Statistical methodology and software to analyse oscillatory signals with applications to biology

Abstract

Many physiological and biological phenomena, such as menstrual cycles (Draper et al. (2018)), reproductive activity (Simonneaux and Bahougne (2015), Caba et al. (2018)), cell cycle (Liu et al. (2017)) or circadian biology (Hughes et al. (2009), Zhang et al. (2014), Andreani et al. (2015), Seney et al. (2019)), are governed by oscillatory systems consisting of numerous signals that exhibit rhythmic patterns over time. For example, the circadian clock is a molecular pacemaker that orchestrates daily functional activity including metabolic state, endocrine activity or neural excitability. Genes involved in those processes that exhibit rhythmic expression patterns along ~24-hour periods are called circadian genes. The study of such signals with temporal rhythmic patterns, and how these patterns change under different conditions, is called chronobiology. Chronobiology has been an active area of research during the past two decades, with major impact on treating cardiovascular disorders like hypertension (Halberg et al. (2013)), detecting genes associated with neurodegenerative disorders (Li et al. (2013)) or depression (Chauhan et al. (2017)), and improving the effectiveness of cancer treatments (Chan et al. (2017)). For instance, Haus (2009) demonstrated that the timing of radiation according to host and/or tumour rhythms improves the toxic/therapeutic ratio of the treatment. These and other findings in biomedical sciences have increased interest in chronobiological experiments. From a statistical point of view, the analysis of rhythmic signals ( ) in chronobiology has several challenges because of: (a) displays a wide variety of rhythmic patterns over time, which are not exactly sinusoidal or even symmetric (Koren£i£ et al. (2012), Zhang et al. (2014), Rueda et al. (2019)); (b) the density of the time points and the number of periods of data is usually very small (Panda et al. (2002), Hughes et al. (2007, 2009), Yang and Su (2010)); (c) the intrinsic circular nature of data from oscillatory systems; (d) the vari- 1 ability in time course expression data due to noisy nature of the data; (e) in some applications, the temporal order among samples may be unknown. For these reasons, standard time series or Fourier models are not convenient for the analysis of chronobiological rhythms (Elkum and Myles (2006), Wijnen et al. (2006), Leise (2013)). Models based on parametric functions of time, such as Cosinor, have been proposed in chronobiology to model these patterns (Tong (1976), Cornelissen (2014)). The main drawback of these approaches is that such parametric functions are too rigid, as signals in oscillatory systems very often exhibit asymmetric patterns. There are several commonly encountered problems in chronobiology. The main problem to solve in this context is rhythmicity detection as not all patterns observed in an oscillatory system display rhythmic patterns. For a given signal ; rhythmicity detection can be formulated as the following hypothesis test: H0 : is a flat signal (1.1) H1 : is rhythmic signal. This problem has been studied extensively in literature, existing a wide variety of procedures to address it including, among others, those based on sinusoidal curve fitting (Liu et al. (2004), Straume (2004), Cornelissen (2014)), autocorrelation (Levine et al. (2002)) or Fourier analyses (Wichert et al. (2004)). Some non-parametric approaches, such as JTK_Cycle (JTK) (Hughes et al. (2010)) and RAIN (Thaben and Westermark (2014)), based on Jonckheere- Terpstra test and Kendall's tau correlation, are widely employed by biologists. However, these two latter approaches do not detect asymmetric rhythmic patterns properly. A fundamental assumption made in the above discussion is that the time corresponding to each biological sample is known. However, in many instances, such as when dealing with samples obtained from human cadavers (Li et al. (2013), Seney et al. (2019)) or human organ biopsies, (Lamb et al. (2011), Bossé et al. (2012)), the exact time corresponding to each biological sample may be unknown. In such cases, one needs to first estimate or determine the time associated with each sample before investigating rhythmicity. This problem, known as temporal order estimation, is other crucial issue in chronobiology. Some recent procedures to cope with this problem are Oscope (Leng et al. (2015)) and CYCLOPS (Anafi et al. (2017)). Oscope was specifically designed to recover cell cycle dynamic, and it is only applicable in single cell RNA-Seq experiments. CYCLOPS is far from a mathematical close-fitting formulation. It is based on a neural network framework (which is like a black box) and uses additional rhythmicity information which is not always available. In addition to the major rhythmicity issues mentioned above, other interesting questions related to the analysis of oscillatory signals, such as peak time 2 estimation or rhythm-pattern comparisons, deserve consideration. For instance, when dealing with circadian genes, time peak estimation reveals crucial information for biologists about the timings at which genes' biological function is carried out. The main motivation of this thesis was to solve appealing rhythmicity questions specifically related to the analysis of circadian gene expression. In particular, the starting problem of this thesis was to identify among the several thousand of genes registered in a genetic study, those that display rhythmic expression patterns.