A lattice-theoretical perspective on adhesive categories

It is a known fact that the subobjects of an object in an adhesive category form a distributive lattice. Building on this observation, in the paper we show how the representation theorem for finite distributive lattices applies to subobject lattices. In particular, we introduce a notion of irreducible object in an adhesive category, and we prove that any finite object of an adhesive category can be obtained as the colimit of its irreducible subobjects. Furthermore we show that every arrow between finite objects in an adhesive category can be interpreted as a lattice homomorphism between subobject lattices and, conversely, we characterize those homomorphisms between subobject lattices which can be seen as arrows

Colorectal Cancer Through Simulation and Experiment

Colorectal cancer has continued to generate a huge amount of research interest over several decades, forming a canonical example of tumourigenesis since its use in Fearon and Vogelstein’s linear model of genetic mutation. Over time, the field has witnessed a transition from solely experimental work to the inclusion of mathematical biology and computer-based modelling. The fusion of these disciplines has the potential to provide valuable insights into oncologic processes, but also presents the challenge of uniting many diverse perspectives. Furthermore, the cancer cell phenotype defined by the ‘Hallmarks of Cancer’ has been extended in recent times and provides an excellent basis for future research. We present a timely summary of the literature relating to colorectal cancer, addressing the traditional experimental findings, summarising the key mathematical and computational approaches, and emphasising the role of the Hallmarks in current and future developments. We conclude with a discussion of interdisciplinary work, outlining areas of experimental interest which would benefit from the insight that mathematical and computational modelling can provide

Cell growth rate dictates the onset of glass to fluid-like transition and long time super-diffusion in an evolving cell colony

Collective migration dominates many phenomena, from cell movement in living systems to abiotic self-propelling particles. Focusing on the early stages of tumor evolution, we enunciate the principles involved in cell dynamics and highlight their implications in understanding similar behavior in seemingly unrelated soft glassy materials and possibly chemokine-induced migration of CD8$^{+}$ T cells. We performed simulations of tumor invasion using a minimal three dimensional model, accounting for cell elasticity and adhesive cell-cell interactions as well as cell birth and death to establish that cell growth rate-dependent tumor expansion results in the emergence of distinct topological niches. Cells at the periphery move with higher velocity perpendicular to the tumor boundary, while motion of interior cells is slower and isotropic. The mean square displacement, $\Delta(t)$, of cells exhibits glassy behavior at times comparable to the cell cycle time, while exhibiting super-diffusive behavior, $\Delta (t) \approx t^{\alpha}$ ($\alpha > 1$), at longer times. We derive the value of $\alpha \approx 1.33$ using a field theoretic approach based on stochastic quantization. In the process we establish the universality of super-diffusion in a class of seemingly unrelated non-equilibrium systems. Super diffusion at long times arises only if there is an imbalance between cell birth and death rates. Our findings for the collective migration, which also suggests that tumor evolution occurs in a polarized manner, are in quantitative agreement with {\it in vitro} experiments. Although set in the context of tumor invasion the findings should also hold in describing collective motion in growing cells and in active systems where creation and annihilation of particles play a role.Comment: 56 pages, 19 figure

Concatenation and other Closure Properties of Recognizable Languages in Adhesive Categories

We consider recognizable languages of cospans in adhesive categories, ofwhich recognizable graph languages are a special case. We show that such languages are closed under concatenation, i.e. under cospan composition, by providing a con-crete construction that creates a concatenation automaton from two given automata.The construction is considerably more complex than the corresponding construction for finite automata. We conclude by showing negative closure properties for Kleene star and substitution

A Unifying Theory for Graph Transformation

The field of graph transformation studies the rule-based transformation of graphs. An important branch is the algebraic graph transformation tradition, in which approaches are defined and studied using the language of category theory. Most algebraic graph transformation approaches (such as DPO, SPO, SqPO, and AGREE) are opinionated about the local contexts that are allowed around matches for rules, and about how replacement in context should work exactly. The approaches also differ considerably in their underlying formal theories and their general expressiveness (e.g., not all frameworks allow duplication). This dissertation proposes an expressive algebraic graph transformation approach, called PBPO+, which is an adaptation of PBPO by Corradini et al. The central contribution is a proof that PBPO+ subsumes (under mild restrictions) DPO, SqPO, AGREE, and PBPO in the important categorical setting of quasitoposes. This result allows for a more unified study of graph transformation metatheory, methods, and tools. A concrete example of this is found in the second major contribution of this dissertation: a graph transformation termination method for PBPO+, based on decreasing interpretations, and defined for general categories. By applying the proposed encodings into PBPO+, this method can also be applied for DPO, SqPO, AGREE, and PBPO

Rewriting Structured Cospans: A Syntax For Open Systems

The concept of a system has proliferated through natural and social sciences. While myriad theories of systems exist, there is no mathematical general theory of systems. In this thesis, we take a first step towards formulating such a theory. Our focus is on developing a syntax for compositional systems equipped with a rewriting theory. We pull from category theory and linguistics to accomplish this. The basic syntactical unit is a structured cospan and rewriting is introduced via the double pushout method. Two versions of rewriting are proposed: one that tracks intermediate steps and another disregards them. Benefits and drawbacks of both versions are discussed. We apply our results to the decomposition of closed systems, obtaining a structurally inductive viewpoint of rewriting such systems

Validity of the Cauchy-Born rule applied to discrete cellular-scale models of biological tissues

The development of new models of biological tissues that consider cells in a discrete manner is becoming increasingly popular as an alternative to PDE-based continuum methods, although formal relationships between the discrete and continuum frameworks remain to be established. For crystal mechanics, the discrete-to-continuum bridge is often made by assuming that local atom displacements can be mapped homogeneously from the mesoscale deformation gradient, an assumption known as the Cauchy-Born rule (CBR). Although the CBR does not hold exactly for non-crystalline materials, it may still be used as a first order approximation for analytic calculations of effective stresses or strain energies. In this work, our goal is to investigate numerically the applicability of the CBR to 2-D cellular-scale models by assessing the mechanical behaviour of model biological tissues, including crystalline (honeycomb) and non-crystalline reference states. The numerical procedure consists in precribing an affine deformation on the boundary cells and computing the position of internal cells. The position of internal cells is then compared with the prediction of the CBR and an average deviation is calculated in the strain domain. For centre-based models, we show that the CBR holds exactly when the deformation gradient is relatively small and the reference stress-free configuration is defined by a honeycomb lattice. We show further that the CBR may be used approximately when the reference state is perturbed from the honeycomb configuration. By contrast, for vertex-based models, a similar analysis reveals that the CBR does not provide a good representation of the tissue mechanics, even when the reference configuration is defined by a honeycomb lattice. The paper concludes with a discussion of the implications of these results for concurrent discrete/continuous modelling, adaptation of atom-to-continuum (AtC) techniques to biological tissues and model classification

Nanoelectronic Applications of Magnetoelectric Nanostructures

The greatly increased interest in magnetoelectric materials over the last decade is due to their potential to enable next-generation multifunctional nanostructures required for revolutionizing applications spanning from energy-efficient information processing to medicine. Magnetoelectric nanomaterials offer a unique way to use a voltage to control the electron spin and, reciprocally, to use remotely controlled magnetic fields to access local intrinsic electric fields. The magnetoelectric coefficient is the most critical indicator for the magnetoelectric coupling in these nanostructures. To realize the immense potential of these materials, it is necessary to maximize the coefficient. Therefore, the goal of this PhD thesis study was to create a new paradigm for the synthesis and characterizations of magnetoelectric materials which would allow to create a new dynasty of nanostructures required for unlocking all their unprecedented capabilities. Coreshell nanostructures with a 0-3 connectivity scheme, i.e. (Co, Ni) Fe2O4-BaTiO3, represent the most studied system. Their relatively low coefficient value is often attributed to the problem known as the dielectric leakage, which is present during the traditional powder form measurements of the coefficient. To overcome this problem, we implemented a novel approach to measure the coefficient at a single-nanoparticle level. Using a scanning probe microscopy, we entirely eliminated the interparticle interaction and thus the leakage problem. The success of this approach was underscored by achieving, for the first time, perfect crystal lattice matching between the magnetostrictive core and the piezoelectric shell of the coreshell configuration, as confirmed via transmission electron microscopy. As a result, this study led to the coefficient value for CoFe2O4-BaTiO3 nanoparticles of above 5 V cm-1 Oe-1, almost two orders of magnitude higher than the highest reported value elsewhere. Additionally, we for the first time demonstrated three different regions which are barium titanate shell, the interfacial transition, and the cobalt ferrite core, respectively, by imaging a half-grown coreshell nanoparticle with atomic force microscopy. Alternating gradient and cryogenic vibrating sample magnetometry were utilized to study the magnetic properties of materials. X-ray diffraction was employed to bespeak that the crystallinity of barium titanate is enhanced along with the increase of cobalt ferrite dopant on account of heterogeneous nucleation

Structural Decomposition of Reactions of Graph-Like Objects

Inspired by decomposition problems in rule-based formalisms in Computational Systems Biology and recent work on compositionality in graph transformation, this paper proposes to use arbitrary colimits to "deconstruct" models of reactions in which states are represented as objects of adhesive categories. The fundamental problem is the decomposition of complex reactions of large states into simpler reactions of smaller states. The paper defines the local decomposition problem for transformations. To solve this problem means to "reconstruct" a given transformation as the colimit of "smaller" ones where the shape of the colimit and the decomposition of the source object of the transformation are fixed in advance. The first result is the soundness of colimit decomposition for arbitrary double pushout transformations in any category, which roughly means that several "local" transformations can be combined into a single "global" one. Moreover, a solution for a certain class of local decomposition problems is given, which generalizes and clarifies recent work on compositionality in graph transformation. Introduction Compositional methods for the synthesis and analysis of computational systems remain a fruitful research topic with potential applications in practice. Though compositionality is most clearly exhibited in semantics for process calculi where structural operational semantics (SOS) can be found in its "pure" form, a slightly broader perspective is appropriate to make use of the fundamental ideas of SOS in interdisciplinary research. The first source of inspiration of the present paper is the κ-calculus [6], which is an influential modelling framework in Computational Systems Biology. The κ-calculus allows to give abstract, formal descriptions of biological systems that can be used to explain the reaction (rate) of complex systems, so-called complexes, in terms of the reaction (rate) of each of its subsystems, which are called partial complexes. Leaving quantitative aspects as a topic for future research, we concentrate on a specific sub-problem, namely the "purely structural" decomposition of reactions. In the κ-calculus, system states are composed of partial complexes and they have an intuitive, graphical representation. Hence, it is natural to investigate the decomposition of (reactions of) system states using concepts from graph transformation. In its simplest form, the idea of composition of graph transformations is by means of coproducts. Intuitively, the coproduct of two graphs models the assembly of two states put side by side and the two (sub-)states react independently of each other. A well-known, related theorem about graph transformations is the so-called Parallelism Theorem (see e.g. [5, Theorem 17]). A more general formalism of compositionality that is based on pushouts has been (re-)considered in In this paper, we shall remove the restriction to pushouts as a composition mechanism and generalize the results of [18] from pushouts to (pullback stable) colimits of arbitrary shape. This considerably enlarges the set of available gluing patterns. As a simple example, we can now equip each sub-state with several interfaces; this would be appropriate for the model of a cell in an organism that is in direct contact with each of its neighbouring cells with some part of its membrane; each area of contact would be modelled by a different interface. Content of the paper After reviewing some basic category theoretical concepts and the definition of adhesive categories in Section 1, we begin Section 2 with the "deconstruction" of models of system states; more precisely, we explain in Section 2.1 how suitably finite objects in adhesive categories arise as the colimit of a diagram of "atomic" objects, namely irreducible objects in the sense of The main problem, which is concerned with the decomposition of a "global" transformation into a family of "local" ones, is addressed in Section 3. We give a formal description of local decomposition problems, which consist of a given decomposition of a state (as a colimit of a certain shape) and a rule that describes a possible reaction of the state; to solve such a problem means to extend the decomposition of the state to a decomposition of the whole reaction (using colimits of the same shape). Section 3.1 presents a "global" solution, which first constructs the whole transformation "globally"; a "more local" solution of the problem is possible if we are given extra information that involve a generalization of the accommodations o