Models for the generation and interpretation of gradients

Abstract

Source regions for morphogen gradients-organizing regions-can be generated if a local self-enhancing reaction is coupled with a long-ranging reaction that acts antagonistically. Resulting gradients can be translated into patterns of stable gene activities using genes whose products have a positive feedback on the activation on themselves. If several autoregulatory genes compete with each other for activity, cells make an unequivocal choice. Although the signal consists of a smoothly graded distribution, the all-or-nothing response of the cells leads to regions of differently determined cells that are delimited by sharp borders. In some systems, it is not the absolute but the relative level of a gradient that matters. The sequence of head, tentacles, and foot formation in hydra is controlled by a head activation gradient and is an example of this widely used but conceptually rather neglected mode. For subpatterns such as legs and wings, different "compartments" cooperate to produce new signaling substances. Here, morphogen production is restricted to the common borders or where they intersect. The model accounts for the formation of substructures in pairs at the correct positions within the embryo and for the correct orientation and handedness with respect to the main body axes.

Figure 1.

Pattern formation by an activator–inhibitor interaction. (A) Reaction scheme: The activator catalyses its own production. The production of its rapidly spreading antagonist, the inhibitor, is also under activator control (Gierer and Meinhardt 1972; Meinhardt 1982, 2008). (B) In such a reaction, the homogeneous distribution of both substances is unstable. Small random fluctuations in the ability to produce these substances (blue squares) are sufficient to initiate pattern formation. A high concentration appears at a marginal position. Thus, although the genetic information is the same in all cells, such a system is able to generate a reproducible polar pattern, appropriate to accomplish space-dependent cell differentiation (see Fig. 3). (C) A biological example: the emerging Nodal gradient in the sea urchin, responsible for the formation of the oral field. (D) Antivin (or Lefty2, left) acts as inhibitor (Duboc et al. 2004). As predicted, it is produced at the same position as the activator. (E–H) Regeneration. After fragmentation, in the nonactivated fragment, the remnant inhibitor disappears (F) until a new activation is triggered (G). The graded profiles are restored (H) as long as the remaining fragment is still large enough. Because the inhibitor can escape only into a smaller nonactivated region, the activations are somewhat reduced (Fig. C kindly provided by Dr. Thierry Lepage).

Figure 2.

Model for the dorsoventral (DV) patterning in vertebrates and the geometry of axis formation. (A) Reaction scheme: The mutual inhibition of Chordin and BMP generates a system with self-enhancing properties as required for pattern formation. ADMP is assumed to act as the inhibitor because it is under the same control as Chordin, and diffuses more rapidly. It blocks the self-enhancing process by undermining the BMP repression exerted by Chordin. (B) Geometry: The ancestral organizer of a hydra-like ancestor became a large ring in vertebrates, e.g., the marginal zone in amphibians (red: Brachyury expression). The Wnt signal (fading blue), produced in marginal zone, accomplishes the anteroposterior (AP) specification. The Spemann-organizer (SO) can only be formed in the marginal zone. (C) With mesoderm ingression, the organizer-derived mesodermal cells form the prechordal plate (yellow) that acts as line of reference for the mediolateral organization. Both organizers together set up a perfect Cartesian coordinate system: the marginal zone for the AP and the prechordal plate for the DV or mediolateral axis. (D) Simulation of pattern formation according to the scheme (A). Even when starting from homogeneous distributions, a sharp Chordin peak (green) emerges. BMP has a complementary distribution and is assumed to provide positional information for the DV axis. (E) After organizer removal, the organizer regenerates (for details, see Meinhardt 2006, 2008).

Figure 3.

Space-dependent activation of several genes under the influence of a morphogenetic gradient. The genes whose gene products have a positive and nonlinear feedback on the activation of their own gene are assumed. They compete with each other for activity. (A) Starting with a default activation of the gene 1 (blue), the genes 2, 3, and 4 become successively activated. Regions with sharp borders are formed. In a given cell, only one of the alternative genes can be active. The sequential activation of genes proceeds faster in regions of high signal concentration, which leads to the apparent wavelike movement of gene activities. Activation of new genes occur only unidirectionally (distal or posterior transformation). With the activation of a further gene, the concentration of a common repressor (grey) increases. In the model, this reduces the sensitivity for the signal, causing the sequential gene switching to come to rest according to the local signal level. (B) A simulation as shown in (A); the gene activities are plotted as densities of pixels, analogous to what is seen in in situ hybridizations. The slowing down of the wavelike movement of particular gene activities and the sharpening of the borders is clearly visible. (C) A short-ranging gradient can control a larger region if cell proliferation is essentially restricted to the source region. Cells leaving the source region because of proliferation enter a region of lower signal strength and attain a stable determination. Evidence for such a mechanism exists for the determination of the digits in the chicken wing bud (Harfe et al. 2004; see also Bénazet and Zeller 2009). (D) Scheme: several autoregulatory genes are assumed that locally compete with each other either by a common repressor R or by a direct negative cross-regulation between alternative gene products g₁, g₂ … (Meinhardt 1978, 1982).

Figure 4.

Patterning in hydra, a system in which relative levels of a gradient are important. (A) Small fragments of hydra always regenerate with the original polarity. (B) During regeneration, the signal for tentacle formation appears first in the cells at the extreme tip and shift later to the appropriate position (Bode et al. 1988; signal for tentacle formation is visualized by antibodies, shown here in red). (C), (D) Model: The signals for the formation of the head (green), tentacles (dark red), and foot (pink) are assumed to be generated by separate activator–inhibitor systems. These systems are positioned by source density gradient (blue) that is elevated by the head-system and down-regulated by the foot-system. The source density gradient corresponds to the head activation gradient in the hydra literature. Tentacles appear in the region of the highest source density that is not occupied by the head signal. (E), (F) Small fragments regenerate all signals with the correct polarity. The head is formed at the relatively highest levels of the residual head activation gradient, the foot at the lowest. The temporary formation of the tentacle signal at the very tip as shown in (B) is correctly described (F) (simulations after Meinhardt 1993; see also Bode 2009).

Figure 5.

Formation of positional information for substructures at boundaries between differently determined cells (Meinhardt 1982, ,b); the formation of insect legs as an example. (A) in the posterior compartment (P, red) a diffusible cofactor (brown, hedgehog) is produced. It enables morphogen production (dpp) in the anterior compartment (A, green). The region of production (blue) restricted to A-cells that are close to the A/P border. Because of diffusion, a bell-shaped distribution arises (pink). The local concentration is a measure for the distance of a cell from the A/P boundary. (B) Geometry of leg formation. Segmentation was proposed to be based on the repetition of at least three cell states, … A/PSA/P … (Meinhardt 1982). Thus, only one A/P border per segment exists. Leg formation requires an intersection of this A/P border with a D/V border. Leg formation occurs in pairs that have opposite handedness (arrow heads). (C, D) A collaboration of all three compartments (AD, AV, and P) for a morphogen production leads to a conical morphogen distribution (dll, EGF) that is appropriate to generate the concentric fate map of the insect leg. (E) Heat shock can lead to supernumerary limbs (Girton 1981). The limbs are formed in a plane; the two outer limbs have the normal, the central limb the opposite handedness. (F, G) Model: The heat shock is assumed to cause a flip of some A-cells to a P-specification. The patch of P-cells located on the DV-border in the A-compartment leads to two new intersections along the DV border and thus to two additional limbs. The model-predicted handedness agrees with the observation.