Liquid crystal phases of biological networks: models and analysis
Cytoskeletal networks consist of rigid, rod-like actin protein units jointed by flexible crosslinks, presenting coupled orientational and deformation effects analogous to liquid crystal elastomers. The alignment properties of the rigid rods influence the mechanical response of the network to applied stress and deformation, affecting functionality of the systems. Parameters that characterize these networks include the aspect ratio of the rods and the average length of the crosslinks, with a large span of parameter values found across in-vivo networks. For instance, cytoskeletal networks of red blood cells have very large linkers and small rod aspect ratio, whereas those of cells of the outer hair of the ear have large aspect ratio and short linkers favoring well aligned nematic, in order to achieve optimal sound propagation. We propose a class of free energy densities consisting of the sum of polyconvex functions of the anisotropic deformation tensor and the Landau-de Gennes energy of lyotropic liquid crystals. The growth conditions of the latter, with respect to the rod density and the nematic order tensor at the limit of the minimum eigenvalue -1/3 are essential to recover the limiting deformation map from the minimizing sequences of the anisotropic deformation gradient. We consider a bulk free energy density encoding properties of the rod and the network based on the Lopatina-Selinger construction for the Maier-Saupe theory. We then analyze the phase transition behavior under uniform expansion, biaxial extension and shear deformation, showing that the nematic-isotropic transition may be accompanied by a change of volume, which manifests itself in the nonconvexity of the stress-strain relation. We also account for the fact that in-vivo networks are found in the gel state. We conclude with some remarks on the roles of active elements in the model.