Anisotropy links cell shapes to tissue flow during convergent extension

Abstract

Within developing embryos, tissues flow and reorganize dramatically on timescales as short as minutes. This includes epithelial tissues, which often narrow and elongate in convergent extension movements due to anisotropies in external forces or in internal cell-generated forces. However, the mechanisms that allow or prevent tissue reorganization, especially in the presence of strongly anisotropic forces, remain unclear. We study this question in the converging and extending Drosophila germband epithelium, which displays planar-polarized myosin II and experiences anisotropic forces from neighboring tissues. We show that, in contrast to isotropic tissues, cell shape alone is not sufficient to predict the onset of rapid cell rearrangement. From theoretical considerations and vertex model simulations, we predict that in anisotropic tissues, two experimentally accessible metrics of cell patterns-the cell shape index and a cell alignment index-are required to determine whether an anisotropic tissue is in a solid-like or fluid-like state. We show that changes in cell shape and alignment over time in the Drosophila germband predict the onset of rapid cell rearrangement in both wild-type and snail twist mutant embryos, where our theoretical prediction is further improved when we also account for cell packing disorder. These findings suggest that convergent extension is associated with a transition to more fluid-like tissue behavior, which may help accommodate tissue-shape changes during rapid developmental events.

Keywords: Drosophila; epithelia; morphogenesis; vertex models.

Fig. 1.

Cell shapes and cell rearrangements in the converging and extending Drosophila germband epithelium during axis elongation. (A) Schematic of Drosophila body axis elongation. The germband epithelium (dark gray) narrows and elongates along the head-to-tail body axis in a convergent extension movement. The tissue is anisotropic, experiencing internal stresses from planar-polarized patterns of myosin II (red) within the tissue as well as external stresses (orange) due to the movements of neighboring tissue. (B) Schematic of oriented cell rearrangement and cell-shape change. (C) The germband epithelium doubles in length along the head-to-tail AP axis in 30 min (black). Cell rearrangements are thought to drive tissue elongation (magenta), and cell-shape changes also contribute (green). Tissue elongation begins at t = 0. The cell rearrangement rate includes cell-neighbor changes through T1 processes and higher-order rosette rearrangements. Relative cell length along the AP axis is normalized by the value at t = −10 min. Mean and SD between embryos is plotted (n = 8 embryos with an average of 306 cells analyzed per embryo per time point).

Fig. 2.

Cell shape and packing disorder alone are not sufficient to predict the onset of cell rearrangements in the Drosophila germband. (A) Confocal images from time-lapse movies of epithelial cell patterns in the ventrolateral region of the germband tissue during Drosophila axis elongation. Cell outlines were visualized by using the fluorescently tagged cell membrane marker gap43:mCherry (53). Anterior left, ventral down. Images with overlaid polygon representations used to quantify cell shapes (green) are shown. (Scale bar, 10 μm.) See SI Appendix, Fig. S1. (B) The average cell shape index $\overline{p}$ in the germband before and during convergent extension. The cell shape index, p, is calculated for each cell from the ratio of cell perimeter to square root of cell area, and the average value for cells in the tissue, $\overline{p}$, is calculated at each time point. The mean and SD between embryos is plotted. Dashed line denotes the reported value for the solid–fluid transition in the isotropic vertex model, $\overline{p}=3.81$. See also SI Appendix, Fig. S2. (C) The instantaneous rate of cell rearrangements per cell versus the average cell shape index $\overline{p}$ from movies of individual embryos at time points before and during convergent extension in eight wild-type embryos (different symbols correspond to different embryos). Small green arrows indicate the values of $\overline{p}$ at the onset of rapid cell rearrangement (>0.02 per cell per min; dashed line) in different embryos. The shaded region denotes values of $\overline{p}$ for which different embryos display distinct behaviors, either showing rapid cell rearrangement or not. Thus, a fixed value of $\overline{p}$ is not sufficient to determine the onset of rearrangement. (D) In vertex model simulations, the solid–fluid transition depends on exactly how cells are packed in the tissue (SI Appendix, Materials and Methods and Fig. S3). In model tissues, we find a linear dependence of the critical cell shape index on the fraction of pentagonal cells f₅, which is a metric for packing disorder. The dashed line represents a linear fit to this transition: $p_o^{\ast }=3.725+0.59f_5$. (E) The relationship between $\overline{p}$ and f₅ for eight wild-type embryos, with each point representing a time point in a single embryo. The dashed line is the prediction from vertex model results (same as in D). (F) The relationship between $\overline{p}$ and vertex coordination number for eight wild-type embryos, with each point representing a time point in a single embryo. The dashed line is the prediction from ref. . (E and F) Instantaneous cell rearrangment rate per cell in the tissue is represented by the color of each point, with blue indicating low rearrangement rates and red to yellow indicating high rearrangement rates.

Fig. 3.

The solid-to-fluid transition in a vertex model of anisotropic tissues. (A) We study the effect of anisotropies on the solid–fluid transition in the vertex model by externally applying an anisotropic strain ε. An initially quadratic periodic box with dimensions L₀ × L₀ is deformed into a box with dimensions e^εL₀ × e^−εL₀. (B) Vertex model tissue rigidity as a function of the average cell shape index with different levels of externally applied strain ε (values for ε, increasing from blue to red: 0, 0.4, and 0.8). For comparison, the strain in the wild-type germband between the times t = 0 min and t = 20 min is ε ∼ 0.6. For every force-balanced configuration, the shear modulus was analytically computed as described in SI Appendix, SI Materials and Methods. For zero strain, we find a transition at an average cell shape index of $\overline{p}=3.94$ from solid behavior to fluid behavior. For increasing strain, the transition from solid to fluid behavior (i.e., the shear modulus becomes zero for a given strain) occurs at higher $\overline{p}$ (approximate positions marked by yellow arrows). Thus, a single critical cell shape index is not sufficient to determine the solid–fluid transition in an anisotropic tissue. (C) Cell shape and cell shape alignment can be used to characterize cell patterns in anisotropic tissues. Cell shape alignment Q characterizes both cell shape anisotropy and cell shape alignment across the tissue. While a high cell shape index $\overline{p}$ correlates with anisotropic cell shapes, the cell shape alignment Q is only high if these cells are also aligned. Conversely, low $\overline{p}$ implies low cell shape anisotropy and, thus, low Q. (D and E) Vertex model simulations for the case of an anisotropic tissue arising due to externally induced deformation (D) (cf. A and B), due to internal active stresses generated by an anisotropic cell–cell interfacial tension combined with externally applied deformation (E), and due to internal active stresses without any externally applied force (E, Inset) (SI Appendix). The fraction of tissue configurations that are fluid is plotted as a function of $\overline{p}$ and Q. For both internal and external sources of anisotropy, the critical shape index $\overline{p}$ marking the transition between solid states (blue) and fluid states (red) is predicted to depend quadratically on Q. White regions denote combinations of $\overline{p}$ and Q for which we did not find force-balanced states. In particular, in the case of finite tension anisotropy, we did not find any stable force-balanced fluid states, and the red fluid states in E all correspond to the limiting value of zero-tension anisotropy. In SI Appendix, Materials and Methods, we explain how the lack of fluid states for finite tension anisotropy can be explained analytically. Our findings quite generally suggest that stationary states of fluid tissues with an anisotropic cell–cell interfacial tension are difficult to stabilize even when preventing overall oriented tissue flow via the boundaries. In D, the solid line shows a fit of the transition to Eq. 1 with $p_o^{\ast }=3.94$ and b = 0.43; E and E, Inset show this same line. In D, a deviation from Eq. 1 is only seen around $\overline{p}\sim 4.15$ and Q ∼ 0.3, where we observe an abundance of solid states, which is likely due to the occurrence of manyfold vertices in this regime (SI Appendix, Fig. S4), which are known to rigidify vertex model tissue (42).

Fig. 4.

Cell shape and cell shape alignment together predict the onset of cell rearrangements during Drosophila convergent extension. (A) Confocal images from time-lapse movies of epithelial cell patterns in the ventrolateral region of the germband during Drosophila axis elongation. Cell outlines were visualized with gap43:mCherry (53). Anterior left, ventral down. (Scale bar, 10 µm.) Images of cells with overlaid triangles that were used to quantify cell shape anisotropy. Cell centers (green dots) are connected with each other by a triangular network (red bonds). Cell shape stretches are represented by triangle stretches (blue bars), and the average cell elongation, Q, is measured (56). (B) The cell shape alignment index Q (red) and average cell shape index $\overline{p}$ (black, same as Fig. 2B) for the germband tissue before and during axis elongation. Q was calculated for each time point, and the mean and SD between embryos is plotted (n = 8 embryos with an average of 306 cells analyzed per embryo per time point). The onset of tissue elongation occurs at t = 0. The dashed line denotes the reported value for the solid–fluid transition in the isotropic vertex model, $\overline{p}=3.81$ (39). (C) The relationship between $\overline{p}$ and Q for eight individual wild-type embryos, with each point representing $\overline{p}$ and Q for a time point in a single embryo. Instantaneous cell rearrangement rate per cell in the tissue is represented by the color of each point, with blue indicating low rearrangement rates and red to yellow indicating high rearrangement rates. The black solid line indicates a fit to Eq. 1 with a rearrangement-rate cutoff of 0.02 min⁻¹ per cell (SI Appendix, Materials and Methods), from which we extract $p_o^{\ast }=3.83$, where b was fixed to the value obtained in vertex model simulations (cf. Fig. 3D). (C, Inset) $\overline{p}$ and Q for individual embryos over time. (D) The relationship between the corrected average cell shape index ${\overline{p}}_{corr}$ and cell shape alignment Q for eight individual wild-type embryos, with each point representing a time point in a single embryo. The cell shape index is corrected by the vertex coordination number z as ${\overline{p}}_{corr}=\overline{p}-\left(z-3\right)/B$, with B = 3.85 (42). Instantaneous cell rearrangement rate per cell in the tissue is represented by the color of each point. The solid line indicates the parameter-free prediction of Eq. 2. (D, Inset) Single embryo fit to Eq. 1. (E) $p_o^{\ast }$ from single embryo fits to Eq. 1 correlate with the fraction of pentagonal cells f₅, a metric for cell packing disorder in the tissue, at the transition point. The dashed line represents a linear fit to the data. When using a rearrangement-rate cutoff of 0.02 min⁻¹ per cell for the single embryo fits, we obtain for this linear fit $p_o^{\ast }=3.755+0.27f_5$. (F) $p_o^{\ast }$ from single-embryo fits to Eq. 1 correlate with the average vertex coordination number, another metric for packing disorder in the tissue, at the transition point. The dashed line represents the previous theoretical prediction for how manyfold vertices influence tissue behavior (42).

Fig. 5.

Cell shape, cell shape alignment, and cell rearrangement rates in the germband of snail twist and bnt mutant embryos. snail twist embryos lack ventral patterning genes required for presumptive mesoderm invagination. bnt embryos lack AP patterning genes required for axis elongation and show severely disrupted myosin planar polarity compared to wild type (SI Appendix, Fig. S10). (A and B) Confocal images from time-lapse movies of cell patterns at t = +2 min and t = +15 min. Cell outlines visualized with fluorescently tagged cell membrane markers: gap43:mCherry in wild type, Spider:GFP in snail twist, and Resille:GFP in bnt. Polygon representations of cell shapes are overlaid (green). (Scale bar, 10 µm.) (C) Tissue elongation is moderately reduced in snail twist and severely reduced in bnt compared to wild type. (D) Cell rearrangement rate is moderately decreased in snail twist and severely reduced in bnt. (E) In snail twist, the average cell shape index $\overline{p}$ is reduced compared to in wild type for −5 min < t < 5 min. In bnt, $\overline{p}$ shows similar behavior to in wild type for t < 5 min, but does not show further increases with time for t > 5 min. (F) In snail twist, the cell alignment index Q is strongly reduced for −5 min < t < 10 min compared to in wild type. In bnt, Q shows similar behavior to in wild type for t < 5 min, but relaxes more slowly to low levels. (C–F) The mean and SD between embryos is plotted (three snail twist and five bnt embryos with an average of 190 cells per embryo per time point). (G–I) Relationship between the corrected cell shape index ${\overline{p}}_{corr}$ and Q for three snail twist (G and H), eight wild-type (H), and five bnt (I) embryos, with each point representing a time point in a single embryo. Instantaneous rearrangement rate is represented by the color of each point. Solid lines represent the prediction of Eq. 2. (H) Tissue behavior in snail twist and wild-type embryos, all of which exhibit rapid cell rearrangement during convergent extension, is well described by the prediction of Eq. 2, which does not require any fitting parameters. Avg., average.