arXiv subjects

Sean T. Vittadello

Publications and source records attributed to Sean T. Vittadello.

10 recordsLinked to original sources

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↗

arXiv2025

Studying self-organisation across the biosphere with process-enablement graphs

At the heart of many contemporary theories of life is the concept of biological self-organisation: organisms have to continuously produce and maintain the conditions of their own existence in order to stay alive. The way in which these accounts articulate this concept, however, differs quite significantly. As a result, it can be difficult to identify self-organising features within biological systems, and to compare different descriptions of such features. In this paper, we develop a graph theoretic formalism -- process-enablement graphs -- to study the organisational structure of living systems. Cycles within these graphs capture self-organising components of a system in a general and abstract way. We build the mathematical tools needed to compare biological models as process-enablement graphs, facilitating a comparison of their corresponding descriptions of self-organisation in a consistent and precise manner. We apply our formalism to a range of classical theories of life and demonstrate exactly how these models are similar, and where they differ, with respect to their organisational structure. While our current framework does not demarcate living systems from non-living ones, it does allow us to better study systems that lie in the grey area between life and non-life.

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↗

arXiv2022

A group theoretic approach to model comparison with simplicial representations

The complexity of biological systems, and the increasingly large amount of associated experimental data, necessitates that we develop mathematical models to further our understanding of these systems. As biological systems are generally not well understood, most mathematical models of these systems are based on experimental data, resulting in a seemingly heterogeneous collection of models that ostensibly represent the same system. To understand the system we therefore need to know how the different models are related, with a view to obtaining a unified mathematical description. This goal is complicated by the fact that distinct mathematical formalisms may be used to represent the same system, making direct comparison of the models very difficult. In previous work we developed an appropriate framework for model comparison where we represent models as labelled simplicial complexes and compare them with two general methodologies: comparison by distance or equivalence. In this article we continue the development of our model comparison methodology in two directions. First, we present a rigorous and automatable methodology for the core process of comparison by equivalence, namely determining the vertices in a simplicial representation, corresponding to model components, that are conceptually related and the identification of these vertices via simplicial operations. Our methodology is based on considerations of vertex symmetry in the simplicial representation, for which we develop the required mathematical theory of group actions on simplicial complexes. This methodology greatly simplifies and expedites the process of determining model equivalence. Second, we provide an alternative mathematical framework for our model-comparison methodology by representing models as groups, which allows for the direct application of group-theoretic techniques within our model-comparison methodology.

q-bio.QM↗

arXiv2022

Open Problems in Mathematical Biology

Biology is data-rich, and it is equally rich in concepts and hypotheses. Part of trying to understand biological processes and systems is therefore to confront our ideas and hypotheses with data using statistical methods to determine the extent to which our hypotheses agree with reality. But doing so in a systematic way is becoming increasingly challenging as our hypotheses become more detailed, and our data becomes more complex. Mathematical methods are therefore gaining in importance across the life- and biomedical sciences. Mathematical models allow us to test our understanding, make testable predictions about future behaviour, and gain insights into how we can control the behaviour of biological systems. It has been argued that mathematical methods can be of great benefit to biologists to make sense of data. But mathematics and mathematicians are set to benefit equally from considering the often bewildering complexity inherent to living systems. Here we present a small selection of open problems and challenges in mathematical biology. We have chosen these open problems because they are of both biological and mathematical interest.

q-bio.QM↗

arXiv2021

Model comparison via simplicial complexes and persistent homology

In many scientific and technological contexts we have only a poor understanding of the structure and details of appropriate mathematical models. We often, therefore, need to compare different models. With available data we can use formal statistical model selection to compare and contrast the ability of different mathematical models to describe such data. There is, however, a lack of rigorous methods to compare different models \emph{a priori}. Here we develop and illustrate two such approaches that allow us to compare model structures in a systematic way {by representing models in terms of simplicial complexes}. Using well-developed concepts from simplicial algebraic topology, we define a distance between models based on their simplicial representations. Employing persistent homology with a flat filtration provides for alternative representations of the models as persistence intervals, which represent the structure of the models, from which we can also obtain the distances between models. We then expand on this measure of model distance to study the concept of model equivalence in order to determine the conceptual similarity of models. We apply our methodology for model comparison to demonstrate an equivalence between a positional-information model and a Turing-pattern model from developmental biology, constituting a novel observation for two classes of models that were previously regarded as unrelated.

math.AT↗

arXiv2021

Bayesian and Algebraic Strategies to Design in Synthetic Biology

Innovation in synthetic biology often still depends on large-scale experimental trial-and-error, domain expertise, and ingenuity. The application of rational design engineering methods promise to make this more efficient, faster, cheaper and safer. But this requires mathematical models of cellular systems. And for these models we then have to determine if they can meet our intended target behaviour. Here we develop two complementary approaches that allow us to determine whether a given molecular circuit, represented by a mathematical model, is capable of fulfilling our design objectives. We discuss algebraic methods that are capable of identifying general principles guaranteeing desired behaviour; and we provide an overview over Bayesian design approaches that allow us to choose from a set of models, that model which has the highest probability of fulfilling our design objectives. We discuss their uses in the context of biochemical adaptation, and then consider how robustness can and should affect our design approach.

q-bio.MN↗

arXiv2020

A novel mathematical model of heterogeneous cell proliferation

We present a novel mathematical model of heterogeneous cell proliferation where the total population consists of a subpopulation of slow-proliferating cells and a subpopulation of fast-proliferating cells. The model incorporates two cellular processes, asymmetric cell division and induced switching between proliferative states, which are important determinants for the heterogeneity of a cell population. As motivation for our model we provide experimental data that illustrate the induced-switching process. Our model consists of a system of two coupled delay differential equations with distributed time delays and the cell densities as functions of time. The distributed delays are bounded and allow for the choice of delay kernel. We analyse the model and prove the non-negativity and boundedness of solutions, the existence and uniqueness of solutions, and the local stability characteristics of the equilibrium points. We find that the parameters for induced switching are bifurcation parameters and therefore determine the long-term behaviour of the model. Numerical simulations illustrate and support the theoretical findings, and demonstrate the primary importance of transient dynamics for understanding the evolution of many experimental cell populations.

q-bio.CB↗

arXiv2010

Co-universal C*-algebras associated to generalised graphs

We introduce P-graphs, which are generalisations of directed graphs in which paths have a degree in a semigroup P rather than a length in N. We focus on semigroups P arising as part of a quasi-lattice ordered group (G,P) in the sense of Nica, and on P-graphs which are finitely aligned in the sense of Raeburn and Sims. We show that each finitely aligned P-graph admits a C*-algebra C*_{min}(Lambda) which is co-universal for partial-isometric representations of Lambda which admit a coaction of G compatible with the P-valued length function. We also characterise when a homomorphism induced by the co-universal property is injective. Our results combined with those of Spielberg show that every Kirchberg algebra is Morita equivalent C*_{min}(Lambda) for some (N^2 * N)-graph Lambda.

math.OA↗

arXiv2009

Exel's crossed product for non-unital C*-algebras

We consider a family of dynamical systems (A,alpha,L) in which alpha is an endomorphism of a C*-algebra A and L is a transfer operator for α. We extend Exel's construction of a crossed product to cover non-unital algebras A, and show that the C*-algebra of a locally finite graph can be realised as one of these crossed products. When A is commutative, we find criteria for the simplicity of the crossed product, and analyse the ideal structure of the crossed product.

math.OA↗