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Primary and Secondary Waves in Developmental Biology

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... To illustrate this, I shall discuss two early examples in detail, both depending on catastrophe theory. In the first, Zeeman (1974) predicted that there could be a wave associated with the formation of a boundary in a previously undifferentiated tissue. In the second, Bazin and Saunders (1978) predicted that an amoeba should modify a certain chemoattractant. ...
... The application that became the centre of the controversy about catastrophe theory was Zeeman's (1974) paper on travelling wave fronts. Zeeman claimed that when a frontier forms in a region that was previously undifferentiated, i.e. either homogeneous or with at most a gradient in properties along it, then this frontier does not first appear in its final position. ...
Chapter
Biologists and social scientists often carry out their research in ways that are quite different from those used by physical scientists. Their results are often different as well; the nature of their subjects makes it less likely that they can produce the firm and generally quantitative predictions that are standard in physics. As mathematics is being applied more and more outside the physical sciences, a new methodology is appearing that better reflects the nature of these other subjects. This is happening only slowly, chiefly because the relevant work is generally seen only in its context of trying to solve a particular problem rather than as a contribution to applied mathematics as such. The aim of this paper is to encourage progress by describing some results that have already been obtained, and by discussing explicitly some of the issues that arise.
... This theory focuses on solving the problem that how and why a dynamic system becomes unstable under the continuous changes with the influence of one or more parameters (Thom, 1972;Wang et al., 2017). Catastrophe Theory can find out the inherent instability in a nonlinear dynamical system (Diks and Wang, 2016), and it has been proven to be a successful tool to investigate the qualitative properties in many fields, e.g., in physics (Berry, 1976;Holmes and Rand, 1976;Poston and Stewart, 1978), in biology (Zeeman, 1974;Bazin and Saunders, 1979), in social sciences (Zeeman, 1976a;Zeeman, 1976b), in engineering and technology (Henley, 1976;John and Brain, 1976), in railway system safety (Wang et al., 2017;Huang et al., 2019b), in traffic flow analysis (Acha-Daza and Hall, 1994) and so on. ...
Article
In this paper, the cusp catastrophe model is applied to analyze the railway dangerous goods transportation system risk state changes. Firstly, a Risk-Accident Catastrophe Tree is proposed to formulate the process that how the risk factors cause the accident. Secondly, the whole risk factors are classified into five categories including failure of human behavior, machine failure, transported materials, environment problems and management problems. Next, the cusp catastrophe model of railway dangerous goods transportation system is established, the Split Coefficient is defined to evaluate bifurcation set. The results of the case study show that the risk state changes of railway dangerous goods transportation system satisfy bimodality, inaccessibility, sudden transitions and divergence, but not satisfy hysteresis. The larger the Split Coefficient is, the easier the trajectory of system control point crosses with bifurcation sets. The reason of the system state changes from a safe state to a risk state is: as long as the trajectory of system control point crosses with bifurcation sets, there must be a cusp catastrophe in the system, and the system control point will cross the fold surface, which makes the risk energy increase sharply, the structure, information and energy of the system will also be destroyed.
... We illustrate the two alternatives by a couple of examples. EXAMPLE 1. AGGRESSION [22]. According to Konrad Lorenz [7] fear and rage are conflicting drives influencing aggression. ...
Article
Catastrophe theory is a method discovered by Thorn [14] of using singularities of smooth maps to model nature. In such models there are often several levels of structure, just as in a geometry problem there can be several levels of structure, for instance the topological, dif-ferential, algebraic, and affine, etc. And, just as in geometry the topological level is generally the deepest and may impose limitations upon the higher levels, so in applied mathematics, if there is a catastrophe level, then it is generally the deepest and likely to impose limitations upon any higher levels, such as the differential equations involved, the asymptotic behaviour, etc. Again, in geometry the com-plexity of the higher levels may render them inaccessible, so that they can only be handled implicitly rather than explicitly, while at the same time the underlying topological invariants may even be computable. Similarly in applied mathematics the complexity of the differential equations may sometimes render them inacces-sible (even to computers), so that they can only be handled irnplicitly rather than explicitly, while the underlying catastrophe can be modeled, possibly even to the extent of providing quantitative prediction. Therefore catastrophe theory pifers two attractions: On the one hand it some-times provides the deepest level of insight and lends a simplicity of understanding. On the other hand, in very complex systems such as occur in biology and the social sciences, it can sQmetimes provide a model where none was previously thought possible. In this paper we discuss various levels of structure that can be superim-posed upon an underlying catastrophe and illustrate them with an assortment of examples. For convenience we shall mostly use the familiar cusp catastrophe (see [5], [13], [14], [24]). <P 1975, Canadian Mathematical Congress 533 534 E. C. ZEEMAN Level 1. Singularities. Level 2. Fast dynamic (homeostasis). Level 3. Slow dynamic (development). Level 4. Feedback. Level 5. Noise. Level 6. Diffusion. Thorn's classification of elementary catastrophes belongs to Level 1. Levels 2,3,4 refer to ordinary differential equations, and Level 6 refers to partial differential equations. Level 1. Singulairties. We begin by recalling the main classification theorem. Let C 9 X be manifolds with dim C ^ 5, and let/e C°°{C x X). Suppose that/is generic in the sense that the related map C -» C°°{X) is transverse to the orbits of the group Diff {X) x Diff {R) acting on C°°{X). (Genericity is open-dense in the Whitney C°°-topology.) Let M c C x X be given by V x f = 0, and let i\M- • C be in-duced by projection C x X -> C.
... In [6] and [63], Cooke and Zeeman propose a clock and wavefront model to explain somite formation. They postulate the existence of a longitudinal positional information gradient down the AP axis of the embryo which determines regional development by setting the time in each cell at which it will undergo a catastrophe. ...
... In many cases, skin organ patterns are actually laid down sequentially, and it is believed that travelling waves of determination (Zeeman (1974)) often initiate morphogenetic processes. For example, stripe pigment patterns on the alligator develop sequentially. ...
Article
001 Abstract. A tissue interaction model for skin organ pattern formation is presented. Possi- ble spatially patterned solutions on rectangular domains are investigated. Linear stability analysis suggests that the model can exhibit pattern formation. A weakly nonlinear two-dimensional pertur- bation analysis is then carried out. This demonstrates that when bifurcation occurs via a simple eigenvalue, patterns such as rolls, squares, and rhombi can be supported by the model equations. Our nonlinear analysis shows that more complex patterns are also possible if bifurcation occurs via a double eigenvalue. Surprisingly, hexagonal patterns could not develop from a primary bifurcation.
... Such waves play an organizing role in the general processes of structure formation in biological systems. Very interesting discussions of similar phenomena can be also found in [12,13]. I n the former, for instance, Zeeman discusses the wave propagation of a frontier in a homogeneous tissue, to its final position, a process which can be described by means of state variables (of the cells) in a n-dimensional space (Rn) and of specific control parameters. ...
Article
A general review is presented of some of the approaches to the general problem of morphogenesis which originate either directly or indirectly in the work of Turing. The main points are the stability analysis of the Gierer-Meinhardt equations and the discussion on the applicability to biological systems of the concepts and techniques of field theory.
... Although the applied theories that have given rise to the MPSTW such as catastrophe theory [20], Hjelmslevian and Greimasian semiotics [19], graph theory and crystalographic groups in its beginnings orbited around the static conception of relation while product, later they have approached the dynamic conception of relation while production which is in tune with an ever emerging Quality. Lourenci has always been aware that if it were possible to show that architectural design could be considered a language, necessarily it should resemble the nature of a Chinese language [3] which is essentially emergent, relational and dynamic. ...
Article
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The tight coupling quantity/quality in nature reflects an efficient use of exergy in constructing biological material leading to a very effective storage of information per unit of mass maintaining homeostatic conditions. It seems the entity called incipient Emergy applied to the modeling of holarchic man-made eco-mimetic systems enables us to evince a sustainable basis for the design of our cities due to its explicitation of the relationship of the quantity/quality. Resembling autocatalytic feedback loops, the balanced trade-off between the knowledge of the knowing subject and the known (phenomenological) object is introduced in the eco-design model and underlying geometric and computational models precisely because its nature tunes in to the premises of incipient Emergy.
... We proved that the wave of determination is not stopped by a cut across its path. Such a 'kinematic' wave (Zeeman, 1974) does not depend upon the propagation of a signal across the tissue, but results from the fact that the cells are timers laid out in the order in which they are preset to change -anterior cells are everywhere in advance of posterior cells. ...
Article
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Following neurulation, the frog segments c.40 somites and concurrently undergoes a striking elongation along the anteroposterior axis. This elongation (excluding the head) is largely the result of a presegmental extension of posterior tissue with a lesser contribution from the extension of segmented tissue. Presegmental extension is entirely the result of activity within a narrow zone of extension that occupies the central region in the tail bud. Within the zone of extension, a minimum of six prospective somites undergo an eight- to ten-fold extension along the axis. The zone passes posteriorly across the tissue of the tail tip. The anterior of the tail bud contains three extended prospective somites in the course of segmentation. The anterior boundary of the zone of extension coincides in space exactly with the anterior boundary of the zone of abnormal segmentation that results from temperature shock. This means that extension ceases immediately before the sudden tissue change associated with segmentation.
... The development of the pattern is, therefore, a strikingly dynamic process. A morphogenetic 'wave' (Zeeman, 1974) sweeps across the skin: ahead of the wave, the tissue lacks any obvious pattern; behind it, primordia have formed in an orderly triangular array. ...
Article
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A regular array of feather primordia covers chick dorsal skin in vivo. The pattern develops over a period of 2 days as a morphogenetic wave sweeps across either side of the back of the chick, forming successive anteroposterior rows of primordia. This paper describes a new method for the culture of chick skin which allows the development of large areas of the feather pattern to be investigated experimentally. Skin is cultured on a substratum of hydrated collagen; since the collagen is transparent, feather primordium development can be observed in detail. The new method has been used to investigate the problem of when the positions of feathers are determined. I show that during the time when the first few rows of primordia are forming, skin taken from just lateral to the most recently formed row can be caused to form an increased number of primordia per row by stretching it anteroposteriorly. This result indicates that the positions of feathers are determined sequentially along an invisible wave which moves just ahead of the visible wave of primordium morphogenesis.
... With the recent experimental findings concerning the existence of a segmentation clock and a wavefront of FGF8 along the AP of vertebrate embryos, the Clock and Wavefront model has received a large amount of support. First proposed by Cooke and Zeeman (Cooke, 1975(Cooke, , 1998Cooke and Zeeman, 1976;Zeeman, 1974), the model assumed the existence of a longitudinal positional information gradient along the AP axis which interacts with a smooth cellular oscillator (the clock) to set the times at which cells undergo a catastrophe. In this context, Cooke and Zeeman were referring to the changes in adhesive and migratory behavior of cells as they form somites. ...
Article
Somitogenesis is the process of division of the anterior-posterior vertebrate embryonic axis into similar morphological units known as somites. These segments generate the prepattern which guides formation of the vertebrae, ribs and other associated features of the body trunk. In this work, we review and discuss a series of mathematical models which account for different stages of somite formation. We begin by presenting current experimental information and mechanisms explaining somite formation, highlighting features which will be included in the models. For each model we outline the mathematical basis, show results of numerical simulations, discuss their successes and shortcomings and avenues for future exploration. We conclude with a brief discussion of the state of modeling in the field and current challenges which need to be overcome in order to further our understanding in this area.
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Catastrophe theory can describe a continuous process that is undergoing abrupt changes. A dynamic process can be considered a cusp catastrophe if it has the following five qualities: bimodality, sudden transitions, hysteresis, inaccessibility, and divergence. In this paper, the cusp catastrophe model is applied to describe the dynamic changing process of railway system safety. This dynamic process is also proved to have the five qualities of a cusp catastrophe. With the use of the cusp catastrophe model, a framework for the system risk of railway systems is constructed, and some ideas on the risk field of railway system safety are presented. On the basis of the framework and ideas, the dynamic changes in the system risk in a railway accident evolution process can be described in a novel approach. The concept of system risk in railway system safety is also discussed in three aspects. We hope that the theoretical description of and discussion on system risk could facilitate a profound analysis of railway system safety.
Chapter
Experimental investigations and theoretical analyses of embryo-genesis have attempted to discover and describe the processes whereby a newly fertilized egg cell is transformed into an adult organism. These investigations have centered on the possible organismic, multicellular, cellular, cytoplasmic and genetic organization responsible for this transformation. Theoretical models, reflecting to some extent contemporary activity in the physical sciences and mathematics, have attempted to provide more or less precise mechanisms for these processes. The present paper reviews some of these studies and their contribution to an understanding of the nature of embryogenesis and the emergence of adult organisms.
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Catastrophe Theory, proposed by Thom (1969), describes the behavior of a dynamical system in terms of the maxima and minima of the associated potential energy function. The minima represent stationary or (quasi-) equilibrium conditions for the energy function and serve as states of attraction for the dynamical system, while the maxima act as repellor states. The potential energy function V is parameterized by a manifold C, the control space, on a manifold X, the behavior space. The set $$M = \left\{ {\left( {c,x \in } \right)CxX\left| {{\nabla _x}\nabla \left( {c,x} \right) = 0} \right.} \right\} $$ defines the catastrophe manifold when V is differentiable everywhere (Woodcock and Poston, 1974a). A catastrophe is a singularity of the map $$X:M \to C$$ $$\left( {c,x} \right) \to c
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The world I grew up in believed that change and development in life are part of a continuous process of cause and effect, minutely and patiently sustained throughout the millenniums. With the exception of the initial act of creation ..., the evolution of life on earth was considered to be a slow, steady and ultimately demonstrable process. No sooner did I begin to read history, however, than I began to have my doubts. Human society and living beings, it seemed to me, ought to be excluded from so calm and rational a view. The whole of human development, far from having been a product of steady evolution, seemed subject to only partially explicable and almost invariably violent mutations. Entire cultures and groups of individuals appeared imprisoned for centuries in a static shape which they endured with long-suffering indifference, and then suddenly, for no demonstrable cause, became susceptible to drastic changes and wild surges of development. It was as if the movement of life throughout the ages was not a Darwinian caterpillar but a startled kangaroo, going out towards the future in a series of unpredictable hops, stops, skips and bounds. Indeed, when I came to study physics I had a feeling that the modern concept of energy could perhaps throw more light on the process than any of the more conventional approaches to the subject. It seemed that species, society and individuals behaved more like thunder-clouds than scrubbed, neatly clothed and well-behaved children of reason. Throughout the ages life appeared to build up great invisible charges, like clouds and earth of electricity, until suddenly in a sultry hour the spirit moved, the wind rose, a drop of rain fell acid in the dust, fire flared in the nerve, and drums rolled to produce what we call thunder and lightening in the heavens and chance and change in human society and personality. LAURENS VAN DER POST, The Lost World of the Kalahari
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Disruption of normal vertebral development results from abnormal formation and segmentation of the vertebral precursors, called somites. Somitogenesis, the sequential formation of a periodic pattern along the antero-posterior axis of vertebrate embryos, is one of the most obvious examples of the segmental patterning processes that take place during embryogenesis and also one of the major unresolved events in developmental biology. We review the most popular models of somite formation: Cooke and Zeeman's clock and wavefront model, Meinhardt's reaction-diffusion model and the cell cycle model of Stern and co-workers, and discuss the consistency of each in the light of recent experimental findings concerning FGF-8 signalling in the presomitic mesoderm (PSM). We present an extension of the cell cycle model to take account of this new experimental evidence, which shows the existence of a determination front whose position in the PSM is controlled by FGF-8 signalling, and which controls the ability of cells to become competent to segment. We conclude that it is, at this stage, perhaps erroneous to favour one of these models over the others.
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Multistable figures show that the stimulus-percept relation is not a single valued function. We therefore propose a tentative nonlinear model on the hypothesis that the graph of this relation is the equilibrium set of a dynamic system. For simplicity and to obtain testable predictions, we consider a system whose bifurcations are gradient-like and thus generically described by the elementary catastrophes. We motivate this general model, and then show how, in conjunction with the principle of minimal singularity, it implies cusp catastrophe geometry in a specific perceptual example. Indeed, we argue for canonical cusp geometry in this case. The model incorporates naturally certain observed features of multistable perception, such as hysteresis and bias effects. Despite being a continuum model it is naturally compatible with the subjective dichotomy of bistable perception. The model makes testable predictions which may easily be extended to other specific examples of multistable perception.
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Catastrophe Theory was developed in an attempt to provide a form of Mathematics particularly apt for applications in the biological sciences. It was claimed that while it could be applied in the more conventional “physical” way, it could also be applied in a new “metaphysical” way, derived from the Structuralism of Saussure in Linguistics and Lévi-Strauss in Anthropology. Since those early beginnings there have been many attempts to apply Catastrophe Theory to Biology, but these hopes cannot be said to have been fully realised. This paper will document and classify the work that has been done. It will be argued that, like other applied Mathematics, applied Catastrophe Theory works best where the underlying laws are securely known and precisely quantified, requiring those same guarantees as does any other branch of Mathematics when it confronts a real-life situation.
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It is argued that all chronic gastroduodenal peptic ulcers result from localised increase in mucosal susceptibility to acid attack at the interface between a segment of gastroduodenitis and gastric fundus or duodenal mucosa. The site is predetermined by the background mucosal pattern. Changes can occur in the differentiated gastroduodenal mucosa that closely resemble cell population transformations described in embryology and regeneration biology. A second pathological process, gastroduodenitis, may develop that does not of itself predispose to ulceration, but the combination of factors can produce a zone of increased acid susceptibility. These complex changes could be generated by immunologically activated gastroduodenitis. Destructive or stimulatory immune reactions, analogous to those seen in the thyroid gland, could affect the gastrin-secreting G cells and other paracrine cells. The resulting tropic and inflammatory reactions would provide the background for peptic ulceration.
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How do animals wiggle and bend? Throughout the animal kingdom may be found a persistent body organization of repeated metameric segments. In practically all metazoans, this meristic body pattern provides an efficient arrangement for body movement and locomotion. This is particularly striking in the annelids, in which this meristic pattern is seen in body segments and organs within the segments. Naturalists as far back as Aristotle (1910, translated by Thompson) have noted the common feature of segmentation in animals as diverse as worms, maggots, and snakes and in chicken embryos. Because of the ready availability of chicken eggs and the accessibility of the developing embryo, the segmentation of the vertebrate embryo was noted as early as the sixteenth century. This segmentation was alluded to as the “division of the embryo,” which could seldom be seen in the early stages of development without aids for magnification. Although lenses for magnification were noted as early as Aristophanes (fifth century BC), Roger Bacon (thirteenth century) is thought to have been the first scientist—naturalist to use lenses for investigative purposes (Needham, 1959). Embryologists were slow to use microscopes in their observations, although occasional use of the “perspicilli” was reported, and undoubtedly this instrument helped Harvey in his pioneer observations (1651) on early chick development (Meyer, 1936). Others in the seventeenth century also represented somites in their drawings of chicken embryos, although they did not remark upon them.
Article
One of the most visually striking patterns in the early developing embryo is somite segmentation. Somites form as repeated, periodic structures in pairs along nearly the entire caudal vertebrate axis. The morphological process involves short- and long-range signals that drive cell rearrangements and cell shaping to create discrete, epithelialized segments. Key to developing novel strategies to prevent somite birth defects that involve axial bone and skeletal muscle development is understanding how the molecular choreography is coordinated across multiple spatial scales and in a repeating temporal manner. Mathematical models have emerged as useful tools to integrate spatiotemporal data and simulate model mechanisms to provide unique insights into somite pattern formation. In this short review, we present two quantitative frameworks that address the morphogenesis from segment to somite and discuss recent data of segmentation and epithelialization.
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This tutorial paper explains some of the basic mathematical ideas involved in René Thom's Catastrophe Theory, in the simplest and most accessible case: the Elementary Catastrophes. Topics discussed include the classification of local behavior of smooth functions, determinacy (how much of a Taylor series is adequate to capture the function's behavior), unfoldings (what are the possible perturbations?), and structural stability (robustness). These and other concepts are exemplified using simple models from science: the buckling of an arch, the response of patients suffering from hyperthyroidism to therapy, and cellular differentiation. These models have been selected to act as illustrative examples, and no attempt is made here to survey the applications of the theory since this has already been done in Stewart (1981,1982).
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