Rigidity theory (physics)

Rigidity theory, or topological constraint theory, is a tool for predicting properties of complex networks (such as glasses) based on their composition. It was introduced by Phillips in 1979[1] and 1981[2], and refined by Thorpe in 1983.[3] Inspired by the study of the stability of mechanical trusses as pioneered by James Clerk Maxwell[4], and by the seminal work on glass structure done by William Houlder Zachariasen[5], this theory reduces complex molecular networks to nodes (atoms, molecules, proteins, etc.) constrained by rods (chemical constraints), thus filtering out microscopic details that ultimately don't affect macroscopic properties. An equivalent theory was developed by P.K. Gupta A.R. Cooper in 1990, where rather than nodes representing atoms, they represented unit polytopes[6]. An example of this would be the SiO tetrahedra in pure glassy silica. This style of analysis has applications in biology and chemistry, such as understanding adaptability in protein-protein interaction networks.[7] Rigidity theory applied to the molecular networks arising from phenotypical expression of certain diseases may provide insights regarding their structure and function.

In molecular networks, atoms can be constrained by radial 2-body bond-stretching constraints, which keep interatomic distances fixed, and angular 3-body bond-bending constraints, which keep angles fixed around their average values. As stated by Maxwell's criterion, a mechanical truss is isostatic when the number of constraints equals the number of degrees of freedom of the nodes. In this case, the truss is optimally constrained, being rigid but free of stress. This criterion has been applied by Phillips to molecular networks, which are called flexible, stressed-rigid or isostatic when the number of constraints per atoms is respectively lower, higher or equal to 3, the number of degrees of freedom per atom in a three-dimensional system.[8] Typically, the conditions for glass formation will be optimal if the network is isostatic, which is for example the case for pure silica.[9] Flexible systems show internal degrees of freedom, called floppy modes,[3] whereas stressed-rigid ones are complexity locked by the high number of constraints and tend to crystallize instead of forming glass during a quick quenching.