Starting from the observation that substances and reactions are the central entities of chemistry, I have structured chemical knowledge into a formal space called a directed hypergraph, which arises when substances are connected by their reactions. I call this hypernet chemical space. In this thesis, I explore different levels of description of this space: its evolution over time, its curvature, and categorical models of its compositionality. The vast majority of the chemical literature focuses on investigations of particular aspects of some substances or reactions, which have been systematically recorded in comprehensive databases such as Reaxys for the last 200 years. While complexity science has made important advances in physics, biology, economics, and many other fields, it has somewhat neglected chemistry. In this work, I propose to take a global view of chemistry and to combine complexity science tools, modern data analysis techniques, and geometric and compositional theories to explore chemical space. This provides a novel view of chemistry, its history, and its current status. We argue that a large directed hypergraph, that is, a model of directed relations between sets, underlies chemical space and that a systematic study of this structure is a major challenge for chemistry. Using the Reaxys database as a proxy for chemical space, we search for large-scale changes in a directed hypergraph model of chemical knowledge and present a data-driven approach to navigate through its history and evolution. These investigations focus on the mechanistic features by which this space has been expanding: the role of synthesis and extraction in the production of new substances, patterns in the selection of starting materials, and the frequency with which reactions reach new regions of chemical space. Large-scale patterns that emerged in the last two centuries of chemical history are detected, in particular, in the growth of chemical knowledge, the use of reagents, and the synthesis of products, which reveal both conservatism and sharp transitions in the exploration of the space. Furthermore, since chemical similarity of substances arises from affinity patterns in chemical reactions, we quantify the impact of changes in the diversity of the space on the formulation of the system of chemical elements. In addition, we develop formal tools to probe the local geometry of the resulting directed hypergraph and introduce the Forman-Ricci curvature for directed and undirected hypergraphs. This notion of curvature is characterized by applying it to social and chemical networks with higher order interactions, and then used for the investigation of the structure and dynamics of chemical space. The network model of chemistry is strongly motivated by the observation that the compositional nature of chemical reactions must be captured in order to build a model of chemical reasoning. A step forward towards categorical chemistry, that is, a formalization of all the flavors of compositionality in chemistry, is taken by the construction of a categorical model of directed hypergraphs. We lifted the structure from a lineale (a poset version of a symmetric monoidal closed category) to a category of Petri nets, whose wiring is a bipartite directed graph equivalent to a directed hypergraph. The resulting construction, based on the Dialectica categories introduced by Valeria De Paiva, is a symmetric monoidal closed category with finite products and coproducts, which provides a formal way of composing smaller networks into larger in such a way that the algebraic properties of the components are preserved in the resulting network. Several sets of labels, often used in empirical data modeling, can be given the structure of a lineale, including: stoichiometric coefficients in chemical reaction networks, reaction rates, inhibitor arcs, Boolean interactions, unknown or incomplete data, and probabilities. Therefore, a wide range of empirical data types for chemical substances and reactions can be included in our model.

Le concept de radio logicielle (SDR) est une solution pertinente pour concevoir des équipements multi-standards. Une façon de réaliser de tels équipements est d'identifier les fonctions et opérateurs communs entre les standards. Cette approche s’appelle la paramétrisation et est divisée en deux catégories : l'approche pragmatique qui est une version pratique pour créer et développer des opérateurs communs à partir d’opérateurs existants, et l'approche théorique dont l’objectif est de réaliser une exploration graphique d’un équipement multi-standards selon différents niveaux de granularité, accompagnée d’un problème d'optimisation. C’est cette dernière approche qui a constitué le sujet de base de cette thèse. Ainsi, une fonction de coût doit être optimisée afin de sélectionner les opérateurs communs entre les différentes normes, ce qui permet de proposer une configuration optimale à partir de laquelle sont déduits les opérateurs communs. Dans notre travail, nous avons dans un premier temps modélisé théoriquement la structure graphique d’un système multi-standards par un hypergraphe orienté. En outre, nous avons fourni une expression mathématique alternative de la fonction de coût suggérée, en utilisant des définitions propres à la théorie des graphes. Ensuite, nous avons montré que le problème d'optimisation associé était un problème NP sous une certaine contrainte, ce qui a entraîné une preuve d'exclusion de certaines configurations dont les coûts ne peuvent être minimaux. Ceci a constitué la deuxième contribution de cette thèse. Enfin, nous avons proposé un nouvel algorithme permettant de résoudre le problème d'optimisation donné, et dont l'intérêt est de donner une solution optimale du problème au lieu d’une solution approchée fournie par les méthodes heuristiques classiques. Un programme associé à cet algorithme a été développé en langage C, puis appliqué à plusieurs exemples de cas génériques afin d’en étudier les performances. / The Software-Defined Radio (SDR) concept is emerging as a potential and efficient solution for designing flexible future-proof multi-standard systems. A way of realizing a multi-standard terminal is to identify the appropriate common functions and operators inside and between the standards. This is what's called the parametrization approach, which can be divided into two categories: the pragmatic approach which is a practical version to create and develop common operators, and the theoretical approach which represents a graphical exploration of the SDR multi-standard system at different levels of granularity accompanied with an optimization problem. It’s in this last approach where our thesis subject dwells. In this context, a suggested cost function (in previous work) has to be optimized in order to select the convenient common operators between the different standards, enabling to construct an optimal design. In our work, we theoretically model a previously proposed graph structure of an SDR multi-standard system as a directed hypergraph as well as provide an alternative mathematical formal expression of the suggested cost function, using various graph theoretical definitions and notations. Afterwards, we prove that the associated optimization problem is an NP-problem under a certain constraint, which entails a proof of exclusion of some particular design options when searching for a minimum cost design. This was the second contribution in this thesis before we finally present a new algorithm (which exploits various modelization aspects of directed hypergraphs) that can solve the optimization problem, whose interest is in it giving an exact-optimal solution to our problem instead of a near-optimal one provided by heuristics. A program code for this algorithm was developed in C-language, and then it was applied on several generic case examples in order to explore its performance skills.

Lors de cette thèse, je me suis intéressée à la modélisation du métabolisme des micro-organismes. Nous nous sommes focalisé sur le métabolisme des petites molécules qui ne prend pas en compte les réactions associées aux macromolécules, telle que la synthèse des protéines.Nous avons ainsi utilisé différents formalismes de modélisation.Tout d'abord, nous avons développé TOTORO où les réseaux métaboliques sont représentés par des hypergraphes dirigés et qui permet d'identifier les réactions ayant participé à une transition métabolique. TOTORO a été utilisé sur un jeu de données sur la levure en présence de cadmium. Nous avons pu montrer que nous retrouvons les mécanismes connus de désintoxication.Ensuite, en utilisant une méthode de modélisation par contraintes, nous discutons d'un développement en cours, KOTOURA, qui propose d'utiliser les connaissances actuelles de concentrations de métabolites entre différentes conditions pour inférer de manière quantitative les possibles asynchronies des réactions lors du passage d'un état stable à un autre. Nous avons testé son implémentation sur des données simulées.Enfin, nous proposons MULTIPUS, une méthode d'extraction d'(hyper)-arbres de Steiner dirigés qui permet de sélectionner les voies métaboliques pour la production de composés au sein d'une communauté bactérienne. Les réseaux métaboliques sont modélisés en utilisant des hypergraphes dirigés et pondérés. Nous proposons un algorithme de programmation dynamique paramétré ainsi qu'une formulation utilisant la programmation par ensemble réponse. Ces deux propositions sont ensuite comparées dans deux cas d'applications / In this PhD work, we proposed to model metabolism. Our focus was to develop generic models, that are not specific to one organism or condition, but are instead based on general assumptions that we tried to validate using data from the literature.We first present TOTORO that uses a qualitative measurement of concentrations in two steady-states to infer the reaction changes that lead to differences in metabolite pools in both conditions.TOTORO enumerates all sub-(hyper)graphs that represent a sufficient explanation for the observed differences in concentrations. We exploit a dataset of Yeast (Saccharomyces cerevisiae) exposed to cadmium and show that we manage to retrieve the known pathways used by the organisms. We then address the same issue, but using a constraint-based programming framework, called KOTOURA, that allows to infer more quantitatively the reaction changes during the perturbed state. We use in this case exact concentration measurements and the stoichiometric matrix, and show on simulated datasets that the overall variations of reaction fluxes can be captured by our formulation.Finally, we propose MULTIPUS, a method to infer microbial communities and metabolic roads to produce specific target compounds from a set of defined substrates. We use in this case a weighted directed hypergraph. We apply MULTIPUS to the production of antibiotics using a consortium composed of an archae and an actinobacteria and show hat their metabolic capacities are complementary. We then infer for another community the excretion of an inhibitory product (acetate) by a 1,3-propanediol (PDO) producer and its consumption by a methanogene archae