Florida State University Seal

Mathematics Colloquium


1. Partial demixing of RNA-protein complexes leads to intra-droplet patterning in phase- separated biological condensates
2. Reeb graphs as metric graph approximations of geodesic spaces
3. Integrable nonlinear pdes and their connections

Date: Friday, September 13, 2019
Place and Time: Room 101, Love Building, 3:35-4:25 pm
Refreshments: Room 204, Love Building, 3:00 pm

Abstracts. 1. An emerging mechanism for intracellular organization is liquid-liquid phase separation (LLPS). Found in both the nucleus and the cytoplasm, liquid-like droplets condense to create compartments that are thought to promote and inhibit specific biochemistry. In this work, a multiphase, Cahn-Hilliard diffuse interface model is used to examine RNA-protein interactions driving LLPS. We create a bivalent system that allows for two different species of protein-RNA complexes and model the competition that arises for a shared binding partner, free protein. With this system we demonstrate that the binding and unbinding of distinct RNA-protein complexes leads to diverse spatial pattern formation and dynamics within droplets. Both the initial formation and transient behavior of spatial patterning are subject to the exchange of free proteins between RNA-protein complexes. This study illustrates that spatiotemporal heterogeneity can emerge within phase-separated biological condensates with simple binding reactions and competition. Intra-droplet patterning may influence droplet composition and, subsequently, cellular organization on a larger scale.

2. A standard result in metric geometry is that every compact geodesic metric space can be approximated arbitrarily well by finite metric graphs in the Gromov-Hausdorff sense. It is well known that the first Betti number of the approximating graphs may blow up as the approximation gets finer. We study what happens if we put an upper bound on the first Betti number of approximating graphs and how can Reeb graphs be used in this situation.

3. The self-induced velocity of a vortex filament is governed by the Biot-Savart law. The localized induction approximation to the Biot-Savart law, which ignores negligible nonlocal effects, is known to be integrable and may be transformed via Hasimoto’s transformation into the nonlinear Schrodinger equation. When effects from within the vortex core are included, the localized induction approximation becomes more complicated, but remains integrable. Through Hasimoto’s transformation it can be associated with a higher order nonlinear Schrodinger equation referred to in the literature as Hirota’s equation. With this connection we are able to study extreme events such as breathers and rogue waves on an isolated vortex filament. We conclude this talk with a brief discussion of current and future research into nonlocal integrable pdes.