Cortical geometry may influence placement of interface between Par protein domains in early Caenorhabditis elegans embryos

During polarization, proteins and other polarity determinants segregate to the opposite ends of the cell (the poles) creating biochemically and dynamically distinct regions. Embryos of the nematode worm Caenorhabditis elegans (C. elegans) polarize shortly after fertilization, creating distinct regions of Par protein family members. These regions are maintained through to first cleavage when the embryo divides along the plane specified by the interface between regions, creating daughter cells with different protein content. In wild type single cell embryos the interface between these Par protein regions is reliably positioned at approximately 60% egg length, however, it is not known what mechanisms are responsible for specifying the position of the interface. In this investigation, we use two mathematical models to investigate the movement and positioning of the interface: a biologically based reaction-diffusion model of Par protein dynamics, and the analytically tractable perturbed Allen-Cahn equation. When we numerically simulate the models on a static 2D domain with constant thickness, both models exhibit a persistently moving interface that specifies the boundary between distinct regions. When we modify the simulation domain geometry, movement halts and the interface is stably positioned where the domain thickness increases. Using asymptotic analysis with the perturbed Allen-Cahn equation, we show that interface movement depends explicitly on domain geometry. Using a combination of analytic and numeric techniques, we demonstrate that domain geometry, a historically overlooked aspect of cellular simulations, may play a significant role in spatial protein patterning during polarization.