Self-organization in and on biological spheres

Three biological settings involving self-organization performed by the Turing-Child field inside a sphere and on its surface are considered. In the first setting the interior of a sphere made up of cells communicating via gap junctions is considered. It is suggested that the Turing-Child self-organization is the cause of radial polarization, the first differentiation of an early mammalian embryo. In the second setting, the Turing example of gastrulation of a hollow cellular sphere is considered. It is shown that Child's experimental patterns are predicted and explained by the Turing-Child theory. The third setting is the interior of a biological cell, and it is suggested that it is the self-organization of the Turing-Child field that causes the formation of the mitotic spindle.     

Symmetry breaking and convergent extension in early chordate development

The initiation of axis, polarity, cell differentiation, and gastrulation in the very early chordate development is due to the breaking of radial symmetry. It is believed that this occurs by an external signal. We suggest instead spontaneous symmetry breaking through the agency of the Turing-Child field. Increased size or decreased diffusivity, both brought about by mitotic activity, cause the spontaneous loss of stability of the homogeneous state and the evolution of the metabolic pattern during development. The polar metabolic pattern is the cause of polar gene expression, polar morphogenesis (gastrulation), and polar mitotic activity. The Turing-Child theory explains not only the spontaneous formation of the invagination in gastrulation but also the coherent cell movement observed in convergence and extension during gastrulation and neurulation. The theory is demonstrated with respect to experimental observations on the early development of fish, amphibian, and the chick. The theory can explain a multitude of experimental details. For example, it explains the splayed polar progression of reduction in the fish blastoderm. Reduction starts on that side of the blastoderm margin, which will initiate invagination several hours later. It progresses toward the blastoderm center and somewhat laterally from this future "dorsal lip". This is precisely as predicted by a Turing-Child system in a circle. And for a fish like zebrafish with a blastoderm that is slightly oval, reduction is observed to progress along the long axis of the ellipse, which is what Turing-Child theory predicts. In general the shape and the chemical nature of the experimental patterns are the same as predicted by the Turing couple (cAMP, ATP). Embryological polarity and convergent extension are based on polar eigenfunction and saddle-shaped eigenfunction, respectively.     

Self-organization in and on biological spheres

Three biological settings involving self-organization performed by the Turing-Child field inside a sphere and on its surface are considered. In the first setting the interior of a sphere made up of cells communicating via gap junctions is considered. It is suggested that the Turing-Child self-organization is the cause of radial polarization, the first differentiation of an early mammalian embryo. In the second setting, the Turing example of gastrulation of a hollow cellular sphere is considered. It is shown that Child's experimental patterns are predicted and explained by the Turing-Child theory. The third setting is the interior of a biological cell, and it is suggested that it is the self-organization of the Turing-Child field that causes the formation of the mitotic spindle.     

Induction and the Turing-Child field in development

The central problem in biological development is the understanding of epigenesis. The dominant theory of development in the last 80 years that also purports to explain epigenesis is induction theory. It suggests that development is driven by sequential inductions where each "induction" (in one sense of the word induction) is effected by the action of an inducing part of the embryo on a responding part of the embryo. The theory stems from Spemann and Mangold (W.Roux' Arch.f.Entw.d.Organis.u.mikrosk.Anat.100 (1924) 599) who transplanted a tissue from the dorsal blastopore lip of Triturus into the ventral ectoderm of another gastrula and thus initiated and "induced" (in another sense of the word induction) gastrulation and embryogenesis in the ventral side of the host that became a double embryo (siamese twins). We explain this induction, i.e. the formation of the double embryo, according to the Child theory and the Turing-Gierer-Meinhardt theory when it is also assumed that cAMP and ATP are the Turing activator and inhibitor, respectively. Spemann and Mangold (W.Roux' Arch.f.Entw.d.Organis.u.mikrosk.Anat.100 (1924) 599) also suggested that the ingressing mesoderm induces the overlying ectoderm to form the neural plate and neural tube. This 'neural induction', the 'primary embryonic induction', became the cornerstone of induction theory, i.e. of the sequential induction concept referred to above. But we argue that the metabolic gradients that precede and accompany neurulation, as obtained by Child, also for Triturus, arise through a Turing self-organization if it is assumed that cAMP and ATP are the Turing morphogens, and these gradients are the cause and primary event of neurulation. Thus there is no need to invoke the 'neural induction'. It is argued that fundamental events such as gastrulation and also organ formation are caused by the Turing-Child field and not by sequential induction. Similar principles, such as bud formation caused by a radial metabolic pattern that transforms to a longitudinal pattern, govern the formation, for example, of the mouth and the gut. The formation and localization of bottle cells is explained according to the Child-Turing field and modern biochemistry. The chemical metabolic pre-pattern precedes, and causes, morphogenesis and differentiation as envisaged by Turing. The Spemann and Mangold (W.Roux' Arch.f.Entw.d.Organis.u.mikrosk.Anat.100 (1924) 599) transplantation experiment when performed on a sea urchin duplicates not only the phenotype but also the metabolic (reduction) pattern. These experimental results, by Horstadius, predicted by Child, follow from the Turing-Gierer-Meinhardt theory if it is assumed that cAMP and ATP are the Turing morphogens. If the transplantation is performed not onto the whole sea urchin but onto only a part of it, that manifests only a part of the metabolic pattern, then from the part a phenotypic whole underlain by a normal and a whole metabolic pattern can be rescued. These experimental results of Horstadius follow from Turing theory if cAMP and ATP are the Turing morphogens. Understanding how to transform a part into a whole can be valuable in regenerative medicine. Unspecific induction of a secondary amphibian embryo is similar to the induction of posterior structures at the anterior pole of an insect, and the "double abdomen" (and Kalthoff's experimental results) of the midge Smittia resulting from UV irradiation of the anterior pole, can be explained by Meinhardt theory of unspecific induction if ATP is the Turing morphogen. When not working on regeneration, Child investigated intact living organisms and his observation method was not disruptive to normal development, whereas workers in induction theory work with pieces and in general disrupt normal development. We conclude that the Turing-Child field causes all development and explains epigenesis. Sequential induction does not explain epigenesis and does not exist in normal development. But induction in the sense of a transplantation leading to double embryo or rescuing a whole phenotype from a part is of interest.     

Segmentation and zooid formation in animals with a posterior growing region: the case for metabolic gradients and Turing waves

The discovery of periodic propagation of anteriorly moving pulses/stripes of gene expression in the presomitic mesoderm (PSM) of vertebrates has given new life to the clock and wavefront model, and other models of morphogenesis based on a molecular oscillator where the time periodicity is translated into spatial periodicity. Instead we suggest that segmentation, somitogenesis and metamerism in vertebrates and in invertebrates with a posterior growing region are based on a Turing-Child metabolic gradient that is progressively shifted posteriorly with the PSM as elongation, segmentation and somitogenesis proceed. This gradient corresponds to anteriorly propagating metabolic front in the PSM that drives the anteriorly propagating mRNA synthesis and which, together with mRNA degradation, explains stripe formation and spatial periodicity.The process of segmentation has been compared to zooid formation. We show that for annelids the metabolic profile behaves as a Turing field in the sense that an increase in the length of the system or a decrease of the Turing wavelength results in an additional peak in the posterior growing region as predicted by Turing theory. In particular, it is shown that the metabolic gradient that drives the segmentation is based on a Turing system.     

The Turing-Child energy field as a driver of early mammalian development

The equivalence of the early mammalian cells, of importance in assisted reproductive technologies (ART), is considered. It is suggested that this controversial topic can be settled by finding whether the cells are distinguished by the Turing-Child (TC) field, as expressed for example by patterns of mitochondrial activity. The division of the pronuclear embryo is driven by a symmetrical bipolar TC pattern whose experimental shape and chemical nature is predicted by TC theory. This bipolar pattern drives the subsequent cell divisions too, and according to present experimental results all cells are equivalent until compaction since they are not distinguished by the TC field in normal development. Interphase cells exhibit homogeneous mitochondrial activity, or perinuclear, or perinuclear and cortical activity, and these patterns too and the rotational symmetry observed are predicted by TC theory. The first differentiation, into an inner mass cell and the trophectoderm, as well as the formation of cell polarity in the trophectoderm are considered. It is suggested that these two events are driven by a peripheral spherical shell of high energy metabolism in the morula; such a shell is predicted by TC theory in a compacted multicellular sphere whose cells are connected by gap junctions. The experimental patterns of mitochondrial activity in unfertilized oocytes exhibit rotational symmetry or polarity. The shape and the chemical nature of these patterns also are predicted and explained by TC theory in a sphere. The change in the spatial pattern of mitochondrial activity with development is attributed to a change in the spatial pattern of mitochondrial activity and not to physical translocation of mitochondria. The experimental finding that these spatial patterns of mitochondrial activity are observed only in live and not in dead biological material is explained by the TC pattern being biology's unique and universal dissipative structure that requires ongoing specific biochemical reactions and energy dissipation.     

How well does Turing's theory of morphogenesis work?

In 1952 Turing published a paper which showed how under restricted conditions a class of chemical reactions could give biological patterns in diffusion-coupled cells. Although this theory has been much discussed, little has been learnt about the range and type of pattern it can generate. In order to do this and to see how stable the patterns are, we have examined the system in detail and written a computer program to simulate Turing's kinetics for two morphogens over various assemblies of cells. We find that on one-dimensional lines of cells, patterns can indeed be produced and that the chemical wavelengths follow all of Turing's predictions. The results show that stable repeating peaks of chemical concentration of periodicity 2–20 cells can be obtained in embryos in periods of time of less than an hour. We do find however that these patterns are not reliable: small variations in initial conditions give small but significant changes in the number and positions of observed peaks. Similar results are observed in two-dimensional assemblies of cells. On rectangles, random blotches are observed whose position cannot be reliably predicted. On cylinders whose circumference is less than the chemical wavelength, annular stripes are produced. For larger cylinders, blotches that lie very approximately on helices are generated; again sharp prediction of the detailed pattern is impossible.The significance of these results for the developing embryo is discussed. We conclude that Turing kinetics, at least in the simple cases that we have studied, are too unreliable to serve as the generating mechanism for features such as digits which are characterized by a consistent number of units. The theory is however more than adequate by these criteria to specify less well-defined developing patterns such as those of hair follicles or leaf organization. It is emphasized however that the Turing theory is quite unable to generate regulative systems, only mosaicpatterns can be produced.     

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.     

Chemical morphogenesis: Turing patterns in an experimental chemical system

Patterns resulting from the sole interplay between reaction and diffusion are probably involved in certain stages of morphogenesis in biological systems, as initially proposed by Alan Turing. Self-organization phenomena of this type can only develop in nonlinear systems (i.e. involving positive and negative feedback loops) maintained far from equilibrium. We present Turing patterns experimentally observed in a chemical system. An oscillating chemical reaction, the CIMA reaction, is operated in an open spatial reactor designed in order to obtain a pure reaction-diffusion system. The two types of Turing patterns observed, hexagonal arrays of spots and parallel stripes, are characterized by an intrinsic wavelength. We identify the origin of the necessary difference of diffusivity between activator and inhibitor. We also describe a pattern growth mechanism by spot splitting that recalls cell division.     

An hypothesis: phosphorylation fields as the source of positional information and cell differentiation--(cAMP, ATP) as the universal morphogenetic Turing couple.

It is hypothesized that (cAMP, ATP) is the elusive, universal Turing morphogenetic couple, which defies the second law of thermodynamics, i.e. the inexorable march towards homogeneity. cAMP and ATP can be distributed nonhomogeneously because the whole of the intermediary metabolism is so organized that they mutually satisfy the Turing bifurcation conditions upon nonlocalized application of an extracellular ligand, in particular a soluble peptide growth factor, which is nature's distinguished universal bifurcation parameter, acting homogeneously in space and removing the substrate inhibition from adenylate cyclase and thus triggering embryonic induction by triggering the (cAMP, ATP) Turing system. The hypothesis predicts that although the extracellular signal, the growth factor, is applied homogeneously, an organized "dissipative structure" will emerge spontaneously in the responding tissue; this "symmetry breaking" in a reaction-diffusion system occurs precisely in the manner envisaged by Turing, where (cAMP, ATP) constitutes the "reaction-diffusion system". This Turing bifurcation explicates the recent experiments where a differentiated embryoid emerges from the mere immersion of frog animal caps in an homogeneous growth factor solution, and similar experiments on chicks. The "metabolic" patterns found by Child and colleagues also reflect dissipative structures arising in a (cAMP, ATP) reaction-diffusion system when interpreted in the light of modern biochemistry: in particular, the localized glycogen depletion reflects localized cAMP; localized redox, respiratory or susceptibility activity reflects localized ATP. The dramatic collapse of organized structure found by Child and colleagues, for example, when Planaria or a section of it is exposed to an homogeneous environment of a narcotic solution, and the reemergence of structure upon return to water, are explained on the basis of the violation or satisfaction of the Turing bifurcation conditions with respect to (cAMP, ATP), respectively. cAMP is the "activator", ATP is the "inhibitor", and together they mutually satisfy the four activator-inhibitor inequalities, including the all-important autocatalytic cAMP production, as well as the lateral inhibition condition. The functional significance of gap junctions is to generate a multicellular purely reaction-diffusion system for (cAMP, ATP) as envisaged by Turing. It is emphasized that localization and pattern formation occur intracellularly in gap junction-coupled cells and not, as often suggested, extracellularly, the latter localization being too fragile to be maintained for long enough, and soon succumbing to the mixing effect of convection and movement. The activator-inhibitor property of (cAMP, ATP) means that the spatial distribution of cAMP and ATP could be not only nonhomogeneous but also of the same shape.(ABSTRACT TRUNCATED AT 400 WORDS)     

Beyond Turing: The response of patterned ecosystems to environmental change

Spatially periodic patterns can be observed in a variety of ecosystems. Model studies revealed that patterned ecosystems may respond in a nonlinear way to environmental change, meaning that gradual changes result in rapid degradation. We analyze this response through stability analysis of patterned states of an arid ecosystem model. This analysis goes one step further than the frequently applied Turing analysis, which only considers stability of uniform states. We found that patterned arid ecosystems systematically respond in two ways to changes in rainfall: (1) by changing vegetation patch biomass or (2) by adapting pattern wavelength. Minor adaptations of pattern wavelength are constrained to conditions of slow change within a high rainfall regime, and high levels of stochastic variation in biomass (noise). Major changes in pattern wavelength occur under conditions of either low rainfall, rapid change or low levels of noise. Such conditions facilitate strong interactions between vegetation patches, which can trigger a sudden loss of half the patches or a transition to a degraded bare state. These results highlight that ecosystem responses may critically depend on rates, rather than magnitudes, of environmental change. Our study shows how models can increase our understanding of these dynamics, provided that analyses go beyond the conventional Turing analysis.     

Components, structure, biogenesis and function of the Hydra extracellular matrix in regeneration, pattern formation and cell differentiation

The body wall of Hydra is organized as an epithelial bilayer (ectoderm and endoderm) with an intervening extracellular matrix (ECM), termed mesoglea by early biologists. Morphological studies have determined that Hydra ECM is composed of two basal lamina layers positioned at the base of each epithelial layer with an intervening interstitial matrix. Molecular and biochemical analyses of Hydra ECM have established that it contains components similar to those seen in more complicated vertebrate species. These components include such macromolecules as laminin, type IV collagen, and various fibrillar collagens. These components are synthesized in a complicated manner involving cross-talk between the epithelial bilayer. Any perturbation to ECM biogenesis leads to a blockage in Hydra morphogenesis. Blockage in ECM/cell interactions in the adult polyp also leads to problems in epithelial transdifferentiation processes. In terms of biophysical parameters, Hydra ECM is highly flexible; a property that facilitates continuous movements along the organism's longitudinal and radial axis. This is in contrast to the more rigid matrices often found in vertebrates. The flexible nature of Hydra ECM can in part now be explained by the unique structure of the organism's type IV collagen and fibrillar collagens. This review will focus on Hydra ECM in regard to: 1) its general structure, 2) its molecular composition, 3) the biophysical basis for the flexible nature of Hydra's ECM, 4) the relationship of the biogenesis of Hydra ECM to regeneration of body form, and 5) the functional role of Hydra ECM during pattern formation and cell differentiation.     

Maternal-effect genes as the recording genes of Turing-Child patterns: sequential compartmentalization in Drosophila

The early embryo is often a two-dimensional surface. The fate map is the subdivision of this surface into regions which give rise to parts of the phenotype. It is shown for Drosophila that the fate map is generated by the spontaneous and sequential formation of Turing-Child (TC) eigenfunction patterns. These patterns are recorded by the maternal-effect genes. The addition of the nodal lines of the TC patterns yields the correct number, positions, sequences and symmetries of regional boundaries. A simplest nontrivial 'homeotic transformation' is suggested and explained. A single mutation converts a region in one end of the fate map to a mirror-symmetric image of a nonadjacent region in the other end of the fate map, and this is attributed to the geometry of the TC patterns. This geometry also determines the initial shape of the zygotic gene expression. The vision of William Bateson that biological form is shaped like Chladni's patterns in acoustics and music is justified. A similar sequence of TC patterns occurs in the normal development of all organisms, and it is suggested that artificial intervention which completes the full sequence of TC patterns can be useful in the context of regenerative medicine and this is illustrated with the sea urchin.     

Forging patterns and making waves from biology to geology: a commentary on Turing (1952) ‘The chemical basis of morphogenesis’

Alan Turing was neither a biologist nor a chemist, and yet the paper he published in 1952, ‘The chemical basis of morphogenesis’, on the spontaneous formation of patterns in systems undergoing reaction and diffusion of their ingredients has had a substantial impact on both fields, as well as in other areas as disparate as geomorphology and criminology. Motivated by the question of how a spherical embryo becomes a decidedly non-spherical organism such as a human being, Turing devised a mathematical model that explained how random fluctuations can drive the emergence of pattern and structure from initial uniformity. The spontaneous appearance of pattern and form in a system far away from its equilibrium state occurs in many types of natural process, and in some artificial ones too. It is often driven by very general mechanisms, of which Turing''s model supplies one of the most versatile. For that reason, these patterns show striking similarities in systems that seem superficially to share nothing in common, such as the stripes of sand ripples and of pigmentation on a zebra skin. New examples of ‘Turing patterns'' in biology and beyond are still being discovered today. This commentary was written to celebrate the 350th anniversary of the journal Philosophical Transactions of the Royal Society.     

整体大于部分之和: 生态自组织斑图及其涌现属性

近30年来, 自组织理论已经发展成为解释生态系统呈现规则空间格局的有效理论。伴随着生态系统自发有序空间格局的生成, 自组织过程产生一系列的涌现属性, 这些特征对生态系统功能至关重要。在此, 我们将介绍这一正蓬勃发展的研究领域的主要理论进展。首先, 叙述了自组织这一概念的发展历程与定义, 详细阐述了自组织理论的两个经典理论框架: 图灵原理与相分离原理。然后, 根据几个典型的生态自组织研究案例, 描述了图灵原理与相分离原理在不同生态系统中的具体数学模型表达形式。接着, 分别阐述了图灵原理的涌现属性对生态系统功能以及相分离原理的涌现属性对细胞功能的作用。最后, 从多尺度自组织斑图、瞬态斑图和生物个体行为自组织3个方面对未来生态自组织理论发展方向进行了探讨。自组织研究在生态学与生物学研究中方兴未艾, 希望更多的学者在未来关注与参与该领域的发展。     

Vegetation patterns generated by a wind driven sand-vegetation system in arid and semi-arid areas

Wind has been proposed as a driving factor in determining vegetation patterns, but there are few dynamic models that include both vegetation and wind. In this paper, we present a dynamic model to investigate how a vegetation pattern is generated and affected by wind. In the model, the effects of prevailing wind and non-prevailing wind on sand and vegetation are modeled respectively as advection terms and diffusion terms. With these considerations and proper parameter values that satisfy Turing bifurcation conditions, labyrinth and banded vegetation patterns are obtained in two situations of wind. By changing wind transportation capacity, we simulate the adaptation process from one vegetation pattern to another. With environmental changes of large amplitude, the width of vegetation bands varies while the wavelength can increase but does not decrease in our simulation. Then we describe the difference between simulated patterns and real patterns. And in the discussion, we explain the mechanism that forming patterns and the consistency of this research with other studies.     

Stochastic amplification of spatial modes in a system with one diffusing species

The problem of pattern formation in a generic two species reaction–diffusion model is studied, under the hypothesis that only one species can diffuse. For such a system, the classical Turing instability cannot take place. At variance, by working in the generalized setting of a stochastic formulation to the inspected problem, spatially organized patterns can develop, seeded by finite size corrections. General conditions are given for the stochastic patterns to occur. The predictions of the theory are tested for a specific case study.     

Spatio-temporal pattern formation in Rosenzweig–MacArthur model: Effect of nonlocal interactions

Spatio-temporal pattern formation in reaction–diffusion models of interacting populations is an active area of research due to various ecological aspects. Instability of homogeneous steady-states can lead to various types of patterns, which can be classified as stationary, periodic, quasi-periodic, chaotic, etc. The reaction–diffusion model with Rosenzweig–MacArthur type reaction kinetics for prey–predator type interaction is unable to produce Turing patterns but some non-Turing patterns can be observed for it. This scenario changes if we incorporate non-local interactions in the model. The main objective of the present work is to reveal possible patterns generated by the reaction–diffusion model with Rosenzweig–MacArthur type prey–predator interaction and non-local consumption of resources by the prey species. We are interested in the existence of Turing patterns in this model and in the effect of the non-local interaction on the periodic travelling wave and spatio-temporal chaotic patterns. Global bifurcation diagrams are constructed to describe the transition from one pattern to another one.     

Mixed-mode pattern in Doublefoot mutant mouse limb--Turing reaction-diffusion model on a growing domain during limb development

It has been suggested that the Turing reaction-diffusion model on a growing domain is applicable during limb development, but experimental evidence for this hypothesis has been lacking. In the present study, we found that in Doublefoot mutant mice, which have supernumerary digits due to overexpansion of the limb bud, thin digits exist in the proximal part of the hand or foot, which sometimes become normal abruptly at the distal part. We found that exactly the same behaviour can be reproduced by numerical simulation of the simplest possible Turing reaction-diffusion model on a growing domain. We analytically showed that this pattern is related to the saturation of activator kinetics in the model. Furthermore, we showed that a number of experimentally observed phenomena in this system can be explained within the context of a Turing reaction-diffusion model. Finally, we make some experimentally testable predictions.