Waves and patterning in developmental biology: vertebrate segmentation and feather bud formation as case studies

Abstract

In this article we will discuss the integration of developmental patterning mechanisms with waves of competency that control the ability of a homogeneous field of cells to react to pattern forming cues and generate spatially heterogeneous patterns. We base our discussion around two well known patterning events that take place in the early embryo: somitogenesis and feather bud formation. We outline mathematical models to describe each patterning mechanism, present the results of numerical simulations and discuss the validity of each model in relation to our example patterning processes.

Figure 1. Schematic illustration of the vertebrate body plan during somitogenesis

The top part of the diagram shows the opposing wavefronts of FGF8 and retinoic acid with the determination front (threshold level of FGF signalling) marked. The middle section of the diagram depicts the AP axis with several pairs of somites at the anterior end, followed by a sequence of five potential somite pairs, marked by a genetic pre-pattern, and the PSM. The bottom part of the diagram illustrates the interaction with the segmentation clock. The light blue blocks mark the position of the next somite pair to be specified: the posterior boundary is fixed by the level of the determination front at the time at which cells at the anterior boundary become able to signal.

Figure 2. Schematic illustration of the processes involved in feather bud formation

Initially the field is incompetent and cells cannot form buds. As a wave of competency passes, cells become able to form microaggregates and begin to secrete bud -promoting and -inhibiting factors. Promoting factors (red) recruit more cells into each aggregate whilst inhibiting factors (green) stop aggregates from becoming too large. Eventually, some aggregates are stabilised and go on to become dermal condensations. Adapted from (Lin et al., 2006).

Figure 3. Schematic illustration of the interactions between activator (u) and inhibitor (v)

Arrowheads indicate catalysis whilst arrowtails indicate inhibition. Dashed lines indicate diffusion, with the length representative of the diffusion rate.

Figure 4. Numerical solution of a reaction-diffusion model in one spatial dimension: Activator

Small initial fluctuations in an otherwise homogeneous system are amplified into a series of peaks and troughs in chemical concentration. See Appendix B for more details.

Figure 5. Numerical solution of a reaction-diffusion model in one spatial dimension: Inhibitor

Small initial fluctuations in an otherwise homogeneous system are amplified into a spotted pattern of peaks and troughs in chemical concentration. See Appendix B for more details.

Figure 6. Numerical solution of a cell-chemotaxis model in two spatial dimensions: Activator

Small initial fluctuations are amplified into a series of peaks and troughs in cell density and chemical concentration. See Appendix C for more details.

Figure 7. Numerical solution of a cell-chemotaxis model in two spatial dimensions: Inhibitor

Small initial fluctuations are amplified into a complicated pattern of peaks and troughs in cell density and chemical concentration. See Appendix C for more details.

Figure 8. Numerical solution of the cell-chemotaxis model in one spatial dimension

Initially, the field is supposed to be homogeneous throughout, with a small perturbation made to the cell density at x = 0. In this case, the pattern propagates across the domain, from left to right.

Figure 9. The feedback mechanisms that lead to patterns in cell density

Reproduced with slight modifications from (Murray, 2003, Oster et al., 1983).

Figure 10. An illustration of the kind of propagating patterns that arise during pattern formation with the simplified model

The grey shading represents the competent region of the patterning domain (τ >τ_c in our model) which expands as time proceeds. Patterning occurs as follows: (i) initially the competent region is too narrow to permit pattern formation; (ii) as the domain expands an initial row of aggregations starts to form; (iii) as the competent domain expands even further, a second row forms, offset from the first. Reproduced with slight modifications from (Perelson et al., 1986).

Figure 11. Numerical solution of the clock and wavefront model in one spatial dimension

Continuous regression of the FGF8 wavefront (c), is accompanied by a series of pulses in signalling molecule (b), and coherent rises in somitic factor concentration (a).

Figure 12. A plot of h(k²) as given by equation (17)

The roots are given approximately by $k_{-}^2=0.0426$ and $k_{+}^2=0.2741$ (red asterisks), which gives a range of admissible modes: n = 6,7,K 13. Parameters are as follows: D = 30 and b = 0.35.

Figure 13. A plot of h(k²) given by equation (33)

The roots are given approximately by $k_{-}^2=0.0493$ and $k_{+}^2=0.8107$ (red asterisks), which gives a range of admissible modes: n = 6,7,K 22. Parameters are as follows: D = 25, χ = 1.9, r = 0.01 and N = 1.0

Figure 14. A plot of b(k²) given by equation (61)

The roots are given approximately by $k_{-}^2=0.0531$ and $k_{+}^2=0.2214$ (red asterisks), which gives a range of possible modes of n = 6,7,K 11. Parameters are as follows: μ = 0.01, τ′ = 15.0, λ = 2.0, γ = 1.7, and s′ = 0.1.