## MSc Thesis Presentation – Turner Silverthorne

Thursday, September 22, 2022 at 12:00 p.m.
BA6183

MSc Candidate: Turner Silverthorne
Thesis title: A mathematical model of promoter methylation in the circadian clock

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Living systems use genetic networks to produce complex dynamics necessary for survival. Examples include limit-cycle oscillations, bistable switching, and the attenuation of molecular noise. The mammalian circadian clock is a particularly interesting example, consisting of two interacting genetic feedback loops that achieve a delicate balance between robustness and plasticity.

The clock’s period (it’s most important output) is vital for normal biological function, and regulated by a variety of factors. Recent experiments have provided strong evidence that DNA methylation plays a role in controlling the period of the circadian clock. The connections between epigenetic factors (such as DNA methylation) and the circadian clock are multifaceted and poorly understood. In this thesis, I investigate epigenetic regulation of the circadian clock from a mathematical perspective.

Building on an earlier model of the primary feedback loop in the circadian clock, we add a layer of epigenetic regulation. From this extended model, we derive a perturbative estimate of the clock’s period that quantifies the influence of DNA methylation. We then use timescale separation arguments to derive an approximate model with the structure of a monotone cyclic feedback system. Such systems obey a generalization of the Poincaré-Bendixson theorem and hence have a relatively-tame bifurcation theory. Using our reduced model, we show that methylation can induce Hopf bifurcations, alter the period, and remove bistability. Together, our analysis of the reduced and full model add a new perspective to epigenetic regulation of one of the most important biological oscillators: the circadian clock.

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A copy of the thesis can be found here: masters_thesis_silverthorne

## Departmental MSc Thesis Presentation – Mahmud Azam

Tuesday, August 23 2022 at 3:00 p.m. (sharp)

MSc Candidate: Mahmud Azam
Supervisor: Alexander Kupers
Thesis title: Semidirect Products of ∞–Operads

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We provide a construction of an $\infty$–operad from a functor $BG \to \Op_\infty$ encoding the action of a group $G$ on a given unital $\infty$–operad whose underlying $\infty$–category is a Kan complex.  This construction, restricted to classical operads in $\Set$ viewed as $\infty$–operads, coincides with the semidirect product construction. Taking this as the definition of semidirect product of $\infty$–operads, we show that the action of $G$ on the given $\infty$–operad is equivalent to the trivial action if and only if the corresponding semidirect products
are equivalent. We then outline how one might generalize this result to operads in $\Top$
and use this to show that the semidirect product of the real version of the little $n$–disks operad with $SO(n – 1)$ or $SO(n – 2)$ for $n$ even or odd respectively corresponding to the usual action is equivalent to the semidirect product corresponding to the trivial action.

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A copy of the thesis can be found here:Operads