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Gene regulatory networks (GRNs) model the controlling interactions between genes, where the ex- pression of some genes activate or inhibit the expression of other genes. In the study of biomedical systems, a better understanding of the system can be achieved by knowing the underlying GRN in different conditions (e.g. health/disease or control/mutant). Generally, the underlying GRN of the system is not known, and it is inferred from transcriptomic data by computational methods. Single- cell transcriptomic measurements have been developed and exponentially improved over the last decade. These recent experimental techniques can measure the expression of almost each gene for most of the individual cells in a sample and have been widely used to study the heterogeneity of biological systems. However, there are not many computational methods available to infer GRNs from this type of data and the existing ones suffer from major limitations. Thus, there is a need for the development of computational approaches to infer GRNs from single-cell transcriptomics. The aim of this thesis is to develop a simple and scalable method that can infer GRNs from single-cell transcriptomic time series data by studying pairwise regulations between genes. The presented method, named single-cell All-to-All (scATA), is based on estimating the parameters of a stochastic linear differential equation that describes the regulation between each pair of regulator and target genes, one pair at a time while ignoring other genes. The parameters are estimated by solving an optimization problem that minimizes the Wasserstein distance between the simulated distribution of the target gene and the corresponding time series data. The simulated distribution is obtained by numerically integrating a stochastic differential equation several times to obtain a distribution of the regulated gene trajectories. The developed method was tested on synthetic data simulated from different network models with different sizes and topologies up to 10 genes, with AUROC between 0.65 and 0.91 for 5- genes networks and between 0.54 and 0.71 for 10-genes networks. The shape of the ROC curves show that, with scATA, we are able to identify a few links with high confidence. To evaluate the applicability and performance of the algorithm on experimental data, the method was applied to infer the GRN of a publicly available, single-cell transcriptomic time series data, with a publicly available GRN compiled from literature. The use of this tool can provide new insights into the regulatory mechanism inside biological systems. It can propose novel key connections between genes to be validated experimentally, that, if verified, could be useful in better understanding the underlying system and in developing targeted treatments. This thesis is as proof of concept that dynamical model-based pairwise approaches, previously used in bulk transcriptomics, can also be used for GRN inference using single-cell time series.

From population collapses to cell-fate decision, critical phenomena are abundant in complex real-world systems. Among modelling theories to address them, the critical transitions framework gained traction for its purpose of determining classes of critical mechanisms and identifying generic indicators to detect and alert them (“early warning signals”). This thesis contributes to such research field by elucidating its relevance within the systems biology landscape, by providing a systematic classification of leading mechanisms for critical transitions, and by assessing the theoretical and empirical performance of early warning signals. The thesis thus bridges general results concerning the critical transitions field – possibly applicable to multidisciplinary contexts – and specific applications in biology and epidemiology, towards the development of sound risk monitoring system.

The present thesis is articulated over two main topics which have in common the modeling of the dynamical properties of complex biological systems from large-scale time-series data. On one hand, this thesis analyzes the inverse problem of reconstructing Gene Regulatory Networks (GRN) from gene expression data. This first topic seeks to reverse-engineer the transcriptional regulatory mechanisms involved in few biological systems of interest, vital to understand the specificities of their different responses. In the light of recent mathematical developments, a novel, flexible and interpretable modeling strategy is proposed to reconstruct the dynamical dependencies between genes from short-time series data. In addition, experimental trade-offs and optimal modeling strategies are investigated for given data availability. Consistent literature on these topics was previously surprisingly lacking. The proposed methodology is applied to the study of circadian rhythms, which consists in complex GRN driving most of daily biological activity across many species. On the other hand, this manuscript covers the characterization of dynamically differentiable brain states in Zebrafish in the context of epilepsy and epileptogenesis. Zebrafish larvae represent a valuable animal model for the study of epilepsy due to both their genetic and dynamical resemblance with humans. The fundamental premise of this research is the early apparition of subtle functional changes preceding the clinical symptoms of seizures. More generally, this idea, based on bifurcation theory, can be described by a progressive loss of resilience of the brain and ultimately, its transition from a healthy state to another characterizing the disease. First, the morphological signatures of seizures generated by distinct pathological mechanisms are investigated. For this purpose, a range of mathematical biomarkers that characterizes relevant dynamical aspects of the neurophysiological signals are considered. Such mathematical markers are later used to address the subtle manifestations of early epileptogenic activity. Finally, the feasibility of a probabilistic prediction model that indicates the susceptibility of seizure emergence over time is investigated. The existence of alternative stable system states and their sudden and dramatic changes have notably been observed in a wide range of complex systems such as in ecosystems, climate or financial markets.

Dynamic network reconstruction refers to a class of problems that explore causal interactions between variables operating in dynamical systems. This dissertation focuses on methods and algorithms that reconstruct/infer network topology or dynamics from observations of an unknown system. The essential challenges, compared to system identification, are imposing sparsity on network topology and ensuring network identifiability. This work studies the following cases: multiple experiments with heterogeneity, low sampling frequency and nonlinearity, which are generic in biology that make reconstruction problems particularly challenging. The heterogeneous data sets are measurements in multiple experiments from the underlying dynamical systems that are different in parameters, whereas the network topology is assumed to be consistent. It is particularly common in biological applications. This dissertation proposes a way to deal with multiple data sets together to increase computational robustness. Furthermore, it can also be used to enforce network identifiability by multiple experiments with input perturbations. The necessity to study low-sampling-frequency data is due to the mismatch of network topology of discrete-time and continuous-time models. It is generally assumed that the underlying physical systems are evolving over time continuously. An important concept system aliasing is introduced to manifest whether the continuous system can be uniquely determined from its associated discrete-time model with the specified sampling frequency. A Nyquist-Shannon-like sampling theorem is provided to determine the critical sampling frequency for system aliasing. The reconstruction method integrates the Expectation Maximization (EM) method with a modified Sparse Bayesian Learning (SBL) to deal with reconstruction from output measurements. A tentative study on nonlinear Boolean network reconstruction is provided. The nonlinear Boolean network is considered as a union of local networks of linearized dynamical systems. The reconstruction method extends the algorithm used for heterogeneous data sets to provide an approximated inference but improve computational robustness significantly. The reconstruction algorithms are implemented in MATLAB and wrapped as a package. With considerations on generic signal features in practice, this work contributes to practically useful network reconstruction methods in biological applications.