Rigidity in mechanical biological networks

Multicellular organisms generate complex morphologies required for their function. Organisms control these morphologies by tuning active forces and by altering the emergent 'material properties' of a tissue, i.e. the rheology of the tissue. In many cases, organisms take advantage of dramatic changes in the rheology that occur when the material undergoes a rigidity transition from a fluid-like or floppy state to a solid-like or rigid state. This transition in turn depends on internal parameters at the scale of cells and molecules. This review highlights recent theoretical work identifying the mechanisms that drive such transitions, so that biologists can look for these mechanisms in in vivo or in vitro systems. We discuss two main types of transition: a first-order rigidity transition that depends on the connectivity of small-scale structures, such as the number of contacts between cells or the number of branch points in a biopolymer network; and a second-order rigidity transition that depends on the geometry of small-scale structures, such as the shape of cells or the distance between crosslinks in a polymer network. We provide examples of each type of transition in model organisms and discuss methods for distinguishing between the mechanisms in future experiments.