A two-dimensional numerical study of spatial pattern formation in interacting Turing systems

For many years Turing systems have been proposed to account for spatial and spatiotemporal pattern formation in chemistry and biology. We extend the study of Turing systems to investigate the rôle of boundary conditions, domain shape, non-linearities, and coupling of such systems. We show that such modifications lead to a wide variety of patterns that bear a striking resemblance to pigmentation patterns in fish, particularly those involving stripes, spots and transitions between them. Using the Turing system as a metaphor for activator—inhibitor models we conclude that such a mechanism, with the aforementioned modifications, may play a rôle in fish patterning.     

Pattern formation in a space- and time-discrete predator–prey system with a strong Allee effect

The spatiotemporal dynamics of a space- and time-discrete predator–prey system is considered theoretically using both analytical methods and computer simulations. The prey is assumed to be affected by the strong Allee effect. We reveal a rich variety of pattern formation scenarios. In particular, we show that, in a predator–prey system with the strong Allee effect for prey, the role of space is crucial for species survival. Pattern formation is observed both inside and outside of the Turing domain. For parameters when the local kinetics is oscillatory, the system typically evolves to spatiotemporal chaos. We also consider the effect of different initial conditions and show that the system exhibits a spatiotemporal multistability. In a certain parameter range, the system dynamics is not self-organized but remembers the details of the initial conditions, which evokes the concept of long-living ecological transients. Finally, we show that our findings have important implications for the understanding of population dynamics on a fragmented habitat.     

Pattern formation in generalized Turing systems

Turing's model of pattern formation has been extensively studied analytically and numerically, and there is recent experimental evidence that it may apply in certain chemical systems. The model is based on the assumption that all reacting species obey the same type of boundary condition pointwise on the boundary. We call these scalar boundary conditions. Here we study mixed or nonscalar boundary conditions, under which different species satisfy different boundary conditions at any point on the boundary, and show that qualitatively new phenomena arise in this case. For example, we show that there may be multiple solutions at arbitrarily small lengths under mixed boundary conditions, whereas the solution is unique under homogeneous scalar boundary conditions. Moreover, even when the same solution exists under scalar and mixed boundary conditions, its stability may be different in the two cases. We also show that mixed boundary conditions can reduce the sensitivity of patterns to domain changes.Supported in part by NIH Grant # GM29123     

Receptor-based models with hysteresis for pattern formation in hydra

In this paper, we propose a new receptor-based model for pattern formation and regulation in a fresh-water polyp, namely hydra. The model is defined in the form of a system of reaction-diffusion equations with zero-flux boundary conditions coupled with a system of ordinary differential equations. The production of diffusible biochemical molecules has a hysteretic dependence on the density of these molecules and is modeled by additional ordinary differential equations. We study the hysteresis-driven mechanism of pattern formation and we demonstrate the advantages and constraints of its ability to explain different aspects of pattern formation and regulation in hydra. The properties of the model demonstrate a range of stationary and oscillatory spatially heterogeneous patterns, arising from multiple spatially homogeneous steady states and switches in the production rates.     

Effects of Hydraulic Soil Properties on Vegetation Pattern Formation in Sloping Landscapes

Current models of vegetation pattern formation rely on a system of weakly nonlinear reaction–diffusion equations that are coupled by their source terms. While these equations, which are used to describe a spatiotemporal planar evolution of biomass and soil water, qualitatively capture the emergence of various types of vegetation patterns in arid environments, they are phenomenological and have a limited predictive power. We ameliorate these limitations by deriving the vertically averaged Richards’ equation to describe flow (as opposed to “diffusion”) of water in partially saturated soils. This establishes conditions under which this nonlinear equation reduces to its weakly nonlinear reaction–diffusion counterpart used in the previous models, thus relating their unphysical parameters (e.g., diffusion coefficient) to the measurable soil properties (e.g., hydraulic conductivity) used to parameterize the Richards equation. Our model is valid for both flat and sloping landscapes and can handle arbitrary topography and boundary conditions. The result is a model that relates the environmental conditions (e.g., precipitation rate, runoff and soil properties) to formation of multiple patterns observed in nature (such as stripes, labyrinth and spots).     

Simple and complex spatiotemporal structures in a glycolytic allosteric enzyme model

Pattern formation in glycolysis is studied with a classical reaction-diffusion allosteric enzyme model. It is found that, similar to recent experimental reports in the yeast extracts, a small magnitude local perturbation can induce transient target waves in a two dimensional oscillatory medium. An above threshold stimulation generates target waves which eventually evolve into spatiotemporal chaos upon collisions with the boundary or other wave activities. Detailed simulation studies show that the studied simple glycolytic reaction-diffusion model can support three types of spatiotemporal behaviors which are independent of the boundary conditions: (1) a spatially uniform stable steady state, (2) periodic global oscillations and (3) spatiotemporal chaos.     

Implications of diffusion and time-varying morphogen gradients for the dynamic positioning and precision of bistable gene expression boundaries

The earliest models for how morphogen gradients guide embryonic patterning failed to account for experimental observations of temporal refinement in gene expression domains. Following theoretical and experimental work in this area, dynamic positional information has emerged as a conceptual framework to discuss how cells process spatiotemporal inputs into downstream patterns. Here, we show that diffusion determines the mathematical means by which bistable gene expression boundaries shift over time, and therefore how cells interpret positional information conferred from morphogen concentration. First, we introduce a metric for assessing reproducibility in boundary placement or precision in systems where gene products do not diffuse, but where morphogen concentrations are permitted to change in time. We show that the dynamics of the gradient affect the sensitivity of the final pattern to variation in initial conditions, with slower gradients reducing the sensitivity. Second, we allow gene products to diffuse and consider gene expression boundaries as propagating wavefronts with velocity modulated by local morphogen concentration. We harness this perspective to approximate a PDE model as an ODE that captures the position of the boundary in time, and demonstrate the approach with a preexisting model for Hunchback patterning in fruit fly embryos. We then propose a design that employs antiparallel morphogen gradients to achieve accurate boundary placement that is robust to scaling. Throughout our work we draw attention to tradeoffs among initial conditions, boundary positioning, and the relative timescales of network and gradient evolution. We conclude by suggesting that mathematical theory should serve to clarify not just our quantitative, but also our intuitive understanding of patterning processes.     

Bursting Regimes in a Reaction-Diffusion System with Action Potential-Dependent Equilibrium

The equilibrium Nernst potential plays a critical role in neural cell dynamics. A common approximation used in studying electrical dynamics of excitable cells is that the ionic concentrations inside and outside the cell membranes act as charge reservoirs and remain effectively constant during excitation events. Research into brain electrical activity suggests that relaxing this assumption may provide a better understanding of normal and pathophysiological functioning of the brain. In this paper we explore time-dependent ionic concentrations by allowing the ion-specific Nernst potentials to vary with developing transmembrane potential. As a specific implementation, we incorporate the potential-dependent Nernst shift into a one-dimensional Morris-Lecar reaction-diffusion model. Our main findings result from a region in parameter space where self-sustaining oscillations occur without external forcing. Studying the system close to the bifurcation boundary, we explore the vulnerability of the system with respect to external stimulations which disrupt these oscillations and send the system to a stable equilibrium. We also present results for an extended, one-dimensional cable of excitable tissue tuned to this parameter regime and stimulated, giving rise to complex spatiotemporal pattern formation. Potential applications to the emergence of neuronal bursting in similar two-variable systems and to pathophysiological seizure-like activity are discussed.     

Pattern Selection by Dynamical Biochemical Signals

The development of multicellular organisms involves cells to decide their fate upon the action of biochemical signals. This decision is often spatiotemporally coordinated such that a spatial pattern arises. The dynamics that drive pattern formation usually involve genetic nonlinear interactions and positive feedback loops. These complex dynamics may enable multiple stable patterns for the same conditions. Under these circumstances, pattern formation in a developing tissue involves a selection process: why is a certain pattern formed and not another stable one? Herein we computationally address this issue in the context of the Notch signaling pathway. We characterize a dynamical mechanism for developmental selection of a specific pattern through spatiotemporal changes of the control parameters of the dynamics, in contrast to commonly studied situations in which initial conditions and noise determine which pattern is selected among multiple stable ones. This mechanism can be understood as a path along the parameter space driven by a sequence of biochemical signals. We characterize the selection process for three different scenarios of this dynamical mechanism that can take place during development: the signal either 1) acts in all the cells at the same time, 2) acts only within a cluster of cells, or 3) propagates along the tissue. We found that key elements for pattern selection are the destabilization of the initial pattern, the subsequent exploration of other patterns determined by the spatiotemporal symmetry of the parameter changes, and the speeds of the path compared to the timescales of the pattern formation process itself. Each scenario enables the selection of different types of patterns and creates these elements in distinct ways, resulting in different features. Our approach extends the concept of selection involved in cellular decision-making, usually applied to cell-autonomous decisions, to systems that collectively make decisions through cell-to-cell interactions.     

Flow-distributed spikes for Schnakenberg kinetics

We study a system of reaction–diffusion–convection equations which combine a reaction–diffusion system with Schnakenberg kinetics and the convective flow equations. It serves as a simple model for flow-distributed pattern formation. We show how the choice of boundary conditions and the size of the flow influence the positions of the emerging spiky patterns and give conditions when they are shifted to the right or to the left. Further, we analyze the shape and prove the stability of the spikes. This paper is the first providing a rigorous analysis of spiky patterns for reaction-diffusion systems coupled with convective flow. The importance of these results for biological applications, in particular the formation of left–right asymmetry in the mouse, is indicated.     

Optogenetic Patterning of Whisker-Barrel Cortical System in Transgenic Rat Expressing Channelrhodopsin-2

The rodent whisker-barrel system has been an ideal model for studying somatosensory representations in the cortex. However, it remains a challenge to experimentally stimulate whiskers with a given pattern under spatiotemporal precision. Recently the optogenetic manipulation of neuronal activity has made possible the analysis of the neuronal network with precise spatiotemporal resolution. Here we identified the selective expression of channelrhodopsin-2 (ChR2), an algal light-driven cation channel, in the large mechanoreceptive neurons in the trigeminal ganglion (TG) as well as their peripheral nerve endings innervating the whisker follicles of a transgenic rat. The spatiotemporal pattern of whisker irradiation thus produced a barrel-cortical response with a specific spatiotemporal pattern as evidenced by electrophysiological and functional MRI (fMRI) studies. Our methods of generating an optogenetic tactile pattern (OTP) can be expected to facilitate studies on how the spatiotemporal pattern of touch is represented in the somatosensory cortex, as Hubel and Wiesel did in the visual cortex.     

Spatio-temporal patterns in a mechanical model for mesenchymal morphogenesis

We present an in-depth study of spatio-temporal patterns in a simplified version of a mechanical model for pattern formation in mesenchymal morphogenesis. We briefly motivate the derivation of the model and show how to choose realistic boundary conditions to make the system well-posed. We firstly consider one-dimensional patterns and carry out a nonlinear perturbation analysis for the case where the uniform steady state is linearly unstable to a single mode. In two-dimensions, we show that if the displacement field in the model is represented as a sum of orthogonal parts, then the model can be decomposed into two sub-models, only one of which is capable of generating pattern. We thus focus on this particular sub-model. We present a nonlinear analysis of spatio-temporal patterns exhibited by the sub-model on a square domain and discuss mode interaction. Our analysis shows that when a two-dimensional mode number admits two or more degenerate mode pairs, the solution of the full nonlinear system of partial differential equations is a mixed mode solution in which all the degenerate mode pairs are represented in a frequency locked oscillation.     

Formation and deformation of patterns through diffusion

Simulating various patterns exhibited on biological forms with mathematical models has become an important supplement to theoretical biology. Models based on a certain mechanism are intended to provide explanations to the formation of a basic pattern. However, in real phenomena, among a basic pattern there always exist some difference between any two individuals. Such differences are consequences of environmental factors posed during the developmental processes. These factors, such as temperature, affect the diffusion rates of corresponding morphogenes which, in turn, alter a basic pattern to certain extent. We provide, in this paper, a quantitative characterization of this effect for a class of reaction-diffusion models.Mathematically, we study the emergence of stationary patterns and their dependence on diffusion rates for this class of models (RD-equations) with no-flux boundary conditions. The results are generalized to systems with homogeneous Dirichlet boundary conditions when the kinetic terms are odd functions. Through an analysis of the phase dynamics, we show that the deformation of stationary patterns, as the diffusion rates change, is governed by the variation of certain plane curves in the phase space. A constructive proof is given which shows explicitly how to obtain such curves.Applications of this study are illustrated with three model examples. We use these models to explain the biological implications of the mathematical features we investigated. Results from computer simulations are presented and compared with physical patterns.     

A clock and wavefront mechanism for somite formation

Somitogenesis, the sequential formation of a periodic pattern along the antero-posterior axis of vertebrate embryos, is one of the most obvious examples of the segmental patterning processes that take place during embryogenesis and also one of the major unresolved events in developmental biology. In this article, we develop a mathematical formulation of a new version of the Clock and Wavefront model proposed by Pourquié and co-workers (Dubrulle, J., McGrew, M.J., Pourquié, O., 2001. FGF signalling controls somite boundary position and regulates segmentation clock control of spatiotemporal Hox gene activation. Cell 106, 219-232). Dynamic expression of FGF8 in the presomitic mesoderm constitutes the wavefront of determination which sweeps along the body axis interacting as it moves with the segmentation clock to gate cells into somites. We also show that the model can mimic the anomalies formed when progression of the wavefront is disturbed and make some experimental predictions that can be used to test the hypotheses underlying the model.     

A Conceptual Model for Milling Formations in Biological Aggregates

Collective behavior of swarms and flocks has been studied from several perspectives, including continuous (Eulerian) and individual-based (Lagrangian) models. Here, we use the latter approach to examine a minimal model for the formation and maintenance of group structure, with specific emphasis on a simple milling pattern in which particles follow one another around a closed circular path. We explore how rules and interactions at the level of the individuals lead to this pattern at the level of the group. In contrast to many studies based on simulation results, our model is sufficiently simple that we can obtain analytical predictions. We consider a Newtonian framework with distance-dependent pairwise interaction-force. Unlike some other studies, our mill formations do not depend on domain boundaries, nor on centrally attracting force-fields or rotor chemotaxis. By focusing on a simple geometry and simple distant-dependent interactions, we characterize mill formations and derive existence conditions in terms of model parameters. An eigenvalue equation specifies stability regions based on properties of the interaction function. We explore this equation numerically, and validate the stability conclusions via simulation, showing distinct behavior inside, outside, and on the boundary of stability regions. Moving mill formations are then investigated, showing the effect of individual autonomous self-propulsion on group-level motion. The simplified framework allows us to clearly relate individual properties (via model parameters) to group-level structure. These relationships provide insight into the more complicated milling formations observed in nature, and suggest design properties of artificial schools where such rotational motion is desired.     

Buffering the savanna: fire regimes and disequilibrium ecology in West Africa

According to contemporary ecological theory, the mechanisms governing tree cover in savannas vary by precipitation level. In tropical areas with mesic rainfall levels, savannas are unstable systems in which disturbances, such as fire, determine the ratio of trees to grasses. Precipitation in these so-called “disturbance-driven savannas” is sufficient to support forest but frequent disturbances prevent transition to a closed canopy state. Building on a savanna buffering model we argue that a consistent fire regime is required to maintain savannas in mesic areas. We hypothesize that the spatiotemporal pattern of fires is highly regular and stable in these areas. Furthermore, because tree growth rates in savannas are a function of precipitation, we hypothesize that savannas with the highest rainfall levels will have the most consistent fire pattern and the most intense fires—thus the strongest buffering mechanisms. We analyzed the spatiotemporal pattern of burning over 11 years for a large subset of the West African savanna using a moderate resolution imaging spectroradiometer active fire product to document the fire regime for three savanna belts with different precipitation levels. We used LISA analysis to quantify the spatiotemporal patterns of fires, coefficient of variance to quantify differences in peak fire dates, and center or gravity pathways to characterize the spatiotemporal patterns of the fires for each area. Our analysis confirms that spatiotemporal regularity of the fire regime is greater for mesic areas that for areas where precipitation is lower and that areas with more precipitation have more regular fire regimes.     

Analytical treatment of pattern formation in the Gierer-Meinhardt model of morphogenesis

Summary We first perform a linear stability analysis of the Gierer-Meinhardt model to determine the critical parameters where the homogeneous distribution of activator and inhibitor concentrations becomes unstable. There are two kinds of instabilities, namely, one leading to spatial patterns and another one leading to temporal oscillations. Focussing our attention on spatial pattern formation we solve the corresponding nonlinear equations by means of our previously introduced method of generalized Ginzburg-Landau equations. We explicitly consider the two-dimensional case and find both rolls and hexagon-like structures. The impact of different boundary conditions on the resulting patterns is also discussed. The occurrence of the new patterns has all the features of nonequilibrium phase transitions.     

A cell cycle model for somitogenesis: mathematical formulation and numerical simulation

After many years of research, the mechanisms that generate a periodic pattern of repeated elements (somites) along the length of the embryonic body axis is still one of the major unresolved problems in developmental biology. Here we present a mathematical formulation of the cell cycle model for somitogenesis proposed in Development105 (1989), 119-130. Somite precursor cells in the node are asynchronous, and therefore, as a population, generate continuously pre-somite cells which enter the segmental plate. The model makes the hypothesis that there exists a time window within the cell cycle, making up one-seventh of the cycle, which gates the pre-somite cells so that they make somites discretely, seven per cycle. We show that the model can indeed account for the spatiotemporal patterning of somite formation during normal development as well as the periodic abnormalities produced by heat shock treatment. We also relate the model to recent molecular data on the process of somite formation.