A Proof of the Molecular Conjecture

Abstract

To identify flexible/rigid region in a protein is one of the central issues in the field of molecular biology as this could provide insight into its function and a means to predict possible changes of structural flexibility by the environmental factors such as temperature and pH. Several methods were developed for this purpose.

One of standard methods is to model the protein as a geometric graph embedded in ℝ³ by regarding an atom as a vertex and a bond between atoms as an edge with a fixed length. It analyzes the protein’s rigidity by using the theory of structural rigidity. Algorithms such as 3D pebble game were developed for this. Computer software like FIRST (Floppy Inclusions and Rigid Substructure Topography) uses this algorithm. These software are used widely and successfully now. Algorithms used by FIRST and other programs rely their correctness upon the theory of structural rigidity. From the mathematical point of view, however, the correctness proof is incomplete because it relies on the so called ”Molecular Conjecture” which has been a long standing open problem over twenty-five years in the field of combinatorial rigidity. In the past years, however, in spite of the absence of the rigorous proof of the Molecular Conjecture, empirical data have been accumulated that support the conjecture. Recently, we were able to settle the Molecular Conjecture affirmatively in ℝ³ and in higher dimensions and provide the theoretical validity of the algorithms behind such software as FIRST, FRODA, etc.

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 $\left({d+1 \choose 2}-1\right)G$ contains ${d+1\choose 2}$ edge-disjoint spanning trees, where $\left({d+1 \choose 2}-1\right)G$ is the graph obtained from G by replacing each edge by $\left({d+1\choose 2}-1\right)$ parallel edges. In 1984 they jointly posed a question about whether their combinatorial characterization can be further applied to a nongeneric case. Specifically, they conjectured that G can be realized as an infinitesimally rigid bofy-and-hinge framework if and only if G can be realized as that with the additional “hinge-coplanar” property, i.e., all the hinges incident to each body are contained in a common hyperplane. This is the definition of “the Molecular Conjecture”.

In this talk, we will first introduce main topics in the field of combinatorial rigidity and then give a brief overview of the proof of the moleclar conjecture.

This is a joint work with Shin-ichi Tanigawa.