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Zebrafish Leopard gene as a component of the putative reaction-diffusion system
Abstract
It has been suggested, on a theoretical basis, that a reaction-diffusion (RD) mechanism underlies pigment pattern formation in animals, but as yet, there is no molecular evidence for the putative mechanism. Mutations in the zebrafish gene, leopard, change the pattern from stripes to spots. Interestingly each allele gives a characteristic pattern, which varies in spot size, density and connectivity. That mutations in a single gene can generate such a variety of patterns can be understood using a RD model. All the pattern variations of leopard mutants can be generated in a simulation by changing a parameter value that corresponds to the reaction kinetics in a putative RD system. Substituting an intermediate value of the parameter makes the patterns similar to the heterozygous fish. These results suggest that the leopard gene product is a component of the putative RD mechanism.
... 12 Time scale cannot be detached from the observer. 13 Which also aects the way that people behave (the eects feed back to the organization). ...
... He According to Turing, the laws of physical chemistry are sucient to explain many aspects of morphogenesis. The reaction-diusion theory may apply to the study of biological phenomena [13,136,205]. From early applications of reaction-diusion theory, the authors has compared mathematical models to biological patterns [206,207]. In [177] authors show that the formation of bans of calcium in a cytosol solution can be stylized by a reaction-diusion mechanism. ...
... 7 reports the robustness of street networks and the six theoretic planar graphs. As we expected, the lower values of robustness are observed12 We do not plot points of theoretic planar graphs because they are too far.13 In statistics, the coecient of determination R 2 ∈ [0, 1] is a measure that indicates how well the data t the regression line. ...
The characteristics, functions and morphogenetic processes of a large number of complex spatial networks are influenced by the position and the geometry of their constituent elements. In this work, we address the computational aspects of the morphogenesis of complex networks by proposing a general model, simulating their formation. The networks are generated under the influence of constraints expressed through a vector field that is determined using a reaction-diffusion system. We use the Gray-Scott model to produce a wide variety of dynamic patterns. The resulting vector field controls the geometry and the growth rate of the constructed network that feeds back the reaction-diffusion process. A study was carried out on the influence of the patterns and feedback processes on the structure of the obtained networks using measures from graph theory and multi-fractality theory. A process of validation and evaluation of the model's behaviour was carried out and applied by comparing the networks obtained to largest French cities and the most relevant geometric planar graphs.
... One of the most spectacular features of many biological organism is their capability to generate various intriguing and visually fascinating complex spatio-temporal patterns. The exotic beauty and bewildering complexity of animal coat patterns, namely, zebra stripes [1,2], patterns on snake skin [2,3], coloration of butterfly wings [4][5][6], coat patterns of the giraffe [7], leopard [8], stripe patterns on fish skin [9][10][11] have attracted every curious mind from ancient times. Many such natural color patterns are treasured objects in certain communities of the world and they are sometimes used for the identification of species. ...
... The existence of morphogens have already been established in experiments by developmental biologists [25]. There are strong experimental evidences that the pigment patterns on the skin of the angelfish [9][10][11] and the hair-follicle patterns in mice [26] could be modeled by a Turing mechanism. All the subsequent theoretical (mathematical) developments are based on Turing revolutionary idea that the mutual interaction of elements under certain conditions results in spontaneous pattern formation. ...
... As they grow in size, these spots gradually transform into rosettes (small spots organized into patterns of six or fewer spots) in leopards and blotches (irregular shaped areas of dark on a usually lighter background) in jaguar at their adult stage [8]. Numerous observations of fish skin markings [9,10,31] also demonstrated the alternation of patterns during growth. Experiments on zebrafish skin [30] suggest that the stripe formation takes place due to an autonomous mechanism that satisfies local self-enhancement and longrange inhibition, and is independent of prepattern. ...
The spontaneous formation of a wide variety of natural patterns with
different shapes and symmetries in many physical and biological systems
is one of the deep mysteries in science. This article describes
the physical principles underlying the formation of various intriguing
spatio-temporal patterns in Nature with special emphasis on some
biological structures. We discuss how the spontaneous symmetry
breaking due to diffusion driven instability in the reaction dynamics
lead to the emergence of such complicated natural patterns. The
mechanism of the formation of various animal coat patterns is explained
via the Turing-type reaction-diffusion models.
... In particular, reaction-diffusion systems are a well-accepted class of models for multiple pigmentation processes. Examples include marine [27] and emperor [26] angelfish, genets [1], plecostomi [12], cheetah [46], zebra fish [2], and many other mammal skin patterns [34]; see Figure 1. Although animal skin pattern simulation through reaction-diffusion systems has been explored in many researches, see the second row of Figure 5 for typical examples, very few of them have taken the skin texture into account. ...
... with the partial derivatives z x and z y computing from (6). We are interested in obtaining the eigenvalues and eigenvectors of A. Rather than computing these quantities directly, it turns out to be easier to first compute the eigenvalues and eigenvectors of the Riemannian metric tensor G in (2). From the characteristic equation of G ...
We are interested in simulating patterns on rough surfaces. First, we consider periodic rough surfaces with analytic parametric equations, which are defined by some superposition of wave functions with random frequencies and angles of propagation. The amplitude of such surfaces is also an important variable in the provided eigenvalue analysis for the Laplace-Beltrami operator and in our numerical studies. Simulations show that the patterns become irregular as the amplitude and frequency of the rough surface increase. Next, for the sake of easy generalization to closed manifolds, we propose another construction method of rough surfaces by using random nodal values and discretized heat filters. We provide numerical evidence that both surface constructions yield comparable patterns to those found in real-life animals.
... Knowledge of the minimal kinds of interactions that could generate the patterns could then serve to guide hypothesis generation for electrophysiological and biochemical studies, which are necessary for obtaining the right physiological realization of the model. This happened in the study of fish chromatic displays, where early theoretical work (Kondo and Asai, 1995) guided work on protein interactions that instantiate the reactions (Watanabe and Kondo, 2014;Asai et al., 1999). ...
... We believe that the present work has import for the empirical study of cephalopod patterns (Osorio, 2014) as the higher-level interaction parameters can be used to robustly quantify pattern similarity and differences. We also believe that the work has implications for electrophysiological and biochemical work on cephalopod chromatic displays, as the reactiondiffusion mechanisms we have discussed could guide the search for the biochemical events instantiating them, as has happened in the study of developmental fish skin patterning (Asai et al., 1999). ...
The skin of a cephalopod forms a dazzling array of patterns made by chromatophores, elastic sacs of pigment that can be expanded by muscles to reveal their color. Tens of thousands of these chromatophores can work together to generate a stable display of stripes, spots, mottled grainy camouflage, or dynamic oscillations and traveling waves of activation. How does a neuromuscular system organize the coactivation of thousands of degrees of freedom through simple central commands? We provide a minimally-complex physiologically-plausible mathematical model, using Turing's morphogenetic equations, that can generate the array of twelve static and four dynamic types of skin displays seen in several cephalopod species. These equations specify how muscle cells on the skin need to locally interact for the global chromatic patterns to be formed. We also demonstrate a link between Turing neural computations and the asynchronous type of computing that has been extensively demonstrated in brain systems: population coding, using bimodal codes, with the relative heights of the modes specifying the kind of global pattern generated. Since Cephalopod skins are a "visible neural net", we believe that the computational principles uncovered through their study may have wider implications for the functioning of other neural systems.
... Independently on the observed variation, once spots are formed, their average shape is similar across individuals (Table SA4), suggesting a strong constraint on this pattern element (EE), which is also weakly correlated to the other indices (Fig. 5). While variation in spot size and density has also been observed in other organisms (e.g., Asai et al., 1999;Morgan et al., 2014;Rudh, Rogell & Höglund, 2007;Balogová & Uhrin, 2015;Druml et al., 2017), less is known about variation in spot shape. Potential genetic or developmental mechanisms may have evolved to ensure maintenance of spot shape and low variability of this trait. ...
... Future research could further investigate if low variation in spot shape also occurs in other spotted vertebrates and if it is similarly achieved across organisms. In zebrafish, different alleles of the leopard gene result in changes in spot size, density, and connectivity among spots, suggesting that this gene may regulate the synthesis of an activator in a model of reaction-diffusion pattern formation (Asai et al., 1999). Later studies identified the role of leopard in regulating interaction among melanophores (or among xanthophores) and in controlling boundary shape for the spots (reviewed in Kondo, Iwashita & Yamaguchi, 2009;Singh & Nüsslein-Volhard, 2015). ...
Animal color patterns are widely studied in ecology, evolution, and through mathematical modeling. Patterns may vary among distinct body parts such as the head, trunk or tail. As large amounts of photographic data is becoming more easily available, there is a growing need for general quantitative methods for capturing and analyzing the full complexity and details of pattern variation. Detailed information on variation in color pattern elements is necessary to understand how patterns are produced and established during development, and which evolutionary forces may constrain such a variation. Here, we develop an approach to capture and analyze variation in melanistic color pattern elements in leopard geckos. We use this data to study the variation among different body parts of leopard geckos and to draw inferences about their development. We compare patterns using 14 different indices such as the ratio of melanistic versus total area, the ellipticity of spots, and the size of spots and use these to define a composite distance between two patterns. Pattern presence/absence among the different body parts indicates a clear pathway of pattern establishment from the head to the back legs. Together with weak within-individual correlation between leg patterns and main body patterns, this suggests that pattern establishment in the head and tail may be independent from the rest of the body. We found that patterns vary greatest in size and density of the spots among body parts and individuals, but little in their average shapes. We also found a correlation between the melanistic patterns of the two front legs, as well as the two back legs, and also between the head, tail and trunk, especially for the density and size of the spots, but not their shape or inter-spot distance. Our data collection and analysis approach can be applied to other organisms to study variation in color patterns between body parts and to address questions on pattern formation and establishment in animals.
... While variation in spot size and density has also been observed in other organisms (e.g. Asai et al., 1999;Morgan et al., 2014;Rudh et al., 2007;Balogová and Uhrin, 2015;Druml et al., 2017), less is known about variation in spot shape. Potential genetic or developmental mechanisms may have evolved to ensure maintenance of spot shape and low variability of this trait. ...
... Future research could further investigate if low variation in spot shape also occurs in other spotted vertebrates and if it is similarly achieved across organisms. In zebrafish, different alleles of the leopard gene result in changes in spot size, density, and connectivity among spots, suggesting that this gene may regulate the synthesis of an activator in a model of reaction-diffusion pattern formation (Asai et al., 1999). Later studies identified the role of leopard in regulating interaction among melanophores (or among xanthophores) and in controlling boundary shape for the spots (reviewed in Kondo et al., 2009;Singh and Nüsslein-Volhard, 2015). ...
Animal color patterns are widely studied in ecology, evolution, and through mathematical modeling. Patterns may vary among distinct body parts such as the head, trunk or tail. As large amounts of photographic data is becoming more easily available, there is a growing need for general quantitative methods for capturing and analyzing the full complexity and details of pattern variation. Detailed information on variation in color pattern elements is necessary to understand how patterns are produced and established during development, and which evolutionary forces may constrain such a variation. Here, we develop an approach to capture and analyze variation in melanistic color pattern elements in leopard geckos. We use this data to study the variation among different body parts of leopard geckos and to draw inferences about their development. We compare patterns using 14 different indices such as the ratio of melanistic versus total area, the ellipticity of spots, and the size of spots and use these to define a composite distance between two patterns. Pattern presence/absence among the different body parts indicates a clear pathway of pattern establishment from the head to the back legs. Together with weakwithin-individual correlation between leg patterns and main body patterns, this suggests that pattern establishment in the head and tail may be independent from the rest of the body. We found that patterns vary greatest in size and density of the spots among body parts and individuals, but little in their average shapes. We also found a correlation between the melanistic patterns of the two front legs, as well as the two back legs, and also between the head, tail and trunk, especially for the density and size of the spots, but not their shape or inter-spot distance. Our data collection and analysis approach can be applied to other organisms to study variation in color patterns between body parts and to address questions on pattern formation and establishment in animals.
... Zebrafish are amenable to observational studies, since all development takes place outside the mother and the skin is transparent. This, combined with the availability of multiple key mutants (affecting, for example, cell-type differentiation and patterning), and the development of innovative in vivo cell ablation and in vitro cell culture techniques, have provided a unique opportunity to investigate the cellular and molecular basis for pigment pattern formation experimentally (Eom et al., 2015;Budi et al., 2011;Ceinos et al., 2015;Yamanaka and Kondo, 2014;Eom and Parichy, 2017;Hamada et al., 2014;Fadeev et al., 2015;Inoue et al., 2014;Irion et al., 2014;Watanabe et al., 2006;Frohnhö fer et al., 2013;Parichy et al., 2009;Svetic et al., 2007;Mellgren and Johnson, 2006;Parichy et al., 2000b;Iwashita et al., 2006;Hirata et al., 2005;Kelsh et al., 1996;Lister et al., 1999;Parichy et al., 2000a;Maderspacher and Nüsslein-Volhard, 2003;Walderich et al., 2016;Patterson et al., 2014;McMenamin et al., 2014;Patterson and Parichy, 2013;Krauss et al., 2014;Parichy and Turner, 2003;Eom et al., 2012;Mahalwar et al., 2016;Mahalwar et al., 2014;Asai et al., 1999;Takahashi and Kondo, 2008). ...
... However, a potential limitation is that parameters do not always have a clear biological interpretation which, can sometimes make it difficult to link parameters to measurable data. In the context of zebrafish stripe formation, these models have not yet incorporated S-iridophores Kondo, 2017;Painter et al., 2015;Bloomfield et al., 2011;Binder and Simpson, 2013;Volkening and Sandstede, 2015;Kondo, 2017;Nakamasu et al., 2009;Moreira and Deutsch, 2005;Bullara and De Decker, 2015;Yamaguchi et al., 2007;Asai et al., 1999). They suggest that the role for iridophores is restricted to simply orienting stripes (Volkening and Sandstede, 2015;Nakamasu et al., 2009;Binder and Simpson, 2013). ...
Pattern formation is a key aspect of development. Adult zebrafish exhibit a striking striped pattern generated through the self-organisation of three different chromatophores. Numerous investigations have revealed a multitude of individual cell-cell interactions important for this self-organisation, but it has remained unclear whether these known biological rules were sufficient to explain pattern formation. To test this, we present an individual-based mathematical model incorporating all the important cell-types and known interactions. The model qualitatively and quantitatively reproduces wild type and mutant pigment pattern development. We use it to resolve a number of outstanding biological uncertainties, including the roles of domain growth and the initial iridophore stripe, and to generate hypotheses about the functions of leopard. We conclude that our rule-set is sufficient to recapitulate wild-type and mutant patterns. Our work now leads the way for further in silico exploration of the developmental and evolutionary implications of this pigment patterning system.
... Zebrafish are amenable to observational studies, since all development takes place outside the mother and the skin is transparent. This, combined with the availability of multiple key mutants (affecting, for example, cell-type differentiation and patterning), and the development of innovative in vivo cell ablation and in vitro cell culture techniques, have provided a unique opportunity to investigate the cellular and molecular basis for pigment pattern formation experimentally (Eom et al., 2015;Budi et al., 2011;Ceinos et al., 2015;Yamanaka and Kondo, 2014;Eom and Parichy, 2017;Hamada et al., 2014;Fadeev et al., 2015;Inoue et al., 2014;Irion et al., 2014;Watanabe et al., 2006;Frohnhö fer et al., 2013;Parichy et al., 2009;Svetic et al., 2007;Mellgren and Johnson, 2006;Parichy et al., 2000b;Iwashita et al., 2006;Hirata et al., 2005;Kelsh et al., 1996;Lister et al., 1999;Parichy et al., 2000a;Maderspacher and Nüsslein-Volhard, 2003;Walderich et al., 2016;Patterson et al., 2014;McMenamin et al., 2014;Patterson and Parichy, 2013;Krauss et al., 2014;Parichy and Turner, 2003;Eom et al., 2012;Mahalwar et al., 2016;Mahalwar et al., 2014;Asai et al., 1999;Takahashi and Kondo, 2008). ...
... However, a potential limitation is that parameters do not always have a clear biological interpretation which, can sometimes make it difficult to link parameters to measurable data. In the context of zebrafish stripe formation, these models have not yet incorporated S-iridophores Kondo, 2017;Painter et al., 2015;Bloomfield et al., 2011;Binder and Simpson, 2013;Volkening and Sandstede, 2015;Kondo, 2017;Nakamasu et al., 2009;Moreira and Deutsch, 2005;Bullara and De Decker, 2015;Yamaguchi et al., 2007;Asai et al., 1999). They suggest that the role for iridophores is restricted to simply orienting stripes (Volkening and Sandstede, 2015;Nakamasu et al., 2009;Binder and Simpson, 2013). ...
Pattern formation is a key aspect of development. Adult zebrafish exhibit a striking striped pattern generated through the self-organisation of three different chromatophores. Numerous investigations have revealed a multitude of individual cell-cell interactions important for this self-organisation, but it has remained unclear whether these known biological rules were sufficient to explain pattern formation. To test this, we present an individual-based mathematical model incorporating all the important cell-types and known interactions. The model qualitatively and quantitatively reproduces wild type and mutant pigment pattern development. We use it to resolve a number of outstanding biological uncertainties, including the roles of domain growth and the initial iridophore stripe, and to generate hypotheses about the functions of leopard. We conclude that our rule-set is sufficient to recapitulate wild-type and mutant patterns. Our work now leads the way for further in silico exploration of the developmental and evolutionary implications of this pigment patterning system.
... Zebrafish are amenable to observational studies, since all development takes place outside the mother and the skin is transparent. This, combined with the availability of multiple key mutants (affecting, for example, cell-type differentiation and patterning), and the development of innovative in vivo cell ablation and in vitro cell culture techniques, have provided a unique opportunity to investigate the cellular and molecular basis for pigment pattern formation experimentally (Eom et al., 2015;Budi et al., 2011;Ceinos et al., 2015;Yamanaka and Kondo, 2014;Eom and Parichy, 2017;Hamada et al., 2014;Fadeev et al., 2015;Inoue et al., 2014;Irion et al., 2014;Watanabe et al., 2006;Frohnhö fer et al., 2013;Parichy et al., 2009;Svetic et al., 2007;Mellgren and Johnson, 2006;Parichy et al., 2000b;Iwashita et al., 2006;Hirata et al., 2005;Kelsh et al., 1996;Lister et al., 1999;Parichy et al., 2000a;Maderspacher and Nüsslein-Volhard, 2003;Walderich et al., 2016;Patterson et al., 2014;McMenamin et al., 2014;Patterson and Parichy, 2013;Krauss et al., 2014;Parichy and Turner, 2003;Eom et al., 2012;Mahalwar et al., 2016;Mahalwar et al., 2014;Asai et al., 1999;Takahashi and Kondo, 2008). ...
... However, a potential limitation is that parameters do not always have a clear biological interpretation which, can sometimes make it difficult to link parameters to measurable data. In the context of zebrafish stripe formation, these models have not yet incorporated S-iridophores Kondo, 2017;Painter et al., 2015;Bloomfield et al., 2011;Binder and Simpson, 2013;Volkening and Sandstede, 2015;Kondo, 2017;Nakamasu et al., 2009;Moreira and Deutsch, 2005;Bullara and De Decker, 2015;Yamaguchi et al., 2007;Asai et al., 1999). They suggest that the role for iridophores is restricted to simply orienting stripes (Volkening and Sandstede, 2015;Nakamasu et al., 2009;Binder and Simpson, 2013). ...
Pattern formation is a key aspect of development. Adult zebrafish exhibit a striking striped pattern generated through the self-organisation of three different chromatophores. Numerous investigations have revealed a multitude of individual cell-cell interactions important for this self-organisation, but it has remained unclear whether these known biological rules were sufficient to explain pattern formation. To test this, we present an individual-based mathematical model incorporating all the important cell-types and known interactions. The model qualitatively and quantitatively reproduces wild type and mutant pigment pattern development. We use it to resolve a number of outstanding biological uncertainties, including the roles of domain growth and the initial iridophore stripe, and to generate hypotheses about the functions of leopard . We conclude that our rule-set is sufficient to recapitulate wild-type and mutant patterns. Our work now leads the way for further in silico exploration of the developmental and evolutionary implications of this pigment patterning system.
... this variation, because related species share many genetic and developmental pathways, like mutants of an individual species. Reaction-diffusion and related models have been applied to simulate pattern formation in several fish species 2,6,7,13,14 . Single component models predict simple patterns, as in the zebra fish 13 . ...
... Reaction-diffusion and related models have been applied to simulate pattern formation in several fish species 2,6,7,13,14 . Single component models predict simple patterns, as in the zebra fish 13 . Two-component Turing systems predict a wide variety of patterns of dots, stripes and combinations of them, as the ones in Pomacanthus imperator and other fish 14 . ...
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.
... A particular, well-studied example in this context is the zebrafish, see Figure 1. Studies have connected genes (Asai et al. (1999)), chemical properties (Wertheim and Roose (2019)), and cell-network interactions (Nakamasu et al. (2009)) to the formation of Turing patterns within the zebrafish. Recently, researchers revealed how mechanical stresses can lead to tissue anisotropies and thus morphogen gradients in the formation of Turing patterns, causing the characteristic stripes of a common zebrafish, see Hiscock and Megason (2015). ...
... Four fully-grown (5-6 months) female zebrafish, with different alleles of the leopard gene, exhibiting striped and dappled patterns respectively. Image taken from Asai et al. (1999). patterns generated by ants, termites, and other subterranean animals are inherently caused by such feedback procedures, see Pringle and Tarnita (2017). ...
In his 1952 paper "The chemical basis of morphogenesis", Alan M. Turing presented a model for the formation of skin patterns. While it took several decades, the model has been validated by finding corresponding natural phenomena, e.g. in the skin pattern formation of zebrafish. More surprising, seemingly unrelated pattern formations can also be studied via the model, like e.g. the formation of plant patches around termite hills. In 1984, David A. Young proposed a discretization of Turing's model, reducing it to an activator/inhibitor process on a discrete domain. From this model, the concept of three-dimensional Turing-like patterns was derived. In this paper, we consider this generalization to pattern-formation in three-dimensional space. We are particularly interested in classifying the different arising sub-structures of the patterns. By providing examples for the different structures, we prove a conjecture regarding these structures within the setup of three-dimensional Turing-like pattern. Furthermore, we investigate - guided by visual experiments - how these sub-structures are distributed in the parameter space of the discrete model. We found two-fold versions of zero- and one-dimensional sub-structures as well as two-dimensional sub-structures and use our experimental findings to formulate several conjectures for three-dimensional Turing-like patterns and higher-dimensional cases.
... Nature offers an immense variety of living as well as nonliving-patterns arising from complex physico-chemical processes. The seminal work of Turing [1] in autonomous generation of very distinctive patterns inspired scientific communities for a long time as it serves as a basis for describing a remarkable variety of naturally occurring patterns in living systems such as coat patterns of giraffe [2], leopard and jaguar [3], zebra stripes [4,5], patterns on snake skin [5,6], coloration of butterfly wings [7][8][9], and stripe patterns on fish skin [10][11][12][13]. Turing considered the reaction of two diffusive chemicals having concentrations u(x, t) and v(x, t) with their corresponding diffusion rates D u and D v , governed by the coupled partial differential equations ...
... The differences in the diffusion rates of activator and inhibitor species destabilize the uniform state and lead to spontaneous emergence of periodic spatial patterns. Afterwards, many Turing-type models have been proposed and showed that they are capable of generating patterns observed in mammals [2,4,7], fish [10,11,13], ladybugs [14], bacterial colonies [15,16], phylotaxis [17,18] and many others. Apart from biological patterns, Turing mechanism has been employed to describe pattern formation in semiconductor physics [19,20], star formation in galaxies [21], and scale distribution of galaxies in the universe [22], to mention just a few. ...
We consider a Turing-type reaction-diffusion system involving quadratic and cubic nonlinearities and numerically investigate the role of nonlinear terms in producing spots, stripes, labyrinths, hexagonal arrangement of spots, blotches and transitions among them. From our numerical experiments performed on a square domain with zero-flux boundary conditions, we observe that the system displays a form of multistability for which different stable spatial distribution of concentrations appear for a same set of control parameters depending upon the initial conditions. For varying values of model parameters both in the first- and the second-stage of simulations, we obtain a number of transition states that are found to be sensitive on the relative strength of the quadratic and cubic coupling terms. We obtain a graphical relationship among such model parameters at which the transitions take place.
... Another approach, the reaction-diffusion model, was introduced by Alan Turing in 1952, demonstrating how a reaction between two morphogens diffusing through tissue could result in self-regulating periodic biological patterns [31]. This mechanism has been used to replicate many natural patterns, such as zebrafish stripes [32,33] and seashell patterns [34], but experimental demonstrations of this mechanism in vivo are few and far between, and none identify both diffusible morphogens. The Clock and Wavefront model, proposed by Cooke and Zeeman in 1976, explains the formation of segments in the growing body axis of vertebrate embryos by a biological clock ticking at the posterior of the elongation embryo. ...
In embryonic development and organogenesis, cells sharing identical genetic codes acquire diverse gene expression states in a highly reproducible spatial distribution, crucial for multicellular formation and quantifiable through positional information. To understand the spontaneous growth of complexity, we constructed a one-dimensional division-decision model, simulating the growth of cells with identical genetic networks from a single cell. Our findings highlight the pivotal role of cell division in providing positional cues, escorting the system toward states rich in information. Moreover, we pinpointed lateral inhibition as a critical mechanism translating spatial contacts into gene expression. Our model demonstrates that the spatial arrangement resulting from cell division, combined with cell lineages, imparts positional information, specifying multiple cell states with increased complexity—illustrated through examples in C.elegans. This study constitutes a foundational step in comprehending developmental intricacies, paving the way for future quantitative formulations to construct synthetic multicellular patterns.
... Another approach, the reaction-85 diffusion model, was introduced by Alan Turing in 1952, demonstrating how a reaction between 86 two morphogens diffusing through tissue could result in self-regulating periodic biological patterns 87 31 . This mechanism has been used to replicate many natural patterns, such as zebrafish stripes 32,33 88 and seashell patterns 34 length of a developing segment is dictated by the distance a wavefront advances across the 93 embryonic axis during a clock cycle 35 . While these mathematical models have merit, they have not 94 yet been systematically applied to early development, which involves not only the emergence of 95 patterns but also the reproducible expression of genes at each exact time and place 36 . ...
In embryonic development and organogenesis, cells sharing identical genetic codes acquire diverse gene expression states in a highly reproducible spatial distribution, crucial for multicellular formation and quantifiable through positional information. To understand the spontaneous growth of complexity, we constructed a one-dimensional division-decision model, simulating the growth of cells with identical genetic networks from a single cell. Our findings highlight the pivotal role of cell division in providing positional cues, escorting the system toward states rich in information. Moreover, we pinpointed lateral inhibition as a critical mechanism translating spatial contacts into gene expression. Our model demonstrates that the spatial arrangement resulting from cell division, combined with cell lineages, imparts positional information, specifying multiple cell states with increased complexity—illustrated through examples in C.elegans. This study constitutes a foundational step in comprehending developmental intricacies, paving the way for future quantitative formulations to construct synthetic multicellular patterns.
... The current work provides a simple mathematical model that, while ignoring the specifics, draws out the logical connections in complex glass-forming liquids. The concept is derived from Turing's reactiondiffusion theory [33,34], which explains how ordered patterns spontaneously develop from random complex systems with only an activator and an inhibitor and has been applied in many other fields, such as biology [35][36][37], chemistry [38][39][40][41], and 2D materials [42,43]. In details, the elemental distribution of Cu 50 Zr 50 metallic glass is firstly visualized through continuous patterning, which is then confirmed as a Turing pattern. ...
... In tissues, a source of noise is the stochasticity of gene expression due to low copy numbers of DNA and regulatory molecules (Elowitz et al. 2002). Turing models can successfully explain pattern formation in many biological systems, including zebrafish stripes, mouse palate ridges, mouse limb digits, and tetrapod limb joints (Asai et al. 1999, Economou et al. 2012, Raspopovic et al. 2014, Scoones & Hiscock 2020). Reviews of Turing pattern formation include those by Bailles et al. (2022) and Hiscock & Megason (2015). ...
Multicellular organisms generate tissues of diverse shapes and functions from cells and extracellular matrices. Their adhesion molecules mediate cell-cell and cell-matrix interactions, which not only play crucial roles in maintaining tissue integrity but also serve as key regulators of tissue morphogenesis. Cells constantly probe their environment to make decisions: They integrate chemical and mechanical information from the environment via diffusible ligand- or adhesion-based signaling to decide whether to release specific signaling molecules or enzymes, to divide or differentiate, to move away or stay, or even whether to live or die. These decisions in turn modify their environment, including the chemical nature and mechanical properties of the extracellular matrix. Tissue morphology is the physical manifestation of the remodeling of cells and matrices by their historical biochemical and biophysical landscapes. We review our understanding of matrix and adhesion molecules in tissue morphogenesis, with an emphasis on key physical interactions that drive morphogenesis.
Expected final online publication date for the Annual Review of Cell and Developmental Biology, Volume 39 is October 2023. Please see http://www.annualreviews.org/page/journal/pubdates for revised estimates.
... The current work provides a simple mathematical model that, while ignoring the specifics, draws out the logical connections in complex glass-forming liquids. The concept is derived from Turing's reaction-diffusion theory [33,34], which explains how ordered patterns spontaneously develop from random complex systems with only an activator and an inhibitor and has been applied in many other fields, such as biology [35][36][37], chemistry [38][39][40][41], and 2D materials [42,43]. In details, the elemental distribution of Cu 50 Zr 50 metallic glass is firstly visualized through continuous patterning, which is then confirmed as a Turing pattern. ...
The formation of bulk metallic glass requires the constituent elements to have a negative heat of mixing but has no restrictions on its magnitude. An understanding of this issue is lacking due to the absence of a valid method for describing chemical ordering of metallic glasses. For example, the radial distribution function is ineffective in identifying the elemental preferences of packed atoms. Here, we show that using molecular-dynamics simulation, the chemical medium-range ordering of liquid alloys can be evaluated from persistent homology. This inherently arising chemical medium-range order in metallic glasses is exclusively regulated by the activation and inhibition of the constituent components, making the topology of metallic glasses a Turing pattern. The connecting schemes of atoms of the same species form three distinct regions, reflecting different correlations at the short and medium length scales, while the difference in the schemes corresponds to chemical ordering. By changing the elemental types, it is demonstrated that the chemical medium-range order strongly depends on the relative depth of the interatomic-potential wells. The study separates metallic glasses from crystals under the condition of negative heat of mixing by emphasizing their fundamental difference in interatomic potentials.
... De l'ensemble de ces théories de l'auto-organisation par des réactions couplées à des phénomènes de transport de la matière s'est développé un gigantesque champ de recherches en biologie théorique [Nicolis, 1977 ;Bard Jonathan, 1981 ;Meinhardt, 1982 ;Harrison, 1987 ;Murray, 1990 ;Helbing, 1997 ;Asai, 1999 ;Sept, 1999 ;Glade, 2002 ;Maini, 2003 ;Suzuki, 2003] : ces phénomènes permettent d'expliquer à moindre coût, c'est-à-dire sans besoin d'un contrôle complexe, la formation de motifs naturels comme les taches et les zébrures des animaux, mais aussi certaines structuration de populations comme les insectes sociaux, ou encore l'auto-organisation de microtubules in vitro [Cohen, 1971 ;Tyson, 1989 ;Goss, 1989 ;Mandelkow, 1989 ;Beckers, 1992 ;Kondo, 1995 ;Bonnabeau, 1999]. ...
[FR] Les origines de la vie, la nature même de la matière vivante et son activité biologique, les représentations théoriques que nous en faisons pour la modéliser, l’appréhender, la contrôler, et les conséquences que l’usage de ces représentations ont sur sur notre façon de penser le vivant sont autant de considérations théoriques, méthodologiques et philosophiques que j’aborde dans ce manuscrit en m’appuyant sur mes recherches expérimentales et théoriques actuelles et passées concernant l’organisation du vivant, ses origines et son évolution.
Cette réflexion et les recherches effectives que nous menons, mes étudiants, collaborateurs et moi, se placent dans le cadre de l’émergence et la complexification de la vie. Les systèmes biologiques depuis les âges prébiotiques (ex : proto-métabolismes, proto-génetique) jusqu’à nos jours (ex : réseaux de régulation génétique, organisations cellulaires, subcellulaires, réseaux de neurones) ont évolué mais dépendent toutefois de systèmes existants. L’étude des propriétés structurelles et dynamiques (complexité, robustesse) mais également évolutives, de systèmes formels comme les réseaux Booléens ou autres systèmes de vie artificielle, et la recherche d’ensembles de systèmes aux propriétés structurelles ou dynamiques communes, permet d’introduire une vision ensembliste des systèmes biologiques et penser l’évolution ou les variations de fonctionnement d’un système biologique comme des trajectoires dans un paysage morphogénétique, un meta-réseau. Le passage, évolutif ou fonctionnel, d’un système à un autre est permis par leurs proximités structurelles et dynamiques. Notre travail utilise de tels systèmes formels en tant qu’abstractions des systèmes biologiques pour étudier (i) comment ils évoluent tout en préservant des caractères et comportements ancestraux, (ii) ce qui détermine l’évolutivité de ces réseaux, leur respective robustesse vis-à-vis de changements structuraux et leur relation avec la complexité structurelle et fonctionnelle, (iii) comment des trajectoires peuvent exister sous contraintes de viabilité dans de tels paysages morphogénétiques, et (iv) comment des systèmes et leurs comportements associés peuvent se combiner au cours de ces évolutions.
L’essai que je propose ici s’inscrit dans le cadre général de la biologie théorique et de la philosophie des sciences. J’y aborde des questions aussi variées que la méthode scientifique, l’humain dans les sciences, l’organisation et le fonctionnement de la vie, la nature des modèles, le contrôle, la finalité du vivant. J’y réfléchis, sur la base des recherches effectives que nous menons, à comment et pourquoi s’éloigner de la pensée mécaniste pour considérer davantage un vivant dont la nature et l’activité, depuis ses origines jusqu’au vivant actuel, sont davantage fondées sur ce que je désigne par rare, faible et amorphe, des qualificatifs qui s’éloignent de la conception que nous en avons, le vivant-machine, et des outils dont nous disposons pour l’étude du monde biologique.
[EN] I address in this manuscript theoretical, methodological and philosophical considerations based on my current and past, experimental and theoretical research on the organisation of life, its origins and its evolution. Of particular interest is how the theoretical representations of the very nature of living matter and the origins of life have consequences when using modeling to understand and control living matter.
These reflection and research are carried out by my students, collaborators and myself in the context of the emergence and complexification of life. Biological systems from prebiotic ages (e.g. proto-metabolisms, proto-genetics) to the present day (e.g. gene regulation networks, cellular and subcellular organisations, neural networks) have evolved but continue to depend on current systems. The study of structural and dynamical properties (complexity, robustness) but also evolutionary properties of formal systems such as Boolean networks or other artificial life systems, and the search for sets of systems with common structural or dynamical properties, allows us to introduce a holistic vision of biological systems and to think of their evolution and variations in terms of trajectories in a morphogenetic landscape, i.e. a meta-network. An evolutionary or functional path from one system to another is allowed by their structural and dynamical proximities. In our work, using such formal systems as abstractions of biological systems, we study (i) how they evolve while preserving ancestral traits and behaviours, (ii) what determines the evolvability of these networks, their respective robustness to structural changes and their relation to structural and functional complexity, (iii) how trajectories can exist under viability constraints in such morphogenetic landscapes, and (iv) how systems and their associated behaviours can combine during these evolutions.
The essay I propose falls within the general framework of theoretical biology and the philosophy of sciences. I try to address questions as various as the scientific methodology, humans in science, the organisation of life, the nature of models, control, purpose in life and many others. While using machine-based formalisms such as Boolean networks, I aim at legitimate the need to move away from a mechanistic thinking and the usual conception of the biological world, i.e. Life as a Machine, and the associated theoretical and experimental tools we use to study it. I actually propose to consider an alternate view of the nature and functioning of living matter - from its origins to the present day - based more on what I call the rare, the weak and the amorphous.
... Turing first proposed that a system with two chemicals, such as an activator and inhibitor, could generate spatial patterns due to the reaction and diffusion between these chemicals from an initial near-homogeneous state in his seminal work on Turing models [1]. Since then, numerous skin patterns found in nature, including the marine angelfish [2], [3], zebrafish [4] and cats [5], have been found to evolve in accordance with this mathematical framework. Similarly, RD processes have been critical to the synthesis of nanostructures [6] and various forms of self-assembly and self-organisation [7], [8]. ...
Reaction-diffusion (Turing) systems are fundamental to the formation of spatial patterns in nature and engineering. These systems are governed by a set of non-linear partial differential equations containing parameters that determine the rate of constituent diffusion and reaction. Critically, these parameters, such as diffusion coefficient, heavily influence the mode and type of the final pattern, and quantitative characterization and knowledge of these parameters can aid in bio-mimetic design or understanding of real-world systems. However, the use of numerical methods to infer these parameters can be difficult and computationally expensive. Typically, adjoint solvers may be used, but they are frequently unstable for very non-linear systems. Alternatively, massive amounts of iterative forward simulations are used to find the best match, but this is extremely effortful. Recently, physics-informed neural networks have been proposed as a means for data-driven discovery of partial differential equations, and have seen success in various applications. Thus, we investigate the use of physics-informed neural networks as a tool to infer key parameters in reaction-diffusion systems in the steady-state for scientific discovery or design. Our proof-of-concept results show that the method is able to infer parameters for different pattern modes and types with errors of less than 10\%. In addition, the stochastic nature of this method can be exploited to provide multiple parameter alternatives to the desired pattern, highlighting the versatility of this method for bio-mimetic design. This work thus demonstrates the utility of physics-informed neural networks for inverse parameter inference of reaction-diffusion systems to enhance scientific discovery and design.
... For incompatible equilibria, the system is homogeneous and compositions are determined by chemical equilibrium (thick orange line). (6) At phase equilibrium eq 4, the partition coefficients can be expressed in terms of the activity coefficients in both phases, γ i I/II , by using eqs 1 and 2 as (7) This expression reveals that partitioning is governed by the composition dependence of activity coefficients γ i I/II in phase separating systems. If all solutes are dilute with respect to the solvent, solutes partition equally with P i = 1 since there is no phase coexistence, that is, ...
The kinetics of chemical reactions are determined by the law of mass action, which has been successfully applied to homogeneous, dilute mixtures. At nondilute conditions, interactions among the components can give rise to coexisting phases, which can significantly alter the kinetics of chemical reactions. Here, we derive a theory for chemical reactions in coexisting phases at phase equilibrium. We show that phase equilibrium couples the rates of chemical reactions of components with their diffusive exchanges between the phases. Strikingly, the chemical relaxation kinetics can be represented as a flow along the phase equilibrium line in the phase diagram. A key finding of our theory is that differences in reaction rates between coexisting phases stem solely from phase-dependent reaction rate coefficients. Our theory is key to interpreting how concentration levels of reactive components in condensed phases control chemical reaction rates in synthetic and biological systems.
... There is a large body of literature focused on the mechanism proposed by Turing to explain the patterns of self-organization during the animal development [3][4][5][6][7][8][9][10][11]. However, the identification of the molecular agents driving Turing patterns remains an unsolved issue in most cases. ...
One of the most surprising mechanisms to explain the symmetry breaking phenomenon linked to pattern formation is known as Turing instabilities. These patterns are self-organising spatial structures resulting from the interaction of at least two diffusive species in specific conditions. The ideas of Turing have been used extensively in the specialised literature both to explain developmental patterns, as well as synthetic biology design. In the present work we study a previously proposed morphogenetic synthetic circuit consisting of two genes controlled by the same regulatory system. The spatial homogeneous version of this simple model presents a rich phase diagram, since it has a saddle-node bifurcation, spirals and limit cycle. Linear stability analysis and numerical simulations of the complete model allow us to determine the conditions for the development of Turing patterns, as well as transient patterns. We found that the parameter region where Turing patterns are found is much smaller than the region where transient patterns occur. We observed that the temporal evolution towards Turing patterns can present one or two different length scales, depending on the initial conditions. Further, we found a parameter region where the persistence time of the transient patterns depends on the distance between the parameters values on which the system is operating and the boundary of Turing patterns. This persistence time has a singularity at a critical distance that gives place to metastable patterns. To the best of our knowledge, transient and metastable patterns associated with Turing instabilities have not been previously reported in morphogenetic models.
... Such models have been used to unravel the minimal principles underlying chemical patterns in non-living [3,4] and living systems. Examples include pattern formation in tissues [5][6][7] or on artificial and cellular membranes [8][9][10][11][12]. In models of such systems, chemical reaction rates and diffusive fluxes are typically considered to be independent [13]. ...
... From Turing's aforementioned seminal article and the study from Gierer & Meinhardt [2], a progressive wealth of reaction-diffusion models were developed, paving the way to become an essential and pivotal concept to understand tissue morphogenesis [3][4][5][6]. The model was extensively used to investigate distributions of morphogens in a variety of tissues and organisms such as Drosophila melanogaster wing imaginal disc [7], chick limb [8] and the stripe pattern of Danio rerio [9], among other examples. ...
The reaction–diffusion model constitutes one of the most influential mathematical models to study distribution of morphogens in tissues. Despite its widespread use, the effect of finite tissue size on model-predicted spatio-temporal morphogen distributions has not been completely elucidated. In this study, we analytically investigated the spatio-temporal distributions of morphogens predicted by a reaction–diffusion model in a finite one-dimensional domain, as a proxy for a biological tissue, and compared it with the solution of the infinite-domain model. We explored the reduced parameter, the tissue length in units of a characteristic reaction–diffusion length, and identified two reaction–diffusion regimes separated by a crossover tissue size estimated in approximately three characteristic reaction–diffusion lengths. While above this crossover the infinite-domain model constitutes a good approximation, it breaks below this crossover, whereas the finite-domain model faithfully describes the entire parameter space. We evaluated whether the infinite-domain model renders accurate estimations of diffusion coefficients when fitted to finite spatial profiles, a procedure typically followed in fluorescence recovery after photobleaching (FRAP) experiments. We found that the infinite-domain model overestimates diffusion coefficients when the domain is smaller than the crossover tissue size. Thus, the crossover tissue size may be instrumental in selecting the suitable reaction–diffusion model to study tissue morphogenesis.
... These models implement different effects depending on the distance from each pigment cell by agent-based models [17,18] and by minimal lattice models [19,20]. Several attempts were made to explain the observed patterns in zebrafish mutants by a general Turing model [21,22]; however, they were not supported experimentally even though there are several paths to cause the expected pattern changes in mutants. ...
Different diffusivities among interacting substances actualize the potential instability of a system. When these elicited instabilities manifest as forms of spatial periodicity, they are called Turing patterns. Simulations using general reaction-diffusion (RD) models demonstrate that pigment patterns on the body trunk of growing fish follow a Turing pattern. Laser ablation experiments performed on zebrafish reveal apparent interactions among pigment cells, which allow for a three-component RD model to be derived. However, the underlying molecular mechanisms responsible for Turing pattern formation in this system remain unknown. A zebrafish mutant with a spotted pattern was found to have a defect in Connexin41.8 (Cx41.8) which, together with Cx39.4, exists in pigment cells and controls pattern formation. Here, molecular-level evidence derived from connexin analyses is linked to the interactions among pigment cells described in previous RD modeling. Channels on pigment cells are generalized as “gates,” and the effects of respective gates were deduced. The model uses partial differential equations (PDEs) to enable numerical and mathematical analyses of characteristics observed in the experiments. Furthermore, the improved PDE model, including nonlinear reaction terms, enables the consideration of the behavior of components realistically.
... Such models have been used to unravel the minimal principles underlying chemical patterns in non-living [3,4] and living systems. Examples include pattern formation in tissues [5][6][7] or on artificial and cellular membranes [8][9][10][11][12]. In models of such systems, chemical reaction rates and diffusive fluxes are typically considered to be independent [13]. ...
The kinetics of chemical reactions are determined by the law of mass action, which has been successfully applied to homogeneous, dilute mixtures. At non-dilute conditions, interactions among the components can give rise to coexisting phases, which can significantly alter the kinetics of chemical reactions. Here, we derive a theory for chemical reactions in coexisting phases at phase equilibrium. We show that phase equilibrium couples the rates of chemical reactions of components with their diffusive exchanges between the phases. Strikingly, the chemical relaxation kinetics can be represented as a flow along the phase equilibrium line in the phase diagram. A key finding of our theory is that differences in reaction rates between coexisting phases stem solely from phase-dependent reaction rate coefficients. Our theory is key to interpret how concentration levels of reactive components in condensed phases control chemical reaction rates in synthetic and biological systems.
... I would like to show the reader an example, even though the animal in question is not butterfly, but fish. This is an experiment-based theoretical paper (Asai et al., 1999), in which the analysis has been done by using reaction-diffusion equations, that is, an activator-inhibitor system. Parameters such as reaction rates and diffusion constants in the equations originate from basic biochemical reaction networks and random mobilities of chemicals in cells. ...
The relationship between people and the cougar Puma concolor and other larger
predators in Canada (brown bears Ursus arctos, black bears Ursus americanus) and Latin
America (jaguars Panthera onca) is examined in a comparative mode for cases in
Kamloops and Vancouver Island, Canada and El Salvador. Literature derived hypotheses
mooting greater sympathy for carnivores from women and young people, greater fear
from women and older people, and greater fear of the larger brown bears and jaguars over
cougars are tested. The results show that the hypotheses are both supported and refuted.
The size of the predator is as important as common ideas about its behaviour, and
troublesome invasive behaviour (mostly from bears) is at least as important as predatory
behaviour (largely from cougars and jaguars). However, consistent with hypotheses,
women and older people were in many instances more afraid of cougars and other
carnivores, and older people were generally less tolerant of carnivores. Differences
included opinions on positive impacts on quality of life and useful functions of cougars
(favoured by men in El Salvador, both men and women in Vancouver Island, and women
in Kamloops), and danger to adults, children and other animals (favoured by women
more than men in El Salvador for all groups, women more than men in Vancouver Island
for children and animals, not adults, and women more than men in Kamloops for adults
and children, not animals). For age, older people in El Salvador were generally less
tolerant than younger people of large carnivores, similar to Vancouver Island, but not
Kamloops, where there were no age-related opinions of the danger from carnivores
towards people. The cougar’s elusive behaviour limited comparisons with bears and
jaguars. Awareness of such issues may assist wildlife conservation.
... I would like to show the reader an example, even though the animal in question is not butterfly, but fish. This is an experiment-based theoretical paper (Asai et al., 1999), in which the analysis has been done by using reaction-diffusion equations, that is, an activator-inhibitor system. Parameters such as reaction rates and diffusion constants in the equations originate from basic biochemical reaction networks and random mobilities of chemicals in cells. ...
Traditional zoogeographical theory focuses on global regions, with less attention to
small spaces shared by people and animals. Two relevant, possibly opposite, research
strands, the statistical analysis of alert/flight distances and the subjective 'actancy' strand
within animal geography are rarely examined as complementary to a new look at microspaces of human/avian relations within the larger theoretical framework of
zoogeography. Alert distances (the distance between a bird and an approaching human
when the bird shows awareness), flight distances (the distance, in such a scenario when
the bird takes flight) and alert periods (the period between first alertness and flight) vary
according to cover structure, elevation and proximity, human and road proximity. This
chapter argues that while these parameters give a generalized contribution to the sharing
of micro spaces, a strand of animal geography that emphasizes the unpredictable,
subjective, individualist behavior of animals (birds in this study) offers a complementary
perspective to the more classical statistical prediction of alert/flight distance-based
analyses. Illustrative examples are four papers published by this author; two on the alert
distances of birds in the green spaces of Stirling, Scotland and Peterborough, Canada;
two papers on the utility of a strand of animal geography (emphasizing avian
individualism) in feeding spaces of Glasgow, Scotland and Peterborough, Canada; and
one paper on the landscape ecology of birds in Ottawa, Canada. It is concluded that for
effective environmental applications, the study of alert and flight distances must
acknowledge bird (intra- and inter-species) adaptation and habituation to human
presence; the lack of such acknowledgment may reduce the validity of distance analysis
in avian studies, with implications for conservation and landscape planning. This
represents an important contribution to the sub-discipline of zoogeography and to its
parent disciplines of zoology and geography.
... Here, differences in iridophore development might drive whether stripes of spots form, much as changes in iridophore development lead to spots in tjp1a mutant D. rerio [52] and a reduced number of stripes in D. nigrofasciatus [17]. Alternatively, changes to the network of interactions among pigment cells could themselves contribute, as inferred for some D. rerio mutants [28,29], and as simulated in models of pattern formation [35,81,82]. Indeed, the quantitative data we provide should help to parameterize agent-based models of pigment pattern development. ...
Vertebrate pigmentation is a fundamentally important, multifaceted phenotype. Zebrafish, Danio rerio, has been a valuable model for understanding genetics and development of pigment pattern formation due to its genetic and experimental tractability, advantages that are shared across several Danio species having a striking array of pigment patterns. Here, we use the sister species D. quagga and D. kyathit, with stripes and spots, respectively, to understand how natural genetic variation impacts phenotypes at cellular and organismal levels. We first show that D. quagga and D. kyathit phenotypes resemble those of wild-type D. rerio and several single locus mutants of D. rerio, respectively, in a morphospace defined by pattern variation along dorsoventral and anteroposterior axes. We then identify differences in patterning at the cellular level between D. quagga and D. kyathit by repeated daily imaging during pattern development and quantitative comparisons of adult phenotypes, revealing that patterns are similar initially but diverge ontogenetically. To assess the genetic architecture of these differences, we employ reduced-representation sequencing of second-generation hybrids. Despite the similarity of D. quagga to D. rerio, and D. kyathit to some D. rerio mutants, our analyses reveal a complex genetic basis for differences between D. quagga and D. kyathit, with several quantitative trait loci contributing to variation in overall pattern and cellular phenotypes, epistatic interactions between loci, and abundant segregating variation within species. Our findings provide a window into the evolutionary genetics of pattern-forming mechanisms in Danio and highlight the complexity of differences that can arise even between sister species. Further studies of natural genetic diversity underlying pattern variation in D. quagga and D. kyathit should provide insights complementary to those from zebrafish mutant phenotypes and more distant species comparisons.
... Here, differences in iridophore development might drive whether stripes of spots form, much as changes in iridophore development lead to spots in tjp1a mutant D. rerio [51] and a reduced number of stripes in D. nigrofasciatus [16]. Alternatively, changes to the network of interactions among pigment cells could themselves contribute, as inferred for some D. rerio mutants [27,28], and as simulated in models of pattern formation [34,80,81]. Indeed, the quantitative data we provide should help to parameterize agent-based models of pigment pattern development. ...
Vertebrate pigmentation is a fundamentally important, multifaceted phenotype. Zebrafish, Danio rerio , has been a valuable model for understanding genetics and development of pigment pattern formation due to its genetic and experimental tractability, advantages that are shared across several Danio species having a striking array of pigment patterns. Here, we use the sister species D. quagga and D. kyathit , with stripes and spots, respectively, to understand how natural genetic variation impacts phenotypes at cellular and organismal levels. We first show that D. quagga and D. kyathit phenotypes resemble those of wild-type D. rerio and several single locus mutants of D. rerio , respectively, in a morphospace defined by pattern variation along dorsoventral and anteroposterior axes. We then identify differences in patterning at the cellular level between D. quagga and D. kyathit by repeated daily imaging during pattern development and quantitative comparisons of adult phenotypes, revealing that patterns are similar initially but diverge ontogenetically. To assess the genetic architecture of these differences, we employ reduced-representation sequencing of second-generation hybrids. Despite the similarity of D. quagga to D. rerio , and D. kyathit to some D. rerio mutants, our analyses reveal a complex genetic basis for differences between D. quagga and D. kyathit , with several quantitative trait loci contributing to variation in overall pattern and cellular phenotypes, epistatic interactions between loci, and abundant segregating variation within species. Our findings provide a window into the evolutionary genetics of pattern-forming mechanisms in Danio and highlight the complexity of differences that can arise even between sister species. Further studies of natural genetic diversity underlying pattern variation in D. quagga and D. kyathit should provide insights complementary to those from zebrafish mutant phenotypes and more distant species comparisons.
Author Summary
Pigment patterns of fishes are diverse and function in a wide range of behaviors. Common pattern themes include stripes and spots, exemplified by the closely related minnows Danio quagga and D. kyathit , respectively. We show that these patterns arise late in development owing to alterations in the development and arrangements of pigment cells. In the closely related model organism zebrafish ( D. rerio ) single genes can switch the pattern from stripes to spots. Yet, we show that pattern differences between D. quagga and D. kyathit have a more complex genetic basis, depending on multiple genes and interactions between these genes. Our findings illustrate the importance of characterizing naturally occuring genetic variants, in addition to laboratory induced mutations, for a more complete understanding of pigment pattern development and evolution.
... Thompson's work inspired continued theoretical work, and in 1952, the modeling of pattern formation took an unexpected turn when Alan Turing demonstrated using a simple mathematical model that reaction and diffusion of two interacting chemical substances (morphogens) at different diffusion rates can induce the spontaneous symmetry breaking of a homogeneous state, leading to patterns [2]. Turing reaction-diffusion system, belonging to the class of nonlinear dissipative system, is one of the successful frames for generating and understanding a great variety of structures in nature from a physicochemical point of view with examples as diverse as biological [3][4][5][6][7][8][9][10][11], chemical [12][13][14][15], and physical systems [16][17][18]. ...
We perform numerical simulations beyond Turing space in an activator–inhibitor system involving quadratic and cubic nonlinearities . We show that while all the three fixed points of the system are stable nodes, it exhibits spatially stable patterns as diverse as labyrinths, worms, negatons, and combination of them. The transition among the patterns is found to be dependent on the relative strength (h) of quadratic and cubic couplings. The labyrinths and worms are formed for small values of h while stable negatons are obtained for \(0.0891\le h\le 0.1106\). The negatons start showing decaying behavior for \(h\ge 0.1135\) and, finally they vanish at random positions by emitting remnant solitary waves, yielding a pattern of stable concentric rings. The spatial extension of the concentric rings is found to depend on the initial concentration profile of the decaying negatons. The resulting concentric rings do not decay further due to limitation of numerical precision. We also find that the transient period for each pattern also depends on h.
... In 1952, Turing proposed a simple model that two morphogens diffusing through a tissue could create self-regulating periodic patterns, the reaction-diffusion model [1]. Simulation of these mechanisms replicates many biological pattern types, such as fish stripes, digits, and feather and hair spacing [2][3][4][5][6]. Turing-type reaction-diffusion mechanisms have thus been shown to play a critical role in regulating periodic patterning in organogenesis [3,5]. ...
Periodic patterning of iterative structures is diverse across the animal kingdom. Clarifying the molecular mechanisms involved in the formation of these structure helps to elucidate the process of organogenesis. Turing-type reaction-diffusion mechanisms have been shown to play a critical role in regulating periodic patterning in organogenesis. Palatal rugae are periodically patterned ridges situated on the hard palate of mammals. We have previously shown that the palatal rugae develop by a Turing-type reaction-diffusion mechanism, which is reliant upon Shh (as an inhibitor) and Fgf (as an activator) signaling for appropriate organization of these structures. The disturbance of Shh and Fgf signaling lead to disorganized palatal rugae. However, the mechanism itself is not fully understood. Here we found that Lrp4 (transmembrane protein) was expressed in a complementary pattern to Wise (a secreted BMP antagonist and Wnt modulator) expression in palatal rugae development, representing Lrp4 expression in developing rugae and Wise in the inter-rugal epithelium. Highly disorganized palatal rugae was observed in both Wise and Lrp4 mutant mice, and these mutants also showed the downregulation of Shh signaling, which was accompanied with upregulation of Fgf signaling. Wise and Lrp4 are thus likely to control palatal rugae development by regulating reaction-diffusion mechanisms through Shh and Fgf signaling. We also found that Bmp and Wnt signaling were partially involved in this mechanism.
... The emergence of multicellularity, together with cell differentiation and the ensuing division of labor, conferred unique advantages to the survival of organisms and paved the way for the evolution of patterned complex forms such as those extant today. Among the remarkable diversity of organismal shapes, nearly periodic structures such as digits in a limb [1], sensory bristles in Drosophila [2], palatal ridges [3], and stripes in zebrafish [4] represent a fundamental and ubiquitous motif, suggesting that common mechanisms may be at play behind these structures' morphogenesis. A striking example of nearly periodic developmental patterns is displayed by cyanobacterial Anabaena sp. ...
Author summary
Multicellular organisms, from simple to complex, often undergo a developmental process in which cells differentiate into various types, improving survivability under adverse conditions. We study experimentally and theoretically the developmental mechanism of pattern formation in Anabaena sp. PCC 7120, a multicellular cyanobacterial organism of ancient origin, which forms one-dimensional patterns of single, nitrogen-fixing cells separated by nearly regular intervals of photosynthetic vegetative cells, under nitrogen-poor conditions. By following the developmental process at the level of single cells in real time, we show directly that two genes involved in the inhibition of a nondiffusing activator have different spatiotemporal roles and discuss why a classical, deterministic Turing mechanism may not describe pattern formation in this system. Our stochastic model, which incorporates inevitable fluctuations in molecular numbers or demographic noise, suggests a much more robust mechanism of pattern formation: Noise can seed the formation of transient, stochastic Turing patterns for parameter values in which deterministic patterns do not form. These patterns can then be fixed by downstream genetic commitment pathways. This robust scenario of pattern formation may apply to a wide range of developmental pattern-forming systems.
... The existence of biological patterns has been confirmed by numerical solution of the RD equations by many authors, see for instance Refs. [13][14][15][16][17][18]. ...
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.
... The key features underlying the Turing model have been discovered in the molecular and cellular mechanisms of biological pattern formation. At a cellular-circuit level, biological examples of the Turing model were reported in pattern formation of animal skin (Asai et al., 1999;Kondo and Asal, 1995) and in the process of vertebrate morphogenesis (Economou et al., 2012;M€ uller et al., 2012;Sheth et al., 2012). At a molecularnetwork level, a system consisting of a group of proteins called MinC, MinD, and MinE is known to show typical spatiotemporal patterns explained based on Turing instability (Raskin and de Boer, 1999;Zieske and Schwille, 2013;Loose et al., 2008). ...
Previous autonomous pattern-formation models often assumed complex molecular and cellular networks. This theoretical study, however, shows that a system composed of one substrate with multisite phosphorylation and a pair of kinase and phosphatase can generate autonomous spatial information, including complex stripe patterns. All (de-)phosphorylation reactions are described with a generic Michaelis-Menten scheme, and all species freely diffuse without pre-existing gradients. Computational simulation upon >23,000,000 randomly generated parameter sets revealed the design motifs of cyclic reaction and enzyme sequestration by slow-diffusing substrates. These motifs constitute short-range positive and long-range negative feedback loops to induce Turing instability. The width and height of spatial patterns can be controlled independently by distinct reaction-diffusion processes. Therefore, multisite reversible post-translational modification can be a ubiquitous source for various patterns without requiring other complex regulations such as autocatalytic regulation of enzymes and is applicable to molecular mechanisms for inducing subcellular localization of proteins driven by post-translational modifications.
... The authors [10] proved that when fish are growing, their skin patterns change, and this change in their patterns could be explained by a simple reaction-diffusion system. Based on the molecular mechanisms, it was suggested that leopard gene production is a component of putative reaction-diffusion system [1,9], which also showed that mutations in the zebrafish gene, leopard, changed the pattern from stripes to spots. All the pattern variations of leopard mutations could be generated in a simulation by changing a parameter value that corresponds to the reaction kinetics in a putative reaction-diffusion system. ...
This paper investigates the spatial dynamics of a zebraflsh model with cross-diffusions. Sufficient conditions for Hopf bifurcation and Turing bi- furcation are obtained by analyzing the associated characteristic equation. In addition, we deduce amplitude equations based on multiple-scale analysis, and further by analyzing amplitude equations five categories of Turing patterns are gained. Finally, numerical simulation results are presented to validate the theo- retical analysis. Furthermore, some examples demonstrate that cross-diffusions have an effect on the selection of patterns, which explains the diversity of ze- braflsh pattern very well.
Background:
Connexin 39.4 (Cx39.4) is involved in zebrafish (Danio rerio) skin patterning; when mutated, zebrafish exhibit a wavy stripe/labyrinth pattern instead of stripes. Cx39.4 is unique in that it has two additional serine/arginine (SR) residues, Ser2 and Arg3, at positions 2 and 3. Here, I investigated the role of these SR residues in Cx39.4 function.
Methods:
To examine the SR residues in Cx39.4, mutants of the SR residues were generated. Voltage-clamp recordings were performed using Xenopus oocytes to characterize the channel properties of the mutants. Transgenic zebrafish expressing each mutant were generated, and the effects of each mutation on fish skin patterning were evaluated.
Results:
The Cx39.4R3K mutant showed essentially the same properties as the wild-type (Cx39.4WT) in both electrophysiological analyses, leading to transgenic, complete phenotype rescue. Both the Cx39.4R3A mutant and deletion mutant of SR residues (Cx39.4delSR) showed a faster decay of gap junction activity and abnormal hemichannel activity, resulting in wide stripes and interstripes that indicate instability. Although the Cx39.4R3D mutant showed no channel activity in gap junctions or hemichannels, it caused unstable phenotypes in the transgene, namely a completely rescued phenotype in some individuals and loss of melanophores in others.
Conclusions:
The SR residues in the NT domain of Cx39.4 are critical for the regulation of channel function, which appears to determine skin patterning.
General significance:
These results elucidate the roles of the two SR residues unique to the NT domain of Cx39.4 in its channel function, which is important for zebrafish stripe pattern formation.
Seed is the offspring of angiosperms. Plants produce large numbers of seeds to ensure effective reproduction and survival in varying environments. Ovule is a fundamentally important organ and is the precursor of the seed. In Arabidopsis and other plants characterized by multi‐ovulate ovaries, ovule initiation determines the maximal ovule number, thus greatly affecting seed number per fruit and seed yield. Investigating the regulatory mechanism of ovule initiation has both scientific and economic significance. However, the genetic and molecular basis underlying ovule initiation remains unclear due to technological limitations. Very recently, rules governing the multiple ovules initiation from one placenta have been identified, the individual functions and crosstalk of phytohormones in regulating ovule initiation have been further characterized, and new regulators of ovule boundary are reported, therefore expanding the understanding of this field. In this review, we present an overview of current knowledge in ovule initiation and summarize the significance of ovule initiation in regulating the number of plant offspring, as well as raise insights for the future study in this field that provide potential routes for the improvement of crop yield.
Background:
Periodic patterning of iterative structures is diverse across the animal kingdom. Clarifying the molecular mechanisms involved in the formation of these structures helps to elucidate the genetic commonality of developmental processes, as organs with these structures are believed to share the same molecular mechanisms and fundamental processes. Palatal rugae are periodic corrugated structures on the hard palate and are conserved in all mammals. Although the numbers and patterns of the palatal rugae are species specific, they are consistent in each mammalian species, except humans.
Highlight:
Palatal rugae development is thus under strict genetic control in most mammals and is an excellent model to investigate the genetic commonality of developmental processes to form periodic patterning.
Conclusion:
This review highlights the current understanding of the molecular mechanisms of palatal rugae development.
The study of zebrafish skin pattern development could lead to a better understanding of how these patterns are generated and how they evolved. To compare and contrast wild-type (WT) striped and leopardt1 mutant spotted patterns, photographs were taken of the developing fish. Initial observations led to the hypothesis that the black melanocyte spots in leopardt1 mutants were not randomly distributed, but rather were located in "dashed" stripes. To test this, melanocyte-spot-sized transparent grids were overlaid onto photographs and the location of melanocyte clusters was recorded. The grid maps were used to identify whether a black, melanocyte positive, grid area was present adjacent to each melanocyte cluster in each cardinal and intercardinal direction. In addition, Python-based computer programs were used to analyze the photographs at the pixel level. When analyzed using analysis of variance and logistic regression models, the striped and spotted patterns expressed more similarities than expected. In the leopardt1 zebrafish, the spots were organized into dashed stripes that had similar locations to the WT stripes. This research suggests that spotted and striped patterns are related. Further, the leopardt1 spots were farther apart along the dorsal-ventral axis than in the anterior-posterior direction, suggesting that different mechanisms control spacing along these two axes.
Periodic patterning of iterative structures is a fundamental process during embryonic development, since these structures are diverse across the animal kingdom. Therefore, elucidating the molecular mechanisms in the formation of these structures promotes understanding of the process of organogenesis. Periodically patterned ridges, palatal rugae (situated on the hard palate of mammals), are an excellent experimental model to clarify the molecular mechanisms involved in the formation of periodic patterning of iterative structures. Primary cilia are involved in many biological events, including the regulation of signaling pathways such as Shh and non-canonical Wnt signaling. However, the role of primary cilia in the development of palatal rugae remains unclear. We found that primary cilia were localized to the oral cavity side of the interplacode epithelium of the palatal rugae, whereas restricted localization of primary cilia could not be detected in other regions. Next, we generated mice with a placodal conditional deletion of the primary cilia protein Ift88, using ShhCre mice (Ift88 fl/fl;ShhCre). Highly disorganized palatal rugae were observed in Ift88 fl/fl;ShhCre mice. Furthermore, by comparative in situ hybridization analysis, many Shh and non-canonical Wnt signaling-related molecules showed spatiotemporal expression patterns during palatal rugae development, including restricted expression in the epithelium (placodes and interplacodes) and mesenchyme. Some of these expression were found to be altered in Ift88 fl/fl;ShhCre mice. Primary cilia is thus involved in development of palatal rugae.
Biological and biochemical systems are manifestations of chemical reaction networks (CRNs). The ability to design and engineer such networks may allow the construction of artificial systems that are as complex as those seen in biology, opening the way to translational possibilities including adaptive materials. One venue for progress is the design of autonomous systems capable of pattern generation; however many synthetic CRNs, such as the Belousov-Zhabotinsky reaction, cannot be rewired to encode more complex interactions and thus lack the capacity for more detailed engineering algorithms. In contrast, DNA is an information-rich molecule with predictable and reliable base-pairing interactions and well-studied kinetics, and the use of DNA has greatly enabled the rational design of much more complex synthetic CRNs. Recent advances in the DNA computing field include circuits for pattern transformation, an example of self-organization. An arsenal of tools for designing DNA circuits to implement various CRNs has been developed by DNA nanotechnologists, including software to reliably program strand-displacement nucleic acid circuits. In addition, DNA walkers can be used to create CRNs with controlled diffusivity, while DNA gels similarly represent a new medium for implementing CRNs that may ultimately lead to the development of smart materials. As we will argue, future endeavors in nucleic acid-based pattern generation will be most greatly advanced by harnessing well-known enzymatic processes to serve as generators and amplifiers. Once nucleic acid computing tools are further developed to expedite the design process of pattern generation, we anticipate a transition from proof-of-concept curiosities to application-driven inquiries.
The striped pigmentation pattern of zebrafish is determined by the interaction between pigment cells with different colors. Recent studies show the behaviors of pigment cells are substantially different according to the environment. Interestingly, the resulting patterns are almost identical, suggesting a robustness of the patterning mechanism. To know how this robustness originates, we investigated the behavior of melanophores in various environments including different developmental stages, different body positions, and different genetic backgrounds. Normally, when embryonic melanophores are excluded from the yellow stripe region in the body trunk, two different cellular behaviors are observed. Melanophores migrate to join the black stripe or disappear (die) in the position. In environments where melanophore migration was restricted, we observed that most melanophores disappeared in their position, resulting in the complete exclusion of melanophores from the yellow stripe. In environments where melanophore cell death was restricted, most melanophores migrated to join the black stripes, also resulting in complete exclusion. When both migration and cell death were restricted, melanophores remained alive in the yellow stripes. These results show that migration and cell death complement each other to achieve the exclusion of melanophores. This flexibility may be the basis of the mechanistic robustness of skin pattern formation. In this study, we observed the process of melanophore exclusion in different developmental stages, different body positions, and different genetic backgrounds. Experimental results suggest that two alternative methods of melanophore exclusion contribute to the flexibility of the patterning mechanism.
Gap junctions are intercellular channels that allow passage of ions and small molecules between adjacent cells. Gap junctions in vertebrates are composed of connexons, which are an assembly of six proteins, connexins. Docking of two connexons on the opposite cell surfaces forms a gap junction between the cytoplasm of two neighboring cells. Connexins compose a family of structurally related four-pass transmembrane proteins. In mammals, there are ~20 connexins, each of which contributes to unique permeability of gap junctions, and mutations of some connexin-encoding genes are associated with human diseases. Zebrafish has been predicted to contain 39 connexin-encoding genes; the high number can be attributed to gene duplication during fish evolution, which resulted in diversified functions of gap junctions in teleosts. The determination of body shapes and skin patterns in animal species is an intriguing question. Mathematical models suggest principle mechanisms explaining the diversification of animal morphology. Recent studies have revealed the involvement of gap junctions in fish morphological diversity, including skin pattern formation and body shape determination. This review focuses on connexins in teleosts, which are integrated in the mathematical models explaining morphological diversity of animal skin patterns and body shapes.
Mutations causing a visible phenotype in the adult serve as valuable visible genetic markers in multicellular genetic model organisms such as Drosophila melanogaster, Caenorhabditis elegans and Arabidopsis thaliana. In a large scale screen for mutations affecting early development of the zebrafish, we identified a number of mutations that are homozygous viable or semiviable. Here we describe viable mutations which produce visible phenotypes in the adult fish. These predominantly affect the fins and pigmentation, but also the eyes and body length of the adult. A number of dominant mutations caused visible phenotypes in the adult fish. Mutations in three genes, long fin, another long fin and wanda affected fin formation in the adult. Four mutations were found to cause a dominant reduction of the overall body length in the adult. The adult pigment pattern was found to be changed by dominant mutations in wanda, asterix, obelix, leopard, salz and pfeffer. Among the recessive mutations producing visible phenotypes in the homozygous adult, a group of mutations that failed to produce melanin was assayed for tyrosinase activity. Mutations in sandy produced embryos that failed to express tyrosinase activity. These are potentially useful for using tyrosinase as a marker for the generation of transgenic lines of zebrafish.
The reliable development of highly complex organisms is an intriguing and fascinating problem. The genetic material is, as a rule, the same in each cell of an organism. How then do cells, under the influence of their common genes, produce spatial patterns? Simple models are discussed that describe the generation of patterns out of an initially nearly homogeneous state. They are based on nonlinear interactions of at least two chemicals and on their diffusion. The concepts of local autocatalysis and of long-range inhibition play a fundamental role. Numerical simulations show that the models account for many basic biological observations such as the regeneration of a pattern after excision of tissue or the production of regular (or nearly regular) arrays of organs during (or after) completion of growth. Very complex patterns can be generated in a reproducible way by hierarchical coupling of several such elementary reactions. Applications to animal coats and to the generation of polygonally shaped patterns are provided. It is further shown how to generate a strictly periodic pattern of units that themselves exhibit a complex and polar fine structure. This is illustrated by two examples: the assembly of photoreceptor cells in the eye of Drosophila and the positioning of leaves and axillary buds in a growing shoot. In both cases, the substructures have to achieve an internal polarity under the influence of some primary pattern-forming system existing in the fly's eye or in the plant. The fact that similar models can describe essential steps in organisms as distantly related as animals and plants suggests that they reveal some universal mechanisms.
Models of pattern formation and possible molecular realizations are discussed and compared with recent experimental observations. In application to the dorsoventral patterning of insects, it is shown that a superposition of two pattern-forming reactions is required. The first system generates the overall dorsoventral polarity of the oocyte, the second generates the positional information proper with a stripe-like region of high concentration along the ventral side of the embryo. A single reaction would be insufficient since the two reactions require different parameters. The model accounts for the orientation of the DV axes of the oocytes in the ovary of Musca domestica and Sarcophaga, independent of the DV axis of the mother, for the formation of several ventral furrows in the absence of the primary gurken/torpedo system in Drosophila, as well as for the good size regulation of the dorsoventral axis as observed in some insect species.
Segregation of a homogeneous cell population into different cell types requires autocatalytic processes that saturate at relatively low concentrations and nondiffusible substances responsible for the autocatalytic feedback loops. Thus, these loops can be realized directly on the gene level via their gene products, for instance, by the mutual repression of two genes. A balance of the two cell types is achieved by a long-ranging substance interfering with the self-enhancing process. This substance is expected to have a more or less homogeneous distribution. This model accounts for the reestablishment of the correct proportion after an experimental interference and the change of determination after transplantation. Applications to the segregation of prestalk and prespore cells in Dictyostelium and of neuroblast cells from the ventral ectoderm in Drosophila are provided.
Neural crest development involves cell-fate specification, proliferation, patterned cell migration, survival and differentiation. Zebrafish neural crest derivatives include three distinct chromatophores, which are well-suited to genetic analysis of their development. As part of a large-scale mutagenesis screen for embryonic/early larval mutations, we have isolated 285 mutations affecting all aspects of zebrafish larval pigmentation. By complementation analysis, we define 94 genes. We show here that comparison of their phenotypes permits classification of these mutations according to the types of defects they cause, and these suggest which process of neural crest development is probably affected. Mutations in eight genes affect the number of chromatophores: these include strong candidates for genes necessary for the processes of pigment cell specification and proliferation. Mutations in five genes remove part of the wild-type pigment pattern, and suggest a role in larval pigment pattern formation. Mutations in five genes show ectopic chromatophores in distinct sites, and may have implications for chromatophore patterning and proliferation. 76 genes affect pigment or morphology of one or more chromatophore types: these mutations include strong candidates for genes important in various aspects of chromatophore differentiation and survival. In combination with the embryological advantages of zebrafish, these mutations should permit cellular and molecular dissection of many aspects of neural crest development.
It is suggested that a system of chemical substances, called morphogens, reacting together and diffusing through a tissue, is adequate to account for the main phenomena of morphogenesis. Such a system, although it may originally be quite homogeneous, may later develop a pattern or structure due to an instability of the homogeneous equilibrium, which is triggered off by random disturbances. Such reaction-diffusion systems are considered in some detail in the case of an isolated ring of cells, a mathematically convenient, though biologically unusual system. The investigation is chiefly concerned with the onset of instability. It is found that there are six essentially different forms which this may take. In the most interesting form stationary waves appear on the ring. It is suggested that this might account, for instance, for the tentacle patterns on Hydra and for whorled leaves. A system of reactions and diffusion on a sphere is also considered. Such a system appears to account for gastrulation. Another reaction system in two dimensions gives rise to patterns reminiscent of dappling. It is also suggested that stationary waves in two dimensions could account for the phenomena of phyllotaxis. The purpose of this paper is to discuss a possible mechanism by which the genes of a zygote may determine the anatomical structure of the resulting organism. The theory does not make any new hypotheses; it merely suggests that certain well-known physical laws are sufficient to account for many of the facts. The full understanding of the paper requires a good knowledge of mathematics, some biology, and some elementary chemistry. Since readers cannot be expected to be experts in all of these subjects, a number of elementary facts are explained, which can be found in text-books, but whose omission would make the paper difficult reading.
IN 1952, Turing proposed a hypothetical molecular mechanism, called the reaction–diffusion system, which can develop periodic patterns from an initially homogeneous state. Many theoretical models based on reaction–diffusion have been proposed to account for patterning phenomena in morphogenesis, but, as yet, there is no conclusive experimental evidence for the existence of such a system in the field of biology. The marine angel fish, Pomacanthus has stripe patterns which are not fixed in their skin. Unlike mammal skin patterns, which simply enlarge proportionally during their body growth, the stripes of Pomacanthus maintain the spaces between the lines by the continuous rearrangement of the patterns. Although the pattern alteration varies depending on the conformation of the stripes, a simulation program based on a Turing system can correctly predict future patterns. The striking similarity between the actual and simulated pattern rearrangement strongly suggests that a reaction–diffusion wave is a viable mechanism for the stripe pattern of Pomacanthus.
1.
The genetics of some colour breeds was investigated. One gene is responsible for the degree of pigmentation of the melanophores (normal pigmentation versus slight pigmentation); another one controls the arrangement of the pigment cells (longitudinal stripes-spot pattern respectively); and a third one regulates the destruction of all the xanthophores and some of the iridophores and melanophores.
2.
The melanophores of the longitudinal stripes and those of the spots lie in the uppermost subcutaneous layer. More melanophores were found in the epidermis above and below the scales; with others making direct contact with the scales or occurring in the loose dermis within the scale pockets.
3.
The ontogenetic sequence of pattern formation in the phenotypesre, fr andgr was investigated.
4.
The reduced pigmentation of melanophores of the light formmlr could not be explained by a lack of chromogen (Dopa) but was found to be based upon the intrinsic formation of numerous intermediary weakly pigmented “melanosome” within these cells.
5.
A phenocopy of themlr type could be obtained by treatment with phenylthiourea (0.002–0.004%).
6.
Factors within the subcutis were proved responsible for the formation of the melanophore stripes. The stability of these factors was demonstrated by auto- and isotransplantation.
7.
Melanoblasts could be found in the xanthophore stripes of the anal fin of the wild form.
8.
The number of xanthophores in the anal fin of the phenotypesre andmlr is nearly twice as high as that of the phenotypefr. ♂♂ and ♀♀ always show the same number of xanthophores.
9.
In the xanthophores offr-animals melanosome-like structures were found which do not occur in there-phenotypes.
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.
It is suggested that a system of chemical substances, called morphogens, reacting together and diffusing through a tissue,
is adequate to account for the main phenomena of morphogenesis. Such a system, although it may originally be quite homogeneous,
may later develop a pattern or structure due to an instability of the homogeneous equilibrium, which is triggered off by random
disturbances. Such reaction-diffusion systems are considered in some detail in the case of an isolated ring of cells, a mathematically
convenient, though biologically unusual system. The investigation is chiefly concerned with the onset of instability. It is
found that there are six essentially different forms which this may take. In the most interesting form stationary waves appear
on the ring. It is suggested that this might account, for instance, for the tentacle patterns onHydra and for whorled leaves. A system of reactions and diffusion on a sphere is also considered. Such a system appears to account
for gastrulation. Another reaction system in two dimensions gives rise to patterns reminiscent of dappling. It is also suggested
that stationary waves in two dimensions could account for the phenomena of phyllotaxis.
The purpose of this paper is to discuss a possible mechanism by which the genes of a zygote may determine the anatomical structure
of the resulting organism. The theory does not make any new hypotheses; it merely suggests that certain well-known physical
laws are sufficient to account for many of the facts. The full understanding of the paper requires a good knowledge of mathematics,
some biology, and some elementary chemistry. Since readers cannot be expected to be experts in all of these subjects, a number
of elementary facts are explained, which can be found in text-books, but whose omission would make the paper difficult reading.
Zebrafish genes and development are being studied in a growing number of laboratories. Given that many other organisms are already being exploited by large numbers of investigators, and that our general knowledge about the zebrafish embryo and genome is at present rather sketchy, why should we now concern ourselves with how this tropical fish develops? Whereas the zebrafish embryo is similar in important ways to other vertebrate embryos, it is relatively simple and unusually accessible for both cellular and genetic analyses.
The paper describes a model of competition that explains the formation of the ocular dominance stripes found in layer IVc of cat and monkey visual cortex. The main proposal is that synapses exert effects on the growth of other synapses, and that these effects extend over distances of at least 600 mu m and vary in magnitude and sign within this distance. Interactions between like type synapses are assumed to be stimulating for distances up to about 200 mu m, and inhibitory for distances of 200-600 mu m. The reverse is true of interactions between synapses of opposite eye type, where the effects are inhibitory for distances up to about 200 mu m and stimulating for longer ones. The interactions are assumed to be circularly symmetric. Growth of, for example, right eye synapses at one point will therefore (a) encourage local growth of right eye synapses and inhibit local growth of left eye synapses and (b) encourage growth of left eye synapses and inhibit growth of right eye synapses in an annular ring surrounding the point of initial increase. At the start of development, right and left eye synapses are assumed to be intermixed randomly within layer IVc. Computer simulations show that a wide variety of conditions incorporating these assumptions will lead to the formation of stripe patterns. These reproduce many of the morphological features of monkey ocular dominance stripes, including Y- and H-type branches and terminations, the tendency for stripes to run at right angles into the boundaries of the pattern, and to narrow at branch points. The model can explain the effects of monocular deprivation on stripe morphology if it is assumed that the effectiveness of deprived eye synapses in determining rates of growth locally is reduced. The existence of a critical period for these effects can be explained if it is assumed that lateral growth of terminals does not occur, and that some factor such as a limited availability of postsynaptic sites decreases the rate of growth of synapses as their density approaches a maximum. The model can be generalized to account for pattern formation in other systems, such as zebra or mackerel skin, where similar striped patterns occur. In this context, the simplest model based on diffusion that will produce a pattern of stripes requires that one cell type should secrete two substances, one of which stimulates growth or differentiation of its parent cell type and has a low rate of diffusion or is rapidly inactivated, and another that inhibits growth or differentiation and either has a higher rate of diffusion or is less rapidly inactivated. A preliminary account of some of these results has appeared elsewhere (Swindale 1979).
Adult zebrafish stripes are formed from stripes of gold iridophores alternating with stripes of black melanocytes lying beneath silver stripes. Analysis of defects in pigment pattern development caused by sparse (spa, rose (ros), and leopard (leo) single and double mutant combinations suggests that spa+ and ros+ functions are required for development of separate populations of pigment cells in the adult and that leo+ functions to control assembly of melanocytes into stripes. Thus, between 2 and 3 weeks of zebrafish development, spa-dependent melanocytes differentiate throughout the flank, followed by leo-dependent assembly of these cells into stripes. Beginning at 3 weeks of development, a distinct ros-dependent population of melanocytes differentiates in the stripe. Both early and late differentiating melanocytes then affect the formation of the silver stripes, ensuring registration of melanocyte and iridophore stripes.
In Drosophila melanogaster and Caenorhabditis elegans, the elucidation of developmental mechanisms has relied primarily on the systematic induction and isolation of mutations in genes with specific functions in development. Such an approach has not yet been possible in a vertebrate species, owing to the difficulty of analyzing and keeping a sufficiently high number of mutagenized lines of animals.
We have developed the methods necessary to perform large-scale saturation screens for mutations affecting embryogenesis in the zebrafish, Danio (Brachydanio) rerio. Firstly, a new aquarium system was developed to raise and keep large numbers of strains of genetically different fish safely and with little maintenance care. Secondly, by placing adult male fish in water containing the chemical mutagen, ethylnitrosourea, we induced point mutations in premeiotic germ cells with a rate of one to three mutations per locus per 1,000 mutagenized haploid genomes. This rate, which is similar to the mutagenesis rates produced by ethylmethanesulfonate in Drosophila, was determined for alleles at four different pigmentation genes. Finally, in a pilot screen in which mutagenized fish were inbred for two generations and scored for embryonic mutants, we isolated 100 recessive mutations with phenotypes visible in the homozygous embryos.
The high rate of induction and recovery of point mutations, in addition to an efficient aquarium system to house large numbers of mutagenized lines, means that it is now possible to perform saturation mutagenesis screens in a vertebrate, similar to those done in invertebrates.
The induction and isolation of mutations in genes playing important roles in pattern formation have been extremely valuable in elucidating the mechanisms governing pattern formation in invertebrates. The recent establishment of genetic methods in the zebrafish, Brachydanio rerio, has permitted systematic mutational studies of vertebrate embryonic development.
The systematic isolation and characterization of mutants in Drosophila has enormously facilitated the analysis of molecular mechanisms underlying developmental pathways in the embryo. A similar approach is presently being used to study embryonic development of the zebrafish, which is becoming a mainstream model organism for vertebrate development. With its genetic versatility and sophisticated embryology, zebrafish offers the possibility to rapidly increase our knowledge of vertebrate development and add to what we have learned from other vertebrate model organisms.
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