Quantitative Relationships Between Growth, Differentiation, and Shape That Control Drosophila Eye Development and Its Variation

Abstract

The size of organs is critical for their function and often a defining trait of a species. Still, how organs reach a species-specific size or how this size varies during evolution are problems not yet solved. Here, we have investigated the conditions that ensure growth termination, variation of final size and the stability of the process for developmental systems that grow and differentiate simultaneously. Specifically, we present a theoretical model for the development of the Drosophila eye, a system where a wave of differentiation sweeps across a growing primordium. This model, which describes the system in a simplified form, predicts universal relationships linking final eye size and developmental time to a single parameter which integrates genetically-controlled variables, the rates of cell proliferation and differentiation, with geometrical factors. We find that the predictions of the theoretical model show good agreement with previously published experimental results. We also develop a new computational model that recapitulates the process more realistically and find concordance between this model and theory as well, but only when the primordium is circular. However, when the primordium is elliptical both models show discrepancies. We explain this difference by the mechanical interactions between cells, an aspect that is not included in the theoretical model. Globally, our work defines the quantitative relationships between rates of growth and differentiation and organ primordium size that ensure growth termination (and, thereby, specify final eye size) and determine the duration of the process; identifies geometrical dependencies of both size and developmental time; and uncovers potential instabilities of the system which might constraint developmental strategies to evolve eyes of different size.

Keywords: IbM computational model; computer simulation; drosophila; evolution; eye development; mathematical modeling; organ growth; size.

Figure 1

(A) Adult eye in Drosophila melanogaster. (B) Schematic representation of the shape of the eye according to the theoretical model. The blue rectangle corresponds to the primordium shape at t = 0. The black contour represents the final shape of the eye. The red curve is the shape of the eye at an intermediate stage. (C) Eye imaginal discs from early (left), mid (middle) and final (right) third larval stage, stained with Rhodamine-Phalloidin, to mark cell contours, and imaged using a Leica SPE confocal set up. Images were processed with Adobe Photoshop. In these figures X₀ and Y₀ correspond to the initial dimensions of the primordium. The morphogenetic furrow (MF) is represented as a vertical red dashed line that separates the anterior A from the posterior P region and it moves with speed v_f. The instantaneous and final posterior-to-anterior dimensions of P area, X_P and X_P,f, as well as the instantaneous dimension of A, X, and Y, are also represented in the figures.

Figure 2

Experimental measurement of P (black circles) and A (red squares) along with least-square fitting results to Equations (15) (black line) and (14) red line for the Drosophila strains GMR>+, Or-R and GMR>Upd. In the three panels, the x-axis represents X_P in μm and the y-axis depicts A and P in 10⁻⁴ · μm².

Figure 3

Middle panel: Dependence of the highest and lowest value of v_f for which eye growth termination does not occur ($v_f^{nc}$, red square and dashed line) or does take place ($v_f^c$, black circles and dashed line) with v_gr as obtained by computer simulation for a circular primordium. The solid blue line is the limiting theoretical value of v_f, v_f,l, from Equation (16). Both v_f and v_gr are expressed in units of σ/τ. Top and bottom panel: From right to left, sequence in the evolution of the eye as obtained with the simulation algorithm. In the top panel an eye that terminates its growth (v_gr = 6σ/τ, v_f = 37σ/τ, X₀ = 14.7σ and $\tilde{F}=0.606$). In the bottom panel the development of an eye that does not terminate growth (v_gr = 6σ/τ, v_f = 25σ/τ, X₀ = 14.7σ and $\tilde{F}=0.409$). In both panels, the number of particles (cells) is printed close to each snapshot. Both cases start with the same circular primordium (top right snapshot).

Figure 4

Time required to complete eye growth multiplied by the constant growth rate (k · t_f), as a function of $\tilde{F}=F/X_0$ obtained by computer simulation for the case of circular primordium and v_gr = 2σ/τ (black line and circles), 4σ/τ (red line and squares), 6σ/τ (green line and diamonds) and 10σ/τ (blue line and triangles). Violet right triangles correspond to oblate primordium with X₀/Y₀ = 0.540, orange left triangles correspond to prolate primordium with X₀/Y₀ = 1.923. In these last two cases v_gr = 6σ/τ. Symbols are simulation results, and solid lines are guidance for the eyes. Additionally, the universal law Equation (17) is plotted as a maroon dashed line. The values of this final time for four Drosphila strains calculated using the parameters of Table 1 in Equation (17) are included as open circles for GMR>+, open squares for Or-R and open diamonds for GMR>Upd. Here, as well as in Figure 5, it can be seen how the results of the simulation model with circular primordium agree with the predictions of the theoretical model, while there are systematic discrepancies for the cases of non-spherical primordium. Note that as $\tilde{F}$ decreases the finalization time t_f increase, and this increase is non-linear.

Figure 5

(A) Relative final size of the eye T_f/T₀ as a function of $\tilde{F}=F/X_0$ obtained by computer simulation for v_gr = 2σ/τ (black line and circles), 4σ/τ (red line and squares), 6σ/τ (green line and diamonds) and 10σ/τ (blue line and triangles). Violet right triangles correspond to oblate primordium with X₀/Y₀ = 0.540, orange left triangles correspond to prolate primordium with X₀/Y₀ = 1.923, in both cases with v_gr = 6σ/τ. Symbols are simulation results, and solid lines are guidance for the eyes. Additionally, the universal law Equation (18) is plotted as a maroon dashed line. The values of this relative final size for four Drospila strains calculated using the parameters of Table 1 in Equation (17) are included as open circles for GMR>+, open squares for Or-R and open diamond for GMR>Upd. Note that as $\tilde{F}$ decreases the relative final eye size increases, and this increase is non-linear. (B) In the inset are plotted measured values of eye head ratio distribution for GMR>Upd (black lines), GMR>+ (red lines) and Or-R strains. Solid and dashed lines correspond, respectively, to females and males. The relative standard deviation of these distributions are 0.9, 0.7, and 2.3 for Or-R, GMR>+ and GMR>Upd strains, respectively, without significant differences between males and females.

Figure 6

Dependence of the highest and lowest value of $\tilde{F}=F/X_0$ for which eye growth termination does not occur (red square) or does take place (black circles) with the shape of the primordium indicated as Y₀/X₀ as obtained by computer simulation. In all the cases v_gr = 6σ/τ. The snapshots are typical examples of primordia with X₀ > Y₀ and X₀ > Y₀. An example of the case X₀ = Y₀ is shown in Figure 3. The dashed line is the theoretical limiting value $\tilde{F}=F/X_0=0.5$.

Figure 7

Typical sequence śof snapshots of the growth of the eye obtained by computer simulation in the case of the circular (top row), oblate (middle row) and prolate (bottom row) primordium. The number of cells is indicated in each case. The cells in P region are colored red. The color of the cells in A region is a combination between blue and green, being the green contribution the square of the component of the force over the cell, while the blue contribution is the square of the vertical component of the force over the cell. For the three cases the number of particles in the primordium was 200 and v_gr = 6σ/τ. (X₀, v_f) = (14.7σ, 35σ/τ), (20σ, 42σ/τ), and (10.6σ, 29σ/τ) for circular, oblate and prolate primordium, respectively.

Figure 8

Relative final size of the eye (T_f/T₀, top panel) and eye termination time (t_f, bottom panel) for primordium width X₀ = 50μm (left column), 100μm (middle column) and 150μm (right column) for values of k ∈ [0.05, 0.4]h⁻¹ and $v_f\in \left[1,10\right]\mu m{h}^{-1}$ calculated with Equations (18, 17). The scale both for T_f/T₀ and t_f is display in the colorbar.