Kavli Affiliate: Mark J. Bowick
| First 5 Authors: Arthur Hernandez, Michael F. Staddon, Mark J. Bowick, M. Cristina Marchetti, Michael Moshe
| Summary:
Vertex Models, as used to describe cellular tissue, have an energy controlled
by deviations from both a target area and a target perimeter. The constrained
nonlinear relation between area and perimeter, as well as subtleties in
selecting the appropriate reference state, lead to a host of interesting
mechanical responses. Here we provide a mean-field treatment of a highly
simplified model: a network of regular polygons with no topological
rearrangements. Since all polygons deform in the same way we need only analyze
the ground states and the response to deformations of a single polygon (cell).
The model exhibits the known transition between a fluid/compatible state, where
the cell can accommodate both target area and perimeter, and a
rigid/incompatible state. The fluid state has a manifold of degenerate zero
energy states. The rigid state has a single gapped ground state. We show that
linear elasticity fails to describe the response of the vertex model to even
infinitesimal deformations in both regimes and that the response depends on the
precise deformation protocol. We give a pictorial representation in
configuration space that reveals that the complex elastic response of the
Vertex Model arises from the presence of two distinct sets of reference
configurations (associated with target area and target perimeter) that may be
either compatible or incompatible, as well as from the underconstrained nature
of the model energy. An important result of our work is that the elasticity of
the Vertex Model cannot be captured by a Taylor expansion of the energy for
small strains, as Hessian and higher order gradients are ill-defined.
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