The Algebraic Geometry of Motions of Bar-and-Body Frameworks

White, Neil; Whiteley, Walter

doi:10.1137/0608001

Cited by 85 publications

(121 citation statements)

References 16 publications

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“…This matrix (which is often written in the language of the projective Grassmann-Cayley algebra [4,6]) has six columns for each body of (G, q) (i.e. one column for each of the six degrees of freedom of the body) and one row for each bar of (G, q) (i.e.…”

Section: Generic Flexibility Of Molecular Structuresmentioning

confidence: 99%

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How does symmetry impact the flexibility of proteins?

Schulze

Sljoka

Whiteley

2014

Phil. Trans. R. Soc. A.

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It is well known that (i) the flexibility and rigidity of proteins are central to their function, (ii) a number of oligomers with several copies of individual protein chains assemble with symmetry in the native state and (iii) added symmetry sometimes leads to added flexibility in structures. We observe that the most common symmetry classes of protein oligomers are also the symmetry classes that lead to increased flexibility in certain three-dimensional structures—and investigate the possible significance of this coincidence. This builds on the well-developed theory of generic rigidity of body–bar frameworks, which permits an analysis of the rigidity and flexibility of molecular structures such as proteins via fast combinatorial algorithms. In particular, we outline some very simple counting rules and possible algorithmic extensions that allow us to predict continuous symmetry-preserving motions in body–bar frameworks that possess non-trivial point-group symmetry. For simplicity, we focus on dimers, which typically assemble with twofold rotational axes, and often have allosteric function that requires motions to link distant sites on the two protein chains.

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Section: Generic Flexibility Of Molecular Structuresmentioning

confidence: 99%

“…one row for each constraint on the body motions). So, R(G, q) is an |E| × 6|B| matrix, which has the following basic structure (see [4][5][6], for example, for details):…”

Section: Generic Flexibility Of Molecular Structuresmentioning

confidence: 99%

How does symmetry impact the flexibility of proteins?

Schulze

Sljoka

Whiteley

2014

Phil. Trans. R. Soc. A.

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“…Instead we use the Grassman-Cayley algebra 7,21,22 developed for weighted objects in affine or projective space. Specifically, given two Euclidean points or equivalently, two projective points we take the six minors of the matrix:…”

Section: General Hinge Structuresmentioning

confidence: 99%

Rigidity of Molecular Structures: Generic and Geometric Analysis

Whiteley

Fundamental Materials Research

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“…In this section we shall provide a formal definition of body-and-hinge frameworks following the description given in [12,27]. Refer to [3,12,26,27] for more detailed descriptions.…”

Section: Body-and-hinge Frameworkmentioning

confidence: 99%

“…Refer to [3,12,26,27] for more detailed descriptions. Throughout the paper, we simply refer to a d-dimensional affine space as a d-affine space for any nonnegative integer d.…”

Section: Body-and-hinge Frameworkmentioning

confidence: 99%

A Proof of the Molecular Conjecture

Katoh

Tanigawa

2011

Discrete Comput Geom

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A d-dimensional body-and-hinge framework is a structure consisting of rigid bodies connected by hinges in d-dimensional space. The generic infinitesimal rigidity of a body-and-hinge framework has been characterized in terms of the underlying multigraph independently by Tay and Whiteley as follows: A multigraph G can be realized as an infinitesimally rigid body-and-hinge framework by mapping each vertex to a body and each edge to a hinge if and only if d+1 2

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The Algebraic Geometry of Motions of Bar-and-Body Frameworks

Cited by 85 publications

References 16 publications

How does symmetry impact the flexibility of proteins?

How does symmetry impact the flexibility of proteins?

Rigidity of Molecular Structures: Generic and Geometric Analysis

A Proof of the Molecular Conjecture

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