Morphological Study of Trichoplax Adhaerens Using a Phase Field Model

Abstract

Understanding the physics of free boundary dynamics and the kinetics of non-equilibrium phase transformation was, and still is, essential to the advancement of technology in terms of engineered materials. Even historically, said physics was paramount during the industrial revolution because it facilitated the wide-scale production of steel and other alloys. It did so because, at the critical points of solidification and precipitation, dynamical free boundaries are formed, which are often described by sharp interface models (e.g., the Stefan problem).

Despite current advances in computational physics, such models are difficult to solve, even numerically. A more convenient model is a phase-field one, where the interface is taken to be continuous rather than sharp. Mathematically, this is achieved by introducing a field (order parameter), which assumes a constant value in the bulk of each phase and varies continuously along the interface. One then solves for this field, among others (temperature, strain, etc.), by writing a free energy functional and minimizing it. The advantages this model provide are illustrated in its application to the Grinfeld instability, where stress-induced forces cause an instability at the interface between a uni-axially strained solid and its melt. Naturally, this model is attractive when thinking about biological systems because they can be thought of as materials with a free boundary and elastic properties.

In this thesis, we propose to apply this model to study the dynamic morphology of a peculiar little creature known as Trichoplax adhaerens (Placozoa). It has a thin flat body composed of a few thousand cells of six types, and it moves using cilia on its external surface. To do so, we numerically model its dynamics as a 2D conserved phase field coupled to an elastic field and subject to a time dependent external forcing. The behavior of an initial random surface is analyzed for a linear and power-law decreasing dependence. This analysis is quantified by introducing the structure factor S(q), defined as the Fourier transform of the height-height correlation function. Initial profiles of different Fourier modes are also analyzed similarly. We find that the power-law dependence is much better in mimicking the out-of-plane buckling observed in the creature. Lastly, we also extend the model to 3D, but only briefly explore it due to time constraints.