Turing Instabilities are Not Enough to Ensure Pattern Formation.

Symmetry-breaking instabilities play an important role in understanding the mechanisms underlying the diversity of patterns observed in nature, such as in Turing's reaction-diffusion theory, which connects cellular signalling and transport with the development of growth and form. Extensive literature focuses on the linear stability analysis of homogeneous equilibria in these systems, culminating in a set of conditions for transport-driven instabilities that are commonly presumed to initiate self-organisation. We demonstrate that a selection of simple, canonical transport models with only mild multistable non-linearities can satisfy the Turing instability conditions while also robustly exhibiting only transient patterns. Hence, a Turing-like instability is insufficient for the existence of a patterned state. While it is known that linear theory can fail to predict the formation of patterns, we demonstrate that such failures can appear robustly in systems with multiple stable homogeneous equilibria. Given that biological systems such as gene regulatory networks and spatially distributed ecosystems often exhibit a high degree of multistability and nonlinearity, this raises important questions of how to analyse prospective mechanisms for self-organisation.

Turing Pattern Formation in Reaction-Cross-Diffusion Systems with a Bilayer Geometry.

Conditions for self-organisation via Turing's mechanism in biological systems represented by reaction-diffusion or reaction-cross-diffusion models have been extensively studied. Nonetheless, the impact of tissue stratification in such systems is under-explored, despite its ubiquity in the context of a thin epithelium overlying connective tissue, for instance the epidermis and underlying dermal mesenchyme of embryonic skin. In particular, each layer can be subject to extensively different biochemical reactions and transport processes, with chemotaxis - a special case of cross-diffusion - often present in the mesenchyme, contrasting the solely molecular transport typically found in the epidermal layer. We study Turing patterning conditions for a class of reaction-cross-diffusion systems in bilayered regions, with a thin upper layer and coupled by a linear transport law. In particular, the role of differential transport through the interface is explored together with the presence of asymmetry between the homogeneous equilibria of the two layers. A linear stability analysis is carried out around a spatially homogeneous equilibrium state in the asymptotic limit of weak and strong coupling strengths, where quantitative approximations of the bifurcation curve can be computed. Our theoretical findings, for an arbitrary number of reacting species, reveal quantitative Turing conditions, highlighting when the coupling mechanism between the layered regions can either trigger patterning or stabilize a spatially homogeneous equilibrium regardless of the independent patterning state of each layer. We support our theoretical results through direct numerical simulations, and provide an open source code to explore such systems further.

Sensitivity of cartilage mechanical behaviour to spatial variations in material properties.

Articular cartilage tissue exhibits a spatial dependence in material properties that govern mechanical behaviour. A mathematical model of cartilage tissue under one dimensional confined compression testing is developed for normal tissue that takes account of these variations in material properties. Modifications to the model representative of a selection of mechanisms driving osteoarthritic cartilage are proposed, allowing application of the model to both physiological and pathophysiological, osteoarthritic tissue. Incorporating spatial variations into the model requires the specification of more parameters than are required in the absence of these variations. A global sensitivity analysis of these parameters is implemented to identify the dominant mechanisms of mechanical response, in normal and osteoarthritic cartilage tissue, to both static and dynamic loading. The most sensitive parameters differ between dynamic and static mechanics of the cartilage, and also differ between physiological and osteoarthritic pathophysiological cartilage. As a consequence changes in cartilage mechanics in response to alterations in cartilage structure are predicted to be contingent on the nature of loading and the health, or otherwise, of the cartilage. In particular the mechanical response of cartilage, especially deformation, is predicted to be much more sensitive to cartilage stiffness in the superficial zone given the onset of osteoarthritic changes to material properties, such as superficial zone increases in permeability and reductions in fixed charge. In turn this indicates that any degenerative changes in the stiffness associated with the superficial cartilage collagen mesh are amplified if other elements of osteoarthritic disease are present, which provides a suggested mechanism-based explanation for observations that the range of mechanical parameters representative of normal and osteoarthritic tissue can overlap substantially.