Rigidity and flexibility of biological networks

The network approach became a widely used tool to understand the behaviour of complex systems in the last decade. We start from a short description of structural rigidity theory. A detailed account on the combinatorial rigidity analysis of protein structures, as well as local flexibility measures of proteins and their applications in explaining allostery and thermostability is given. We also briefly discuss the network aspects of cytoskeletal tensegrity. Finally, we show the importance of the balance between functional flexibility and rigidity in protein-protein interaction, metabolic, gene regulatory and neuronal networks. Our summary raises the possibility that the concepts of flexibility and rigidity can be generalized to all networks.Comment: 21 pages, 4 figures, 1 tabl

Iterative Universal Rigidity

A bar framework determined by a finite graph $G$ and configuration $\bf p$ in $d$ space is universally rigid if it is rigid in any ${\mathbb R}^D \supset {\mathbb R}^d$. We provide a characterization of universally rigidity for any graph $G$ and any configuration ${\bf p}$ in terms of a sequence of affine subsets of the space of configurations. This corresponds to a facial reduction process for closed finite dimensional convex cones.Comment: 41 pages, 12 figure

Super Stable Tensegrities and Colin de Verdi\`{e}re Number ν\nu

A super stable tensegrity introduced by Connelly in 1982 is a globally rigid discrete structure made from stiff bars or struts connected by cables with tension. In this paper we show an exact relation between the maximum dimension that a multigraph can be realized as a super stable tensegrity and Colin de Verdi\`{e}re number~$\nu$ from spectral graph theory. As a corollary we obtain a combinatorial characterization of multigraphs that can be realized as 3-dimensional super stable tensegrities

Geometry of Point-Hyperplane and Spherical Frameworks

In this thesis we show that the infinitesimal rigidity of point-hyperplane frameworks in Euclidean spaces is equivalent to the infinitesimal rigidity of bar-joint frameworks in spherical spaces with a set of joints (corresponding to the hyperplanes) located on a hyperplane. This is done by comparing the rigidity matrix of Euclidean point-hyperplane frameworks and the rigidity matrix of spherical frameworks. This result clearly shows how the first-order rigidity in projective spaces and Euclidean spaces are globally connected. This geometrically significant result is central to the thesis. This result leads to the equivalence of the first-order rigidity of point-hyperplane frameworks with that of bar-joint frameworks with a set of joints in a hyperplane in a Euclidean space (joint work). We also study the rigidity of point-hyperplane frameworks and characterize their rigidity in Euclidean spaces. We next highlight the relationship between point-line frameworks and slider mechanisms in the plane. Point-line frameworks are used to model various types of slider mechanisms. A combinatorial characterization of the rigidity of pinned-slider frameworks in the plane is derived directly as an immediate consequence of the analogous result for pinned bar-joint frameworks in the plane. Using fixed-normal point-line frameworks, we model a second type of slider system in which the slider directions do not change. Also, a third type of slider mechanism is introduced in which the sliders may only rotate around a fixed point but do not translate. This slider mechanism is defined using point-line frameworks with rotatory lines (no translational motion of the lines is allowed). A combinatorial characterization of the generic rigidity of these frameworks is coauthored in a joint work. Then we introduce point-hyperplane tensegrity frameworks in Euclidean spaces. We investigate the rigidity and the infinitesimal rigidity of these frameworks using tensegrity frameworks in spherical spaces. We characterize these different types of rigidity for point-hyperplane tensegrity frameworks and show how these types of rigidity are linked together. This leads to a characterization of the rigidity of a broader class of slider mechanisms in which sliders may move under variable distance constraints rather than fixed-distance constraints. Finally we investigate body-cad constraints in the plane. A combinatorial characterization of their generic infinitesimal rigidity is given. We show how angular constraints are related to non-angular constraints. This leads to a combinatorial result about the rigidity of a specific class of body-bar frameworks with point-point coincidence constraints in the space

Applications of combinatorics to statics—rigidity of grids

AbstractThe infinitesimal rigidity (or briefly rigidity) of a bar-and-joint framework (in any dimension) can be formulated as a rank condition of the so-called rigidity matrix. If there are n joints in the framework then the size of this matrix is O(n), so the time complexity of determining its rank is O(n3). But in special cases we can work with graph and matroid theoretical models from which very fast and effective algorithms can be obtained. At first the case of planar square grids will be presented where they can be made rigid with diagonal rods and cables in the squares, and with long rods and cables which may be placed between any two joints of the grid. Then we will consider the one- and multi-story buildings, and finally some other results and algorithms

Rigidity and volume preserving deformation on degenerate simplices

Given a degenerate $(n+1)$-simplex in a $d$-dimensional space $M^d$ (Euclidean, spherical or hyperbolic space, and $d\geq n$), for each $k$, $1\leq k\leq n$, Radon's theorem induces a partition of the set of $k$-faces into two subsets. We prove that if the vertices of the simplex vary smoothly in $M^d$ for $d=n$, and the volumes of $k$-faces in one subset are constrained only to decrease while in the other subset only to increase, then any sufficiently small motion must preserve the volumes of all $k$-faces; and this property still holds in $M^d$ for $d\geq n+1$ if an invariant $c_{k-1}(\alpha^{k-1})$ of the degenerate simplex has the desired sign. This answers a question posed by the author, and the proof relies on an invariant $c_k(\omega)$ we discovered for any $k$-stress $\omega$ on a cell complex in $M^d$. We introduce a characteristic polynomial of the degenerate simplex by defining $f(x)=\sum_{i=0}^{n+1}(-1)^{i}c_i(\alpha^i)x^{n+1-i}$, and prove that the roots of $f(x)$ are real for the Euclidean case. Some evidence suggests the same conjecture for the hyperbolic case.Comment: 27 pages, 2 figures. To appear in Discrete & Computational Geometr

Growing super stable tensegrity frameworks

This paper discusses methods for growing tensegrity frameworks akin to what are now known as Henneberg constructions, which apply to bar-joint frameworks. In particular, the paper presents tensegrity framework versions of the three key Henneberg constructions of vertex addition, edge splitting and framework merging (whereby separate frameworks are combined into a larger framework). This is done for super stable tensegrity frameworks in an ambient two or three-dimensional space. We start with the operation of adding a new vertex to an original super stable tensegrity framework, named vertex addition. We prove that the new tensegrity framework can be super stable as well if the new vertex is attached to the original framework by an appropriate number of members, which include struts or cables, with suitably assigned stresses. Edge splitting can be secured in R2 (R3) by adding a vertex joined to three (four) existing vertices, two of which are connected by a member, and then removing that member. This procedure, with appropriate selection of struts or cables, preserves super-stability. In d dimensional ambient space, merging two super stable frameworks sharing at least d + 1 vertices that are in general positions, we show that the resulting tensegrity framework is still super stable. Based on these results, we further investigate the strategies of merging two super stable tensegrity frameworks in IRd; (d 2 f2; 3g)that share fewer than d + 1 vertices, and show how they may be merged through the insertion of struts or cables as appropriate between the two structures, with a super stable structure resulting from the merge

Design of freeform membrane -tensegrity structure

Inspired by a lightweight, and tectonically decent pavilion, MOOM pavilion in Japan, this thesis study explores a digital approach to transform an existing physical structure into a digital computational model by following the principle behind in order to explore and generalize the principle and develop a generic digital tool for designing freeform membrane-tensegrity structures in architecture. Through generalize the regulations behind the existing structure and the generic digital tool development, the way of designing the same type of structure could be more efficient, logical and free. In this thesis, a generic digital tool for constructing membrane-tensegrity structure will be developed by referring to the analysis of MOOM pavilion and the generic freeform tensegrity algorithm proposed by Tomohiro Tachi and his team in The University of Tokyo. Through analysis and tool development process, the digital modeling and simulation programs are required. Here the used programs are Rhinoceros 6; Rhinoceros plug-in Grasshopper and Kangaroo Physics; Kangaroo 2; Weaverbird etc. in grasshopper. Furthermore, two demonstrators of freeform membrane-tensegrity structures would be proposed as two possible approaches to apply the developed digital tool in architectural and structural design. Since then, this thesis study will be an inspiring starting point for the further researches and designs of membrane-tensegrity structures