Triangles bridge the scales: Quantifying cellular contributions to tissue deformation

In this article, we propose a general framework to study the dynamics and topology of cellular networks that capture the geometry of cell packings in two-dimensional tissues. Such epithelia undergo large-scale deformation during morphogenesis of a multicellular organism. Large-scale deformations emerge from many individual cellular events such as cell shape changes, cell rearrangements, cell divisions, and cell extrusions. Using a triangle-based representation of cellular network geometry, we obtain an exact decomposition of large-scale material deformation. Interestingly, our approach reveals contributions of correlations between cellular rotations and elongation as well as cellular growth and elongation to tissue deformation. Using this triangle method, we discuss tissue remodeling in the developing pupal wing of the fly Drosophila melanogaster.

Figure 1

(a) The developing fly wing is an important model system to study epithelial morphogenesis. This panel shows the wing blade at the developmental time of 23 h after puparium formation (hAPF). (b) Magnified region of membrane-stained wing tissue overlaid with the corresponding polygonal network. Cells are represented by polygons (green), cell-cell interfaces correspond to polygon edges (blue), and polygon corners correspond to vertices (red). (c) We consider four kinds of cell-scale processes. (d) Two examples for pure shear of a piece of cellular material: (i) pure shear by cell shape change and (ii) pure shear by T1 transitions. Colors in panels (c) and (d) indicate cell identities.

Figure 2

Triangulation of the cellular network. (a) Each threefold vertex $n$ (red dot) gives rise to a single triangle (red), which is also denoted by $n$. The corners of the triangle are defined by the centers of the three abutting cells (green dots). (b) Triangulation (red) on top of membrane-stained biological tissue (white). There are no gaps between the triangles.

Figure 3

Infinitesimal displacement gradients $\delta {\mathsf{U}}_{ij}$ can be decomposed into trace $\delta {\mathsf{U}}_{kk}$ describing isotropic expansion, symmetric, traceless part $\delta {\tilde{\mathsf{U}}}_{ij}$ describing pure shear, and antisymmetric part $\delta \mathrm{\Psi }$ describing rotation.

Figure 4

Characterization of triangle deformation and shape. (a) Deformation of a triangle $n$ from an initial state to a final state. The deformation is characterized by the linear transformation tensor ${\mathsf{m}}_{ij}^n$ mapping the initial side vectors of the triangle to the final side vectors (blue arrows). (b) The shape of a triangle $n$ in a given state is characterized by the tensor ${\mathsf{s}}_{ij}$. Tensor ${\mathsf{s}}_{ij}$ maps the side vectors of a virtual equilateral reference triangle to the side vectors of triangle $n$ (blue arrows). (c) Shape properties of a triangle $n$. The transformation tensor ${\mathsf{s}}_{ij}$ is decomposed into a counterclockwise rotation by the triangle orientation angle $\theta$, a pure shear deformation characterized by the triangle elongation tensor ${\tilde{\mathsf{q}}}_{ij}$, and an isotropic rescaling to match the actual triangle area $a$. (d) Connection between triangle shape and triangle deformation. A triangle deforms from an initial state to a final state. Deformation, initial state, and final state are characterized by the tensors ${\mathsf{m}}_{ij}, {\mathsf{s}}_{ij}$, and ${\mathsf{s}}_{ij}^{\prime }$, respectively.

Figure 5

Correlation contributions to pure shear. (a) Inhomogeneous isotropic expansion that is correlated with elongation creates a change in the average elongation ${\tilde{\mathsf{Q}}}_{ij}$, which is due to the area weighting in the definition of ${\tilde{\mathsf{Q}}}_{ij}$. This contribution to the time derivative of ${\tilde{\mathsf{Q}}}_{ij}$ is compensated for by the growth correlation term in $\delta {\tilde{\mathsf{K}}}_{ij}$. (b) Inhomogeneous rotation that is correlated with elongation creates a change in the average elongation ${\tilde{\mathsf{Q}}}_{ij}$. This contribution to the time derivative of ${\tilde{\mathsf{Q}}}_{ij}$ is compensated for by the rotational correlation term in $\delta {\tilde{\mathsf{K}}}_{ij}$.

Figure 6

The elongation ${\tilde{\mathsf{q}}}_{ij}^{\alpha }$ of a cell $\alpha$ (green) is defined by the average elongation of the triangles belonging to $\alpha$ (red). The triangles belonging to $\alpha$ are those that have one of their corners defined by the center of $\alpha$.

Figure 7

Effects of a single topological transition on the triangulation. (a) A T1 transition removes two triangles ($m$ and $n$) and creates two new ones ($p$ and $q$). (b) A cell division creates two triangles ($p$ and $q$, yellow). All other triangles shown (red) change their shape instantaneously. (c) A T2 transition removes three triangles ($m, n, p$) and creates a new one ($q$).

Figure 8

A T1 transition induces an instantaneous change of the average triangle elongation. The average triangle elongation before and after the T1 transition only depends on the position of the four involved cell centers (green dots).

Figure 9

Illustration of the shear rate contributions by average triangle elongation change and T1 transitions. (a) Between the time points 0 and $T$, a triangular network is continuously sheared along the horizontal axis. At time point $t_k$, a T1 transition occurs, which instantaneously changes the average triangle elongation. (b) For the process shown in (a), schematic time-dependent plots of shear rate [blue, (i)], the average triangle elongation [green, (ii)], its derivative [green, (iii)], and the shear rate by T1 transitions [red, (iv)]. For each tensor, the respective $xx$ component, i.e., the horizontal component, is plotted. The arrows in (iii) and (iv) indicate Dirac $\delta$ peaks. Their magnitude corresponds to the step in ${\tilde{\mathsf{Q}}}_{xx}$ at $t_k$.

Figure 10

Patterns of tissue shear and contributions to shear in the pupal wing of the fruit fly at different times in hours after puparium formation (hAPF). Local rate of pure shear (blue), corotational time derivative of the cell elongation (center), and shear rate by T1 transitions (right). The bars indicate the axis and norm of the tensors. Shown are averages over squares with size ${\left(33\mu \mathrm{m}\right)}^2$ and over time intervals of about $2\mathrm{h}$. The scale bars correspond to $100\mu \mathrm{m}$.

Figure 11

Contributions to tissue shear as a function of time during pupal development of the fly wing. Shown are the data for one wing. (a) The fly wing undergoes complex tissue remodeling, which we recorded between 15 and 32 hAPF. The colored areas mark regions of tissue in which all cells were tracked during this time interval. (b) Schematic representation of the coordinate system used to describe tissue deformations. The $x$ axis points towards the tip of the wing and is aligned parallel to the axis of cell elongation averaged within the interval between 24 and 32 hAPF and over all four regions. The average cell elongation computed for a single region deviates at most by $5^{\circ }$ from this $x$ axis. (c) Legend specifying different contributions to tissue shear. (d) Cellular contributions to shear and total shear rate averaged over regions 1–4 in (a) as a function of time. Plotted are the projections of the tensors on the $x$ axis, for example, the component ${\tilde{\mathsf{V}}}_{xx}$ of the tissue shear rate. (e) Cumulative tissue shear and cellular contributions, projected on the $x$ axis. (f), (g) Same plots as in (d) and (e), but for the subregions 1 to 4 indicated in (a). In (d)–(g), data were averaged over 10 subsequent interframe intervals.

Figure 12

Contributions to isotropic tissue expansion as a function of time during pupal development of the fly wing. Shown are the data for one wing. (a) Large-scale isotropic expansion rate and cellular contributions to it averaged over regions 1–4 as a function of time. (b) Cumulative isotropic expansion rate and cellular contributions to it. (c), (d) Same plots as in (a) and (b), but for the subregions 1 to 4 indicated in Fig. 11. The legend in (a) applies to (b)–(d), too. In all panels, data were averaged over 10 subsequent interframe intervals.

Figure 13

Decomposition of deformations in cellular contributions. Key quantities and their relationships are shown as a schematic diagram. (a) Decomposition of the isotropic deformation rate into cellular contributions. (b) Decomposition of the pure shear rate into cellular contributions. Red numbers indicate the definition of the respective quantity (or the equation where the quantity first appears). Black numbers indicate important relations between the quantities.

Figure 14

Geometrical interpretation of the elongation tensor ${\tilde{\mathsf{q}}}_{ij}$ for a given triangle. (a) Shows an equilateral triangle (red) with circumscribed circle (blue) and centroid (i.e., center of mass, yellow). (b) This triangle is deformed by the pure shear deformation given by $exp\left(\tilde{\mathsf{q}}\right)$, where ${\tilde{\mathsf{q}}}_{ij}$ is the elongation tensor of the so-created triangle. The former circumscribed circle is transformed to an ellipse (blue), and the former centroid is still the centroid of both the triangle and the ellipse (yellow). (c) Long and short axes of the ellipse with lengths $l$ and $s$, respectively.

Figure 15

Illustration of the shear-induced rotation effect $\delta \xi$ appearing in Eq. (A22). For clarity, we use a Minerva head in place of a triangle. (a) The definitions of orientation angle $\theta$ and elongation tensor ${\tilde{\mathsf{q}}}_{ij}$ are analogous to the triangle quantities [Fig. 4]. Roughly, the orientation angle $\theta$ corresponds to the direction in which the Minerva head looks. The isotropic scaling has been set to one for simplicity ($a=a_0$). (b) The elongated Minerva head is subject to a continuous pure shear deformation with varying shear axis. The pure shear axis is at each time point oriented with an angle of $\pi /4$ with respect to the elongation axis $\varphi$, such that the elongation norm $|\tilde{\mathsf{q}}|$ does not change but only the angle $\varphi$. Here, snapshots of such a deformation are shown. Alternatively, Movie M5 shows this deformation more smoothly [30]. Strikingly, the head orientation angle $\theta$ changes by this deformation although the deformation never includes any rotation component $\delta \psi =0$. Here, we have set $|\tilde{\mathsf{q}}|=(ln2)/2$ such that $\delta \theta =\delta \xi =0.2\delta \varphi$.

Figure 16

Path dependence of the cumulative pure shear. Shown are two finite deformations with the same initial and final states, but with a different cumulative shear. Initial and final states are isosceles triangles, with the same elongation norm. (a) The triangle is sheared along the $y$ axis. (b) The triangle is sheared such that the elongation tensor is rotated but the norm stays constant.

Figure 17

Illustration of the virtual intermediate states $I_1^k, I_2^k$, and $I_3^k$ introduced between two observed states $O^k$ and $O^{k+1}$.