Mathematics

Viral infections are social processes. Viral replication requires shared gene products that can be used by multiple viral genomes within the same cell, and hence act as public goods. This gives rise to viral cheats, a type of molecular parasite formed by large deletions, that spread by exploiting public goods encoded by full-length viruses. Cheats exist across the viral universe, arise frequently in laboratory infections, and reflect the emergence of evolutionary conflict at the molecular level. In this talk, I will explore two evolutionary consequences of viral cheating that play out at different timescales. Firstly, we will consider the evolution of multipartite viruses, in which the genome is fragmented, and each fragment must separately infect a host. This genome structure comes with clear costs, but has nevertheless evolved multiple times, and today accounts for nearly 40% of known plant viral species. Previous explanations for the evolution of multipartitism have focused on group benefits, but typically require unrealistic rates of coinfection, especially for multipartite viruses with more than two segments. We will argue that cheating provides a contrasting explanation. By combining evolutionary game theory models with agent-based simulations, we will show that the invasion of mutually complementing viral cheats can drive the evolution of multipartitism under far more permissive conditions, including transitions to highly multipartite viruses. This framework shows that multipartitism need not be a group-level adaptation, but can instead emerge as the evolutionary endpoint of the tragedy of the commons. Secondly, we will consider the evolution of cheat-driven extinction in viruses. Cheats emerge spontaneously in laboratory infections of almost all known viruses, driving drastic reductions in viral population sizes. As a result, virologists have long argued that viral infections may be ‘self-limiting’, a claim supported by recent discoveries of cheats in natural viral infections. However, it is unclear whether viral infections provide enough time for viral cheats to emerge, spread, and drive cooperator extinction. Here, we present a birth-death model that incorporates mutation, demographic noise, and a frequency-dependent selective advantage to cheating. We identify qualitatively different dynamical regimes and the timescales under which they lead to viral extinction. We further show that our model can produce characteristic signatures of selection, opening the door to evolutionary biomarkers for predicting the outcome of viral infections from sequencing data. This approach argues that cheating may not only be relevant over long evolutionary timescales, but may also shape viral dynamics in clinically relevant ways, analogous to the emergence of cancer in multicellular organisms.

We consider a variant of a problem first introduced by Hughes and Rudnick (2003) and generalized by Bernard (2015) concerning conditional bounds for small first zeros in a family of L-functions. Here we seek to estimate the size of the smallest intervals centered at a low-lying height for which we can guarantee the existence of a zero in a family of L-functions. This leads us to consider an extremal problem in analysis which we address by applying the framework of de Branges spaces, introduced in this context by Carneiro, Chirre, and Milinovich (2022).

This talk will discuss two inference problems in dynamical systems, both motivated by applications in mathematical biology. First, we will discuss the classical gene regulatory network inference problem for time-stamped single-cell datasets and recent advances in optimal transport-based methods for this task. Second, if time permits, I will present an algorithm for bifurcation tracing, which aims to identify interfaces in parameter space. Applications to agent-based models and spatially extended reaction-diffusion equations will be demonstrated, both of which simulate Turing patterns commonly observed in animal skin, vegetation patterns, and more.

It has been known since the 80s, thanks to Conrey and Ghosh, that the average of the square of the Riemann zeta function, summed over the extreme points of zeta up to a height $T$, is $\frac{1}{2} (e^2-5) \log T$ as $T \rightarrow \infty$. This problem and its generalisations are closely linked to evaluating asymptotics of joint moments of the zeta function and its derivatives, and for a time was one of the few cases in which Number Theory could do what Random Matrix Theory could not. RMT then managed to retake the lead in calculating these sorts of problems, but we may now tell the story of how Number Theory is fighting back, and in doing so, describe how to find a full asymptotic expansion for this problem, the first of its kind for any nontrivial joint moment of the Riemann zeta function. This is joint work with Chris Hughes and Solomon Lugmayer.

Individual and collective cell polarity has fascinated mathematical modelers for a long time. Recently, a more subtle type of symmetry breaking started to attract attention of experimentalists and theorists alike - emergence of chirality in single cells and in cell groups. I will describe a joint project with Bershadsky/Tee lab to understand collective cell chirality on adhesive islands. From the initial microscopy data, two potential models emerged: in one, cells elongate and slowly rotate, and neighboring cells align with each other. When the collective rotation is stopped by the island boundaries, chirality emerges. In an alternative model, cells become chiral due to stress fibers turns inside the cells on the boundary, and then the polarity pattern propagates inward into the cellular groups. We used agent-based modeling to simulate these two hypotheses. The models make many predictions, and I will show how we discriminated between the models by comparing the data to these predictions.