Topological Defects as Morphogenetic Factors
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Biological systems of cells and cytoskeletal elements can form a nematic phase where elongated constituents align parallel to each other, inducing partial orientational order. This order is described by a macroscopic director field that reflects the local orientation of the system. Topological defects in the nematic order are singularities in the director field; they are quite common and classified by their topological charge. In morphogenesis, the process of biological shape formation, defects act as organising centres enabling organisms to grow persistent protrusions or deplete material to relieve stress. An ideal model organism for studying body shape dynamics in the realm of soft condensed matter physics is Hydra, a freshwater basal marine invertebrate. In Hydra's ectoderm, topological defects align with morphological features; for instance, the +1 defect is localised at the tentacle's tip, while two -1/2 defects reside at its base. The aim of this talk is to address the open question of modeling mathematically the coupling between mesogens disclination and polymeric network by providing a mathematical framework describing the out-of-plane shape changes of initially flat LCN sheets containing a central topological defect. Adopting a variational approach, we define an energy associated with the deformations consisting of two contributions: an elastic energy term accounting for spatial director variations, and a strain-energy function describing the elastic response of the polymer network. The interplay between nematic elasticity, which seeks to minimize distortions in the director field, variations in the degree of order, with the consequent tendency of monomers in the polymer chains to distribute anisotropically in response to an external stimulus, and mechanical stiffness, which resists deformation, determines the resulting morphology. Building on this framework, we introduce a continuum mechanical model aimed at describing developing biological systems, accounting for the dynamic and adaptive changes that occur in living tissues over time. To this end, we incorporate concepts from the theory of solid plasticity and study the temporal evolution of a non-equilibrium system: a folded spheroid with localized topological defects, representing an intermediate stage in the regeneration of an excised Hydra tissue fragment. Our goal is to understand how topological defects relate to the formation of morphological features in the regenerated organism.