Lionel G. Harrison

What is the status of reaction-diffusion theory thirty-four years after turing?

Physicochemical explanations of phenomena are divisible into three classes: structure, equilibrium and kinetics. For the phenomena of biological development, many physical scientists have the preconception that the explanations must turn out to be principally kinetic. In this class of theory, reaction-diffusion is by far the most extensively developed, and is worthy of attention both for its own sake and because many of its features well exemplify the nature of the broader field of kinetic theory. Reaction-diffusion should be thought of as a class, rather than a species, of theory. This review addresses three aspects: first, the general nature of the two-morphogen interaction as first proposed by Turing and incorporated in many later models; second, the specifics of these later models and their probable relative scope; third, the current state of attempts to identify the chemical nature of morphogens. It is concluded that reaction-diffusion in particular, and kinetic theory in general, are now slowly emerging from the almost total neglect by biologists which reaction-diffusion suffered for its first 20 years.

Analysis of pattern precision shows that Drosophila segmentation develops substantial independence from gradients of maternal gene products

We analyze the relation between maternal gradients and segmentation in Drosophila, by quantifying spatial precision in protein patterns. Segmentation is first seen in the striped expression patterns of the pair‐rule genes, such as even‐skipped (eve). We compare positional precision between Eve and the maternal gradients of Bicoid (Bcd) and Caudal (Cad) proteins, showing that Eve position could be initially specified by the maternal protein concentrations but that these do not have the precision to specify the mature striped pattern of Eve. By using spatial trends, we avoid possible complications in measuring single boundary precision (e.g., gap gene patterns) and can follow how precision changes in time. During nuclear cleavage cycles 13 and 14, we find that Eve becomes increasingly correlated with egg length, whereas Bcd does not. This finding suggests that the change in precision is part of a separation of segmentation from an absolute spatial measure, established by the maternal gradients, to one precise in relative (percent egg length) units. Developmental Dynamics 235:2949–2960, 2006.

Complex morphogenesis of surfaces: theory and experiment on coupling of reaction–diffusion patterning to growth

Reaction-diffusion theory for pattern formation is considered in relation to processes of biological development in which there is continuous growth and shape change as each new pattern forms. This is particularly common in the plant kingdom, for both unicellular and multicellular organisms. In addition to the feedbacks in the chemical dynamics, there is then another loop linking size and shape changes with the reaction-diffusion patterning of growth controllers in the growing region. In studies by computation, the codes must incorporate, alongside the usual solvers of the partial differential dynamic equations, a versatile growth code, to express any kind of shape change. We have found that regulation of shape change in particular ways (e.g. to make narrow-angle branchings) demands new features in our chemical mechanisms. Our growth algorithm is for a surface growing tangentially, but moving outward and changing shape to accommodate the extra area. This is potentially applicable both to the tunica layer of multicellular plant meristems and to the growing tip of the cell surface, e.g. in the morphogenesis of single-celled chlorophyte algae which display branching processes: whorl formation in Acetabularia (Dasycladales) and repeated dichotomous branching in Micrasterias (Desmidiaceae). For computational studies, a hemispherical shell is a reasonable idealization of the initial shape. We describe results of two types of study: (1) Pattern formation by three reaction-diffusion models, with contrasted nonlinearities, on the hemispherical shell, particularly to find conditions for robust formation of annular pattern or pattern for dichotomous branching, both of which are common in plants. (2) Sequential dichotomous branchings in a system growing and changing in shape from the hemispherical start.

Hair morphogenesis inAcetabularia mediterranea: Temperature-dependent spacing and models of morphogen waves

SummaryWhorls of sterile hairs inA. mediterranea show, at the moment of first appearance of hair initials, a spacing independent of number of hairs in the whorl but dependent on temperature. By changing the temperature at various times before appearance of hair initials, the pattern-forming event can be located at about 3–4 hours before initials become visible.The temperature dependence of spacing is like that of a chemical rate parameter: In (spacing)versus 1/T is linear. This suggests that the spacing is controlled by kinetic rather than structural factors, and correlates well with reaction-diffusion theory.Mathematical analysis and computer simulation have been used to show that the observed sequence of tip-flattening followed by whorl initiation can be interpreted in terms of published models for generation of “dissipative structures” by reaction and diffusion, and that at least two sequential processes must occur, the first of which shifts growth activity from extremity to circumference of the growing tip, permitting the second to operate around the circumference.

The regulatory capacity of Turing's model for morphogenesis, with application to slime moulds

Abstract For a one-dimensional system (elongated organism), regulation of a pattern of alternating regions of two cell types against disturbance by growth or or by cutting is discussed. Turings linear two-morphogen model requires the number of regions to be that which maximizes the exponential growth rate of morphogen concentrations. On this basis, a diagram is constructed to show ranges of system length for patterns with various numbers of regions. Experimental features of pattern in the Dictyostelium slug, especiially the wide size range having regulatory capacity and the inequality of numbers of the two cell types, are compared with the simple model and with extensions of it to include dependence of diffusivity on cell type and morphogen transport by movement of whole cells.

Quantitative control of Acetabularia morphogenesis by extracellular calcium: A test of kinetic theory

The unicellular marine alga Acetabularia repeatedly produces whorls of hairs at its growing tip. The spacing between hair initials, measured when they first become visible, depends linearly on the reciprocal of calcium concentration in the culture medium. Some hypothetical mechanisms of reaction-diffusion theory predict that pattern spacings will depend on inverse powers of the concentrations of precursors to the morphogenetic substances. Accordingly, we plot spacing as ordinates vs 1/[Ca 2+ ], and examine the consequences of treating this as a plot of 1/(bound Ca) vs 1/(free Ca), i.e. as being analogous to a Lineweaver-Burk plot. Thus we calculate values of binding constant, Δ H 0 and Δ S 0 resembling those for endothermic, entropy-driven binding of calcium to anion paris in small ligands and (binding constant known, not Δ H 0 and Δ S 0 to phospholipid bilayers and some instances of protein side-chains. The probable regions of localization of calcium during morphogenesis are discussed in relation to a two-stage model for whorl formation published earlier from this laboratory. The first stage forms an annular distribution of a precursor which feeds into the whorl-forming second stage. Preliminary data on calcium-chlorotetracycline fluorescence give tentative support to the identification of this precursor with the bound-calcium species.

Coupling between reaction-diffusion prepattern and expressed morphogenesis, applied to desmids and dasyclads

A complete self-organizing mechanism for extension of a cell surface by tip growth, including cases in which the growing tip branches, e.g. in desmids such as Micrasterias and dasyclads such as Acetabularia , requires (if it is to be within the general class of kinetic theory) the following feedback loops: o 1. Molecular-kinetic feedback involving at least two morphogens, if the prepattern generation is to be expressed in terms of reaction-diffusion theory. 2. Catalysis of extension of the cell surface by one of the morphogens, leading to a change in shape of the region on which the chemical prepattern continues to readjust. This chemistry-geometry feedback can be programmed for computation without analytical expression in equations. 3. Control by the morphogenetic mechanism of its own boundaries. The boundary of the growing tip must move up in pace with its growth, for tip growth to be established at all. When growing tips branch, by the same token the growth habit of each branch is not defined as tip growth until the self-organizing mechanism has, without external help, drawn a new boundary around the morphogenetic region of each branch. Here we use feedback between the average local age of the cell surface and the input of a reactant into the reaction-diffusion system to form this loop. A mechanism including these three kinds of feedback, and using the Brusselator as the reaction-diffusion mechanism, is used first to show how a spherical cell may terminate its own growth. Reduced in dimensionality to a closed loop initially of circular shape, the model is used to produce: (a) development to non-circular shapes which are neither tip-growing nor branched; (b) development to an elongated shape with non-branching growing tip; (c) development to branching tip growth giving shapes suggesting those of semicells of the desmids Euastrum and Micrasterias .

The methylation of arsenate by a marine alga Polyphysa Peniculus in the presence of L-methionine-methyl-d3

Abstract Polyphysa peniculus growing in artificial seawater methylates arsenate to produce a dimethylarsenic derivative, probably dimethylarsinate. When L-methionine-methyl-d 3 is added to the culture the CD 3 label is incorporated intact in the dimethylarsenic compound to a considerable extent, indicating that S-adenosylmethionine, or some related sulphonium compound, is involved in the biological methylation. Conclusive evidence of the CD 3 incorporation was provided by using a specially developed hydride generation-gas chromatography-mass spectrometry methodology.

Turing's conditions and the analysis of morphogenetic models☆

Abstract The conditions under which changes in parameters lead to changes in the pattern-generating behaviour of Turings two-morphogen linear model can be expressed in terms of two reduced rate constants, k ′ 1 and k ′ 4 which represent autocatalytic and self-inhibitory rates in relation to cross-catalysis and cross-inhibition, and the ratio of the diffusivities for the two morphogens. This allows a new type of diagram to be drawn in which a two-dimensional k ′ 1 k ′ 4 space is divided by Turings conditions into regions where the various morphogenetic behaviours occur. An analysis using this type of diagram is applied to the linear limit of two non-linear models, those of Gierer and Meinhardt and of Tysons modification of the Brusselator, and is used to clarify what is happening in their non-linear development. Several possible applications are mentioned; to the stability of a simple binary pattern in slime moulds, to the development and decay of a succession of patterns in the imaginal wing discs of Drosophila as treated by Kauffman et al., and to the apparent ease of disturbing the sea urchin blastula to produce a three-part, rather than a two-part pattern.

ORDER AND LOCALIZATION IN REACTION-DIFFUSION PATTERN

The present work is concerned with two aspects of pattern formation: pattern localization and degree of pattern order. In reaction-diffusion models, there are three major effects. These stem from the reaction terms, the diffusion terms and the presence or absence of precursor gradients. Global analysis of reaction terms at late stages of pattern formation is at present unavailable. Therefore, we study the effect of the diffusion terms and of precursor gradients with numerical solution in two models: the Brusselator and the Gierer-Meinhardt. Differences in response to changes in the diffusion terms and the precursor gradients are related to contrasts between the nonlinear kinetics of the two models. These models both have Hill coefficient 2; effects of higher cooperativities have recently been discussed by Hunding and Engelhardt [1].