A Model of the Spatio-temporal Dynamics of Drosophila Eye Disc Development

Abstract

Patterning and growth are linked during early development and have to be tightly controlled to result in a functional tissue or organ. During the development of the Drosophila eye, this linkage is particularly clear: the growth of the eye primordium mainly results from proliferating cells ahead of the morphogenetic furrow (MF), a moving signaling wave that sweeps across the tissue from the posterior to the anterior side, that induces proliferating cells anterior to it to differentiate and become cell cycle quiescent in its wake. Therefore, final eye disc size depends on the proliferation rate of undifferentiated cells and on the speed with which the MF sweeps across the eye disc. We developed a spatio-temporal model of the growing eye disc based on the regulatory interactions controlled by the signals Decapentaplegic (Dpp), Hedgehog (Hh) and the transcription factor Homothorax (Hth) and explored how the signaling patterns affect the movement of the MF and impact on eye disc growth. We used published and new quantitative data to parameterize the model. In particular, two crucial parameter values, the degradation rate of Hth and the diffusion coefficient of Hh, were measured. The model is able to reproduce the linear movement of the MF and the termination of growth of the primordium. We further show that the model can explain several mutant phenotypes, but fails to reproduce the previously observed scaling of the Dpp gradient in the anterior compartment.

Fig 1. Model of regulatory network during eye disc development.

(a) During eye disc development the MF is initiated at the posterior margin and sweeps across the eye disc in an anterior direction. In front of the MF progenitor cells proliferate (arrow (A)1), while behind the MF cells differentiate and eventually form the ommatidia. Hh is expressed in the posterior margin (marked in orange), from where it diffuses into the eye disc (A2), and initiates expression of dpp in the MF (A3). Dpp signals by phosphorylation of Mad to pMad (A4). pMad (A5) as well as Hh (A6) enhance the expression of eya. Both pMad-mediated Dpp signaling (A7) as well as Hh signaling (A8) repress the expression of hth. Initially Hth is present throughout the disc. As the Hth levels decline, progenitor cells can transit into MF cells (A9). Hh supports the differentiation process by inducing the differentiation of MF cells (A10). Thus, Hh and Dpp/pMad together drive the progression of the MF. (b,c) A representative single (x,y) confocal section of a late third instar eye-antennal disc with only pMad staining (b) or merged channels (Hth: green, Eya: red; PMad: white) (c). The eye primordium is marked with an ellipse in (b,c). (d) A magnified sector of a similar eye disc along the A-P axis (top) and a z-section through this sector as in (bottom). Same color code as in (c).

Fig 2. Determination of the Hth degradation rate.

(a) View of the eye/antenna disc fragment. The recovery rate that was obtained on the antenna side (Fig 2a'') was within the standard deviation of the recovery rate obtained on the eye side (Fig 2a'). This implies that the degradation rate does not differ between these two tissues. (b) Time series of the logarithm of the relative Hth intensity between unbleached (red rectangle in Fig 2A) and bleached (yellow rectangle in Fig 2A) regions. The slope of the curve represents the degradation rate, δ_Hth = (6.97 ± 5.00) 10⁻⁵ s^-1; the linear fit is shown as red line; R² = 0.89.

Fig 3. Determination of the Hh diffusion coefficient.

(a) Part of the wing disc where the FRAP was made. The yellow solid circle shows the ROI selected to perform the photobleaching. The yellow dashed circle shows the photobleaching effective area. (b-d) Different time (T) points of the photobleached area. (b'-d') Detailed photobleached area where the differences in fluorescence can be observed. (b,b') Frame before the photobleaching. (c,c') Frame just after photobleaching. (d,d') T = 10 min. (e,e') T = 60 min. (f) Bleaching profile of the normalized mean intensity. Here can be observed the nominal diameter (2r_n) between yellow solid lines (correspondent to the photobleached ROI) and the effective diameter (2r_e), between yellow dashed lines, correspondent to the effective photobleached area. (g) FRAP recovery profile. The grey dashed lines show the half recovery time corresponding to one of the samples used in the experiment. The mean half recovery time is determined as τ_1/2 = 7.12 min (D_Hh = 0.033 ± 0.006 μm² s^-1).

Fig 4. Eye disc model captures linear MF movement and growth termination.

(a) Simulated progression of MF is approximately linear over time (black line). The speed of the MF movement is determined by a linear fit and is ≈3.4 μm h^-1. (b-d) Simulated total area (b), posterior area (c) and anterior area (d) over time in the eye disc model show growth termination towards the end of development.

Fig 5. Effects of parameter changes on MF speed, nonlinearity and final eye disc area.

Blue bars indicate a 1% decrease and red bars a 1% increase in parameter values compared with the wildtype (black line). In panels a, c, and e, the absolute values for speed, nonlinearity and area of the resulting eye discs are shown whereas in b, d, and f, the same data is shown in relation to the wildtype values.

Fig 6. Eye disc model can reproduce several mutant phenotypes.

(a,b) Comparison of the MF movement (a) and the total area over time (b) between a simulation using the original parametrization (indicated by the black line) and a simulation using a reduced hh production rate (indicated by the red line). In the case of the simulation representing the hypomorphic hh mutant the MF is clearly slower and eventually stops (a), while the total area is overgrowing (b). (c) Simulation of a clone where the hh production rate is set to zero at t = 55h. Towards the center of the clone the tissue is not differentiated (blue color) despite residing within the posterior area (colored in orange). (d) Posterior (indicated by the black line) and total (indicated by the green line) eye disc sizes in relation to the wildtype (indicated by the dashed grey line) at different relative influxes of Hh from the posterior margin. (e,f) Comparison of the total area over time (e) and the movement of the MF (f) between a simulation using the original parametrization (black line) and a simulation using a reduced (red line) dpp production rate. In panel e an additional simulation with increased dpp production rate (blue line) is compared. (g) Simulation of a clone where the hth production rate is increased at t = 20h. The MF (indicated by orange color) shows a retardation in the clone. (h) Total area over time in a numerical simulation representing the wildtype phenotype (black line) and a simulation using a reduced (red line) or increased (blue line) hth production rate.

Fig 7. The simulated Dpp gradient is not scaling in the anterior compartment.

(a,b) The simulated spatial profiles of Dpp (a) and pMad (b) in a developing eye disc at 20h (light blue), 40h (dark blue) and 60h (black). (c) Anterior length over time (black solid line). Vertical dashed lines indicate time points corresponding to (a,b).