Quantitative analysis and modeling probe polarity establishment in C. elegans embryos

Abstract

Cell polarity underlies many aspects of metazoan development and homeostasis, and relies notably on a set of PAR proteins located at the cell cortex. How these proteins interact in space and time remains incompletely understood. We performed a quantitative assessment of polarity establishment in one-cell stage Caenorhabditis elegans embryos by combining time-lapse microscopy and image analysis. We used our extensive data set to challenge and further specify an extant mathematical model. Using likelihood-based calibration, we uncovered that cooperativity is required for both anterior and posterior PAR complexes. Moreover, we analyzed the dependence of polarity establishment on changes in size or temperature. The observed robustness of PAR domain dimensions in embryos of different sizes is in agreement with a model incorporating fixed protein concentrations and variations in embryo surface/volume ratio. In addition, we quantified the dynamics of polarity establishment over most of the viable temperatures range of C. elegans. Modeling of these data suggests that diffusion of PAR proteins is the process most affected by temperature changes, although cortical flows appear unaffected. Overall, our quantitative analytical framework provides insights into the dynamics of polarity establishment in a developing system.

Figure 1

Quantifying polarity establishment. (A–D) Automated segmentation using ASSET. ASSET first (A) identifies the eggshell (green) in the DIC image, and then (B) segments the cell membrane (red) in the mCherry::PH image, and finally (C) transposes these two segmentations (green and red) to the GFP::PAR-2 channel and smoothes them (light green and orange). The white box delimits the area magnified in (D). Numbers in (C) and (E) denote corresponding locations. (D) Magnification of area delimited in (C) by the white box. (E) Perpendicular quantification of GFP::PAR-2 fluorescence in frame C (Fig. S1, H–M). (F) Kymograph of the recording containing the frame shown in (A)–(C) (see Movie S1), overlaid with automatic detection of the domain center (dashed line) and of domain expansion (gray line). Horizontal lines: cell cycle events (see Materials and Methods); arrows: position of the frame shown in (A)–(E). Black areas correspond to locations outside the embryo. (G) Final kymograph after centering and cropping, overlaid with landmarks as in (F). CYK, cytokinesis; PCF, pseudocleavage furrow; PNM, pronuclear meeting.To see this figure in color, go online.

Figure 2

Model with newly measured cortical flows faithfully captures the experimental data. (A) Average GFP::PAR-2 protein distribution during polarity establishment at 24°C (n = 47), overlaid with segmentation of domain center (dashed line) and domain expansion (dark gray). Time zero in this figure: onset of polarity establishment, as identified by domain segmentation. (B) Polarity establishment as predicted by M1 (Table S1 (13)), overlaid with the segmentation of this prediction (light gray) and of our experimental data (dark gray). Arrowheads indicate leading front enrichment not observed in (A). (C) Cortical flows measured using VIT-2::GFP particle tracking (Fig. S4 and the Supporting Material). (D) Polarity establishment as predicted by M2 (i.e., M1 with flows shown in C), overlaid with landmarks similar to (B). Note the strong dampening of leading edge enrichment. To see this figure in color, go online.

Figure 3

Cooperativity of both anterior and posterior components is required for polarity establishment. (A–D) Best values for the unmeasured mutual inhibition parameters {(A) k_AP, (B) α, (C) k_PA, (D) β} was identified in each embryo by an optimization procedure. Values detected as outliers (see the Supporting Material) are depicted using crosses and removed from the calculation of the mean as well as from the parameter distribution (Fig. S8). Values are color-coded with respect to their recording condition and overlaid with the corresponding average and standard deviation. Shown are the linear regression (dark gray line) with the corresponding 95% confidence interval (light gray area), the correlation coefficient R² that reports the fraction of variance explained by the linear regression, the slope of the correlation m and the p-value (Student’s two-tailed t-test for the coefficient m). To see this figure in color, go online.

Figure 4

Fixed protein concentration and changes in surface/volume ratio underlie domain scaling. (A, C, and E) DIC images representing the median embryo of each recording condition and (B, D, and F) corresponding kymographs, overlaid with landmarks similar to Fig. 1. The white dashed line in E delimits the boundary of the raw image. (G) Linear regression (dark gray line) performed on the length of the embryo versus the fraction of the membrane spanned by GFP::PAR-2, overlaid with landmarks similar to Fig. 3. The darker circles indicate the locations of the median recordings depicted in (A)–(F). (H) Same as (G), but quantifying the extent of the posterior domain as predicted by M3 (i.e., using the values calibrated with the median recordings). To see this figure in color, go online.

Figure 5

Diffusion is the process most affected by temperature. (A–C) Average kymographs of polarity establishment at the indicated temperatures, overlaid with landmarks similar to Fig. 1. Arrows: portion of the maintenance stage (white line) averaged for the profiles displayed in (E). (D) Quantification of duration of the maintenance phase (see the Supporting Material) at the three temperatures, overlaid with tests of significance (Student’s two-tailed t-test). (E) Profiles of GFP::PAR-2 signal during maintenance (arrows in A–C) at different temperatures. Thin lines: individual profiles; thick lines: average of each condition. Each set of curves (i.e., each recording temperature) is statistically different from each other (p < 0.001, adaptive Neyman test (40)). Arrowheads: variations predicted by the model (see G). (F) Quantification of the time required for polarity maintenance as in (D) when the values of the diffusion parameters D_A and D_P are scaled by the factor λ. Note that the x axes in (D) and (F) cannot be directly compared; however, a similar trend is observed. (G) Profiles simulated by the model as in (F) and aligned as in (E). To see this figure in color, go online.

Figure 6

Predictions of the mathematical model obtained from size and temperature perturbations. (A) Performance (negative log-likelihood score) of the successive modifications of the mathematical model with an indication of the model used (above) and of the corresponding figure panel (below). The right-most gray bar corresponds to optimization of each embryo individually (Fig. 3) and is displayed for comparison. (B) Comparison between the experimental data and the predictions of the optimized model (M4) when the size of each embryo is used (color-coded as indicated). (C) Comparison between the experimental quantification of the duration of the maintenance phase and the predictions of M4. (D) Maintenance profiles simulated by the optimized model at different temperatures, to be compared with Fig. 5E. To see this figure in color, go online.