Diffusion-driven instabilities and emerging spatial patterns in patchy landscapes

Spatial variation in population densities across a landscape is a feature of many ecological systems, from self-organised patterns on mussel beds to spatially restricted insect outbreaks. It occurs as a result of environmental variation in abiotic factors and/or biotic factors structuring the spatial distribution of populations. However the ways in which abiotic and biotic factors interact to determine the existence and nature of spatial patterns in population density remain poorly understood. Here we present a new approach to studying this question by analysing a predator–prey patch-model in a heterogenous landscape. We use analytical and numerical methods originally developed for studying nearest- neighbour (juxtacrine) signalling in epithelia to explore whether and under which conditions patterns emerge. We find that abiotic and biotic factors interact to promote pattern formation. In fact, we find a rich and highly complex array of coexisting stable patterns, located within an enormous number of unstable patterns. Our simulation results indicate that many of the stable patterns have appreciable basins of attraction, making them significant in applications. We are able to identify mechanisms for these patterns based on the classical ideas of long-range inhibition and short-range activation, whereby landscape heterogeneity can modulate the spatial scales at which these processes operate to structure the populations

The isolation of spatial patterning modes in a mathematical model of juxtacrine cell signalling

Juxtacrine signalling mechanisms are known to be crucial in tissue and organ development, leading to spatial patterns in gene expression. We investigate the patterning behaviour of a discrete model of juxtacrine cell signalling due to Owen \& Sherratt (\emph{Math. Biosci.}, 1998, {\bf 153}(2):125--150) in which ligand molecules, unoccupied receptors and bound ligand-receptor complexes are modelled. Feedback between the ligand and receptor production and the level of bound receptors is incorporated. By isolating two parameters associated with the feedback strength and employing numerical simulation, linear stability and bifurcation analysis, the pattern-forming behaviour of the model is analysed under regimes corresponding to lateral inhibition and induction. Linear analysis of this model fails to capture the patterning behaviour exhibited in numerical simulations. Via bifurcation analysis we show that, since the majority of periodic patterns fold subcritically from the homogeneous steady state, a wide variety of stable patterns exists at a given parameter set, providing an explanation for this failure. The dominant pattern is isolated via numerical simulation. Additionally, by sampling patterns of non-integer wavelength on a discrete mesh, we highlight a disparity between the continuous and discrete representations of signalling mechanisms: in the continuous case, patterns of arbitrary wavelength are possible, while sampling such patterns on a discrete mesh leads to longer wavelength harmonics being selected where the wavelength is rational; in the irrational case, the resulting aperiodic patterns exhibit `local periodicity', being constructed from distorted stable shorter-wavelength patterns. This feature is consistent with experimentally observed patterns, which typically display approximate short-range periodicity with defects

Waves and propagation failure in discrete space models with nonlinear coupling and feedback

Many developmental processes involve a wave of initiation of pattern formation, behind which a uniform layer of discrete cells develops a regular pattern that determines cell fates. This paper focuses on the initiation of such waves, and then on the emergence of patterns behind the wavefront. I study waves in discrete space differential equation models where the coupling between sites is nonlinear. Such systems represent juxtacrine cell signalling, where cells communicate via membrane bound molecules binding to their receptors. In this way, the signal at cell j is a nonlinear function of the average signal on neighbouring cells. Whilst considerable progress has been made in the analysis of discrete reaction-diffusion systems, this paper presents a novel and detailed study of waves in juxtacrine systems. I analyse travelling wave solutions in such systems with a single variable representing activity in each cell. When there is a single stable homogeneous steady state, the wave speed is governed by the linearisation ahead of the wave front. Wave propagation (and failure) is studied when the homogeneous dynamics are bistable. Simulations show that waves may propagate in either direction, or may be pinned. A Lyapunov function is used to determine the direction of propagation of travelling waves. Pinning is studied by calculating the boundaries for propagation failure for sigmoidal and piecewise linear feedback functions, using analysis of 2 active sites and exact stationary solutions respectively. I then explore the calculation of travelling waves as the solution of an associated n-dimensional boundary value problem posed on [0, 1], using continuation to determine the dependence of speed on model parameters. This method is shown to be very accurate, by comparison with numerical simulations. Furthermore, the method is also applicable to other discrete systems on a regular lattice, such as the discrete bistable reaction-diffusion equation. Finally, I extend the study to more detailed models including the reaction kinetics of signalling, and demonstrate the same features of wave propagation. I discuss how such waves may initiate pattern formation, and the role of such mechanisms in morphogenesis

Coupling dynamics of 2D Notch-Delta signalling

Understanding pattern formation driven by cell–cell interactions has been a significant theme in cellular biology for many years. In particular, due to their implications within many biological contexts, lateral-inhibition mechanisms present in the Notch-Delta signalling pathway led to an extensive discussion between biologists and mathematicians. Deterministic and stochastic models have been developed as a consequence of this discussion, some of which address long-range signalling by considering cell protrusions reaching non-neighbouring cells. The dynamics of such signalling systems reveal intricate properties of the coupling terms involved in these models. In this work, we investigate the advantages and drawbacks of a single-parameter long-range signalling model across diverse scenarios. By employing linear and multi-scale analyses, we discover that pattern selection is not only partially explained but also depends on nonlinear effects that extend beyond the scope of these analytical techniques

Oscillations and patterns in spatially discrete models for developmental intercellular signalling

We extend previous models for nearest neighbour ligand-receptor binding to include both lateral induction and inhibition of ligand and receptor production, and different geometries (strings of cells and hexagonal arrays, in addition to square arrays). We demonstrate the possibility of lateral inhibition giving patterns with a characteristic length scale of many cell diameters, when receptor production is included. In contrast, lateral induction combined with inhibition of receptor synthesis cannot give rise to a patterning instability under any circumstances. Interesting new dynamics include the analytical prediction and consequent numerical observation of spatiotemporal oscillations—this depends crucially on the production terms and on the relationship between the decay rates of ligand and free receptor. Our approach allows for a detailed comparison with the model for Delta-Notch interactions of Collier et al. [4], and we find that a formal reduction may be made only when the ligand receptor binding kinetics are very slow. Without such very slow receptor kinetics, spatial pattern formation via lateral inhibition in hexagonal cellular arrays requires significant activation of receptor production, a feature that is not apparent from previous analyses

Multiscale analysis of pattern formation via intercellular signalling

Lateral inhibition, a juxtacrine signalling mechanism by which a cell adopting a particular fate inhibits neighbouring cells from doing likewise, has been shown to be a robust mechanism for the formation of fine-grained spatial patterns (in which adjacent cells in developing tissues diverge to achieve contrasting states of differentiation), provided that there is sufficiently strong feedback. The fine-grained nature of these patterns poses problems for analysis via traditional continuum methods since these require that significant variation takes place only over lengthscales much larger than an individual cell and such systems have therefore been investigated primarily using discrete methods. Here, however, we apply a multiscale method to derive systematically a continuum model from the discrete Delta-Notch signalling model of Collier \emph{et al.} (Pattern formation by lateral inhibition with feedback: a mathematical model of Delta-Notch intercellular signalling, \emph{J. Theor. Biol.}, 183, 1996, 429--446) under particular assumptions on the parameters, which we use to analyse the generation of fine-grained patterns. We show that, on the macroscale, the contact-dependent juxtacrine signalling interaction manifests itself as linear diffusion, motivating the use of reaction-diffusion-based models for such cell-signalling systems. We also analyse the travelling-wave behaviour of our system, obtaining good quantitative agreement with the discrete system

Signal Processing during Developmental Multicellular Patterning

Developing design strategies for tissue engineering and regenerative medicine is limited by our nascent understanding of how cell populations self-organize into multicellular structures on synthetic scaffolds. Mechanistic insights can be gleaned from the quantitative analysis of biomolecular signals that drive multicellular patterning during the natural processes of embryonic and adult development. This review describes three critical layers of signal processing that govern multicellular patterning: spatiotemporal presentation of extracellular cues, intracellular signaling networks that mediate crosstalk among extracellular cues, and finally, intranuclear signal integration at the level of transcriptional regulation. At every level in this hierarchy, the quantitative attributes of signals have a profound impact on patterning. We discuss how experiments and mathematical models are being used to uncover these quantitative features and their impact on multicellular phenotype

Mathematical modelling of pattern formation in developmental biology

The transformation from a single cell to the adult form is one of the remarkable wonders of nature. However, the fundamental mechanisms and interactions involved in this metamorphic change still remain elusive. Due to the complexity of the process, researchers have attempted to exploit simpler systems and, in particular, have focussed on the emergence of varied and spectacular patterns in nature. A number of mathematical models have been proposed to study this problem with one of the most well studied and prominent being the novel concept provided by A.M. Turing in 1952. Turing's simple yet elegant idea consisted of a system of interacting chemicals that reacted and di used such that, under certain conditions, spatial patterns can arise from near homogeneity. However, the implicit assumption that cells respond to respective chemical levels, di erentiating accordingly, is an oversimpli cation and may not capture the true extent of the biology. Here, we propose mathematical models that explicitly introduce cell dynamics into pattern formation mechanisms. The models presented are formulated based on Turing's classical mechanism and are used to gain insight into the signi cance and impact that cells may have in biological phenomena. The rst part of this work considers cell di erentiation and incorporates two conceptually di erent cell commitment processes: asymmetric precursor di erentiation and precursor speci cation. A variety of possible feedback mechanisms are considered with the results of direct activator upregulation suggesting a relaxation of the two species Turing Instability requirement of long range inhibition, short range activation. Moreover, the results also suggest that the type of feedback mechanism should be considered to explain observed biological results. In a separate model, cell signalling is investigated using a discrete mathematical model that is derived from Turing's classical continuous framework. Within this, two types of cell signalling are considered, namely autocrine and juxtacrine signalling, with both showing the attainability of a variety of wavelength patterns that are illustrated and explainable through individual cell activity levels of receptor, ligand and inhibitor. Together with the full system, a reduced two species system is investigated that permits a direct comparison to the classical activator-inhibitor model and the results produce pattern formation in systems considering both one and two di usible species together with an autocrine and/or juxtacrine signalling mechanism. Formulating the model in this way shows a greater applicability to biology with fundamental cell signalling and the interactions involved in Turing type patterning described using clear and concise variables

Local and non-local mathematical modelling of signalling during embryonic development

Embryonic development requires cells to communicate as they arrange into the adult organs and tissues. The ability of cells to sense their environment, respond to signals and self-organise is of crucial importance. Patterns of cells adopting distinct states of differentiation arise in early development, as a result of cell signalling. Furthermore, cells interact with each other in order to form aggregations or rearrange themselves via cell-cell adhesion. The distance over which cells can detect their surroundings plays an important role to the form of patterns to be developed, as well as the time necessary for developmental processes to complete. Cells achieve long range communication through the use of extensions such as filopodia. In this work we formulate and analyse various mathematical models incorporating long-range signalling. We first consider a spatially discrete model for juxtacrine signalling extended to include filopodial action. We show that a wide variety of patterns can arise through this mechanism, including single isolated cells within a large region or contiguous blocks of cells selected for a specific fate. Cell-cell adhesion modelling is addressed in this work. We propose a variety of discrete models from which continuous models are derived. We examine the models’ potential to describe cell-cell adhesion and the associated phenomena such as cell aggregation. By extending these models to consider long range cell interactions we were able to demonstrate their ability to reproduce biologically relevant patterns. Finally, we consider an application of cell adhesion modelling by attempting to reproduce a specific developmental event, the formation of sympathetic ganglia