A model of gradient interpretation based on morphogen binding

French flag gradients and Turing reaction-diffusion versus differentiation waves as models of morphogenesis

The Turing reaction-diffusion model and the French Flag Model are widely accepted in the field of development as the best models for explaining embryogenesis. Virtually all current attempts to understand cell differentiation in embryos begin and end with the assumption that some combination of these two models works. The result may become a bias in embryogenesis in assuming the problem has been solved by these two-chemical substance-based models. Neither model is applied consistently. We review the differences between the French Flag, Turing reaction-diffusion model, and a mechanochemical model called the differentiation wave/cell state splitter model. The cytoskeletal cell state splitter and the embryonic differentiation waves was first proposed in 1987 as a combined physics and chemistry model for cell differentiation in embryos, based on empirical observations on urodele amphibian embryos. We hope that the development of theory can be advanced and observations relevant to distinguishing the embryonic differentiation wave model from the French Flag model and reaction-diffusion equations will be taken up by experimentalists. Experimentalists rely on mathematical biologists for theory, and therefore depend on them for what parameters they choose to measure and ignore. Therefore, mathematical biologists need to fully understand the distinctions between these three models.

The organelle of differentiation in embryos: the cell state splitter

The cell state splitter is a membraneless organelle at the apical end of each epithelial cell in a developing embryo. It consists of a microfilament ring and an intermediate filament ring subtending a microtubule mat. The microtubules and microfilament ring are in mechanical opposition as in a tensegrity structure. The cell state splitter is bistable, perturbations causing it to contract or expand radially. The intermediate filament ring provides metastability against small perturbations. Once this snap-through organelle is triggered, it initiates signal transduction to the nucleus, which changes gene expression in one of two readied manners, causing its cell to undergo a step of determination and subsequent differentiation. The cell state splitter also triggers the cell state splitters of adjacent cells to respond, resulting in a differentiation wave. Embryogenesis may be represented then as a bifurcating differentiation tree, each edge representing one cell type. In combination with the differentiation waves they propagate, cell state splitters explain the spatiotemporal course of differentiation in the developing embryo. This review is excerpted from and elaborates on "Embryogenesis Explained" (World Scientific Publishing, Singapore, 2016).

Genesis of two-dimensional patterns in cross-gradient fields

Tissue morphogenesis is controlled by the two-dimensional patterning of gene expression in epithelial layers, that determines cell fates. The mechanisms of pattern formation involve intracellular regulatory networks controlled by paracrine and autocrine signaling. We develop a general logical scheme to deduce the morphology of two-dimensional patterns in the field of two crossed morphogen gradients enriched by the action of autocrine signaling that may subdivide expression domains in nontrivial ways. A variety of persistent patterns, either stationary or oscillatory, are generated using the various interaction schemes, some of which have been generated by a special algorithm including random inputs and selected according to suitable criteria. We give the full classification of a variety of stationary and oscillatory expression patterns in the presence of a single autocrine signal based on logical arguments. These results are further confirmed by numerical computations based on reaction-diffusion equations for morphogens and ligands and the discrete (cell-level) description of intracellular dynamics. Model simulations also elucidate transient processes, in particular interaction schemes. Different internal schemes may lead to identical persistent patterns, although relaxation may proceed in distinct ways. As an illustration of the general method, we test a particular scheme suggested by genetic studies of dynamic gene expression patterns in the follicular epithelium of the Drosophila eggshell.

A gradient of bicoid protein in Drosophila embryos

The maternal gene bicoid (bcd) organizes anterior development in Drosophila. Its mRNA is localized at the anterior tip of the oocyte and early embryo. Antibodies raised against bcd fusion proteins recognize a 55-57 kd doublet band in Western blots of extracts of 0-4 hr old embryos. This protein is absent or reduced in embryonic extracts of nine of the 11 bcd alleles. The protein is concentrated in the nuclei of cleavage stage embryos. It cannot be detected in oocytes, indicating temporal control of bcd mRNA translation. The bcd protein is distributed in an exponential concentration gradient with a maximum at the anterior tip, reaching background levels in the posterior third of the embryo. The gradient is probably generated by diffusion from the local mRNA source and dispersed degradation.

Extravascular fluid dynamics of the embryonic chick wing bud

While a number of models of positional information in the chick wing bud have involved the diffusion of morphogens to establish chemical gradients of morphogenetic activity, only recently have there been attempts to characterize the milieu in which such diffusion must take place. We report an analysis of the fluid dynamics of the extravascular (interstitial) spaces of stage 22-25 chick wing buds, into which aqueous aniline blue dye was microinjected as a visible, unreactive tracer. Six sites along the antero-posterior (A-P) and proximo-distal (P-D) axes were chosen for study. Injections of dye into the posterior half of the wing bud exhibited marked directionality of extravascular transport (mean of all posterior sites = 68%), while anterior injections showed little or no directionality (mean of all anterior sites = 13%). All instances of directed dye movement were disto-proximal, the same direction as the blood flow through the marginal veins. In embryos gassed in situ with CO2, which reversibly stopped the heartbeat and vascular flow, directionality was abolished, yet diffusion rates were unaffected. Posterior disto-proximal extravascular dye movement was correlated with the greater diameter, flow velocity, and volume flow rate of the posterior marginal vein, compared to the anterior marginal vein. Radial diffusion rates were measured, and posterior disto-proximal rates were corrected for measured disto-proximal directionality by the use of a simple diffusion-translation model. Three-way analysis of variance showed that directionality-uncorrected disto-proximal rates in posterior sites were not significantly different from anterior radial rates, but that directionality-corrected posterior rates did differ significantly (P less than 0.0001). A significant stage effect (P less than 0.005) and a significant interaction between the A-P axis and stage (P less than 0.05) were also found. We hypothesize that the spatial arrangement and flow patterns of the vasculature physically determine the fluid dynamics of the interstitium. Based on these observations, we also suggest that disto-proximal extravascular fluid movement in the posterior wing bud presents a barrier to the free diffusion of aqueous molecules, including morphogens originating in the "zone of polarizing activity."

The anatomy, ultrastructure and fluid dynamics of the developing vasculature of the embryonic chick wing bud

The spatiotemporal sequence of vascular pattern development in the embryonic chick wing bud and surrounding shoulder, flank and belly regions is detailed for Hamburger-Hamilton (1951) stages 20-25. Vasculature was microinjected with an unreactive aqueous tracer (aniline blue), and major traffic patterns were visualized. Formation of extensive avascular regions and the emergence of chondrogenic phenotypes are correlated with the retreat of the vasculature from the wing core. Ultrastructural studies of vascular cells show that vessels remain monolayered and undifferentiated until stage 25, after the adult vascular pattern has been laid down. Vascular cytodifferentiation occurs only in the cells of the brachial artery until stage 35, with the veins and capillaries retaining an 'early' morphology. This vascular pattern may be an important component reflecting or directing limb pattern development.

A reaction-diffusion theory of morphogenesis with inherent pattern invariance under scale variations

In the framework of reaction-diffusion theory we deal with the problem of pattern regulation in morphogenesis. A generic model is proposed where the kinetic terms follow constraints imposed by scale invariance considerations. These constraints allow a class of kinetic schemes to be formulated so that, starting with an initially homogeneous morphogen distribution in the field, a stable gradient is established of the form: S(chi,L) = Lpf(chi/L). Here L is the length of the morphogenetic field, chi is the position variable and f(chi/L) is some monotonic function of the relative distance. With this distribution a scale invariant gradient can be constructed which leads to pattern regulation. A linear stability analysis of the model permits the definition of the parameter values enabling the system to abandon the homogeneous state spontaneously. Simulations of the evolution of the system towards its final stable state result in approximate pattern invariance for different field lengths. The accuracy of this invariance is in agreement with some recent quantitative experimental findings in both developing and regenerating systems.

A morphogen gradient model for pattern regulation

A model for pattern formation is proposed based on two concentration gradients S and Sigma. S is a local morphogen generated by a reaction-diffusion mechanism while Sigma is a by-product of the S-decomposition. Under certain conditions S is well approximated by S(x,L) = alpha(L)f(x L ), where alpha(L) is a scaling function of the length L and f(x L ) is a monotonie function of the relative distance x L from the origin. Sigma degradates and diffuses in the field, reaching a stable L-dependent homogeneous distribution. An allosteric protein P with several active sites reacts with S and Sigma and separates the field into segments. To every segment a corresponding active state of P is dominant. Pattern regulation is automatically achieved since the compartmerttal separation depends explicitly only on x L . For the case of repetitive patterns, a supplementary Gierer-Meinhardt mechanism is introduced for activator X and inhibitor Y. The level of Sigma can affect the decomposition rate of X or Y, e.g. via a second order degradation reaction, hence making the chemical wavelength lambda size-dependent. For a particular decay scheme of Y, a variation of L induces a change of lambda so that finally the number of repetitive structures becomes independent of the field size.

The prestalk-prespore pattern in cellular slime molds

In cellular slime molds the slugs become divided into two regions with different properties, and anterior prestalk-zone and a posterior prespore zone. Although the cells in these zones are normally destined to form the stalk cells and spores of the fruiting body, respectively, they are not irreversibly committed to one sort of differentiation or the other during the slug stage. The volume ratio of the two zones remains almost constant over a wide range of slug sizes. If the prestalk-prespore pattern is distrubed by removing tissue from the slug, conversion of tissue from prestalk to prespore or vice versa occurs as necessary to restore a normal pattern with normal proportions. Conversions also occur in both directions during normal development. The initial formation of the prestalk-prespore pattern may well involve sorting-out, but other mechanisms must be invoked to account for regulation. We describe three different models of the generation of the prestalk-prespore pattern, the'cell-contact model' of McMahon, in which pattern is created by interactions of cells with their immediate neighbors, the 'positional-information model' of various authors, in which pattern formation involves an overall gradient and a gradient-reading mechanism, and the 'activator-inhibitor model' of Gierer and Meinhardt, in which the prestalk-prespore pattern is formed by a system of diffusible substances that affect one another's production. The activator-inhibitor model is the most successful of the models at describing the known features of the prestalk-prespore pattern. The various models lead to a number of distinctive predictions. According to the cell-contact model, small transplants may cause gross changes in the prestalk-prespore pattern, and mutants may exist which severely disrupt pattern formation even if diluted with a large excess of wild-type cells. Positional-information models predict the existence of 'gradient-reading mutants'; slugs that are a mixture of such mutants and wild-type cells would show two prestalk-prespore boundaries, one at the mutant and one at the normal position. Both the activator-inhibitor model and some versions of the positional-information model predict that small transplants will sometimes induce accessory prestalk or prespore zones; the quantitative characteristics of these effects may allow one to make a case in favor of one or other of the two models. Finally, the activator-inhibitor model leads one to expect that mutants may be isolated which normally show accessory prestalk or prespore zones. A search for these phenomena may help determine whether the activator-inhibitor model will continue to enjoy its present preeminent position.

Dynamics of skeletal pattern formation in developing chick limb

During development of the embryonic chick limb the skeletal pattern is laid out as cartilaginous primordia, which emerge in a proximodistal sequence over a period of 4 days. The differentiation of cartilage is preceded by changes in cellular contacts at specific locations in the precartilage mesenchyme. Under realistic assumptions, the biosynthesis and diffusion through the extracellular matrix of a cell surface protein, such as fibronectin, will lead to spatial patterns of this molecule that could be the basis of the emergent primordia. As cellular differentiation proceeds, the size of the mesenchymal diffusion chamber is reduced in descrete steps, leading to sequential reorganizations of the morphogen pattern. The successive patterns correspond to observed rows of skeletal elements, whose emergence, in theory and in practice, depends on the maintenance of a unique boundary condition at the limb bud apex.