When data are available for all nodes of a Gaussian graphical model, then, it is possible to use sample correlations and partial correlations to test to what extent the conditional independencies that encode the structure of the model are indeed verified by the data. In this paper, we give a heuristic rule useful in such a validation process: When the correlation subgraph involved in a conditional independence is balanced (i.e., all its cycles have an even number of negative edges), then a partial correlation is usually a contraction of the corresponding correlation, which often leads to conditional independence. In particular, the contraction rule can be made rigorous if we look at concentration subgraphs rather than correlation subgraphs. The rule is applied to real data for elementary gene regulatory motifs.

Motivation: The simultaneous availability of ATAC-seq and RNA-seq experiments allows to obtain a more in-depth knowledge on the regulatory mechanisms occurring in gene regulatory networks. In this article, we highlight and analyze two novel aspects that leverage on the possibility of pairing RNA-seq and ATAC-seq data. Namely we investigate the causality of the relationships between transcription factors, chromatin and target genes and the internal consistency between the two omics, here measured in terms of structural balance in the sample correlations along elementary length-3 cycles. Results: We propose a framework that uses the a priori knowledge on the data to infer elementary causal regulatory motifs (namely chains and forks) in the network. It is based on the notions of conditional independence and partial correlation, and can be applied to both longitudinal and non-longitudinal data. Our analysis highlights a strong connection between the causal regulatory motifs that are selected by the data and the structural balance of the underlying sample correlation graphs: strikingly, >97% of the selected regulatory motifs belong to a balanced subgraph. This result shows that internal consistency, as measured by structural balance, is close to a necessary condition for 3-node regulatory motifs to satisfy causality rules.

Model predictive control (MPC) represents nowadays one of the main methods employed for process control in industry. Its strong suits comprise a simple algorithm based on a straightforward formulation and the flexibility to deal with constraints. On the other hand, it can be questioned its robustness regarding model uncertainties and external noises. Thus, a lot of efforts have been spent in the past years into the search of methods to address these shortcomings. In this study, the authors propose a robust MPC controller which stems from the idea of adding robustness in the prediction phase of the algorithm while leaving the core of MPC untouched. More precisely, they consider a robust Kalman filter that has been recently introduced and they further extend its usability to feedback control systems. Overall the proposed control algorithm allows to maintain all of the advantages of MPC with an additional improvement in performance and without any drawbacks in terms of computational complexity. To test the actual reliability of the algorithm, they apply it to control a servomechanism system characterised by non-linear dynamics.

One of the scopes of Systems Biology is to propose mathematical models that best capture the dynamic behavior of intra-cellular processes. In this regard, the last two decades have brought up a shift in the field, with technological advances now allowing researchers to access a wide range of high-throughput technologies at an affordable cost. These techniques allow to simultaneously interrogate thousands of variables, such as genome-wide transcriptomics and proteomics. However, parallel to these technological advances, there is a growing need for mathematical models that are suited to integrate measurements obtained from different cellular processes.

In this thesis we aim to model combinations of three commonly used high-throughput data: epigenetic (namely ATAC-seq and DNA methylation), transcriptomic (RNA-seq) and proteomic data (MASS-spectrometry). In the first work we analyze paired ATAC-seq and RNA-seq data to integrate measurements of (i) chromatin openness, (ii) transcription factors (TFs) availability and (iii) gene expression. To model these data, we use elementary causal motifs, a class of mathematical models which is suited to represent causal interactions between three nodes. Indeed, our analysis shows that the elementary causal motifs in the data are enriched for biologically relevant TF-gene interactions. Moreover, a significant overlap is observed between the causal motifs identified in datasets representing similar cell stimuli, suggesting that causal motifs represent a robust biological signal.

This work is then extended to include another class of high-throughput data: MASS-spectrometry. More precisely, we propose a framework to model the flow of events that goes from chromatin remodeling to splice variants expression, and from splice variants to protein synthesis. As the underlying graph becomes more complex than the previous case, a more general mathematical framework is considered: Bayesian networks. Interestingly, this work shows that most putative associations between chromatin regions, splice variants and proteins that have been gathered by scientific community so far, are supported by the data. Moreover, similarly to the previous work, the causal interactions identified in the data highlight relevant biological features; more precisely, causal chains between chromatin regions, splice variants and proteins are enriched for splice variants that have a major role in protein synthesis.

From a technical point of view, causal motifs are characterized by a property known as conditional independence, which can be used to identify causal interactions in the data. However, particularly when the data available is limited, it is challenging to assess conditional independencies in the data. It is therefore of interest to investigate the existence of properties that allow us to predict conditional independence. In particular, in our work we propose two properties: structural balance and inverse balance, which are closely connected to what is known in the literature as positive association and multivariate total positivity of order 2 (MTP₂), respectively. Our analysis shows that both heuristics are useful in predicting conditional independence, both from a theoretical perspective and in experimental data.

Lastly, a network-based approach is used to integrate DNA methylation and RNA-seq in a case-control study centered around multiple sclerosis, in order to identify common regulatory patterns in DNA methylation and gene expression during the course of pregnancy. The strategy is based on the rationale that proteins that are interconnected in the protein-protein network are more likely to be involved in similar cellular functions. Indeed, the analysis highlights that similar pathways are altered at epigenetic and transcriptomic level, leading to a set of genes that are likely involved in the modification of the disease symptoms that is observed during pregnancy.