This work revolves around the mathematical modelling of biological oscillators and seeks to explore how their dynamics shapes the functionality of the associated systems; the thesis comprises three main parts.

In the first part, we investigated the ability of biological oscillators, in particular neurons, to synchronize their activities when coupled to each other. Understanding the spontaneous emergence of synchronization is of the utmost importance, as several neuronal diseases, such as Parkinson’s disease or epileptic seizures, are believed to arise as a result of an abnormal synchronization in certain brain regions. We studied a network made of two subpopulations of neurons, an excitatory and an inhibitory one, interconnected through a negative feedback loop with time-delayed couplings. We observed that with strong inter-population couplings, neurons could not only internally synchronize within each population, but also portray a type of complex dynamics known as chimera state. In this state, one population is synchronized while the other remains unsynchronized, with the time delay ensuring the stability of this state. Indeed, with instantaneous connections, the chimera state became neutrally stable, with a family of so-called breathing chimeras around it, namely with one population synchronized and the other periodically switching between more coherent and incoherent states.

In the second part, we analysed the dynamics of oscillating transcription factors, in particular, p53, and its downstream effect in the process of DNA repair. p53 is also known as "guardian of the genome" for its role in cancer prevention and maintenance of genomic stability. As a result of severe DNA damage, p53 nuclear abundance starts to oscillate, with a periodicity of 5.5 h, whose role is still highly debated. At the same time, the formation of microenvironments rich of repair proteins has been observed around the sites of damage, which are believed to arise from liquid-liquid phase separation, effectively giving rise to biomolecular condensates. In this context, the question we sought to address was: how can the cell regulate the distribution of repair material in the presence of multiple DNA damages? Does p53 oscillatory dynamics, and its specific frequency, play a role in this process? We found that p53 oscillations ensure the optimal repair rate in the presence of multiple DNA breaks, by regulating the formation and coarsening of the condensates around the sites of damage, thus ensuring a spatio-temporal resource distribution within the cell.

In the third part, we explored the possibility of entraining p53 oscillations to an external pulsing signal, in order to regulate its frequency. Through a microfluidic device coupled with live-cell imaging, we were able to track the dynamics of p53 oscillations after administering single or repeated pulses of small molecule nutlin-3a. By examining the response of p53 to single perturbations, we successfully predicted the system’s behaviour in response to a train of pulses, leading to the identification of Arnold tongues - an essential characteristic of entrainment. Our investigations revealed that the system becomes entrained to the external signal, giving rise to not only higher order entrainment, but also multi-stability and period-doubling. Finally, we highlighted a potential relationship between p53 frequency and its downstream target, p21. Remarkably, we observed that the natural p53 frequency minimizes p21 accumulation, potentially serving as a mechanism to prevent rapid commitment of cells to specific fates. Collectively, our findings provide valuable insights into the dynamic interplay between p53 oscillations, external stimuli, and downstream cellular processes, contributing to our understanding of cellular regulation and decision-making.