Activator-inhibitor dynamics of vertebrate limb pattern formation

The development of the vertebrate limb depends on an interplay of cellular differentiation, pattern formation, and tissue morphogenesis on multiple spatial and temporal scales. While numerous gene products have been described that participate in, and influence, the generation of the limb skeletal pattern, an understanding of the most salient feature of the developing limb--its quasiperiodic arrangement of bones, requires additional organizational principles. We review several such principles, drawing on concepts of physics and chemical dynamics along with molecular genetics and cell biology. First, a "core mechanism" for precartilage mesenchymal condensation is described, based on positive autoregulation of the morphogen transforming growth factor (TGF)-beta, induction of the extracellular matrix (ECM) protein fibronectin, and focal accumulation of cells via haptotaxis. This core mechanism is shown to be part of a local autoactivation-lateral inhibition (LALI) system that ensures that the condensations will be regularly spaced. Next, a "bare-bones" model for limb development is described in which the LALI-core mechanism is placed in a growing geometric framework with predifferentiated "apical," differentiating "active," and irreversibly differentiated "frozen" zones defined by distance from an apical source of a fibroblast growth factor (FGF)-type morphogen. This model is shown to account for classic features of the developing limb, including the proximodistal (PD) emergence over time of increasing numbers of bones. We review earlier and recent work suggesting that the inhibitory component of the LALI system for condensation may not be a diffusible morphogen, and propose an alternative mechanism for lateral inhibition, based on synchronization of oscillations of a Hes mediator of the Notch signaling pathway. Finally, we discuss how viewing development as an interplay between molecular-genetic and dynamic physical processes can provide new insight into the origin of congenital anomalies.     

On the evolution of morphogenetic models: mechano-chemical interactions and an integrated view of cell differentiation, growth, pattern formation and morphogenesis

In the 1950s, embryology was conceptualized as four relatively independent problems: cell differentiation, growth, pattern formation and morphogenesis. The mechanisms underlying the first three traditionally have been viewed as being chemical in nature, whereas those underlying morphogenesis have usually been discussed in terms of mechanics. Often, morphogenesis and its mechanical processes have been regarded as subordinate to chemical ones. However, a growing body of evidence indicates that the biomechanics of cells and tissues affect in striking ways those phenomena often thought of as mainly under the control of cell-cell signalling. This accumulation of data has led to a revival of the mechano-transduction concept in particular, and of complexity in general, causing us now to consider whether we should retain the traditional conceptualization of development. The researchers' semantic preferences for the terms 'patterning', 'pattern formation' or 'morphogenesis' can be used to describe three main 'schools of thought' which emerged in the late 1970s. In the 'molecular school', the term patterning is deeply tied to the positional information concept. In the 'chemical school', the term 'pattern formation' regularly implies reaction-diffusion models. In the 'mechanical school', the term 'morphogenesis' is more frequently used in relation to mechanical instabilities. Major differences among these three schools pertain to the concept of self-organization, and models can be classified as morphostatic or morphodynamic. Various examples illustrate the distorted picture that arises from the distinction among differentiation, growth, pattern formation and morphogenesis, based on the idea that the underlying mechanisms are respectively chemical or mechanical. Emerging quantitative approaches integrate the concepts and methods of complex sciences and emphasize the interplay between hierarchical levels of organization via mechano-chemical interactions. They draw upon recent improvements in mathematical and numerical morphogenetic models and upon considerable progress in collecting new quantitative data. This review highlights a variety of such models, which exhibit important advances, such as hybrid, stochastic and multiscale simulations.     

Pattern formation in integrative biology--a marriage of theory and experiment

The interdisciplinary challenge to discover the underlying mechanisms in the generation of biological pattern and form are central issues in development. In this review we briefly discuss the philosophy of such an integrative biology approach. We then describe one pattern formation approach which has intimate ties to experiment, namely the mechano-chemical theory. We discuss, by way of example, the successful use of such a framework in the formation of cell-matrix networks, intimately associated with angiogenesis. All of the model parameters are estimated from experiment and the results of the model analysis compare well with experiment. We conclude with some general views on the use of models in biology.     

The role of diffusible molecules in regulating the cellular differentiation of Dictyostelium discoideum

A central problem in developmental biology is to understand how morphogenetic fields are created and how they act to direct regionalized cellular differentiation. This goal is being pursued in organisms as diverse as moulds, worms, flies, frogs and mice. Each organism has evolved its own solution to the challenge of multicellularity but there appear to be common underlying principles and, once pattern formation is fully understood in any system, some general truths seem certain to be revealed. As a non-obligate metazoan, Dictyostelium discoideum has proven a particularly tractable system in which to identify and characterize cellular morphogens. Cyclic AMP and ammonia stimulate prespore cell differentiation and ammonia plays an additional role in repressing terminal cellular differentiation. Differentiation Inducing Factor (DIF) acts to direct prestalk cell differentiation and adenosine may play a synergistic role in repressing prespore cell differentiation. This review summarizes the evidence for these interactions and describes a number of models which show how this small repertoire of diffusible molecules, acting in concert, may direct the formation of a differentiated structure.     

Stripes and belly-spots -- a review of pigment cell morphogenesis in vertebrates

Pigment patterns in the integument have long-attracted attention from both scientists and non-scientists alike since their natural attractiveness combines with their excellence as models for the general problem of pattern formation. Pigment cells are formed from the neural crest and must migrate to reach their final locations. In this review, we focus on our current understanding of mechanisms underlying the control of pigment cell migration and patterning in diverse vertebrates. The model systems discussed here - chick, mouse, and zebrafish - each provide unique insights into the major morphogenetic events driving pigment pattern formation. In birds and mammals, melanoblasts must be specified before they can migrate on the dorsolateral pathway. Transmembrane receptors involved in guiding them onto this route include EphB2 and Ednrb2 in chick, and Kit in mouse. Terminal migration depends, in part, upon extracellular matrix reorganization by ADAMTS20. Invasion of the ectoderm, especially into the feather germ and hair follicles, requires specific signals that are beginning to be characterized. We summarize our current understanding of the mechanisms regulating melanoblast number and organization in the epidermis. We note the apparent differences in pigment pattern formation in poikilothermic vertebrates when compared with birds and mammals. With more pigment cell types, migration pathways are more complex and largely unexplored; nevertheless, a role for Kit signaling in melanophore migration is clear and indicates that at least some patterning mechanisms may be highly conserved. We summarize the multiple factors thought to contribute to zebrafish embryonic pigment pattern formation, highlighting a recent study identifying Sdf1a as one factor crucial for regulation of melanophore positioning. Finally, we discuss the mechanisms generating a second, metamorphic pigment pattern in adult fish, emphasizing recent studies strengthening the evidence that undifferentiated progenitor cells play a major role in generating adult pigment cells.     

干旱半干旱区斑块状植被格局形成模拟研究进展

斑块状植被格局是世界上干旱半干旱区常见的景观类型,它们的形成、组成结构和演替过程研究,对于揭示区域生态系统变化的关键过程具有重要意义。鉴于基于地面调查和遥感技术的方法难以全面刻画斑块状植被格局的形成过程及机制,借助于模型模拟成为解决这一问题的有效方法。自20世纪90年代初至今,斑块状植被格局形成的连续和离散模拟研究不断涌现,然而,连续模拟侧重于植被格局形成的一般机理,缺乏与现实格局的对比和验证,离散模拟单元选择与规则制定等仍需不断研究。在简要回顾斑块状格局形成的反馈机制基础上,重点综述了斑块状植被格局形成的连续和离散模拟的最新研究进展,并指出了现有研究的不足。干旱半干旱区小尺度上植物和水的反馈作用决定了大尺度的斑块状植被格局,充分揭示植被-土壤水分相互作用机理是模型模拟研究的关键,放牧强度和降水格局等外部环境对干旱半干旱区斑块状植被格局特征具有重要影响。在未来研究中,应加强模型模拟结果与实际观测的植被格局比较和验证,重视局域环境条件、生态系统功能在模型中的表达,构建综合连续和离散模型各自优点的混合模型,注重斑块状植被格局形成过程中的标准子模型及模型开发和集成平台的研发,同时强调面向格局模拟和构建空间显式的斑块状植被格局形成模型。     

Reaction and diffusion on growing domains: Scenarios for robust pattern formation

We investigate the sequence of patterns generated by a reaction—diffusion system on a growing domain. We derive a general evolution equation to incorporate domain growth in reaction—diffusion models and consider the case of slow and isotropic domain growth in one spatial dimension. We use a self-similarity argument to predict a frequency-doubling sequence of patterns for exponential domain growth and we find numerically that frequency-doubling is realized for a finite range of exponential growth rate. We consider pattern formation under different forms for the growth and show that in one dimension domain growth may be a mechanism for increased robustness of pattern formation.     

Reaction-Diffusion Patterns in Plant Tip Morphogenesis: Bifurcations on Spherical Caps

We study a chemical reaction-diffusion model (the Brusselator) for pattern formation on developing plant tips. A family of spherical cap domains is used to represent tip flattening during development. Applied to conifer embryos, we model the chemical prepatterning underlying cotyledon (“seed leaf”) formation, and demonstrate the dependence of patterns on tip flatness, radius, and precursor concentrations. Parameters for the Brusselator in spherical cap domains can be chosen to give supercritical pitchfork bifurcations of patterned solutions of the nonlinear reaction-diffusion system that correspond to the cotyledon patterns that appear on the flattening tips of conifer embryos.     

Control of chemical pattern formation by a clock-and-wavefront type mechanism

The segmentation of many animals ranging from insects to mammals involves the sequential formation of stationary stripes of gene expression that are perpendicular to the growth axis of the developing embryo. This process has been accounted for by a variety of theoretical “clock-and-wavefront” type models that involve the arrest of an oscillation (the clock) at a moving boundary (the wavefront). Here, we demonstrate experimentally that progressive arrest of a homogeneous oscillation can control the symmetry as well as the wavelength of spatial structures in a chemical system. We show how a spontaneously formed, labyrinthine pattern can be converted into a pattern composed of ordered, parallel stripes and confirm a previously predicted proportionality between the wavelength and the period of the homogeneous oscillation. Our experiments provide the first experimental demonstration of a general mechanism for the control of pattern formation that has been hypothesized to operate in the context of biological morphogenesis.     

Pigmentation pattern formation in butterflies: experiments and models

Butterfly pigmentation patterns are one of the most spectacular and vivid examples of pattern formation in biology. They have attracted much attention from experimentalists and theoreticians, who have tried to understand the underlying genetic, chemical and physical processes that lead to patterning. In this paper, we present a brief review of this field by first considering the generation of the localised, eyespot, patterns and then the formation of more globally controlled patterns. We present some new results applied to pattern formation on the wing of the mimetic butterfly Papilio dardanus.     

Turing pattern with proportion preservation

Although Turing pattern is one of the most universal mechanisms for pattern formation, in its standard model the number of stripes changes with the system size, since the wavelength of the pattern is invariant. It fails to preserve the proportionality of the pattern, i.e. the ratio of the wavelength to the size, that is often required in biological morphogenesis. To get over this problem, we show that the Turing pattern can preserve proportionality by introducing a catalytic chemical whose concentration depends on the system size. Several plausible mechanisms for such size dependence of the concentration are discussed. Following this general discussion, two models are studied in which arising Turing patterns indeed preserve the proportionality. Relevance of the present mechanism to biological morphogenesis is discussed from the viewpoint of its generality, robustness, and evolutionary accessibility.     

Instabilities and spatiotemporal patterns behind predator invasions with nonlocal prey competition

We study the influence of nonlocal intraspecies prey competition on the spatiotemporal patterns arising behind predator invasions in two oscillatory reaction–diffusion integro-differential models. We use three common types of integral kernels as well as develop a caricature system, to describe the influence of the standard deviation and kurtosis of the kernel function on the patterns observed. We find that nonlocal competition can destabilize the spatially homogeneous state behind the invasion and lead to the formation of complex spatiotemporal patterns, including stationary spatially periodic patterns, wave trains and irregular spatiotemporal oscillations. In addition, the caricature system illustrates how large standard deviation and low kurtosis facilitate the formation of these spatiotemporal patterns. This suggests that nonlocal competition may be an important mechanism underlying spatial pattern formation, particularly in systems where the competition between individuals varies over space in a platykurtic manner.     

Instabilities and spatiotemporal patterns behind predator invasions with nonlocal prey competition

We study the influence of nonlocal intraspecies prey competition on the spatiotemporal patterns arising behind predator invasions in two oscillatory reaction–diffusion integro-differential models. We use three common types of integral kernels as well as develop a caricature system, to describe the influence of the standard deviation and kurtosis of the kernel function on the patterns observed. We find that nonlocal competition can destabilize the spatially homogeneous state behind the invasion and lead to the formation of complex spatiotemporal patterns, including stationary spatially periodic patterns, wave trains and irregular spatiotemporal oscillations. In addition, the caricature system illustrates how large standard deviation and low kurtosis facilitate the formation of these spatiotemporal patterns. This suggests that nonlocal competition may be an important mechanism underlying spatial pattern formation, particularly in systems where the competition between individuals varies over space in a platykurtic manner.     

The long-term evolution of multilocus traits under frequency-dependent disruptive selection

Frequency-dependent disruptive selection is widely recognized as an important source of genetic variation. Its evolutionary consequences have been extensively studied using phenotypic evolutionary models, based on quantitative genetics, game theory, or adaptive dynamics. However, the genetic assumptions underlying these approaches are highly idealized and, even worse, predict different consequences of frequency-dependent disruptive selection. Population genetic models, by contrast, enable genotypic evolutionary models, but traditionally assume constant fitness values. Only a minority of these models thus addresses frequency-dependent selection, and only a few of these do so in a multilocus context. An inherent limitation of these remaining studies is that they only investigate the short-term maintenance of genetic variation. Consequently, the long-term evolution of multilocus characters under frequency-dependent disruptive selection remains poorly understood. We aim to bridge this gap between phenotypic and genotypic models by studying a multilocus version of Levene's soft-selection model. Individual-based simulations and deterministic approximations based on adaptive dynamics theory provide insights into the underlying evolutionary dynamics. Our analysis uncovers a general pattern of polymorphism formation and collapse, likely to apply to a wide variety of genetic systems: after convergence to a fitness minimum and the subsequent establishment of genetic polymorphism at multiple loci, genetic variation becomes increasingly concentrated on a few loci, until eventually only a single polymorphic locus remains. This evolutionary process combines features observed in quantitative genetics and adaptive dynamics models, and it can be explained as a consequence of changes in the selection regime that are inherent to frequency-dependent disruptive selection. Our findings demonstrate that the potential of frequency-dependent disruptive selection to maintain polygenic variation is considerably smaller than previously expected.     

Reaction-diffusion models of growing plant tips: bifurcations on hemispheres

We study two chemical models for pattern formation in growing plant tips. For hemisphere radius and parameter values together optimal for spherical surface harmonic patterns of index l = 3, the Brusselator model gives an 84% probability of dichotomous branching pattern and 16% of annular pattern, while the hyperchirality model gives 88% probability of dichotomous branching and 12% of annular pattern. The models are two-morphogen reaction-diffusion systems on the surface of a hemispherical shell, with Dirichlet boundary conditions. Bifurcation analysis shows that both models give possible mechanisms for dichotomous branching of the growing tips. Symmetries of the models are used in the analysis.     

Proceedings of the fifty-fourth Annual Meeting of the American Association of physical anthropologists to be held in Knoxville,Tennessee—Holiday Inn World's fair April 10–13, 1985

Empirical analyses and models of the lineal effects of fission indicate that considerable genetic differentiation may occur at the time of group formation, thus confusing the usual positive relationship between historical affiliation and genetic differentiation. We analyze the effects of fission pattern on variation in highly heritable morphological traits among eight social groups on Cayo Santiago. The analysis is performed using general network autocorrelation methods that quantitatively and directly measure the amount of variation in social group mean morphology that can be explained by fission. All of the fission autocorrelation coefficients are strongly negative, indicating that groups most recently formed by fission are most dissimilar. Also, most of the variation between groups can be explained by the fission pattern, indicating that lineal fission is the most important process generating between-group variation on Cayo Santiago.     

Motional instabilities in prey-predator systems

Differential fluxes can destabilize the locally stable stationary density distributions in interaction systems with diffusion, advection, and/or locomotion of chemical or biological species. By this method they can cause the formation of stationary or travelling spatial structures. Different scenarios of this general mechanism of spatio-temporal pattern formation in reaction-diffusion-advection systems will be demonstrated, using a simple two-species predator-prey system as an example model.     

A model of primitive streak initiation in the chick embryo

Initiation of the primitive streak in avian embryos provides a well-studied example of a pattern-forming event that displays a striking capacity for regulation. The mechanisms underlying the regulative properties are, however, poorly understood and are not easily accounted for by traditional models of pattern formation, such as reaction-diffusion models. In this paper, we propose a new activator-inhibitor model for streak initiation. We show that the model is consistent with experimental observations, both in its pattern-forming properties and in its ability to form these patterns on the correct time-scales for biologically realistic parameter values. A key component of the model is a travelling wave of inhibition. We present a mathematical analysis of the speed of such waves in both diffusive and juxtacrine relay systems. We use the streak initiation model to make testable predictions. By varying parameters of the model, two very different types of patterning can be obtained, suggesting that our model may be applicable to other processes in addition to streak initiation.     

Study of encapsulation of silicone rubber implants in animals. A foreign-body reaction.

It has long been known that the formation of a fibrous capsule around an implant is a general phenomenon in nature, an inevitable result of the tissue defense mechanism called the foreign body reaction. We have investigated this reaction in animals and find it consists of a series of interrelated processes in which the final result may vary, depending on the susceptibility of the foreign material to phagocytosis, incorporation by giant cells, or isolation by fibrosis. This susceptibility depends, in turn, on the physical and chemical properties of the implant. The process of capsule formation, as well as the structure of the final capsule, is similar in animal models and in humans--so that investigations of human encapsulation may be done in animals. The cause of abnormal induration around human breast implants is still unknown, and the question of whether normal induration in the animal model can be used to elucidate abnormal clinical induration depends on further investigation of both phenomena.