I'm currently working in applied probability and statistics, often in the context of biology.
Intracellular transport of E-cadherin in Drosophila, with Lyubov Chumakova and Natalia Bulgakova's lab at the University of Sheffield. We are investigating the interplay between the stochastic dynamics of molecular motors, and the dynamic instability of microtubules. You can see the qualitatively different transport outcomes of motors which move parallel and antiparallel to the direction of a MT's growth in the video below:
This effect is described in-depth in:
- The walkoff effect: cargo distribution implies motor type in bidirectional microtubule bundles, with Victor Alfred, Natalia A Bulgakova, Lyubov Chumakova (preprint)
- Notes on estimating absorption times
Modeling and detecting trait evolution on small phylogenetic trees, with Jay McEntee. We are investigating general methods for detecting different modes of evolution by looking at comparative data, and in particular, have found pulsed evolution in some social traits of birds:
- Inferring punctuated evolution in the learned songs of African sunbirds, with Jay P McEntee, Chacha Werema, Nadje Najar, Joshua V Penalba, Elia Mulungu, Maneno Mbilinyi, Sylvester Karimi, Lyubov Chumakova, J. Gordon Burleigh, Rauri C K Bowie (preprint)
Stochastic particle systems and the Keller-Segel model with Ibrahim Fatkullin. This is a system of singularly-interacting stochastic particles of varying masses, which experience inelastic ("sticky") collisions. This system can be viewed as 2D overdamped gravity. A preprint is available:
- Coalescing particle systems and applications to nonlinear Fokker-Planck equations, with Ibrahim Fatkullin, CMS 16(2) 2018 (preprint on arXiv)
When scaled appropriately, the hydrodynamic limit of the empirical mass density converges to the solution of a nonlinear Fokker-Planck equation, such as the Keller-Segel model of chemotaxis, or its generalization, the multispecies Keller-Segel model. These PDEs blow up in finite time, and form Dirac-type singularities. Such blow-ups correspond to coalescence in the particle system.
Using a combination of analytical estimates and grid-particle numerical methods, this particle system can be simulated quickly, thereby allowing for the numerical solution of some nonlinear Fokker-Planck PDEs. Here's a coarsening system of ~40k pairwise interacting particles, approximating blow-up in a many-species Keller-Segel system: