Léo Diaz | Searcharxiv

arXiv2026

Abstract relational structures in models of biology

The mathematical formalisms used to model biological systems induce both latent and ambiguous assumptions that can limit or distort their representational capabilities. Developing formalisms that can represent systems more precisely is fundamental to comprehending their intricacies and complexities. Here we introduce the systems hypergraph, a general and extendable formalism for representing abstract relational systems. A systems hypergraph combines a hypergraph, representing multidimensional relations among objects, with a hierarchical system of attributes representing system properties and their interdependencies. The attribute structure ensures that dependencies between system properties are patent and unambiguous, thereby clarifying assumptions and avoiding redundancy in data association. As an application we consider two formalisms widely used in systems biology - chemical reaction networks and stochastic Petri nets - and study their natural representation as systems hypergraphs. This allows us to relate the two formalisms rigorously, demonstrating in particular that stochastic Petri nets are strictly more general than chemical reaction networks in contrast to their commonly assumed equivalence. More broadly our work demonstrates the power of abstraction, and in particular its role in mediating between objects and relations in mathematical representations of biological complexity.

q-bio.QM↗

arXiv2024

Towards a mathematical framework for modelling cell fate dynamics

An adult human body is made up of some 30 to 40 trillion cells, all of which stem from a single fertilized egg cell. The process by which the right cells appear to arrive in their right numbers at the right time at the right place -- development -- is only understood in the roughest of outlines. This process does not happen in isolation: the egg, the embryo, the developing foetus, and the adult organism all interact intricately with their changing environments. Conceptual and, increasingly, mathematical approaches to modelling development have centred around Waddington's concept of an epigenetic landscape. This perspective enables us to talk about the molecular and cellular factors that contribute to cells reaching their terminally differentiated state: their fate. The landscape metaphor is however only a simplification of the complex process of development; it for instance does not consider environmental influences, a context which we argue needs to be explicitly taken into account and from the outset. When delving into the literature, it also quickly becomes clear that there is a lack of consistency and agreement on even fundamental concepts; for example, the precise meaning of what we refer to when talking about a `cell type' or `cell state.' Here we engage with previous theoretical and mathematical approaches to modelling cell fate -- focused on trees, networks, and landscape descriptions -- and argue that they require a level of simplification that can be problematic. We introduce random dynamical systems as one natural alternative. These provide a flexible conceptual and mathematical framework that is free of extraneous assumptions. We develop some of the basic concepts and discuss them in relation to now `classical' depictions of cell fate dynamics, in particular Waddington's landscape.

q-bio.SC↗