Periodic patterning of the Drosophila eye is stabilized by the diffusible activator Scabrous

Abstract

Generation of periodic patterns is fundamental to the differentiation of multiple tissues during development. How such patterns form robustly is still unclear. The Drosophila eye comprises ∼750 units, whose crystalline order is set during differentiation of the eye imaginal disc: an activation wave sweeping across the disc is coupled to lateral inhibition, sequentially selecting pro-neural cells. Using mathematical modelling, here we show that this template-based lateral inhibition is highly sensitive to spatial variations in biochemical parameters and cell sizes. We reveal the basis of this sensitivity, and suggest that it can be overcome by assuming a short-range diffusible activator. Clonal experiments identify Scabrous, a previously implicated inhibitor, as the predicted activator. Our results reveal the mechanism by which periodic patterning in the fly eye is stabilized against spatial variations, highlighting how the need to maintain robustness shapes the design of patterning circuits.

Figure 1. A model of eye disc patterning.

Here and in all figures posterior is to the right. (a) Confocal image of the developing disc. Cell outlines are visualized with FasIII (red), and differentiating cells with the nuclear differentiation marker Sens (blue). Inset: a magnified view of the boxed region, including and immediately posterior to the MF (red line). Scale bar, 10 μm. (b) Schematic representation of sens expression. ato is first expressed uniformly (blue stripe) anterior to the MF (red line), upon which it is refined into pro-neural clusters and expressed together with sens. (c) Equations and interactions are shown for the template-based lateral-inhibition patterning model. h denotes the long-range activator (representing Hh and Dpp), a the cell-autonomous activator (representing Ato and Sens) and u the diffusible inhibitor (representing Sca and the Egfr-dependent signal). Simulations are performed by discretizing the equations on a 42 × 42 cell grid, with h value approximated by its analytical value (Supplementary equation (12.3)) whose approximation is shown here (v represents wave velocity). τ_a and τ_u denote the typical timescales characterizing the dynamics of a and u, respectively; λ_a and λ_u are the respective degradation rates; P_a and P_u are the respective production rates; D_u is the diffusion constant of u. θ(a) is the Heaviside step function, which is equal to 1 for positive values and zero otherwise. The upper-script indexes (i) indicate the grid site and ∇² is the grid two-dimensional Laplacian operator. More details are given in Supplementary Note 1. (d,e) Parameters defining the inhibition range (effective diffusion, D_u, and normalized production rate P_u/u₁) were varied systematically, as shown. White regions denote parameter combinations where no periodic solution was found. Colour scale quantifies noise sensitivity by the maximal spatial variations that can be added without pattern failure. We examined patterns of solitary cells (d) and of clusters (e). Panel 3′ demonstrates pattern failure after adding less than 5% noise to parameters generating the pattern in panel 3. For each parameter configuration, maximal noise is reported after simulating all possible initial conditions. The parameter space enclosed below the dashed line (N.S.S; non-sufficient spacing) yields patterns in which the area surrounding each cluster does not comprise the required 20 cells forming the ommatidium. See Supplementary Methods for details of how noise was defined and Supplementary Table 1 for the parameters used.

Figure 2. Incompatibility of linear activation propagation and radial inhibition.

(a) Pattern formation in one dimension. Time is on the vertical axis and cell position on the horizontal axis. Cluster size is defined by the number of cells that become refractory before the first cell in the cluster produces inhibition (red horizontal line). is the time when cell i receives sufficient h to induce a expression, provided that it was not yet inhibited. is the time when an activated cell becomes refractory. is the time when an activated cell begins secreting the inhibitor u. Δt^ac is the time gap between activation of two adjacent cells. n is the size of the cluster. R is the distance between clusters. The dependence of those times on the model parameters is derived in Supplementary Note 2. (b) Shown is the noise sensitivity of the simplified model in two dimensions for different values of inhibition radii and cluster sizes. Simulations were performed on a grid by sequentially selecting clusters of the desired size and drawing inhibition radii around them. Noise sensitivity was defined by the maximal noise level that can be introduced before pattern failure, for optimized initial conditions as we did in Fig. 1c. Noise was introduced by selecting each inhibition radius from a uniform distribution that was centred at the indicated values R and whose width was , defining the noise level. A pattern of cluster size 1 was considered destroyed when a cluster of size larger than 3 was formed. Similarly, a cluster of 3, 6 and 10 was considered destroyed when clusters of 6, 10 and 15 formed, respectively. All simulations were run until cell differentiation reached the end of the grid. N.S.S. stands for non-sufficient spacing as in Fig. 1d. (c) Selection of a long, uninhibited cell line (catastrophe) is the main source of noise sensitivity. See Supplementary Fig. 2 for more details. (d) Same as (b) for the extended model including an activator. In addition to adding noise to the inhibition radii, noise was added to the activation radii in a similar manner. Pattern failure was determined as in b. (e) Since cluster size is now defined by the short-range activator, rather than propagation of h, sensitivity to catastrophes is reduced.

Figure 3. An extended model including a short-range activator.

(a,b) The extended model including a diffusible activator s. Shown are the model's equations, the scheme of the interactions and illustration of the activating dynamics. τ_s, P_s, λ_s, D_s are the respective timescale, production rate, degradation and diffusion constants of the short-range activator added to the template-based lateral-inhibition patterning model. In b, black arrows depict progression of the short-range activator signal and the red arrow depicts that of the inhibitor. (c) Reduced sensitivity of the model to spatial heterogeneity. Sensitivity analysis for the extended model is shown, similar to the one conducted for the original model. See Supplementary Note 6 and Supplementary Fig. 3 for details and Supplementary Tables 1 and 2 for parameters. (d) Quantitative comparison of the robustness with or without the addition of the short-range activator. Left panel compares the maximal noise that could be added in simulations shown in c to that in simulations in Fig. 1e (no activator). Right panel compares the area of the coloured regions in these simulations, where patterns could be obtained.

Figure 4. Simulating sca function as a competitive inhibitor of Notch activity.

(a) Model scheme of the full network including Notch and Delta. Model equations are given in the Methods and in Supplementary Note 7. (b) Introducing the Notch–Delta interactions enables simulating cluster formation together with cluster refinement. Simulations of the same eye disc are shown at different times (indicated by horizontal arrow), capturing the different stages in eye development. (c,d) An analysis of the sensitivity of the model to spatial heterogeneity for the extended model. Quantitative comparison of the robustness with or without the addition of the short-range activator is shown in d. Left panel compares the maximal noise that could be added in simulations of the full network shown in c to that in simulations in Fig. 1e (no activator). Right panel compares the area of the coloured regions in these simulations, where patterns could be obtained. Parameters can be found in Supplementary Table 3. (e) In sca loss of function mutants, the first distinct phenotype is impaired cluster formation because of increased sensitivity to spatial heterogeneity. The second distinct phenotype is reflected in many cases of impaired cluster refinement leading to a high frequency of twining. Simulations are shown at different times, similar to those shown in b.

Figure 5. Clonal analysis supports Sca role as activator of ato expression.

(a) Simulations of shallow clones (enclosed by broken line) result in smaller clusters. In grey are cluster cells that were inhibited by the selected cells in each cluster (in black). (b) The nuclei stained for Sens are in blue, and shallow clones that lack sca are visualized by lack of GFP signal in green indicated by the white broken line. In the right panel (b′′), clusters are marked in 3D in red using the Imaris imaging software that quantifies the volume of each cluster. White asterisks indicate smaller clusters inside the clones. Between the indicated clusters there is a gap where no cluster has formed. Inset shows the clusters after 90° rotation. Scale bar, 10 μm. (c,d) Same as a,b, but for a deep clone. White arrows in d indicate a larger cluster, while red arrows indicate a smaller cluster in the shallow region of the clone. Scale bar, 10 μm. (e) Shown are the average cluster sizes within sca mutant clones as a function of clone depth. Clusters larger than 600 μm³ where not included. Number of clones analysed are n=55,13,10,6,4,4,4,4 for clones of depth one to eight columns, respectively. Error bars denote s.e. (f) Simulation of Sca gain-of-function (GOF) discs. (g) Cluster density in sca−/− mutants and in sca gain-of-function flies carrying two or four (2 × and 4 ×) copies of roE-sca (an enhancer fragment of the rough gene that drives sca expression) was quantified in ref. (black bars). Regions containing 17 or 18 clusters in at least four discs from each genotype were subjected to this quantitative analysis. Cluster density in our simulations was quantified by dividing the number of cells in the clusters by the number of total cells in 72 simulations for each genotype. Error bars denote s.e. (h,i) Same as a,b for clones depleted of pnt. (j) The percentage of larger sca−/− and pnt−/− mutant clusters as a function of clone depth. Number of sca−/− clones analysed is the same as in e. At least three pnt−/− clones were analysed for each depth.