Quantifying the Gurken morphogen gradient in Drosophila oogenesis

Abstract

Quantitative information about the distribution of morphogens is crucial for understanding their effects on cell-fate determination, yet it is difficult to obtain through direct measurements. We have developed a parameter estimation approach for quantifying the spatial distribution of Gurken, a TGFalpha-like EGFR ligand that acts as a morphogen in Drosophila oogenesis. Modeling of Gurken/EGFR system shows that the shape of the Gurken gradient is controlled by a single dimensionless parameter, the Thiele modulus, which reflects the relative importance of ligand diffusion and degradation. By combining the model with genetic alterations of EGFR levels, we have estimated the value of the Thiele modulus in the wild-type egg chamber. This provides a direct characterization of the shape of the Gurken gradient and demonstrates how parameter estimation techniques can be used to quantify morphogen gradients in development.

Figure 1. Biophysical Model Gurken Secretion, Diffusion, Binding, and Degradation

(A) The model is formulated in a spheroidal coordinates. H denotes the width of the extracellular gap where ligand transport takes place; L_DV and L_AP the equatorial and the polar radii of the spheroid modeling the oocyte. (B) The model includes localized secretion of Gurken from the oocyte (with a constant flux V), ligand transport (with diffusion coefficient D), ligand-receptor binding (with rate constants k_on and k_off), and ligand-induced endocytosis (with rate constant k_e). The three variables in the model are: G, the concentration of the Gurken molecules; R, the surface density of empty EGF receptors; and C, the surface density of Gurken-EGFR complexes. (C) The Gurken gradient computed at Φ = 1. D, dorsal; V, ventral; P, posterior; A, anterior. (D) DV and AP concentration profiles computed for Φ = 1 along the broken lines. (E) The ratio of Gurken concentrations at the ventralmost and dorsalmost positions at the anterior boundary (denoted by the broken line in [C]), computed as a function of the Thiele modulus. g_D and g_V denote the Gurken concentration at the dorsalmost and ventralmost points, respectively.

Figure 2. Model of Gurken-Mediated pipe Repression

(A) The Gurken-mediated pipe repression is modeled as switch-like response (C_T, the concentration of Gurken/EGFR complex at the threshold of pipe expression; D, dorsal; V, ventral; L, half circumference of the egg chamber at the anterior boundary). (B) Comparison between the observed pipe staining in wt and the pipe domain solved using the procedure described in the text. For an arbitrary value of Thiele modulus (Φ) the threshold parameter γ is computed from the Gurken/EGFR complex concentration at 40% ventral along the anterior cross-section (i.e., the pipe domain observed in the wild-type). By using the computed γ at the anterior end, the boundary of pipe expression along the AP-direction is determined by tracing the line of equal Gurken/EGFR complex concentration.

Figure 3. Model Predictions

(A) Graphical illustration of Equation 2: the wild-type location of the boundary of pipe expression domain (θ₀) defines the threshold parameter γ for an arbitrarily set starting Thiele modulus (Φ₀). Demanding that γ remain constant as the Thiele modulus changes, it is possible to compute the new domain of pipe expression (θ₁) corresponding to the new Thiele modulus (Φ₁). See text for more details. (B) The Thiele modulus depends on EGFR level in the follicle cells (Φ ~ R^0.5). The model predicts that the domain of pipe expands/contracts as the receptor level increases/decreases, respectively. Shown are the response curves computed for three different values of the starting Thiele modulus (Φ₀).

Figure 4. Measurements of the Width of the pipe Expression Domain in Genetic Backgrounds with Different Levels of EGFR Expression

Measurements were performed on the cross-sections of egg chambers from late stage 9 to early stage 10B. The differences in the measured pipe domains are statistically significant (p < 0.001); error bars in the bar graphs correspond to 1.96SE. No significant correlation was observed between the fractional domain of pipe and the size of the egg chambers. The size of the egg chambers is not affected by changes in the receptor level (see Table S1).

Figure 5. The Thiele Modulus of the Wild-Type Gurken Gradient

(A) A contour plot of the residual from the optimization procedure for finding the estimate of the Thiele modulus. The color bar corresponds to the log₁₀ value of the residual. Based on the medians of the data, the estimate for the Thiele modulus is found to be 2.7. (B and C) Cross-sections through the minimum of the residual function. (D) The wild-type Gurken gradient along the anterior circumference, computed with the estimated Thiele modulus. Plotted in the y axis is the Gurken concentration scaled by the maximum concentration at the dorsalmost point. Dotted lines show the profiles computed for Thiele moduli at the boundaries of the 90% confidence interval. (E) Histogram for the fitted values of the Thiele modulus computed by Bootstrap. (F) Bootstrap quantiles of the estimated Thiele modulus.

Figure 6. Summary of the Quantitative Analysis of the Gurken Gradient

Gurken is locally secreted from the dorsal anterior cortex of the oocyte and forms a shallow gradient with a Thiele modulus of 2.7 (1.5, 5.1). When normalized to the maximal concentration at the dorsal side, the Gurken gradient drops to 63% (73%, 56%) at one-third dorsal, which coincides roughly with the boundary of the dorsal genes (such as kekkon and sprouty; blue), 22% (42%, 8%) at the boundary of pipe expression (red), and 10% (31%, 1%) at the ventral midline. Numbers in brackets correspond to 90% confidence intervals of the derived estimates.