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Mathematical Analysis of Liquid Crystal Models in Biology

Maria-Carme Calderer

Professor of Mathematics
School of Mathematics
University of Minnesota

In this lecture, I present and analyze models of anisotropic crosslinked polymers employing tools from the theories of nematic liquid crystals and liquid crystal elastomers. The anisotropy of these systems stems from the presence of rigid-rod molecular units in the network. Energy functionals of compressible, incompressible elastomers as well as rod-fluid networks are addressed. The theorems on the minimization of  the energies combine methods of isotropic nonlinear elasticity with the theory of lyotropic liquid crystals. A main feature of the systems is the coupling between the Eulerian Landau-de Gennes liquid crystal energy and the Lagrangian anisotropic elastic energy of deformation. Predictions of cavities in the minimizing configurations follow as a result of the nature of the coupling. I will also present a mixed finite element  analysis of the incompressible elastomer. 
We apply the theory to the study of phase transitions in networks of rigid rods, in order to model the behavior of actin filament systems found in the cytoskeleton of the interstitial tissue and in the inner cell.
The phase transition behavior depends on geometric and physical parameters of the network as well as on the aspect ratio, length and density of the rigid groups. In particular, we focus on the formation of chevron structures in the case that the aspect ratio of the rods is small. 
The results show good agreement with the molecular dynamics experiments reported in the literature as well as with laboratory experiments.  
Finally, we address the problem of polymer encapsulation to study configurations of bacteriophages.