Comparison of Methods for Determining Boundaries of Lipid Patches in a Phase Separation Problem on a Torus

Abstract

Determining lipid configurations on a curved surface in a phase separation problem is a problem that has gained interest in the mathematical biology community recently. We consider the case in which two different lipid types are present, and we locate lipid patch boundaries to understand the lipid configurations. While a variety of methods can be used to solve such problems, we actually compare two different methods in this work: a diffuse interface approach and an initial value problem approach. In the latter approach, we directly compute, albeit approximately, the patch boundary curve on a ring torus. Solving with Euler’s method, we find good agreement between the two approaches and provide results that relate different parameters from the two approaches.