New Mechanism For Pattern Formation Research Articles

Pattern formation and front stability for a moving-boundary model of biological invasion and recession

We investigate pattern formation in a two-dimensional (2D) Fisher–Stefan model, which involves solving the Fisher–KPP equation on a compactly-supported region with a moving boundary. The boundary evolves analogously to the classical one-phase Stefan problem, such that boundary speed is proportional to the local population density gradient. By combining the Fisher–KPP and classical Stefan theory, the Fisher–Stefan model alleviates two limitations of the Fisher–KPP equation for biological populations. Unlike the Fisher–KPP equation, solutions to the Fisher–Stefan model have compact support, explicitly defining the region occupied by the population. Furthermore, the Fisher–Stefan model admits travelling wave solutions with non-negative density for all wave speeds c∈(−∞,∞), and can thus model both population invasion and population recession. In this work, we investigate whether the 2D Fisher–Stefan model predicts pattern formation, by analysing the linear stability of planar travelling wave solutions to sinusoidal transverse perturbations. Planar fronts of the Fisher–KPP equation are linearly stable. Similarly, we demonstrate that invading planar fronts (c>0) of the Fisher–Stefan model are linearly stable to perturbations of all wave numbers. However, our analysis demonstrates that receding planar fronts (c<0) of the Fisher–Stefan model are linearly unstable for all wave numbers. This is analogous to unstable solutions for planar solidification in the classical Stefan problem. Introducing a surface tension regularisation stabilises receding fronts for short-wavelength perturbations, giving rise to a range of unstable modes and a most unstable wave number. We supplement linear stability analysis with level-set numerical solutions that corroborate theoretical results. Overall, front instability in the Fisher–Stefan model suggests a new mechanism for pattern formation in receding biological populations.

Pattern formation in multiplex networks.

The advances in understanding complex networks have generated increasing interest in dynamical processes occurring on them. Pattern formation in activator-inhibitor systems has been studied in networks, revealing differences from the classical continuous media. Here we study pattern formation in a new framework, namely multiplex networks. These are systems where activator and inhibitor species occupy separate nodes in different layers. Species react across layers but diffuse only within their own layer of distinct network topology. This multiplicity generates heterogeneous patterns with significant differences from those observed in single-layer networks. Remarkably, diffusion-induced instability can occur even if the two species have the same mobility rates; condition which can never destabilize single-layer networks. The instability condition is revealed using perturbation theory and expressed by a combination of degrees in the different layers. Our theory demonstrates that the existence of such topology-driven instabilities is generic in multiplex networks, providing a new mechanism of pattern formation.

Domain nucleation and confinement in agent-controlled bistable systems.

We report a mechanism of pattern formation in growing bistable systems coupled indirectly. A modified Fujita etal. model is studied as an example of a reaction-diffusion system of nondiffusive activator and inhibitor molecules immersed in the medium of a fast diffusive agent. Here we show that, as the system grows, a new domain nucleates spontaneously in the area where the local level of the agent becomes critical. Newly nucleated domains are stable and the pattern formation is different from Turing's mechanism in monostable systems. Domains are spatially confined by the agent even if the activator and inhibitor molecules diffuse. With the spatial extension of the system, a larger domain may undergo a wave number instability, and the concentrations of active molecules within the neighboring elements of a domain can become sharply different. The mechanism reported in this work could be generic for pattern formation systems involving multistability, growth, and indirect coupling.

Quasi-Local Competition in Stage-Structured Metapopulations: A New Mechanism of Pattern Formation

A central question of ecology is what determines the presence and abundance of species at different locations. In cases of ecological pattern formation, population sizes are largely determined by spatially distributed interactions and may have very little to do with the habitat template. We find pattern formation in a single-species metapopulation model with quasi-local competition, but only if the populations have (at least) two age or stage classes. Quasi-local competition is modeled using an explicit resource competition model with fast resource dynamics, and assuming that adults, but not juveniles, spend a fraction of their foraging time in habitat patches adjacent to their home patch. Pattern formation occurs if one stage class depletes the common resource but the shortage of resource affects mostly the other stage. When the two stages are spatially separated due to quasi-local competition, this results in competitive exclusion between the populations. We find deep similarity between spatial pattern formation and population cycles due to competitive exclusion between cohorts of biennial species, and discuss the differences between the present mechanism and established ways of pattern formation such as diffusive instability and distributed competition with local Allee-effects.

Morphological instability in freezing colloidal suspensions

We present a linear stability analysis of a planar ice interface during unidirectional solidification of a hard-sphere colloidal suspension. We find that the interface can become unstable due to constitutional supercooling, yielding a new mechanism for pattern formation in colloidal systems. The interfacial stability is shown to depend strongly on the size and concentration of the particles. Increasing the particle radius tends to stabilize the interface, while increasing the concentration has a destabilizing effect. Additional effects that may influence the stability and morphology of such a system are described.

Global cell sorting is mediated by local cell–cell interactions in the C. elegans embryo

The Caenorhabditis elegans embryo achieves pattern formation by sorting cells into coherent regions before morphogenesis is initiated. The sorting of cells is coupled to their fate. Cells move extensively relative to each other to reach their correct position in the body plan. Analyzing the mechanism of cell sorting in in vitro culture experiments using 4D microscopy, we show that all AB-derived cells sort only according to their local neighbors, and that all cells are able to communicate with each other. The directions of cell movement do not depend on a cellular polarity but only on local cell–cell interactions; in experimental situations, this allows even the reversal of the polarity of whole regions of the embryo. The work defines a new mechanism of pattern formation we call “cell focusing”.

Global cell sorting in the C. elegans embryo defines a new mechanism for pattern formation

4D microscopic observations of Caenorhabditis elegans development show that the nematode uses an unprecedented strategy for development. The embryo achieves pattern formation by sorting cells, through far-ranging movements, into coherent regions before morphogenesis is initiated. This sorting of cells is coupled to their particular fate. If cell identity is altered by experiment, cells are rerouted to positions appropriate to their new fates even across the whole embryo. This cell behavior defines a new mechanism of pattern formation, a mechanism that is also found in other animals. We call this new mechanism “cell focusing”. When the fate of cells is changed, they move to new positions which also affect the shape of the body. Thus, this process is also important for morphogenesis.

Stationary and oscillatory spatial patterns induced by global periodic switching.

We propose a new mechanism for pattern formation based on the global alternation of two dynamics, neither of which exhibits patterns. When driven by either one of the separate dynamics, the system goes to a spatially homogeneous state associated with that dynamics. However, when the two dynamics are globally alternated sufficiently rapidly, the system exhibits stationary spatial patterns. Somewhat slower switching leads to oscillatory patterns. We support our findings by numerical simulations and discuss the results in terms of the symmetries of the system and the ratio of two relevant characteristic times, the switching period and the relaxation time to a homogeneous state in each separate dynamics.

Nonadiabatic effects in optical pattern formation

Pattern formation in cavity nonlinear optical systems subjected to a periodic modulation of frequency detuning is studied analytically and numerically with particular reference to the models of optical parametric oscillator and two-level laser. Owing to the nonautonomous dynamics, a new mechanism for pattern formation as a result of a primary instability, rather distinct from the most common tilted-wave mechanism found in autonomous systems, is predicted and explained in detail by means of a phase integral (WKB) analysis of the underlying field equations. This mechanism for pattern formation can be traced back to the existence of coherent field oscillations in the two-field dynamics and associated to the existence of turning points in the WKB expansion, which break the adiabatic following. In particular, it is shown that nonadiabatic effects are likely when the decay rates of interacting fields are equal, corresponding to the existence of real turning points. For different relaxation rates of interacting fields, the turning points are complex and nonadiabatic effects vanish at low modulation frequencies. Above threshold, weakly nonlinear analysis and numerical simulations indicate that traveling waves are selected by the nonlinearity.

Modeling Pattern Formation

Pattern formation in early development is a well-established process. One mechanism for the induction of these patterns is juxtacrine signaling in which the receptor and the ligand are both membrane anchored so that the signal is limited to adjacent cells. Lateral inhibition, exemplified by the Notch-Delta interaction, provides one method for establishing patterns that alternate with a frequency of exactly two cells. Owen et al . have developed a mathematical model for lateral induction that could explain how lateral induction can lead to patterns that are not restricted to the two-cell alternating pattern. Their model provides a mechanism by which juxatracine signaling can lead to patterns that occur with frequencies greater than two cells. Owen, M.R., Sherratt, J.A., and Wearing, H.J. (2000) Lateral induction by juxtacrine signaling is a new mechanism for pattern formation. Dev. Biol . 217 : 54-61 [Abstract]

Lateral Induction by Juxtacrine Signaling Is a New Mechanism for Pattern Formation

Many signaling molecules in epithelia are now known to function in a membrane-bound form, binding to receptors on immediately neighbouring cells. This “juxtacrine” mode of communication has been well studied in the case of lateral inhibition, where ligand binding at the cell surface downregulates ligand and receptor expression, and is known to generate spatial patterns with a wavelength of exactly two cells. However, recent evidence shows that a number of juxtacrine signals can lead to the opposite phenomenon of lateral induction. Here, we use mathematical modeling to show that such positive feedback, in combination with juxtacrine communication, provides a novel mechanism for the generation of spatial patterns, with wavelengths that vary with parameters and can be many cell lengths.

Absolute and convective instabilities in a one-dimensional Brusselator flow model

The paper considers a one-dimensional Brusselator model with a uniform flow of the mixture of reaction components. An absolute as well as a convective instability can arise for both the Hopf and the Turing modes. The corresponding linear stability analysis is presented and supported by the results of computer simulations of the nonlinear equations. Finally, the condition for spatially undamped tails (the Cherenkov condition) is obtained. This represents a new mechanism for pattern formation in chemical reaction-diffusion systems.