}$ such that $v _ 1\cup{v _ 2}\in\Gamma^k$ then we require that $v _ 1\cap{v _ 2}\in\Gamma^{k-2}$. $\Gamma^n _ * $ is a set of oriented n-simplices, whose set of underlying unoriented n-simplices is $\Gamma^n$.

    If $e\in\Gamma^k$ and R in $(R^n)^N$ then let $R _ e=\{R _ i\ |\ i\in{e}\}$ be the associated geometrical simplex. Each element $\{i,j\}$ of $\Gamma^1$ is associated with the Euclidean distance in $R^n$ between points $R _ i$ and $R _ j$ of the mapping R in $(R^n)^N$. Each element $e\in\Gamma^k,$ $k=2,\ldots,$ $n-1$ is associated with an angle (taking values in $(0,\pi)$) between $R _ {v _ 1}$ and $R _ {v _ 2}$ associated to the two $k-1$ simplices $v _ 1,v _ 2\in\Gamma^{k-1}$ on which e is incident, as measured in a plane perpendicular to the affine subspace of dimension $k-2$ spanned by $R _ {v _ 1\cap{v _ 2}}$ and within the affine subspace of dimension k spanned by $R _ e$. Each element $e^ * \in\Gamma^n _ * $ is likewise associated with a signed angle (taking values in $(-\pi,\pi]$), where the sign of the angle and the orientation of the underlying abstract simplex $e\in\Gamma^n$ are compatible in a certain natural fashion. Coordinates are assigned to those mappings in $(R^n)^N$ for which every element $e\in\Gamma^{n-1}$ determines a geometrically independent $n-1$ simplex $R _ e$ in $R^n$; such mappings are said to be in the coordinate domain $D _ C(\Gamma)$. We prove that our coordinate system establishes a diffeomorphism between the orbit space ($G _ a\!\setminus\!D _ C(\Gamma)$ and an explicitly given parameter domain $D _ P(\Gamma)$ of dimension $Nn-\frac{n(n+1)}{2}$. We also prove that B is the union of the coordinate domains $D _ C(\Gamma)$, where $\Gamma$ ranges over all Z-systems on N.

    When $n=3$ our results give an axiomatization of a rigorous mathematical theory for what chemists call valence coordinates, or Z-matrix internal coordinates. Z-systems can be simply manipulated like n-dimensional building blocks; such manipulations are quite complex if one uses Z-matrices. We briefly discuss applications of Z-systems to the study of biomolecular conformation.

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