Mechanical state, material properties and continuous description of an epithelial tissue

Abstract

During development, epithelial tissues undergo extensive morphogenesis based on coordinated changes of cell shape and position over time. Continuum mechanics describes tissue mechanical state and shape changes in terms of strain and stress. It accounts for individual cell properties using only a few spatially averaged material parameters. To determine the mechanical state and parameters in the Drosophila pupa dorsal thorax epithelium, we severed in vivo the adherens junctions around a disc-shaped domain comprising typically a hundred cells. This enabled a direct measurement of the strain along different orientations at once. The amplitude and the anisotropy of the strain increased during development. We also measured the stress-to-viscosity ratio and similarly found an increase in amplitude and anisotropy. The relaxation time was of the order of 10 s. We propose a space-time, continuous model of the relaxation. Good agreement with experimental data validates the description of the epithelial domain as a continuous, linear, visco-elastic material. We discuss the relevant time and length scales. Another material parameter, the ratio of external friction to internal viscosity, is estimated by fitting the initial velocity profile. Together, our results contribute to quantify forces and displacements, and their time evolution, during morphogenesis.

Figure 1.

Large-size annular severing in a fly dorsal thorax. (a) Developmental stages of a fruitfly Drosophila melanogaster. Top: larva. Middle: pupa, with pupal case removed (see electronic supplementary material, figure S1) and cuticle kept intact. Bottom: adult. Dashed lines represent the midline (symmetry axis). The x-axis is antero-posterior: anterior (head) towards the left, posterior (abdomen) towards the right. The y-axis is medio-lateral. Crosses: approximate positions of severing, along the midline in the scutellum (blue) and off-axis in the scutum (black). (b) Epithelial cell apical junctions marked by E-cadherin : GFP just before severing, t = 0⁻, here in an old pupa (see text for classification). Blue circles: two concentric circles define the annular severed region; the distance between circles corresponds to approximately one cell size. Yellow arrows: macrochaete used as spatial references to position the severed region. (c) First image after severing, t = 1 s. Yellow: fitted ellipse [35] (see §5). (d) Time t = 30 s after severing, showing a larger opening along y than x. Scale bars: (a) 1 mm, (b–d) 10 µm.

Figure 2.

Model-independent measurement of strain , stress-to-viscosity ratio and relaxation time τ. The wound margin position (ellipse semi-axis) is plotted versus time after severing; data from figure 1b–d, see electronic supplementary material, movie S3, along the y-axis. The difference between the initial (L) and final () positions (blue dashed lines) directly yields the value of The initial velocity (slope of the orange dashed line) yields an estimate of . Inset: velocity, estimated by finite differences of successive positions, versus the position during the first 30 s. An arrow indicates the direction of increasing time t. The slope of a linear fit (purple line) yields the inverse of the relaxation time, (see equation (3.2)).

Figure 3.

Mechanical state and material properties. Colour code according to pupa development ages: green, young; red, middle-aged; blue, old. The edges of the rectangular regions represent the mean values ± standard deviations for each group. The experiments in the scutum are in yellow. (a) Strain anisotropy: versus . Note the difference in horizontal and vertical scales; the solid line is the first bisectrix y = x, indicating the reference for isotropy. (b) Same for the severed stress-to-viscosity ratio . (c) Relaxation time τ and dimensionless friction-to-viscosity ratio ξ; values are the averages of the measurements along the x- and y-axes (see electronic supplementary material, figure S3). Note the semi-log scale. The blue rectangle takes into account two very small values of ξ, of order 10⁻³ and 10⁻⁴ (below the plotted range).

Figure 4.

Tracking of feature positions. (a) Positions of features (blue squares), old pupa, data of figure 1c after image denoising. The logarithms of intensity levels are represented in grey scale, and contrast is inverted for clarity. (b) Tracking for the first 30 s (time colour-coded from blue to red), inside a rectangle along the y-axis. (c) Same for the x-axis.

Figure 5.

Model-dependent measurement of the friction-to-viscosity ratio ξ. Initial velocity profiles immediately after severing are plotted versus initial position prior to severing, for three typical pupae: young (green), middle-aged (red) and old (blue). Open squares: features, from figure 4b. Closed triangles: same, spatially averaged in eight bins. Lines: fit by a sinh function (equation (3.6)), yielding for ξ a value above, in, or below the measurable range: (green), 5 (red), (blue), respectively.

Figure 6.

Validation of the model. Blue triangles: positions of features from figure 4b, spatially averaged in bins versus time after severing; bars: standard deviation. Red lines: numerical resolution of equation (3.3), using the measured values , s.