Modeling the skin pattern of fishes

Effects of square spatial periodic forcing on oscillatory hexagon patterns in coupled reaction-diffusion systems

One of the central issues in pattern formation is understanding the response of pattern-forming systems to an external stimulus. While significant progress has been made in systems with only one instability, much less is known about the response of complex patterns arising from the interaction of two or more instabilities. In this paper, we consider the effects of square spatial periodic forcing on oscillatory hexagon patterns in a two-layer coupled reaction diffusion system which undergoes both Turing and Hopf instabilities. Two different types of additive forcings, namely direct and indirect forcing, have been applied. It is shown that the coupled system exhibits different responses towards the spatial forcing under different forcing types. In the indirect case, the oscillatory hexagon pattern transitions into other oscillatory Turing patterns or resonant Turing patterns, depending on the forcing wavenumber and strength. In the direct forcing case, only non-resonant Turing patterns can be obtained. Our results may provide new insight into the modification and control of spatio-temporal patterns in multilayered systems, especially in biological and ecological systems.

Synthetic Morphogenesis: introducing IEEE journal readers to programming living mammalian cells to make structures

Synthetic morphogenesis is a new engineering discipline, in which cells are genetically engineered to make designed shapes and structures. At least in this early phase of the field, devices tend to make use of natural shape-generating processes that operate in embryonic development, but invoke them artificially at times and in orders of a technologist's choosing. This requires construction of genetic control, sequencing and feedback systems that have close parallels to electronic design, which is one reason the field may be of interest to readers of IEEE journals. The other reason is that synthetic morphogenesis allows the construction of two-way interfaces, especially opto-genetic and opto-electronic, between the living and the electronic, allowing unprecedented information flow and control between the two types of 'machine'. This review introduces synthetic morphogenesis, illustrates what has been achieved, drawing parallels wherever possible between biology and electronics, and looks forward to likely next steps and challenges to be overcome.

Universal model for the skin colouration patterns of neotropical catfishes of the genus Pseudoplatystoma

Fish skin colouration has been widely studied because it involves a variety of processes that are important to the broad field of the developmental biology. Mathematical modelling of fish skin patterning first predicted the existence of morphogens and helped to elucidate the mechanisms of pattern formation. The catfishes of the genus Pseudoplatystoma offer a good biological study model, since its species exhibit the most spectacular and amazing variations of colour patterns on the skin. They present labyrinths, closed loops (or cells), alternate spots and stripes, only spots and combinations of these. We have extended a well known mathematical model to study the skin of Pseudoplatystoma. The basic model is a two component, non-linear reaction diffusion system that presents a richness of bifurcations. The extended model assumes that there are two interacting cell/tissue layers in which morphogens diffuse and interact giving rise to the skin colouration pattern. We have found that by varying only two parameters we are able to accurately reproduce the distinct patterns found in all species of Pseudoplatystoma. The histological analysis of skin samples of two species of this genus, with different patterns, revealed differences on the disposition of the colouration cells that are consistent with our theoretical predictions.

Isolating and quantifying the role of developmental noise in generating phenotypic variation

Genotypic variation, environmental variation, and their interaction may produce variation in the developmental process and cause phenotypic differences among individuals. Developmental noise, which arises during development from stochasticity in cellular and molecular processes when genotype and environment are fixed, also contributes to phenotypic variation. While evolutionary biology has long focused on teasing apart the relative contribution of genes and environment to phenotypic variation, our understanding of the role of developmental noise has lagged due to technical difficulties in directly measuring the contribution of developmental noise. The influence of developmental noise is likely underestimated in studies of phenotypic variation due to intrinsic mechanisms within organisms that stabilize phenotypes and decrease variation. Since we are just beginning to appreciate the extent to which phenotypic variation due to stochasticity is potentially adaptive, the contribution of developmental noise to phenotypic variation must be separated and measured to fully understand its role in evolution. Here, we show that variation in the component of the developmental process corresponding to environmental and genetic factors (here treated together as a unit called the LALI-type) versus the contribution of developmental noise, can be distinguished for leopard gecko (Eublepharis macularius) head color patterns using mathematical simulations that model the role of random variation (corresponding to developmental noise) in patterning. Specifically, we modified the parameters of simulations corresponding to variation in the LALI-type to generate the full range of phenotypic variation in color pattern seen on the heads of eight leopard geckos. We observed that over the range of these parameters, variation in color pattern due to LALI-type variation exceeds that due to developmental noise in the studied gecko cohort. However, the effect of developmental noise on patterning is also substantial. Our approach addresses one of the major goals of evolutionary biology: to quantify the role of stochasticity in shaping phenotypic variation.

Diffusion dynamics of a conductance-based neuronal population

We study the spatiotemporal dynamics of a conductance-based neuronal cable. The processes of one-dimensional (1D) and 2D diffusion are considered for a single variable, which is the membrane voltage. A 2D Morris-Lecar (ML) model is introduced to investigate the nonlinear responses of an excitable conductance-based neuronal cable. We explore the parameter space of the uncoupled ML model and, based on the bifurcation diagram (as a function of stimulus current), we analyze the 1D diffusion dynamics in three regimes: phasic spiking, coexistence states (tonic spiking and phasic spiking exist together), and a quiescent state. We show (depending on parameters) that the diffusive system may generate regular and irregular bursting or spiking behavior. Further, we explore a 2D diffusion acting on the membrane voltage, where striped and hexagonlike patterns can be observed. To validate our numerical results and check the stability of the existing patterns generated by 2D diffusion, we use amplitude equations based on multiple-scale analysis. We incorporate 1D diffusion in an extended 3D version of the ML model, in which irregular bursting emerges for a certain diffusion strength. The generated patterns may have potential applications in nonlinear neuronal responses and signal transmission.

The interplay between phenotypic and ontogenetic plasticities can be assessed using reaction-diffusion models : The case of Pseudoplatystoma fishes

Every morphological, behavioral, or even developmental character expression of living beings is coded in its genotype and is expressed in its phenotype. Nevertheless, the interplay between phenotypic and ontogenetic plasticities, that is, the capability to manifest trait variations, is a current field of research that needs morphometric, numerical, or even mathematical modeling investigations. In the present work, we are searching for a phenotypic index able to identify the underlying correlation among phenotypic, ontogenetic, and geographic distribution of the evolutionary development of species of the same genus. By studying the case of Pseudoplatystoma fishes, we use their skin patterns as an auxiliary trait that can be reproduced by means of a reaction diffusion (RD) model. From this model, we infer the phenotypic index in terms of one of the parameters appearing in the mathematical equations. To achieve this objective, we perform extensive numerical simulations and analysis of the model equations and link the parameter variations with different environmental and physicochemical conditions in which the individuals develop, and which may be regulated by the ontogenetic plasticity of the species. Our numerical study indicates that the patterns predicted by a set of reaction diffusion equations are not uniquely determined by the value of the parameters of the equation, but also depend on how the process is initiated and on the spatial distribution of values of these parameters. These factors are therefore significant, since they show that an individual's growth dynamics and apparent secondary transport processes, like advection, can be determinant for the alignment of motifs in a skin pattern. Our results allow us to discern the correlation between phenotypic, ontogenetic, and geographic distribution of the different species of Pseudoplatystoma fishes, thus indicating that RD models represent a useful taxonomic tool able to quantify evolutionary indexes.

A new mechanochemical model: coupled Ginzburg-Landau and Swift-Hohenberg equations in biological patterns of marine animals

In this work the skin coating of some vertebrate marine animals is modeled considering only dermis, epidermis and basal layers. The biological process takes into account: cellular diffusion of the epidermis, diffusion inhibition and long-range spatial interaction (nonlocal effect on diffusive dispersal) for cells of dermal tissue. The chemical and physical interactions between dermis and epidermis are represented by coupling quadratic terms and nonlinear terms additional. The model presents an interesting property associated with their gradient form: a connection between some physical, chemical and biological systems. The model equations proposed are solved with numerical methods to study the spatially stable emergent configurations. The spatiotemporal dynamic obtained of the numerical solution of these equations, present similarity with biological behaviors that have been found recently in the cellular movement of chromatophores (as contact-dependent depolarization and repulsion movement between melanophores, xanthophores and iridophores). The numerical solution of the model shows a great variety of beautiful patterns that are robust to changes of boundary condition. The resultant patterns are very similar to the pigmentation of some fish.

A selection criterion for patterns in reaction-diffusion systems

BACKGROUND

Alan Turing's work in Morphogenesis has received wide attention during the past 60 years. The central idea behind his theory is that two chemically interacting diffusible substances are able to generate stable spatial patterns, provided certain conditions are met. Ever since, extensive work on several kinds of pattern-generating reaction diffusion systems has been done. Nevertheless, prediction of specific patterns is far from being straightforward, and a great deal of interest in deciphering how to generate specific patterns under controlled conditions prevails.

RESULTS

Techniques allowing one to predict what kind of spatial structure will emerge from reaction-diffusion systems remain unknown. In response to this need, we consider a generalized reaction diffusion system on a planar domain and provide an analytic criterion to determine whether spots or stripes will be formed. Our criterion is motivated by the existence of an associated energy function that allows bringing in the intuition provided by phase transitions phenomena.

CONCLUSIONS

Our criterion is proved rigorously in some situations, generalizing well-known results for the scalar equation where the pattern selection process can be understood in terms of a potential. In more complex settings it is investigated numerically. Our work constitutes a first step towards rigorous pattern prediction in arbitrary geometries/conditions. Advances in this direction are highly applicable to the efficient design of Biotechnology and Developmental Biology experiments, as well as in simplifying the analysis of morphogenetic models.

Nonlinear effects on Turing patterns: time oscillations and chaos

We show that a model reaction-diffusion system with two species in a monostable regime and over a large region of parameter space produces Turing patterns coexisting with a limit cycle which cannot be discerned from the linear analysis. As a consequence, the patterns oscillate in time. When varying a single parameter, a series of bifurcations leads to period doubling, quasiperiodic, and chaotic oscillations without modifying the underlying Turing pattern. A Ruelle-Takens-Newhouse route to chaos is identified. We also examine the Turing conditions for obtaining a diffusion-driven instability and show that the patterns obtained are not necessarily stationary for certain values of the diffusion coefficients. These results demonstrate the limitations of the linear analysis for reaction-diffusion systems.

Regulation of Turing patterns in a spatially extended chlorine-iodine-malonic-acid system with a local concentration-dependent diffusivity

A chlorine-iodine-malonic-acid Turing system involving a local concentration-dependent diffusivity (LCDD) has fundamental significance for physical, chemical, and biological systems with inhomogeneous medium. We investigated such a system by both numerical computation and mathematical analysis. Our research reveals that a variable local diffusivity has an evident effect on regulating the Turing patterns for different modes. An intrinsic square-root law is given by λ ∼ (c(1)+c(2)k)(1/2), which relates the pattern wavelength (λ) with the LCDD coefficient (k). This law indicates that the system pattern has the properties of an equivalent Turing pattern. The current study confirms that, for the Turing system with LCDD, the system pattern form retains the basic characteristics of a traditional Turing pattern in a wide range of LCDD coefficients.

Towards an integrated experimental-theoretical approach for assessing the mechanistic basis of hair and feather morphogenesis

In his seminal 1952 paper, 'The Chemical Basis of Morphogenesis', Alan Turing lays down a milestone in the application of theoretical approaches to understand complex biological processes. His deceptively simple demonstration that a system of reacting and diffusing chemicals could, under certain conditions, generate spatial patterning out of homogeneity provided an elegant solution to the problem of how one of nature's most intricate events occurs: the emergence of structure and form in the developing embryo. The molecular revolution that has taken place during the six decades following this landmark publication has now placed this generation of theoreticians and biologists in an excellent position to rigorously test the theory and, encouragingly, a number of systems have emerged that appear to conform to some of Turing's fundamental ideas. In this paper, we describe the history and more recent integration between experiment and theory in one of the key models for understanding pattern formation: the emergence of feathers and hair in the skins of birds and mammals.

Stochastic reaction and diffusion on growing domains: understanding the breakdown of robust pattern formation

Many biological patterns, from population densities to animal coat markings, can be thought of as heterogeneous spatiotemporal distributions of mobile agents. Many mathematical models have been proposed to account for the emergence of this complexity, but, in general, they have consisted of deterministic systems of differential equations, which do not take into account the stochastic nature of population interactions. One particular, pertinent criticism of these deterministic systems is that the exhibited patterns can often be highly sensitive to changes in initial conditions, domain geometry, parameter values, etc. Due to this sensitivity, we seek to understand the effects of stochasticity and growth on paradigm biological patterning models. In this paper, we extend spatial Fourier analysis and growing domain mapping techniques to encompass stochastic Turing systems. Through this we find that the stochastic systems are able to realize much richer dynamics than their deterministic counterparts, in that patterns are able to exist outside the standard Turing parameter range. Further, it is seen that the inherent stochasticity in the reactions appears to be more important than the noise generated by growth, when considering which wave modes are excited. Finally, although growth is able to generate robust pattern sequences in the deterministic case, we see that stochastic effects destroy this mechanism for conferring robustness. However, through Fourier analysis we are able to suggest a reason behind this lack of robustness and identify possible mechanisms by which to reclaim it.

Modeling parr-mark pattern formation during the early development of Amago trout

This paper studies the formation of the large dark patterns, known as parr marks, that form on the Amago trout as it grows from the early larval stages to adulthood. The Amago trout, known as Oncorhynchus masou ishikawa, exhibits stripes during the early stages of development that in turn evolve (through reorientation and peak insertion) to form zigzag spot patterns as the fish grows to adulthood. By considering a standard representation of the Turing model for biological self-organization via interacting and diffusing morphogens, we illustrate that a diffusively driven instability can generate transient patterns consistent with those experimentally observed during the process of parr-mark formation in the early development of the Amago trout. Surface evolution is modeled through an experimentally driven growth function. Our studies conclude that the surface evolution profile, the surface geometry, and the curvature are key factors that play a pivotal role in reaction-diffusion systems in a study motivated by observations of Amago trout parr-mark pattern formation.

Power spectra methods for a stochastic description of diffusion on deterministically growing domains

A central challenge in developmental biology is understanding the creation of robust spatiotemporal heterogeneity. Generally, the mathematical treatments of biological systems have used continuum, mean-field hypotheses for their constituent parts, which ignores any sources of intrinsic stochastic effects. In this paper we consider a stochastic space-jump process as a description of diffusion, i.e., particles are able to undergo a random walk on a discretized domain. By developing analytical Fourier methods we are able to probe this probabilistic framework, which gives us insight into the patterning potential of diffusive systems. Further, an alternative description of domain growth is introduced, with which we are able to rigorously link the mean-field and stochastic descriptions. Finally, through combining these ideas, it is shown that such stochastic descriptions of diffusion on a deterministically growing domain are able to support the nucleation of states that are far removed from the deterministic mean-field steady state.

Analysis of stationary droplets in a generic Turing reaction-diffusion system

Solitonlike structures called "droplets" are found to exist within a paradigm reaction-diffusion model that can be used to describe patterning in a number of biological systems, for example, on the skin of various fish species. They have also been found in many other systems that can be modeled with a complex Ginzburg-Landau system. These droplets can be analyzed in the biological paradigm model because the system has two nonzero stable steady states that are symmetric; however, the asymmetric case is more challenging. We first review the properties of the paradigm system and then extend a recently developed perturbation technique [D. Gomila, J. Opt. B: Quantum Semiclassical Opt. 6, S265 (2004)] to investigate the weakly asymmetric case. We compare the results of our mathematical analysis with numerical simulations and show good agreement in the region where the assumptions hold.

Blending of animal colour patterns by hybridization

Biologists have long been fascinated by the amazing diversity of animal colour patterns. Despite much interest, the underlying evolutionary and developmental mechanisms contributing to their rich variety remain largely unknown, especially the vivid and complex colour patterns seen in vertebrates. Here, we show that complex and camouflaged animal markings can be formed by the 'blending' of simple colour patterns. A mathematical model predicts that crossing between animals having inverted spot patterns (for example, 'light spots on a dark background' and 'dark spots on a light background') will necessarily result in hybrid offspring that have camouflaged labyrinthine patterns as 'blended' intermediate phenotypes. We confirmed the broad applicability of the model prediction by empirical examination of natural and artificial hybrids of salmonid fish. Our results suggest an unexplored evolutionary process by means of 'pattern blending', as one of the possible mechanisms underlying colour pattern diversity and hybrid speciation.

Many animals have complex body patterns, which are fixed in some species and flexible in others. Here, using reaction-diffusion mathematical models, together with salmonid fish crosses, intermediate patterns are shown to occur in hybrid animals produced by mating species with different flexible patterns.

Reaction-diffusion systems with stochastic time delay in kinetics

Time delay is an important dynamical issue in a multistep chemical reaction. We present a theoretical scheme for delayed reaction-diffusion systems where the time delay in kinetics is stochastic in nature. It has been shown that a small but finite strength of delay may result in generic change in the nature of spatiotemporal instability in the dynamics. Our theoretical analysis has been corroborated by numerical simulations on two reaction-diffusion systems.

Time-delay-induced instabilities in reaction-diffusion systems

Time delay in the kinetic terms of reaction-diffusion systems has been investigated. It has been shown that short delay beyond a critical threshold may induce spatiotemporal instabilities. For unequal diffusivities and appropriate parameter space delay may induce Turing instability resulting in stationary patterns and also interesting Turing-Hopf transition with the formation of spirals. The theoretical scheme has been numerically explored in two different prototypical reaction-diffusion systems.