Theory and applications of motions of anisotropic interfacial networks

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Considering the gradient flow of anisotropic surface energy constrained by volume preservation, we developed a level set-based method for realizing anisotropic motion of an interface network. First, we considered a variational approach to a vector-type Merriman-Bence-Osher (MBO) thresholding scheme with penalization technique, which realized volume-preserving crystalline motion with a smoothing technique but required an ad hoc scheme to satisfy the correct junction angles. Motivated by cellular pattern formations in sensory epithelium, we considered anisotropic motions driven by cell-cell adhesion. Due to the complexity of tricellular junctions, we then developed an MBO-type convolution scheme with localized auction dynamics which realizes correct cell contact angles and features good compatibility with level set representation and a solid mathematical background including stability and convergence. Here, cell-cell junctions are implicitly represented using level sets and succeeding dynamics are numerically determined by thresholding. We addressed several aspects pertinent to cell biology, e.g., our localization step prevents cells from unnatural splitting during rearrangement. Our method successfully reproduced development of cellular patterns in sensory systems, including cell sorting, total engulfment, and cell internalization. These simulations were implemented using C programming language and Matlab software. Results have been submitted for publication, as well as, presented and well received in local and overseas conferences.