Spatial and spatio-temporal patterns in a cell-haptotaxis model

We investigate a cell-haptotaxis model for the generation of spatial and spatio-temporal patterns in one dimension. We analyse the steady state problem for specific boundary conditions and show the existence of spatially hetero-geneous steady states. A linear analysis shows that stability is lost through a Hopf bifurcation. We carry out a nonlinear multi-time scale perturbation procedure to study the evolution of the resulting spatio-temporal patterns. We also analyse the model in a parameter domain wherein it exhibits a singular dispersion relation.     

Nonlinear pattern selection in a mechanical model for morphogenesis

We present a numerical study of the nonlinear mechanical model for morphogenesis proposed by Oster et al. (1983) with the aim of establishing the pattern forming capability of the model. We present a technique for mode selection based on linear analysis and show that, in many cases, it is a reliable predictor for nonlinear mode selection. In order to determine the set of model parameters that can generate a particular pattern we develop a technique based on nonlinear least square fitting to a dispersion relation. As an application we present a scenario for sequential pattern formation of dermal aggregations in chick embryos which leads to the hexagonal array of cell aggregations observed in feather germ formation in vivo.     

Bifurcation Analysis of Reaction Diffusion Systems on Arbitrary Surfaces

In this paper, we present computational techniques to investigate the effect of surface geometry on biological pattern formation. In particular, we study two-component, nonlinear reaction–diffusion (RD) systems on arbitrary surfaces. We build on standard techniques for linear and nonlinear analysis of RD systems and extend them to operate on large-scale meshes for arbitrary surfaces. In particular, we use spectral techniques for a linear stability analysis to characterise and directly compose patterns emerging from homogeneities. We develop an implementation using surface finite element methods and a numerical eigenanalysis of the Laplace–Beltrami operator on surface meshes. In addition, we describe a technique to explore solutions of the nonlinear RD equations using numerical continuation. Here, we present a multiresolution approach that allows us to trace solution branches of the nonlinear equations efficiently even for large-scale meshes. Finally, we demonstrate the working of our framework for two RD systems with applications in biological pattern formation: a Brusselator model that has been used to model pattern development on growing plant tips, and a chemotactic model for the formation of skin pigmentation patterns. While these models have been used previously on simple geometries, our framework allows us to study the impact of arbitrary geometries on emerging patterns.     

Nonlinear dynamics and pattern bifurcations in a model for vegetation stripes in semi-arid environments

In many semi-arid environments, vegetation is self-organised into spatial patterns. The most striking examples of this are on gentle slopes, where striped patterns are typical, running parallel to the contours. Previously, Klausmeier [1999. Regular and irregular patterns in semiarid vegetation. Science 284, 1826-1828.] has proposed a model for vegetation stripes based on competition for water. Here, we present a detailed study of the patterned solutions in the full nonlinear model, using numerical bifurcation analysis of both the pattern odes and the model pdes. We show that patterns exist for a wide range of rainfall levels, and in particular for much lower rainfall than have been considered by previous authors. Moreover, we show that for many rainfall levels, patterns with a variety of different wavelengths are stable, with mode selection dependent on initial conditions. This raises the possibility of hysteresis, and in numerical solutions of the model we show that pattern selection depends on rainfall history in a relatively simple way.     

Interactions Between Pattern Formation and Domain Growth

In this paper we develop a theoretical framework for investigating pattern formation in biological systems for which the tissue on which the spatial pattern resides is growing at a rate which is itself regulated by the diffusible chemicals that establish the spatial pattern. We present numerical simulations for two cases of interest, namely exponential domain growth and chemically controlled growth. Our analysis reveals that for domains undergoing rapid exponential growth dilution effects associated with domain growth influence both the spatial patterns that emerge and the concentration of chemicals present in the domain. In the latter case, there is complex interplay between the effects of the chemicals on the domain size and the influence of the domain size on the formation of patterns. The nature of these interactions is revealed by a weakly nonlinear analysis of the full system. This yields a pair of nonlinear equations for the amplitude of the spatial pattern and the domain size. The domain is found to grow (or shrink) at a rate that depends quadratically on the pattern amplitude, the particular functional forms used to model the local tissue growth rate and the kinetics of the two diffusible species dictating the resulting behaviour.     

Model and analysis of chemotactic bacterial patterns in a liquid medium

 A variety of spatial patterns are formed chemotactically by the bacteria Escherichia coli and Salmonella typhimurium. We focus in this paper on patterns formed by E. coli and S. typhimurium in liquid medium experiments. The dynamics of the bacteria, nutrient and chemoattractant are modeled mathematically and give rise to a nonlinear partial differential equation system. We present a simple and intuitively revealing analysis of the patterns generated by our model. Patterns arise from disturbances to a spatially uniform solution state. A linear analysis gives rise to a second order ordinary differential equation for the amplitude of each mode present in the initial disturbance. An exact solution to this equation can be obtained, but a more intuitive understanding of the solutions can be obtained by considering the rate of growth of individual modes over small time intervals. Received: 10 March 1998 / Revised version: 7 June 1998     

Event Networks and the Identification of Crime Pattern Motifs

In this paper we demonstrate the use of network analysis to characterise patterns of clustering in spatio-temporal events. Such clustering is of both theoretical and practical importance in the study of crime, and forms the basis for a number of preventative strategies. However, existing analytical methods show only that clustering is present in data, while offering little insight into the nature of the patterns present. Here, we show how the classification of pairs of events as close in space and time can be used to define a network, thereby generalising previous approaches. The application of graph-theoretic techniques to these networks can then offer significantly deeper insight into the structure of the data than previously possible. In particular, we focus on the identification of network motifs, which have clear interpretation in terms of spatio-temporal behaviour. Statistical analysis is complicated by the nature of the underlying data, and we provide a method by which appropriate randomised graphs can be generated. Two datasets are used as case studies: maritime piracy at the global scale, and residential burglary in an urban area. In both cases, the same significant 3-vertex motif is found; this result suggests that incidents tend to occur not just in pairs, but in fact in larger groups within a restricted spatio-temporal domain. In the 4-vertex case, different motifs are found to be significant in each case, suggesting that this technique is capable of discriminating between clustering patterns at a finer granularity than previously possible.     

Symmetries,non-Euclidean metrics,and patterns in a Swift–Hohenberg model of the visual cortex

The aim of this work is to investigate the effect of the shift-twist symmetry on pattern formation processes in the visual cortex. First, we describe a generic set of Riemannian metrics of the feature space of orientation preference that obeys properties of the shift-twist, translation, and reflection symmetries. Second, these metrics are embedded in a modified Swift-Hohenberg model. As a result we get a pattern formation process that resembles the pattern formation process in the visual cortex. We focus on the final stable patterns that are regular and periodic. In a third step we analyze the influences on pattern formation using weakly nonlinear theory and mode analysis. We compare the results of the present approach with earlier models.     

Parametric Pattern Selection in a Reaction-Diffusion Model

We compare spot patterns generated by Turing mechanisms with those generated by replication cascades, in a model one-dimensional reaction-diffusion system. We determine the stability region of spot solutions in parameter space as a function of a natural control parameter (feed-rate) where degenerate patterns with different numbers of spots coexist for a fixed feed-rate. While it is possible to generate identical patterns via both mechanisms, we show that replication cascades lead to a wider choice of pattern profiles that can be selected through a tuning of the feed-rate, exploiting hysteresis and directionality effects of the different pattern pathways.     

Beyond tandem repeats: complex pattern structures and distant regions of similarity

MOTIVATION: Tandem repeats (TRs) are associated with human disease, play a role in evolution and are important in regulatory processes. Despite their importance, locating and characterizing these patterns within anonymous DNA sequences remains a challenge. In part, the difficulty is due to imperfect conservation of patterns and complex pattern structures. We study recognition algorithms for two complex pattern structures: variable length tandem repeats (VLTRs) and multi-period tandem repeats (MPTRs). RESULTS: We extend previous algorithmic research to a class of regular tandem repeats (RegTRs). We formally define RegTRs, as well as two important subclasses: VLTRs and MPTRs. We present algorithms for identification of TRs in these classes. Furthermore, our algorithms identify degenerate VLTRs and MPTRs: repeats containing substitutions, insertions and deletions. To illustrate our work, we present results of our analysis for two difficult regions in cattle and human data which reflect practical occurrences of these subclasses in GenBank sequence data. In addition, we show the applicability of our algorithmic techniques for identifying Alu sequences, gene clusters and other distant regions of similarity. We illustrate this with an example from yeast chromosome I.     

Glucose induced fractal colony pattern of Bacillus thuringiensis

Growing colonies of bacteria on the surface of thin agar plates exhibit fractal patterns as a result of nonlinear response to environmental conditions, such as nutrients, solidity of the agar medium and temperature. Here, we examine the effect of glucose on pattern formation by growing colonies of Bacillus thuringiensis isolate KPWP1. We also present the theoretical modeling of the colony growth of KPWP1 and the associated spatio-temporal patterns. Our experimental results are in excellent agreement with simulations based on a reaction-diffusion model that describes diffusion-limited aggregation and branching, in which individual cells move actively in the periphery, but become immotile in the inner regions of the growing colony. We obtain the Hausdorff fractal dimension of the colony patterns: D_H.Expt=1.1969 and D_{H, R.D.=}1.1965, for experiment and reaction-diffusion model, respectively. Results of our experiments and modeling clearly show how glucose at higher concentration can prove to be inhibitory for motility of growing colonies of B. thuringiensis cells on semisolid support and be responsible for changes in the growth pattern.     

Spatio-temporal self-organization of bone mineral metabolism and trabecular structure of primary bone

A nonlinear two-variable reaction-diffusion model of bone mineral metabolism, built from an overall self-oscillatory compartmental model of calcium metabolism in vivo, has been studied for its ability to generate spatial and spatio-temporal self-organizations in a two-dimensional space. Analytical and numerical results confirm the theoretical properties previously described for this kind of model. In particular, it is shown that, for a given set of reactional parameter values and certain values of the ratio of the two diffusion coefficients, there exists a set of unstable wavenumbers leading spontaneously to the development, from the homogeneous steady state, of either different types of stationary spatial patterns (hexagonal, striped and re-entrant hexagonal patterns) or more or less complex spatio-temporal expressions. We discuss the relevance of analogies established between some spatial or spatio-temporal structures predicted by the model and some peculiar features of the primary bone trabecular architecture which appear during embryonic ossification.     

Pacemaker and network mechanisms of rhythm generation: cooperation and competition

The origin of rhythmic activity in brain circuits and CPG-like motor networks is still not fully understood. The main unsolved questions are (i) What are the respective roles of intrinsic bursting and network based dynamics in systems of coupled heterogeneous, intrinsically complex, even chaotic, neurons? (ii) What are the mechanisms underlying the coexistence of robustness and flexibility in the observed rhythmic spatio-temporal patterns? One common view is that particular bursting neurons provide the rhythmogenic component while the connections between different neurons are responsible for the regularisation and synchronisation of groups of neurons and for specific phase relationships in multi-phasic patterns. We have examined the spatio-temporal rhythmic patterns in computer-simulated motif networks of H-H neurons connected by slow inhibitory synapses with a non-symmetric pattern of coupling strengths. We demonstrate that the interplay between intrinsic and network dynamics features either cooperation or competition, depending on three basic control parameters identified in our model: the shape of intrinsic bursts, the strength of the coupling and its degree of asymmetry. The cooperation of intrinsic dynamics and network mechanisms is shown to correlate with bistability, i.e., the coexistence of two different attractors in the phase space of the system corresponding to different rhythmic spatio-temporal patterns. Conversely, if the network mechanism of rhythmogenesis dominates, monostability is observed with a typical pattern of winnerless competition between neurons. We analyse bifurcations between the two regimes and demonstrate how they provide robustness and flexibility to the network performance.     

A new tool for analysis of root growth in the spatio-temporal continuum

? Quantification of overall growth and local growth zones in root system development is key to understanding the biology of plant growth, and thus to exploring the effects of environmental, genotypic and mutational variations on plant development and productivity. ? We introduce a methodology for analyzing growth patterns of plant roots from two-dimensional time series images, treating them as a spatio-temporal three-dimensional (3D) image volume. The roots are segmented from the images and then two types of analysis are performed: 3D spatio-temporal reconstruction analysis for simultaneous assessment of initiation and growth of multiple roots; and spatio-temporal pixel intensity analysis along root midlines for quantification of the growth zones. ? The test measurements show simultaneous emergence of basal roots but sequential emergence of lateral roots in Phaseolus vulgaris, while lateral roots of Cicer arietinum emerge in a rhythmic pattern. Local growth analysis reveals multimodal transient growth zone in basal roots. At the initial stages after emergence, the roots oscillate rapidly, which slows down with time. ? The methodology presented here allows detailed characterization of the phenomenology of roots, providing valuable information of spatio-temporal development, with applications in a wide range of growing plant organs.     

Instabilities in spatially extended predator-prey systems: spatio-temporal patterns in the neighborhood of Turing-Hopf bifurcations

We investigate the emergence of spatio-temporal patterns in ecological systems. In particular, we study a generalized predator-prey system on a spatial domain. On this domain diffusion is considered as the principal process of motion. We derive the conditions for Hopf and Turing instabilities without specifying the predator-prey functional responses and discuss their biological implications. Furthermore, we identify the codimension-2 Turing-Hopf bifurcation and the codimension-3 Turing-Takens-Bogdanov bifurcation. These bifurcations give rise to complex pattern formation processes in their neighborhood. Our theoretical findings are illustrated with a specific model. In simulations a large variety of different types of long-term behavior, including homogenous distributions, stationary spatial patterns and complex spatio-temporal patterns, are observed.     

A Nonlinear Stability Analysis of Vegetative Turing Pattern Formation for an Interaction–Diffusion Plant-Surface Water Model System in an Arid Flat Environment

The development of spontaneous stationary vegetative patterns in an arid flat environment is investigated by means of a weakly nonlinear diffusive instability analysis applied to the appropriate model system for this phenomenon. In particular, that process can be modeled by a partial differential interaction–diffusion equation system for the plant biomass density and the surface water content defined on an unbounded flat spatial domain. The main results of this analysis can be represented by closed-form plots in the rate of precipitation versus the specific rate of plant density loss parameter space. From these plots, regions corresponding to bare ground and vegetative patterns consisting of parallel stripes, labyrinth-like mazes, hexagonal arrays of gaps, irregular mosaics, and homogeneous distributions of vegetation, respectively, may be identified in this parameter space. Then those theoretical predictions are compared with both relevant observational evidence involving tiger and pearled bush patterns and existing numerical simulations of similar model systems as well as placed in the context of the results from some recent nonlinear vegetative pattern formation studies.     

Pattern formation in morphogenesis

We first treat the Gierer-Meinhardt equations by linear stability analysis to determine the critical parameter, at which the homogeneous distributions of activator and inhibitor concentrations become unstable. We find two types of instabilities: one leading to spatial pattern formation and another one leading to temporal oscillations. We consider the case where two instabilities are present. Using the method of generalized Ginzburg-Landau equations introduced earlier we then analyze the nonlinear equations. As we are mainly interested in spatial pattern formation on a sphere we consider the problem under an appropriate constraint. Combining the two occurring solutions we find patterns well-known in biology, such as a gradient system and temporal oscillations.     

Modification patterns in germinating barley--malting II

Modification refers to breaking down of cell walls and the conversion of starch-to-sugars in the endosperm of grains by the action of enzymes released from the aleurone layer and possibly the scutellum during germination. Experimentalists have observed two completely different modification patterns in germinating barley. Based on an enzyme reaction, strongly nonlinear diffusion model developed in Part I of this series of papers on malting we examine simple models which predict modifications patterns similar to both observed patterns. We show that one of the observed patterns represents a natural propagation mode that acts as an attractor for the system. The speed of approach to this mode is strongly effected by initial conditions, a consequence of the enzymic nature of the reaction and the dramatic change of diffusivity brought about by the reaction.     

Perception of odors by a nonlinear model of the olfactory bulb

The behavior of the olfactory bulb is modeled as a network of interconnected cells with nonlinear dynamics. External inputs from sensory neurons are introduced as perturbations to subsets of cells within the network. We describe the attractors of the system and show how they can be classified and ordered according to their varying degrees of symmetry. By studying networks of attractors in the system's phase space, it is shown how different perturbations may evoke specific switches between various patterns of behavior. This ensures that different odors, even if present at extremely low concentrations, are able to evoke a specific spatio-temporal behavior in the olfactory bulb, permitting their unique perception. The model incorporates many of the processes proposed to mediate perception, such as the topographic organisation of sensory systems, destabilization of cortex by sensory input and synchronisation between neurons. It is also consistent with the character of the olfactory electroencephalogram.     

Phase Transitions in a Logistic Metapopulation Model with Nonlocal Interactions

The presence of one or more species at some spatial locations but not others is a central matter in ecology. This phenomenon is related to ecological pattern formation. Nonlocal interactions can be considered as one of the mechanisms causing such a phenomenon. We propose a single-species, continuous time metapopulation model taking nonlocal interactions into account. Discrete probability kernels are used to model these interactions in a patchy environment. A linear stability analysis of the model shows that solutions to this equation exhibit pattern formation if the dispersal rate of the species is sufficiently small and the discrete interaction kernel satisfies certain conditions. We numerically observe that traveling and stationary wave-type patterns arise near critical dispersal rate. We use weakly nonlinear analysis to better understand the behavior of formed patterns. We show that observed patterns arise through both supercritical and subcritical bifurcations from spatially homogeneous steady state. Moreover, we observe that as the dispersal rate decreases, amplitude of the patterns increases. For discontinuous transitions to instability, we also show that there exists a threshold for the amplitude of the initial condition, above which pattern formation is observed.