A densitometrical method for the study of pattern formation in a ciliateChilodonella

Summary An easy and sensitive method is reported here for testing the similarities of individual patterns by photographically transforming maps of these patterns to given, deductively chosen conventions involving constant distances between selected reference points. A cumulative map is produced by loading all landmarks from a set of individual maps on to one sheet of paper. The use of various a priori conventions results in variable cumulative maps, which are then optically transformed on an analog digital converter, with additional input for optical picture processing. The densitometrical maps thus obtained may be compared as to the cumulative degree of areas of maximal and minimal density of landmarks. The best conventions are those that yield the map with the most contrast.Maps of spatial patterns of the sites of contractile vacuole pore (CVP) primordia in an early stage of divisional morphogenesis of the ciliateChilodonella steini were compared after four different transformations and adjustments of the same set of individual maps. The best focusing of the sites of CVP differentiation was achieved by use of the postoral axis, defined by the center of the oral apparatus and the posterior end of the cell as the scaling parameter. The composite domain map obtained by optical transformation of this cumulative map could distinguish the specific CVP territories observed in earlier work (Kaczanowska 1981). These results confirm earlier findings that indicated the site of the oral apparatus is an important reference point in CVP primordia positioning. They also strongly suggest the existence of an overriding scaling factor governing the positioning of sites of differentiation in both dimensions of the developmental field. The method of superposition and scaling of pattern maps is generally applicable to situations in which pattern elements appear at discrete points on a flat surface.     

Multiscale analysis of pattern formation via intercellular signalling

Lateral inhibition, a juxtacrine signalling mechanism by which a cell adopting a particular fate inhibits neighbouring cells from doing likewise, has been shown to be a robust mechanism for the formation of fine-grained spatial patterns (in which adjacent cells in developing tissues diverge to achieve contrasting states of differentiation), provided that there is sufficiently strong feedback. The fine-grained nature of these patterns poses problems for analysis via traditional continuum methods since these require that significant variation takes place only over lengthscales much larger than an individual cell and such systems have therefore been investigated primarily using discrete methods. Here, however, we apply a multiscale method to derive systematically a continuum model from the discrete Delta-Notch signalling model of Collier et al. (J.R. Collier, N.A.M. Monk, P.K. Maini, J.H. Lewis, Pattern formation by lateral inhibition with feedback: a mathematical model of Delta-Notch intercellular signalling, J. Theor. Biol., 183, 1996, 429-446) under particular assumptions on the parameters, which we use to analyse the generation of fine-grained patterns. We show that, on the macroscale, the contact-dependent juxtacrine signalling interaction manifests itself as linear diffusion, motivating the use of reaction-diffusion-based models for such cell-signalling systems. We also analyse the travelling-wave behaviour of our system, obtaining good quantitative agreement with the discrete system.     

On a simplified model for pattern formation in honey bee colonies

We present a simplified version of a previously presented model (Camazine et al. (1990)) that generates the characteristic pattern of honey, pollen and brood which develops on combs in honey bee colonies. We demonstrate that the formation of a band of pollen surrounding the brood area is dependent on the assumed form of the honey and pollen removal terms, and that a significant pollen band arises as the parameter controlling the rate of pollen input passes through a bifurcation value. The persistence of the pollen band after a temporary increase in pollen input can be predicted from the model. We also determine conditions on the parameters which ensure the accumulation of honey in the periphery and demonstrate that, although there is an important qualitative difference between the simplified and complete models, an analysis of the simplified version helps us understand many biological aspects of the more complex complete model. Corresponding author     

Travelling waves and pattern formation in a model for fungal development

 Under a variety of conditions, the hyphal density within the expanding outer edge of growing fungal mycelia can be spatially heterogeneous or nearly uniform. We conduct an analysis of a system of reaction-diffusion equations used to model the growth of fungal mycelia and the subsequent development of macroscopic patterns produced by differing hyphal and hence biomass densities. Both local and global results are obtained using analytical and numerical techniques. The emphasis is on qualitative results, including the effects of changes in parameter values on the structure of the solution set. Received 22 November 1995; received in revised form 17 May 1996     

Generation of bilateral symmetry in Anthozoa: a model

Polyps of Anthozoa usually display bilateral symmetry with respect to their mouth opening, to their pharynx, and in particular to the arrangement of their mesenteries. Mesenteries, which are endodermal folds running from the apical to the basal end of the body, subdivide the gastric cavity into pouches. They form in a bilateral symmetric sequence. In this article I propose that early in polyp development the endoderm subdivides successively into three different types of compartments. A mesentery forms at the border between compartments. Two of the compartments are homologous to those of Scyphozoa. They form by mutual activation of cell states that locally exclude each other. The third compartment leads to siphonoglyph formation and is an evolutionary innovation of the Anthozoa. The mechanism that controls the number and spatial arrangement of the third type of compartment changes the radial symmetry into a bilateral one and occasionally into a different one. The dynamics of its formation indicate an activator-inhibitor mechanism. Computer models are provided that reproduce decision steps in the generation of the mesenteries.     

Dynamic heterogeneous spatio-temporal pattern formation in host-parasitoid systems with synchronised generations

In this paper we develop a general mathematical model describing the spatio-temporal dynamics of host-parasitoid systems with forced generational synchronisation, for example seasonally induced diapause. The model itself may be described as an individual-based stochastic model with the individual movement rules derived from an underlying continuum PDE model. This approach permits direct comparison between the discrete model and the continuum model. The model includes both within-generation and between-generation mechanisms for population regulation and focuses on the interactions between immobile juvenile hosts, adult hosts and adult parasitoids in a two-dimensional domain. These interactions are mediated, as they are in many such host-parasitoid systems, by the presence of a volatile semio-chemical (kairomone) emitted by the hosts or the hosts food plant. The model investigates the effects on population dynamics for different host versus parasitoid movement strategies as well as the transient dynamics leading to steady states. Despite some agreement between the individual and continuum models for certain motility parameter ranges, the model dynamics diverge when host and parasitoid motilities are unequal. The individual-based model maintains spatially heterogeneous oscillatory dynamics when the continuum model predicts a homogeneous steady state. We discuss the implications of these results for mechanistic models of phenotype evolution.P. Schofield gratefully acknowledges the financial support of the BBSRC and The Wellcome Trust.     

Chemosensory pathways in the capitate tentacles of the hydroid Cladonema

The introduction of an extract of Artemia into the sea water bathing tentacles from the hydroid Cladonema triggers a burst of electrical activity that can be recorded intracellularly from cnidocytes in the capitate tentacles. These bursts, which are composed of a variety of events, including action potentials and EPSPs, are Ca²⁺ dependent, and are abolished by pretreatment with NiCl₂, suggesting that voltage-gated Ca²⁺ channels are involved in their generation or transmission. Intracellular injection of Lucifer Yellow and recordings from pairs of cnidocytes reveal that the cnidocytes are electrically coupled to one another, but that they are not uncoupled by heptanol. The role of these chemosensory pathways in priming the cnidocytes for discharge is discussed.     

Metamorphosis ofHydractinia echinata Insights into pattern formation in Hydroids

Summary Hydractinia echinata is a marine, colony-forming coelenterate. Fertilized eggs develop into freely swimming planula larvae, which undergo metamorphosis to a sessile (primary) polyp. Metamorphosis can be triggered by means of certain marine bacteria and by Cs⁺. Half a day after this treatment a larva will have developed into a polyp. The induction of metamorphosis can be prevented by addition of inhibitor I, a substance partially purified from tissue ofHydra. The larvae ofH. echinata also appear to contain this substance. Inhibitor I appliedafter the onset of metamorphosis blocks its continuation as long as it remains in the culture medium. Cs⁺ applied within the same period of time also blocks the continuation of metamorphosis. However, these two agents have opposite effects on the body pattern of the resultant polyps. The experiments indicate that application of Cs⁺ triggers the generation of the pre-pattern. Inhibitor I appears to be a factor of this prepattern. A model is proposed which describes the basic features of head and foot/stolon formation not only forHydractinia but also for other related hydroids.     

Global cell sorting in the C. elegans embryo defines a new mechanism for pattern formation

4D microscopic observations of Caenorhabditis elegans development show that the nematode uses an unprecedented strategy for development. The embryo achieves pattern formation by sorting cells, through far-ranging movements, into coherent regions before morphogenesis is initiated. This sorting of cells is coupled to their particular fate. If cell identity is altered by experiment, cells are rerouted to positions appropriate to their new fates even across the whole embryo. This cell behavior defines a new mechanism of pattern formation, a mechanism that is also found in other animals. We call this new mechanism "cell focusing". When the fate of cells is changed, they move to new positions which also affect the shape of the body. Thus, this process is also important for morphogenesis.     

One-dimensional daisyworld: spatial interactions and pattern formation

The zero-dimensional daisyworld model of Watson and Lovelock (1983) demonstrates that life can unconsciously regulate a global environment. Here that model is extended to one dimension, incorporating a distribution of incoming solar radiation and diffusion of heat consistent with a spherical planet. Global regulatory properties of the original model are retained. The daisy populations are initially restricted to hospitable regions of the surface but exert both global and local feedback to increase this habitable area, eventually colonizing the whole surface. The introduction of heat diffusion destabilizes the coexistence equilibrium of the two daisy types. In response, a striped pattern consisting of blocks of all black or all white daisies emerges. There are two mechanisms behind this pattern formation. Both are connected to the stability of the system and an overview of the mathematics involved is presented. Numerical experiments show that this pattern is globally determined. Perturbations in one region have an impact over the whole surface but the regulatory properties of the system are not compromised by transient perturbations. The relevance of these results to the Earth and the wider climate modelling field is discussed.     

The role of trans-membrane signal transduction in turing-type cellular pattern formation

The Turing mechanism (Phil. Trans. R. Soc. B 237 (1952) 37) for the production of a broken spatial symmetry in an initially homogeneous system of reacting and diffusing substances has attracted much interest as a potential model for certain aspects of morphogenesis (Models of Biological Pattern Formation, Academic Press, London, 1982; Nature 376 (1995) 765) such as pre-patterning in the embryo. The two features necessary for the formation of Turing patterns are short-range autocatalysis and long-range inhibition (Kybernetik 12 (1972) 30) which usually only occur when the diffusion rate of the inhibitor is significantly greater than that of the activator. This observation has sometimes been used to cast doubt on applicability of the Turing mechanism to cellular patterning since many messenger molecules that diffuse between cells do so at more-or-less similar rates. Here we show that Turing-type patterns will be able to robustly form under a wide variety of realistic physiological conditions though plausible mechanisms of intra-cellular chemical communication without relying on differences in diffusion rates. In the mechanism we propose, reactions occur within cells. Signal transduction leads to the production of messenger molecules, which diffuse between cells at approximately equal rates, coupling the reactions occurring in different cells. These mechanisms also suggest how this process can be controlled in a rather precise way by the genetic machinery of the cell.     

Pattern formation in plant epidermis through inhibition of immediately adjacent cells by pattern elements

Summary Examples of short-range inhibition in plant development were sought which satisfy four criteria for simplicity of expression. The range of inhibition extends only to adjacent cells and these criteria are (1) pattern of expression is within a two-dimensional array of cells, (2) cells of the array do not proliferate during pattern formation, (3) spacing between special cells extends only for a distance of one cell, and (4) no unequal cell divisions occur to complicate the pattern of cell arrangement. In one case of hairs or trichomes, the cells are evenly dispersed over the adaxial surface of the sepal ofSalvia splendens L. The R value of Clark and Evans is 1.64 for hair arrangement and hairs are formed from 21.1% of the epidermal cells. Based on cell behavior and an assumed method of inhibition, ordinary differential equations were written for formation of three types of cells in the tissue, free cells (F), trichome-bearing cells (T), and inhibited cells (I). Computer generated numerical solutions for these equations give cell type frequencies of 0.056 (F), 0.198 (T), and 0.745 (I), close to the observed values of 0.057 (F), 0.211 (T), and 0.731 (I). A similar approach was employed in analysis for the deployment of the cotton fibers in the ovular epidermis ofGossypium hirsutum L. and for that of the floating stomata of the fernAnemia phyllitidis (L). Sw. In general, these three patterns of epidermal deriviatives are most easily explained by one-cell wide inhibition fields between inhibiting special cells and inhibited contiguous cells. Other patterns in plants cannot be explained by this mechanism.     

Spatio-temporal pattern formation on spherical surfaces: numerical simulation and application to solid tumour growth

In this paper we examine spatio-temporal pattern formation in reaction-diffusion systems on the surface of the unit sphere in 3D. We first generalise the usual linear stability analysis for a two-chemical system to this geometrical context. Noting the limitations of this approach (in terms of rigorous prediction of spatially heterogeneous steady-states) leads us to develop, as an alternative, a novel numerical method which can be applied to systems of any dimension with any reaction kinetics. This numerical method is based on the method of lines with spherical harmonics and uses fast Fourier transforms to expedite the computation of the reaction kinetics. Numerical experiments show that this method efficiently computes the evolution of spatial patterns and yields numerical results which coincide with those predicted by linear stability analysis when the latter is known. Using these tools, we then investigate the r?le that pre-pattern (Turing) theory may play in the growth and development of solid tumours. The theoretical steady-state distributions of two chemicals (one a growth activating factor, the other a growth inhibitory factor) are compared with the experimentally and clinically observed spatial heterogeneity of cancer cells in small, solid spherical tumours such as multicell spheroids and carcinomas. Moreover, we suggest a number of chemicals which are known to be produced by tumour cells (autocrine growth factors), and are also known to interact with one another, as possible growth promoting and growth inhibiting factors respectively. In order to connect more concretely the numerical method to this application, we compute spatially heterogeneous patterns on the surface of a growing spherical tumour, modelled as a moving-boundary problem. The numerical results strongly support the theoretical expectations in this case. Finally in an appendix we give a brief analysis of the numerical method. Received: 27 July 2000 / Revised version: 15 August 2000 / Published online: 16 February 2001     

The effects of local application of retinoids to different positions along the proximo-distal axis of embryonic chick wings

Summary Two retinoids, all-trans-retinoic acid and a synthetic analog, TTNPB, were locally applied to different positions along the proximo-distal axis of embryonic chick wing buds using controlled release carriers. Truncations or limbs with duplicated structures across the antero-posterior axis develop after retinoid application to distal positions in buds from stage 20–24 embryos. Phocomelic limbs develop when the retinoids are applied more proximally to buds of stage 23–24 embryos. Duplications of the pattern of structures along the proximo-distal axis never occur.Using TTNPB that is relatively stable, the amount of retinoid in the wing tissue when phocomelia is induced was measured. There is twice as much retinoid per cell in the proximal half of the bud as in the distal half of the bud. The concentration of TTNPB in proximal tissue is estimated to be three times higher than in distal tissue in which pattern formation and cartilage morphogenesis are relatively normal.At early stages in the development of phocomelia, the shape of the bud changes and the indentation that marks the elbow does not arise. Neither retinoid-induced cell killing nor effects on the pattern of programmed cell death were detected.The induction of phocomelia by retinoids appears to be based on effects on proximal cells, whereas retinoids produce pattern changes by acting on distal cells. Furthermore, compared with pattern changes, higher concentrations of retinoid in the bud tissue are required to produce phocomelia.     

Structure-activity relationship of retinoids on the differentiation of cultured chick foot skin

Summary The influence of retinoids (vitamin A and its analogues) on epithelial differentiation was examined in explants of foot skin from chick embryos. In the presence of retinoic acid (10 M) keratinization and differentiation of scale-like structures, which occurred in tarsometatarsal skin explants, was inhibited and a mucous metaplasia developed. Retinoic acid caused club-shaped feathers in skin explants taken from the anterior surface of the tibia — skin which was determined to differentiate into feathers. In skin explants taken from a breed with feathered feet, the differentiation of tarsometatarsal skin was completely blocked; in tibial skin, club-shaped feathers resulted in response to retinoic acid. These findings indicated that skin of the two origins reacted differently to the retinoid, as was noted in the breed with scaly feet. The structure-activity relationship of 22 retinoids with marked differences in their biological activity was investigated in tarsometatarsal skin explants. Comparing the concentration of various retinoids needed to completely inhibit the differentiation of scale-like structures, retinoids containing tetramethylated indane or tetraline were 100 and 1,000 times more active than retinoic acid. Retinoids with a sulphur-containing end group were also active but less so than the corresponding compound with a carboxylic acid end group. The inactive ethyl, ester analogue, etretinate, was activated in the presence of esterase, indicating that the free carboxylic acid group was important for the activity of retinoids. The retinoid-induced inhibition of keratinization followed by mucous metaplasia in cultured chick embryo skin is a simple and useful model system to test new retinoids which may be helpful in the treatment of dermatological and oncological diseases.     

Finding the center of a phyllotactic pattern

The calculation of divergence angles between primordia in a plant apex depends on the point used as the center of the apex. In mathematically ideal phyllotactic patterns, the center is well defined but there has not been a precise definition for the center of naturally occurring phyllotactic patterns. A few techniques have been proposed for estimating the location of the center but without a precise definition for the center the accuracy of these methods cannot be known. This paper provides a precise definition that can be used as the center of a phyllotactic pattern and a numerical method which can accurately find it. These tools will make it easier to compare theory against experiment in phyllotaxis.     

A model for budding in hydra: pattern formation in concentric rings

Current models of pattern formation in Hydra propose head-and foot-specific morphogens to control the development of the body ends and along the body length axis. In addition, these morphogens are proposed to control a cellular parameter (positional value, source density) which changes gradually along the axis. This gradient determines the tissue polarity and the regional capacity to form a head and a foot, respectively, in transplantation experiments. The current models are very successful in explaining regeneration and transplantation experiments. However, some results obtained render problems, in particular budding, the asexual way of reproduction is not understood. Here an alternative model is presented to overcome these problems. A primary system of interactions controls the positional values. At certain positional values secondary systems become active which initiate the local formation of e.g. mouth, tentacles, and basal disc. (i) A system of autocatalysis and lateral inhibition is suggested to exist as proposed by Gierer and Meinhardt (Kybernetik 12 (1972) 30). (ii) The activator is neither a head nor a foot activator but rather causes an increase of the positional value. (iii) On the other hand, a generation of the activator leads to its loss from cells and therewith to a (local) decrease of the positional value. (iv) An inhibitor is proposed to exist which antagonizes an increase of the positional value. External conditions like the gradient of positional values in the surroundings and interactions with other sites of morphogen production decide whether at a certain site of activator generation the positional value will increase (head formation), decrease (foot formation) or increase in the centre and decrease in the periphery thereby forming concentric rings (bud formation). Computer-simulation experiments show basic features of budding, regeneration and transplantation.     

Genetic and developmental analyses of chaetae pattern formation in Drosophila tergites

Summary The role of the achaete-scute complex and extramacrochaetae, Notch, Delta, Enhancer of split and Hairless genes in chaeta patterning in Drosophila tergites was studied in genetic mosaics and in mutant combinations. The mutant phenotypes of different alleles of each gene can be ordered in characteristic topographical seriations. These seriations are related to the pattern of proliferation of histoblasts and the time of singularization of sensory organ mother cells from surrounding epidermal cells. Genetic mosaics of lethal alleles show that these genes are fundamentally involved in this singularization and subsequent differentiation. The study of mutant combinations of alleles of these genes reveals specific relationships of epistasis and synergism between them. The results suggest that spatial and temporal variations in achaete-scute complex functional products in cells, modulated by the activity of other genes involved in signal transduction, define the patterned differentiation of sensory organs in tergites. Offprint requests to: A. García-Bellido     

Analytical treatment of pattern formation in the Gierer-Meinhardt model of morphogenesis

Summary We first perform a linear stability analysis of the Gierer-Meinhardt model to determine the critical parameters where the homogeneous distribution of activator and inhibitor concentrations becomes unstable. There are two kinds of instabilities, namely, one leading to spatial patterns and another one leading to temporal oscillations. Focussing our attention on spatial pattern formation we solve the corresponding nonlinear equations by means of our previously introduced method of generalized Ginzburg-Landau equations. We explicitly consider the two-dimensional case and find both rolls and hexagon-like structures. The impact of different boundary conditions on the resulting patterns is also discussed. The occurrence of the new patterns has all the features of nonequilibrium phase transitions.