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Results for “Spatial pattern formation”

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PubMed2000

Modulation instability and pattern formation in spatially incoherent light beams.

We report on the experimental observation of modulation instability of partially spatially incoherent light beams in noninstantaneous nonlinear media and show that in such systems patterns can form spontaneously from noise. Incoherent modulation instability occurs above a specific threshold that depends on the coherence properties (correlation distance) of the wave packet and leads to a periodic train of one-dimensional filaments. At a higher value of nonlinearity, the incoherent one-dimensional filaments display a two-dimensional instability and break up into self-ordered arrays of light spots. This discovery of incoherent pattern formation reflects on many other nonlinear systems beyond optics. It implies that patterns can form spontaneously (from noise) in diverse nonlinear many-body systems involving weakly correlated particles, such as atomic gases at (or near) Bose-Einstein condensation temperatures and electrons in semiconductors at the vicinity of the quantum Hall regime.

Journal Article↗

PubMed2002

Space and the persistence of male-killing endosymbionts in insect populations.

Male-killing bacteria are bacteria that are transmitted vertically through the females of their insect hosts. They can distort the sex ratio of their hosts by killing infected male offspring. In nature, male-killing endosymbionts (male killers) often have a 100% efficient vertical transmission, and multiple male-killing bacteria infecting a single population are observed. We use different model formalisms to study these observations. In mean-field models a male killer with perfect transmission drives the host population to extinction, and coexistence between multiple male killers within one population is impossible; however, in spatially explicit models, both phenomena are readily observed. We show how the spatial pattern formation underlies these results. In the case of high transmission efficiencies, waves with a high density of male killers alternate with waves of mainly wild-type hosts. The male killers cause local extinction, but this creates an opportunity for uninfected hosts to re-invade these areas. Spatial pattern formation also creates an opportunity for two male killers to coexist within one population: different strains create spatial regions that are qualitatively different; these areas then serve as different niches, making coexistence possible.

Animals↗

PubMed2002

Transient dynamics and pattern formation: reactivity is necessary for Turing instabilities.

The theory of spatial pattern formation via Turing bifurcations - wherein an equilibrium of a nonlinear system is asymptotically stable in the absence of dispersal but unstable in the presence of dispersal - plays an important role in biology, chemistry and physics. It is an asymptotic theory, concerned with the long-term behavior of perturbations. In contrast, the concept of reactivity describes the short-term transient behavior of perturbations to an asymptotically stable equilibrium. In this article we show that there is a connection between these two seemingly disparate concepts. In particular, we show that reactivity is necessary for Turing instability in multispecies systems of reaction-diffusion equations, integrodifference equations, coupled map lattices, and systems of ordinary differential equations.

Animals↗

PubMed2001

Pattern formation and spatial self-entrainment in bistable chemical systems.

We describe the formation of spatial structures generated by diffusive instabilities in bistable systems. The coupling between the different spatial modes emanating from the two homogeneous steady states can then give rise to self-parametric instabilities favoring the occurrence of resonant rhombic or quasiperiodic structures such as superlattices or quasicrystalline patterns.

Journal Article↗

PubMed1998

Pattern formation in a generalized chemotactic model.

Many models have been proposed for spatial pattern formation in embryology and analyzed for the standard case of zero-flux boundary conditions. However, relatively little attention has been paid to the role of boundary conditions on the form of the final pattern. Here we investigate numerically, the effect of nonstandard boundary conditions on a model pattern generator, which we choose to be of a cell-chemotactic type. We specifically focus on the role of boundary conditions and the effects of scale and aspect ratio, and study the spatiotemporal dynamics of pattern formation. We illustrate the properties of the model by application to the spatiotemporal sequence of skeletal development.

Animals↗

PubMed2006

Liesegang pattern formation in kappa-carrageenan gel.

We report a new class of the spatial pattern formation process in which the gel plays essential roles. The system studied here is the solution of kappa-carrageenan in which potassium chloride is diffused. The solution transforms into the gel state with the diffusion of potassium chloride. Then the stripe pattern, which is perpendicular to the direction of the diffusion of potassium chloride, appears within the gel. The pattern thus formed in the gel is studied as a function of the concentration of the solution of potassium chloride. We find that the dense region of the stripe pattern consists of the liquid crystalline gel, whereas the dilute region is the amorphous gel. The transition from the amorphous gel to the liquid crystalline gel, hence, occurs in the gel state of kappa-carrageenan. The gel behaves as a pattern-forming substance as well as the supporting medium of the pattern in this system. The period and the thickness of the layers of liquid crystalline gel are analyzed. Both the period and the thickness of the layers are found to depend strongly on the concentration of the solution of potassium chloride.

Journal Article↗

PubMed2001

Pattern formation and localization in the forced-damped Fermi-Pasta-Ulam lattice.

We study spatial pattern formation and energy localization in the dynamics of an anharmonic chain with quadratic and quartic intersite potential, subject to an optical, sinusoidally oscillating field and a weak damping. The zone-boundary mode is stable and locked to the driving field below a critical forcing that we determine analytically using an approximate model, which describes mode interactions. Above such a forcing, a standing modulated wave forms for driving frequencies below the band edge, while a "multibreather" state develops at higher frequencies. Of the former, we give an explicit approximate analytical expression, which compares well with numerical data. At higher forcing, space-time chaotic patterns are observed.

Journal Article↗

PubMed1997

Pattern formation triggered by rare events: lessons from the spread of rabies.

Understanding of large-scale spatial pattern formation is a key to successful management in ecology and epidemiology. Neighbourhood interactions between local units are known to contribute to large-scale patterns, but how much do they contribute and what is the role of regional interactions caused by long-distance processes? How much long-distance dispersal do we need to explain the patterns that we observe in nature? There seems to be no way to answer these questions empirically. Therefore, we present a modelling approach that is a combination of a grid-based model describing local interactions and an individual-based model describing dispersal. Applying our approach to the spread of rabies, we show that in addition to local rabies dynamics, one long-distance infection per 14000 km2 per year is sufficient to reproduce the wave-like spread of this disease. We conclude that even rare ecological events that couple local dynamics on a regional scale may have profound impacts on large-scale patterns and, in turn, dynamics. Furthermore, the following results emerge: (i) Both neighbourhood infection and long-distance infection are needed to generate the wave-like dispersal pattern of rabies; (ii) randomly walking rabid foxes are not sufficient to generate the wave pattern; and (iii) on a scale of less than 100 km x 100 km, temporal oscillations emerge that are independent from long-distance dispersal.

Animals↗

PubMed1992

Sequential pattern formation in a model for skin morphogenesis.

During morphogenesis regular patterns often develop behind a frontier of pattern formation which travels across the prospective tissue. Here the authors consider the propagating patterns exhibited in a two-dimensional domain by a tissue interaction mechanochemical model for skin pattern formation. It is shown that the model can exhibit travelling waves of complex spatial pattern formation. Two alternative mechanisms that can produce such sequential patterning are presented. In particular, it is demonstrated that the specification of a simple quasi-one-dimensional pattern is all that is required to determine a complex two-dimensional pattern. Finally, the model solutions are related to actual pattern propagation during chick feather primordia initiation.

Animals↗

PubMed2006

Instability criteria and pattern formation in the complex Ginzburg-Landau equation with higher-order terms.

We study the modulational instability and spatial pattern formation in extended media, taking the one-dimensional complex Ginzburg-Landau equation with higher-order terms as a perturbation of the nonlinear Schrödinger equation as a model. By stability analysis for the original partial differential equation, we derive its stability condition as well as the threshold for amplitude perturbations and we show how nonlinear higher-order terms qualitatively change the behavior of the system. The analytical results are found to be in agreement with numerical findings. Modulational instability mediates pattern formation through the lattice. The main feature of the traveling plane waves is its disintegration in pulse train during the propagation through the system.

Journal Article↗

PubMed2007

The interplay between a self-organized process and an environmental template: corpse clustering under the influence of air currents in ants.

Many spatial patterns observed in nature emerge from local processes and their interactions with the local environment. The clustering of objects by social insects represents such a pattern formation process that can be observed at both the individual and the collective level. In this paper, we study the interaction between air currents and clustering behaviour in order to address the coordinating mechanisms at the individual level that underlie the spatial pattern formation process in a heterogeneous environment. We choose the corpse clustering behaviour of the ant Messor sanctus as an experimental paradigm. In a specifically designed experimental set-up with a well-controlled laminar air flow (approx. 1 cm s-1), we first quantify the modulation of the individual corpse aggregation behaviour as a function of corpse density, air flow intensity and the ant's position with respect to corpse piles and air flow direction. We then explore by numerical simulation how the forming corpse piles modify the laminar air flow around them and link this result with the individual behaviour modulation. Finally, we demonstrate on the collective level that this laminar air flow leads to an elongation and a slow displacement of the formed corpse piles in the direction of the air current. Both the individual behaviour modulated by air flow and the air flow modulated by the forming corpse piles can explain the pile patterns observed on the collective level as a stigmergic process. We discuss the generality of this coordinating mechanism to explain the clustering phenomena in heterogeneous environments reported in the literature.

Animals↗

PubMed2000

Mathematical modelling of juxtacrine patterning.

Spatial pattern formation is one of the key issues in developmental biology. Some patterns arising in early development have a very small spatial scale and a natural explanation is that they arise by direct cell-cell signalling in epithelia. This necessitates the use of a spatially discrete model, in contrast to the continuum-based approach of the widely studied Turing and mechanochemical models. In this work, we consider the pattern-forming potential of a model for juxtacrine communication, in which signalling molecules anchored in the cell membrane bind to and activate receptors on the surface of immediately neighbouring cells. The key assumption is that ligand and receptor production are both up-regulated by binding. By linear analysis, we show that conditions for pattern formation are dependent on the feedback functions of the model. We investigate the form of the pattern: specifically, we look at how the range of unstable wavenumbers varies with the parameter regime and find an estimate for the wavenumber associated with the fastest growing mode. A previous juxtacrine model for Delta-Notch signalling studied by Collier et al. (1996, J. Theor. Biol. 183, 429-446) only gives rise to patterning with a length scale of one or two cells, consistent with the fine-grained patterns seen in a number of developmental processes. However, there is evidence of longer range patterns in early development of the fruit fly Drosophila. The analysis we carry out predicts that patterns longer than one or two cell lengths are possible with our positive feedback mechanism, and numerical simulations confirm this. Our work shows that juxtacrine signalling provides a novel and robust mechanism for the generation of spatial patterns.

Animals↗

PubMed1991

Bifurcating spatially heterogeneous solutions in a chemotaxis model for biological pattern generation.

We consider a simple cell-chemotaxis model for spatial pattern formation on two-dimensional domains proposed by Oster and Murray (1989, J. exp. Zool. 251, 186-202). We determine finite-amplitude, steady-state, spatially heterogeneous solutions and study the effect of domain growth on the resulting patterns. We also investigate in-depth bifurcating solutions as the chemotactic parameter varies. This numerical study shows that this deceptively simple-chemotaxis model can produce a surprisingly rich spectrum of complex spatial patterns.

Animals↗

PubMed1986

Differentiation, cell sorting and proportion regulation in the slug stage of Dictyostelium discoideum.

Recent experimental work suggests that under normal conditions cell sorting plays an important part in maintaining and re-establishing the axial pattern of cell types in the slug stage of the cellular slime mold Dictyostelium discoideum. Following removal of the anterior zone of the slug, anterior-like cells that are normally distributed throughout the posterior of the slug rapidly migrate to the anterior end of the transected slug, and new anterior-like cells appear in the posterior portion. These results provide evidence that the direct linkage between spatial location and differentiation hypothesized in positional information models of spatial pattern formation is not universal. In this paper we develop and analyze a class of mathematical models of the slug in which cell determination can be less rigidly tied to spatial location, and which involve chemotactic cell sorting to re-establish and maintain the spatial pattern of cell types. We show that these models can reproduce the qualitative aspects of the experimental observations and that sorting takes place on the observed time scale when reasonable values of the parameters are used.

Cell Differentiation↗

PubMed1992

From stripes to spots: prepatterns which can be produced in the skin by a reaction-diffusion system.

A key question in the area of spatial pattern formation in developmental biology is: how do groups of cells in a homogeneous tissue suddenly differentiate along entirely different developmental paths compared to neighbouring cells? Although experiments are now beginning to provide answers to this question, the mechanisms responsible for the development of repeated or periodic structures and spatial patterns, e.g., hair follicles and pigmentation patterns, are still unknown. Theoretical biologists and applied mathematicians have suggested various prepattern mechanisms as the primary cause of repeated or periodic spatial patterns. A class of biochemical reactions referred to here as reaction-diffusion (RD) systems, having the capacity to spontaneously generate stable stationary wavelike spatial patterns (Turing, 1952), has been suggested as a possible prepattern mechanisms, e.g., during hair follicle initiation and development (Nagorcka, 1989), and pigmentation patterns (Murray, 1989). Spatial patterns arising during development of the vertebrate skin are frequently complex. Spatial patterns in the skin can be seen to vary within an individual from one region of the skin to another. One pattern change commonly observed across the skin is from stripes to spots. An RD system is defined which is able to generate different spatial patterns depending on the value of a single parameter. The parameter varied controls the transport of the chemical components of the RD system across the basement membrane separating the epidermis and dermis. The patterns produced range from stripes to an irregular array of spots. Not only are different patterns produced, but a different time sequence of prepatterns is expected to arise in the different skin regions depending on whether the first prepattern in an array of spots or stripes. As a consequence it is possible to account for hair follicle initiation in the hair-bearing regions of the mammalian skin as well as the sequence of events required for the formation of dermatoglyphics in the volar regions.

Animals↗

PubMed2004

A Malthusian curb on spatial structure in microorganism populations.

That all organisms are born in the company of a parent but die alone is a fundamental biological asymmetry. It has been suggested that this provides a deep-rooted source of spatial pattern formation for microorganisms even at the scale of the population. Such a theory, however, neglects the strong influence in nature of the limited and spatially variable availability of food. The tendency, first recognized by Thomas Malthus in the 18th century, of a population to out-strip its food resources will eventually lead, through local starvation, to the suppression of a heterogeneity growing within a population. Using a generic model it is demonstrated that including local food limitation of breeding strongly dampens spatial structure otherwise resulting from birth and death. The extent of this damping is shown to be a function of the strength of the coupling between organisms and their food and of the total abundance of organic material. Moreover, this work provides an example of a density-dependent process acting to diminish spatial structure rather than to create it and highlights the rich variety of behaviour that is missed by continuum models which fail to represent such local dynamics.

Animals↗

PubMed1990

Wing formation in Drosophila melanogaster requires decapentaplegic gene function along the anterior-posterior compartment boundary.

Previous analyses of the decapentaplegic (dpp) gene in Drosophila melanogaster have suggested that its product, a polypeptide of the transforming growth factor-beta family of secreted factors, acts at the level of intercellular communication to control several events in spatial pattern formation. In this report, we use clonal analysis to demonstrate a localized requirement for wild-type dpp expression along the anterior-posterior (A/P) compartment boundary of the developing wing primordium. Clonal analysis reveals that normal wing blade development is solely dependent on dpp+ function in those anterior compartment cells that border the anterior-posterior (A/P) compartment boundary of the wing imaginal disk. Conversely, the wing blade will not develop if these boundary cells lack dpp activity. The localized requirement for dpp coincides with the spatial distribution of dpp transcripts, which accumulate in a stripe of cells at or near the known A/P compartment boundary of the wing imaginal disk. Thus, only a small subset of the cells that normally comprise the wing must express dpp to permit development of the entire structure. We propose that this localized expression of dpp is essential to proximal-distal appendage development. We discuss the possibility that dpp expression serves as a landmark for establishing and/or maintaining positional information in imaginal disks.

Animals↗

PubMed1998

On rugged shape of skin tumor (basal cell carcinoma).

Basal cell carcinoma (BCC) is the most common cancer of the skin that is made of transformed basal cells of the human epidermis, and spreads along the epidermis--dermis junction. It often forms tumor cell mass that protrudes towards the dermal connective tissue with many branches. Histopathological examination of the cancer demonstrates the cell mass of a rugged shape appearing as many regular spaced islands in a two-dimensional section. We study the process of cell proliferation and spatial pattern formation of the skin tumor by a pair of partial differential equations of tumor cells and nutrients. The assumptions are: (1) proliferation rate of tumor cells depends on the availability of nutrients, which simply diffuse out of capillaries through connective tissue; (2) nutrients are consumed by active tumor cells; (3) cell diffusion coefficient expressing tumor cell movements increase with the cell density and the nutrient availability. Starting from the initial condition with a single layer of tumor cells. the model develops a smooth colony if n'0 is large, but a characteristic rugged spatial pattern of tumor cell mass if n'0 is small, in which n'0 is nutrient concentration multiplied by square root of growth efficiency divided by diffusion coefficient of nutrients. The proportion of the area occupied by tumor cells increases with n'0. The coefficient of variation in the width of "islands" of tumor cell mass is rather small (0.2 reverse similar0.6), implying the regularity of the spatial pattern. We also analyses the photographs of a two-dimensional section of tumor cell mass and compare the spatial patterns generated by the model.

Aged↗