On Gaussian curvature and membrane fission.

We propose a three-dimensional mathematical model to describe dynamical processes of membrane fission. The model is based on a phase field equation that includes the Gaussian curvature contribution to the bending energy. With the addition of the Gaussian curvature energy term numerical simulations agree with the predictions that tubular shapes can break down into multiple vesicles.

Curvature-driven spatial patterns in growing 3D domains: A mechanochemical model for phyllotaxis.

Here we discuss the formation of phyllotactic patterns in the shoot apical meristem (SAM) of plants, where the spatial distribution of the phytohormone auxin determines phyllotaxis in a domain that is growing and changing in time. We assume that the concentration of auxin modifies the mechanical properties of the domain and that the mechanical stress field in the SAM orients the flux of auxin. To study this problem we propose a mechanism for pattern formation in growing domains with variable curvature.

A Method to Categorize 2-Dimensional Patterns Using Statistics of Spatial Organization.

We developed a measurement framework of spatial organization to categorize 2-dimensional patterns from 2 multiscalar biological architectures. We propose that underlying shapes of biological entities can be approached using the statistical concept of degrees of freedom, defining it through expansion of area variability in a pattern. To help scope this suggestion, we developed a mathematical argument recognizing the deep foundations of area variability in a polygonal pattern (spatial heterogeneity).