Figures and data
Light sheet imaging, segmentation, and tracking [15] provides a global picture of the cell level contributions to tissue flow (“tissue tectonics” [24]) during Drosophila gastrulation.
A Segmented and tracked cells on the ellipsoidal surface of the early Drosophila embryo in 3D (top) and projected into the plane (bottom) using a cartographic projection [16]; imaging, segmentation and tracking data from Ref. [15]. Trunk cells are colored in bands along the DV axis to illustrate the major tissue deformations during early development of Drosophila: ventral furrow invagination (VFI, 25 min) and germ band extension (GBE, 45 min). The purple to green regions constitute the lateral ectoderm (germ band) which contracts along the DV axis and extends along the AP axis. The posterior part of the germ band moves over the pole as it extends. The red and orange regions constitute the amnioserosa, which contracts along the AP axis and extends along the DV axis. Cells that are internalized in folds are shaded in dark gray (CF: Cephalic furrow; DF: Dorsal folds; VF: Ventral furrow; PMG: posterior midgut). Only one side of the left-right symmetric embryo is shown but both sides were analyzed throughout the manuscript. B Tissue deformation is the sum of cell shape changes (top) and cell rearrangements (bottom). The elementary cell rearrangement is a T1 transition in a quartet of cells. The interface between the red cells collapses, giving rise to a transient four-fold vertex configuration (center). The four-fold vertex then resolves to form a new interface between the blue cells. C Colored, tracked cells illustrate the cell rearrangements and shape changes in the amnioserosa (top) and lateral ectoderm (bottom). While amnioserosa cells show large deformations and little coordination in their rearrangement, cell intercalations in the lateral ectoderm appear highly choreographed. (ROI size 40 × 40 μm2.) D Convergence and extension of the lateral ectoderm (x-fold change defined relative to the minimum length and maximum width respectively). DuringVFI, the lateral ectoderm is stretched along DV axis and slightly contracts along the AP axis. GBE has an initial fast phase before slowing down at around 40 min. E and E’ Rate of interface collapses serves as a measure for the cell intercalation rate. During VFI, the are few intercalations. During GBE, the majority of intercalations are T1 transitions, while rosettes, rearrangements involving more than four cells, contribute significantly less to tissue deformation (E’). At 40 min, there is a noticeable drop in the T1 rate, marking the transition to the slow phase of GBE. Intercalation events before t = 12 min where excluded from the subsequent analysis.
Inferred tension dynamics distinguishes active and passive T1s.
A The adherens-junctional cytoskeleton maintains tension in the cortex at cell-cell interfaces via active motors (inset). In force balance, the forces Tab exerted on a vertex (red arrows) must sum to zero. This allows inferring the relative cortical tensions from the angles at which the interfaces meet at a vertex (see colorbar). B and B’ The angles in the tension triangles formed by the force vectors (rotated by 90°) are complementary to the interface angles at the vertex. Tension triangles corresponding to adjacent vertices share an edge and therefore fit together to form a tension triangulation (B’). C and C’ Relative tensions inferred (bottom) from cell membrane images (top) in the amnioserosa (C) and in the lateral ectoderm (C’). In the lateral ectoderm, high tension interfaces contract. The regular pattern of alternating high and low tension interfaces therefore leads to coordinated T1s (cf. Fig. 1C, bottom). Blue and orange dots mark an intercalating quartet. D and E While active and passive T1s show similar dynamics of the length of the inner edge (top), they are markedly different in their tension dynamics (bottom). For lateral cells (D), the tension increases as the interface contracts, providing evidence that positive tension feedback actively drives the edge contraction. For dorsal cells (E), the relative tension remains constant on the contracting edge, indicating that the intercalations are passive. Tension jumps at time zero result from the relation between the angles before and after the neighbor exchange. Collapsing and emerging interfaces were tracked and analyzed separately (see SI Sec. 1.3). Bands and fences show SD and SEM respectively; the SEM in (D) is smaller than the line thickness. F The increasing tension of the contracting edge during an active T1 causes the angles opposite of the central interface to become increasingly acute. G The hallmark of passive T1s are constant cortical tensions (and thus vertex angles) before the neighbor exchange. This geometrically fixes the vertex angles after the neighbor exchange, such that the emerging edge is under high tension. H and J In the lateral ectoderm, relative tension predicts time until a interface collapses (H) and high relative tension predicts which interfaces collapse (J); see Fig. S4 for analogous plots for the amnioserosa where no such correlation is observed. Relative tensions were averaged from minute 20 to 21 (over four timepoints), i.e. at the end of VFI (cf. Fig. 1E).
Tension–isogonal decomposition of epithelial geometry identifies active (tension-driven) and passive contributions to tissue deformation.
A The angles in the tension triangulation (red) are complementary to those in the cell tessellation (black) (left). The triangulation acts as a scaffold which leaves freedom for isogonal (angle preserving) deformations which encompass both dilation (center) and shear (right). B Deformations of the physical cell tessellation (black, right) can be decomposed into deformations of the tension triangulation (red, left) and isogonal deformations (blue). The former reflect the dynamics of cortical tensions while the latter reflect the effect of external forces and cell shape elasticity. A reference cell tessellation (purple, e.g. a Voronoi tessellation) constructed from the tension triangulation serves as an intermediate relative to which the isogonal deformations are defined. C and C’ Quartet shape (aspect ratio) and isogonal mode plotted against the length of the quartet’s inner interface, which serves as a pseudo-time parametrization. An aspect ratio of 1 indicates an isotropic quartet shape and no isogonal deformation, respectively. Active T1s (left), are driven by a deformation of the tension triangulation while the isogonal mode remains constant. Passive T1s (right), are driven by isogonal deformations while the shape of the tension triangulation remains constant. Bands indicate SD; SEM is smaller than the line width. D A symmetric pair of tension triangles is characterized by a single angle ϕ. The cell quartet’s central interface in the Voronoi reference configuration (purple) connects the centers of the triangles’ circumcircles (dashed gray circles). The general case of asymmetric triangles, characterized by two internal angles, is discussed in the companion paper [36]. D’ The circumcircles coincide when 
A minimal model for positive tension feedback reproduces the signatures of active T1s, and creates passive T1s when feedback is turned off.
A Single cell quartet of identical cells forms the elementary setting for modelling T1s. This geometry is characterized by two vertex angles (ϕ0, ϕ1, and ϕ2 = 2π − ϕ0 − ϕ1) and three interface lengths (ℓi). The interface angles are determined by the pair of identical tension triangles corresponding to the cell quartet. To avoid boundary effects, the cell quartet and tension triangles are set up to tile the plane periodically as a regular lattice. B Positive tension feedback causes the longest edge in a tension triangle to grow at the expense of the shorter two edges, thus deforming the triangle to become increasingly obtuse. (We fix the total tension scale. In real cells, the overall tension scale is set by the available myosin pool. Relative tensions change as myosin is redistributed between the cortex at different interfaces.) C The tension triangle shape determines the vertex angles, ϕi. To fix the interface lengths ℓi, we determine the cell shape by minimizing an elastic energy while keeping angles fixed (see SI for details). D Two-sided architecture of junctional cortex determines myosin level on newly formed interface. Sketch of intercalating quartet with myosin in each cell’s cortex color-coded. After a neighbor exchange, the active tension (i.e. myosin level) on the new edge is determined by a “handover” mechanism that assumes continuity of myosin concentration at vertices within each cell. As consequence, the active tension on the new edge right after the neighbor exchange is below the total tension that is determined by geometry. This tension imbalance causes the new edge to extend by remodeling. To capture the remodeling, we introduce a passive viscoelastic tension Tpassive. due to passive cortical crosslinkers. Tpassive decays exponentially with a characteristic remodeling time τp. Notably, no additional active ingredients (like medial myosin contractility) are required to drive extension of the new edge. E and F The model reproduces the signatures of active and passive T1s observed in the Drosophila embryo. The tension feedback rate and passive relaxation rate are fitted to match the observed timescales. (Bands show the standard deviation from an ensemble of simulations with initial angles drawn from the experimental vertex angle distribution at 0 min.)
Tissue scale tension anisotropy orients convergent-extension flow.
A Local anisotropy of tension (double-ended arrow) at a single tri-cellular vertex. B Tension anisotropy at the end of VF invagination/onset of GBE (25 min) locally averaged on a grid with 20 μm spacing. Line segments indicating the local orientation and magnitude (length and color of the line segments) of tension anisotropy. C Mean DV component of locally averaged anisotropic tension in the trunk (green region in B). (DV component measured along a fixed axis orthogonal to the long axis of the embryo; Shading shows SD.) D Significant DV alignment of the tension orientation in the trunk precedes any tissue flow, while the tension in the head shows no orientation bias (0 min). The DV alignment of tension slightly increases during VF invagination (25 min) and decreases during GBE (37 min).
Tissue scale quantification of isogonal strain identifies regions of passive tissue deformation.
A Tension–isogonal decomposition at a single tri-cellular vertex. The isogonal strain tensor (illustrated by blue arrows) transforms the tension triangle (solid red lines) into the centroidal triangle (dashed black lines). B and B’ Isogonal strain at the end of VF invagination (25 min, B) and during GBE (37 min, B’) averaged over vertices in a grid with 20 μm spacing. High isogonal strain in the tissue adjacent to the VF at 25 min and in the amnioserosa at 37 min indicate passive tissue deformations in these regions (highlighted by dashed rectangles). High isogonal strain is also found near the posterior midgut (dotted rectangle) Crosses indicate the principal axes of isogonal strain. Bar lengths indicate the magnitude of strain (green: extensile, magenta: contractile). Colored tissue regions are quantified in (C). C Time traces of the DV component of isogonal strain. The isogonal (i.e. passive) stretching of the tissue adjacent to the VF (purple) is transient. The lateral ectoderm as a whole (green and blue) is stretched weakly, but persistently. The amnioserosa (red) is strongly stretched as the lateral ectoderm contracts along the DV axis during GBE. (DV component is defined with respect to the local co-rotating frame, see SI; Shaded bands show one SD; SEM is comparable to the line width.)
Emergence and loss of order in local tension configurations. A Distinct configurations of tension are found on the cell scale: “Tension bridges”, characterized by a high-tension interface connected to four low-tension interfaces, are the local motif of an alternating pattern of high and low tensions. This alternating pattern gives rise to coordinated T1s as the high-tension interfaces collapse, driven by positive tension feedback. By contrast, tension cables, characterized by multiple adjacent high-tension interfaces, cause frustrated or incoherent T1s which manifest as rosettes. B Space of local tension configurations at a single vertex (quantified by the tension triangle shape, see Ref. [36] and SI Sec. 1.7 for details). The dashed line indicates the “T1 threshold” calculated for the average isogonal strain in the germ band (cf. Fig. 3G). This threshold is at lower anisotropy magnitude for tension bridges than for tension cables, indicating that the former are more efficient at driving active intercalations. C Distribution of tension configurations defines an order parameter that quantifies the relative abundance of tension cables and bridges in the lateral ectoderm. Arrows highlight the increasing fraction of tension bridges before GBE (0–25 min) and its decrease during GBE (25–50 min). D Median of the obtuse-acute triangle shape parameter in the lateral ectoderm. (Shading shows SD; SEM is smaller than the line width.)
Mechanical coordination of convergent–extension flow on the tissue and cell scale. A Dorso-ventral patterning of mechanical properties and tension anisotropy (red lines) organize and orient tissue flow. B A cell scale pattern of tensions coordinates active cell intercalations that drive convergent extension of the lateral ectoderm. From relatively uniform but weakly anisotropic initial tensions (i), an alternating pattern of high and low tensions emerges (ii). Subsequently, high-tension interfaces fully contract thereby driving parallel active T1 transitions (iii). As T1s proceed, order in the tension configurations is lost and convergent–extension flow cedes (iv).
Tissue tectonics analysis. A and A’ During VF invagination, the tissue strain rate (A) and cell deformation rate (A’) are high in the lateral ectoderm adjacent to the ventral furrow, where cells get stretched along the DV axis. Crosses show the principal axis of strain averaged in a grid with 20 μm spacing. Bar length is proportional to the strain rate and color indicates extension (green) and convergence (magenta). B and B’ During GBE, the tissue strain rate (B) is high in the lateral ectoderm which undergoes convergent extension. The cell deformation rate (B’) there is low, and incoherent implying that the tissue deforms via cell rearrangements. In contrast, near the dorsal pole, both cell deformation and cell rearrangements contribute to tissue strain rate. C Cell elongation measured by the Beltrami coefficient μ from the best-fit ellipse. Cells near the dorsal pole become highly elongated. D Total number of interface collapses per cell. Note that in Ref. [15] (the source of the segmentation data), the number of intercalations per cell is instead calculated as the sum of collapsed interfaces and new interfaces per cell, leading to numbers higher by a factor of 2. (Invaginating cells are shown in gray. Strain rates are only shown for cells that do not invaginate.)
Orientation of collapsing interfaces. A and B interfaces collapsing early (before 42 min) colored by their orientation at time 0 min (A) and histograms showing the interface orientation (B). Collapsing interfaces near the dorsal pole (shaded blue) are predominantly oriented along the AP axis. In the lateral ectoderm, the orientation is predominantly along the DV axis. C–D Interfaces collapsing late (after 42 min) have low initial orientational bias (C) but progressively align with the co-rotated DV axis before they start contracting (C’).
Length and relative tension of the central interface in intercalating cell quartets resolved by DV position. Two distinct classes of behavior are found: in the dorsal-most region (1), the relative tension remains constant at one on the contracting interface and jumps to a high value 
A In the amnioserosa relative tension is not correlated with the time until a interface collapses. B The distribution of relative tensions on interfaces that collapse is similar to those that persist. Compare to Fig. 2H and J for the quantification of interface collapses in the lateral ectoderm.
Tension–isogonal decomposition: Single-vertex analysis and coarse graining. A To a given tension triangulation (red), a corresponding dual cell tessellation is obtained via the Voronoi construction (purple), which is based on the circumcircles of the triangles. Because the tension triangulation fixes only the angles, it leaves freedom for isogonal (angle-preserving) modes that collectively change the interface lengths. Deformations of the tissue can be decomposed into deformations of the tension triangulation and isogonal deformations. A’ The tension–isogonal decomposition at a single vertex based on the tension triangulation vectors 
A Tension–isogonal decomposition for a cell quartet. The isogonal deformation is defined as the tensor that deforms the “kite” formed by the tension vectors Tij into the corresponding centroidal kite, formed from the cell centroids ci. The indices (ij) label the four cell pairs going around the kite: (12), (23), (34), (41). B–C’ Quartet aspect ratio and eigenvalue ratio of the isogonal deformation tensor as a function of time relative to the T1 event (B and B’) and as a function of the central interface’s length (C and C’). The isogonal deformation seen in B from −20 to −10 min and in C for collapsing edge lengths > 4 μm is driven by the VF invagination which stretches the germ band.) D Passive intercalation of an idealized cell quartet composed of regular hexagons: (i) initial configuration; (ii) four-fold vertex configuration after area-preserving isogonal shear changing the aspect ratio by a factor 3; (iii) further isogonal deformation after the neighbor exchange. Black disks mark the cell centroids used to define the quartet shape (see A).
Quantification of local tension configurations (LTC) in terms of tension triangle shape.
A Tension triangle shape characterizes the local tension configuration. B Space of triangles in in barycentric coordinates using the angles α + β + γ = π. Color indicates the degree of acuteness vs obtuseness by hue from cyan to magenta and the magnitude of anisotropy by brightness (see text for details). A single fundamental domain is highlighted. The remaining shape space is composed of rotated and reflected copies of this fundamental domain, corresponding to permutations of the angles α, β, γ. Dashed white lines indicate the threshold for neighbor exchanges based on the (generalized) Delaunay condition for pairs of identical tension triangles [36]. C and C’ Triangle shape space spanned by the anisotropy magnitude |Ψ| and acute-vs-obtuse parameter 
Shape statistics of random a Delaunay triangulation. A Example of a random Delaunay triangulation seeded by a Ginibre random point process. Triangles have been colored according to the shape parameters 
Additional quantification of single-quartet simulations. A and B Tensionisogonal decomposition of cell quartet shape during active (A) and passive (B) T1s in the single-quartet model (same simulations as Fig. 4E–F). C Geometry of a quartet of identical cells and the corresponding tension triangles (red) parametrized by the angles α, β. D Comparison of different cell shape elastic energies which determine the cell interface lengths ℓi as a function of the angles α, β. The plot shows the inner interface length for a left-right symmetric quartet (i.e. an isosceles tension triangle) where α = (π − β)/2). For reference, the inner interface length ℓref for a Voronoi tessellation based on the tension triangles is shown (dashed black line). The interface length obtained by minimizing elastic energy Eq. (3) with isotropic target shape (solid black line) closely follows the reference Voronoi length and vanishes at the same critical angle. By contrast, minimizing the “area-perimeter” energy of the vertex-model Ecell ∼ (A−A0)2 +(P −P0)2 gives a interface length (dot-dashed black line) that vanishes only for β → π and the elastic energy diverges in this limit (dashed green line). This implies that tension dynamics, changing the angles at vertices, cannot drive T1s for this choice of elastic energy in the absence of noise.
