Statistical physics of periodic biological processes

Earth rotation around its axis imposes a 24-hour rhythmicity to all life on the planet. Rather than passively responding to these periodic changes, nature has given us an internal timekeeper, the circadian clock, to anticipate to our advantage the fluctuations in the environment. The circadian clock is a cell-autonomous pervasive molecular oscillator with a period of about a day. The core of the clock is a transcriptional-transaltional negative feedback loop involving a few dozen genes. This set of genes induces 24-hour rhythms in many downstream processes which are responsible for the daily oscillations in behaviour and physiology. A functioning timekeeper has been associated with well-being while disruptions of the clock have been linked with a variety of diseases, including cancer. To study the properties and behaviour of the circadian clock in mammals the field has relied heavily on animal models and timed animal omics experiments, due to the difficulty of performing relevant human experiments. However, there is a vast set of human data coming from the clinic without a time stamp, so not directly exploitable to study circadian oscillations. Many methods have been proposed to assign time stamps to a set of omics snapshots; we take inspiration from their strengths and flaws to develop a new probabilistic method of circadian phase inference, CHIRAL. The physiological impact of the circadian clock in humans is not fully characterised. We exploited the Genotype-Tissue Expression project (GTEx), assigning time stamps to the samples using CHIRAL. We used existing relationships among samples to both robustly infer one time stamp per donor, and transfer time information from robust clocks to weaker ones. This procedure allowed us to study human mRNA rhythms in 46 tissues and compare circadian behaviour across sexes and ages. Clock transcripts showed highly conserved phase and amplitude relationships across tissues, and were tightly synchronised across the body. Tissue rhythmic gene expression programs differed in breadth, covering global and tissue-specific functions, including metabolic pathways and systemic responses. The circadian clock structure and amplitude was conserved across sexes and age groups. However, overall gene expression rhythms were highly sex-dimorphic and more sustained in females. Moreover, rhythmic programs dampened with age across the body. Together, our stratified analysis unveiled a rich organization of sex- and age-specific circadian gene expression rhythms in humans. To study the clock we took advantage of its low dimensional structure. In fact, projecting high dimensional biological data onto low dimensional manifolds is a widespread technique in biology. In particular, it has proved very useful to exploit single cell transcriptome data. In the single cell framework the RNA velocity technique arose. RNA velocity tackles the idea that cells move in the low dimensional manifolds; it infers the future state of a cell combining current spliced and unspliced RNA counts. However, it is still a very new technique, so not all of the possible adjustments have been made. We develop an analytical framework to add constrains to RNA velocity dynamical systems and differential geometry, allowing for trajectory reconstruction and calculation of time distances on the low dimensional manifold. We apply our method to the cell cycle and infer its period across different brain regions in mouse.