A species tree is a rooted binary tree with unique species names on the leaves, representing the evolutionary relationship between the species:

```
.
/ \
/ \
+ +
/ / \
/ / \
A B C
```

When attempting to reconstruct the evolutionary tree using any one gene, it’s possible to obtain *discordant* trees with conflicting topologies:

```
. . . .
/ \ / \ / \ / \
/ \ / \ / \ / \
+ + + + + + + +
/ / \ / / \ / / \ / / \
/ / \ / / \ / / \ / / \
A B C A B C B A C C A B
```

Resolving this discordance quickly using clustering is ongoing work and the topic of the following preprint:

*Trying Out a Million Genes to Find the Perfect Pair with RTIST*, with James Degnan, Bioinformatics 2022 (published article)- Python/C implementation of the algorithm is available on GitHub
- Watch a recorded talk given at the Algorithms and Complexity in Phylogenetics Seminar

Although most people imagine evolution occurring gradually, other modes of evolution are possible, too. Indeed, there’s evidence that some traits evolve in sudden ``bursts,’’ and then remain static for a long time. To infer which of these two models is correct for an inputted dataset and species tree, we developed a fast, information-theoretic method which can handle very noisey data. As application to a dataset (obtained by Jay McEntee over many summers…) is available here:

*Punctuated evolution in the learned songs of African sunbirds*, with Jay McEntee et al., Proc Roy Soc B 288(1963) 2021 (preprint)- Statistical model, fitting and model selection techniques are described in the supplementary information

This application was described in this very nice video:

Physical and social traits (wingspan, song pitch) can evolve in a variety of ways.

My PhD focused on a system of singularly-interacting stochastic particles of varying masses which experience inelastic 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)

When scaled appropriately, the hydrodynamic limit of the empirical mass density converges to the solution $\pho$ of a nonlinear Fokker-Planck equation, such as the Keller-Segel model of chemotaxis:
`$$ \begin{cases} \partial_t \rho &= \nabla \cdot (\mu \nabla \rho - \chi \rho \nabla c)\\ \Delta c &= -\rho \end{cases}, $$`

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:

This numerical method can be used to solve the original PDE near the blow-up time and observe various properties of the solution, e.g., see the linear change in the system’s second moment:

…in Drosophila, with Lyubov Chumakova and Natalia Bulgakova’s lab at the University of Sheffield. We investigated 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:

The effect is described in-depth here:

*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