Inference methods for the study of interacting biological oscillators in single-cells

Biological oscillators are pervasive in biology, covering all aspects of life from enzyme kinetics reactions to population dynamics. Although their behaviour has been intensively studied in the last decades, the recent advances of high-throughput experimental technologies in the fields of omics and microscopy has called for the development of new analysis methods. Among the many types of models and quantitative analyses, parametric approaches are promising as they enable for a mechanistic or physical explanation of the phenomena under study. In particular, dynamical systems theory seems particularly adapted as the vast majority of oscillators can be modelled through differential equations. Dynamical systems parameters can also be easily opti-mized via maximum likelihood approaches. The validity of the inferred model can then be assessed from the quality of its predictions. We here present three different scientific questions regarding noisy biological oscillators, which are answered using maximum-likelihood inference approaches applied to parametric models. We first take interest in the characterisation of the influence of the cell-cycle over the circadi-an clock in individual mammalian cells. To this end, we develop a method combining a Hidden Markov Model with an Expectation-Maximization algorithm to infer their coupling from single-cell microscopy traces. We show that this coupling predicts multiple phase-locked states exhib-iting different degrees of robustness against molecular fluctuations inherent to cellular scale bio-logical oscillators. We then try to understand how the mammalian transcriptome behaves in the liver. Thence, we use single-cell RNA sequencing (scRNA-seq) along with mixed-models to investigate the interplay between gene regulation in space and time. Categorising mRNA expression profiles using mixed-effect models and smFISH validations, we find that many genes in the liver are both zonated and rhythmic, most of them showing multiplicative space-time effects. Finally, we look more closely at the cell-cycle, as it is one of the main drivers of gene expres-sion cell-to-cell heterogeneity in otherwise homogeneous cell populations. Here, we would like to understand if and how cell-cycle velocity changes depending on the phase of the cycling cells. To that end, we formulate the problem in terms of an autonomous dynamical system and use this to infer consistent dynamics for the cell-cycle from scRNA-seq data. Phase inference being paramount in all of these three studies, a short technical review on the topic is also provided at the end of this thesis, along with Julia scripts for the main inference methods presented. Various computational tools assisting the understanding of the scientific questions at stake are also presented, including a Python Dash web-app, a D3 widget and many Matplotlib animations and widgets