Current & Upcoming

Since the invention of the compound microscope in the early seventeenth century, scientists have marvelled over red blood cells and their surprising shape. An influential model of Canham predicts the shapes of blood cells and similar biomembranes come from a variational problem minimizing the “bending energy” of these surfaces. Because observed cells have the same shape in humans, it is natural to ask whether the model admits a unique solution. Yu and Chen reduced solution uniqueness for the genus one Canham problem (for a range of isoperimetric ratios) to proving positivity of a P-recursive sequence defined by an explicit linear recurrence relation with polynomial coefficients. In this talk we discuss a method of proving this positivity property, joint with Marc Mezzarobba, and its generalization, with Mezzarobba and Ruiwen Dong, to a wide range of P-recursive sequences. We combine rigorous numeric analytic continuation of D-finite functions with classic bounds from singularity analysis to derive an effective index where the asymptotic behaviour of the sequence, which is positive, dominates the sequence behaviour. Positivity of the finite number of remaining terms can then be checked computationally. Our work has been incorporated into the SageMath ore_algebra package, and can be used by researchers to automatically prove positivity for “generic” positive P-recursive sequences.

Communication between single cells or higher organisms by means of diffusive compounds is an important phenomenon in biological systems. A straightforward model is by a diffusion equation with suitable flux boundary conditions at the cell boundaries. Such a model will become computationally inefficient and analytically complex when there are many cells, even more so when they are moving. We propose to consider also a point source model. Each cell is virtually reduced to a point and appears in the diffusion equation for the compound on the full spatial domain as a singular reaction term in the form of a Dirac delta measure located at the cell’s centre. The amplitude of the Dirac delta measure is a nonlocal term of the compound’s concentration near the virtual cell boundary so as to preserve the essential biological features. To investigate the positivity of the solution and the structure of steady states, we employ the Laplace transform. In addition, the theory of elliptic curve is also involved.

TBA

The flows of tissues of epithelial cells often involve neighbour exchanges called T1 transitions. Mechanically, they are irreversible rearrangements crossing an energy barrier. In this talk, I will deploy geometric constructions of classical Euclidean geometry to calculate this energy barrier for general, isolated T1 transitions dominated by line tensions. I will show how regularity of cell packings, tension fluctuations, and nonlinear tensions increase this energy barrier, providing the basis for coarse-grained understanding of cell neighbour exchanges in continuum models of epithelia.

TBA