Topological defects in two-dimensional polar and nematic fields. Grey arrows denote polar vectors, while grey lines denote nematic directions with head–tail symmetry.

a Looping counter-clockwise (blue circle) around the defect core (blue dot), the polar vectors rotate counterclockwise by 2π, giving a topological charge of +1. b Looping counterclockwise, the vectors rotate clockwise by 2π, corresponding to a charge of −1. c In a nematic field, head–tail symmetry allows a half-turn of the field; the vectors rotate by π, giving a charge of . d The same configuration as in panel c but with clockwise rotation, yielding a charge of . A uniform rotation of all vectors leaves the topological charge unchanged, making all spirals +1 topological defects.

Spiral corneal striping of the murine eye.

a LacZ-mosaic (‘XLacZ’) mouse cornea with radial spiral striping patterns of XGal staining, visualising patterns of epithelial cell migration during normal life. Image taken from (Collinson et al., 2002). b Beta-III-tubulin immunofluorescence staining of sensory axons in the spiral region of the basal layer of corneal epithelial cells. The axons follow the migratory flow of the basal epithelium. c A rare (≈1 in 100) occurrence of a double-swirl near the corneal centre. In both immunofluorescence images, integer topological defects (e.g. centres of swirls) can be readily seen. In b there is a single +1 defect (white circle). The unusual pattern, shown in c, shows two +1 defects and a single −1 defect (magenta circle). Scale bars: a 1000 µm. b, c 50 µm.

Experimentally observed corneal flow profiles.

a Projection of the inferred flow field of experiment 1 onto the plane, where the colour corresponds to the velocity direction (see colourwheel legend). b Azimuthally (ϕ) averaged spiral angle α as a function of polar cornea angle θ from the central defect for n = 8 wild-type eyes (θ = 0 represents cornea center). c Local spherical coordinates around the location of the central topological defect (black). The angle α between the velocity vector and the −êθ direction defines the spiral angle as a function of the cornea polar angle θ. d Finding topological defects. From a Delaunay triangulation with the velocity arrows at the nodes, we sum up the angle differences Δ around each triangle and compute c = Δ/2π. These charges identify topological defects, e.g. c = 0 and no defect in the triangle labelled A, and c = 1 and a +1 defect in the triangle labelled B. Figure 2–Figure supplement 1. Fixed, stained and dissected LacZ-mosaic corneas used for flow inference. Figure 2–Figure supplement 2. Inferred flow fields for corneas 1-8.

a-c Angles of centrosome position and mitotic spindle in the cornea during migration and cell division, respectively, showing no global angular correlations. a Definition of the division angle ϕ with respect to the corneal centre and edge. b Angles ϕ of centrosome position in relation to the nucleus and the centre of the cornea. The image shows nuclei stained with DAPI (blue), with the centrosome stained with pericentrin (pink). The position of the centrosome with respect to the nucleus may indicate an intracellular planar polarisation: our data suggested that centrosomes are slightly but significantly biased to lie in front of the nucleus p = 0.0047 (1-way ANOVA with Tukey’s posthoc multiple comparisons test). c Angles of mitotic spindle in relation to the centre of the cornea. The image shows a dividing cell with its mitotic spindle on a DAPI stained cornea. No preferential angle of division was observed. d Corneal epithelium flat mount immunostained for Keratin-12 intermediate filaments, highlighting the cell cytoplasm and boundaries. Cells show no preferred axis of elongation (i.e. nematic order).

Matching procedure of simulated cell flows to experiment, for corneal-derived cells on the stromal substrate within the explant (labelled ‘explant’, n = 3), and for cells proliferating on tissue culture plastic substrate (labelled ‘plastic’, n = 4).

For both conditions, we included 3 simulations with all parameters fixed except for the alignment strength, J, and one simulation with additional pair friction (labelled ‘f’). a Root mean squared cell velocity magnitude in simulations (shaded regions) and for experiments (white regions). Error bars represent one standard deviation. If not shown, the error bar is smaller than the symbol size. b Flocking order parameter ϕ = |⟨v⟩| /υrms, with shading and symbols same as in a. c and d Normalised velocity correlation function ⟨v(r) ⋅ v(0) ⟩ / ⟨υ2⟩ for the ‘explant’ and ‘plastic’ conditions, simulation curves are for the hydrodynamic velocity (see text). e and f Normalised velocity autocorrelation function ⟨v(t) ⋅ v(0) ⟩ / ⟨υ2⟩ for ‘explant’ and ‘plastic’ conditions. g Corneal explant overlaid with velocity arrows from PIV analysis and h simulated explant at J = 0.15. i Cells on the plastic substrate overlaid with velocity arrows and j simulated plastic substrate at J = 0.15. Arrows for the simulations are the hydrodynamic velocity.

Model parameters and their values inferred from experiment and used in simulations.

Bare values for the cell-cell interaction potential strength k and the friction coefficient ζ are not shown since only their ratio k/ζ matters in our model.

Key resources table

In silico corneal model. Cells were constrained to move on the surface of a spherical cap with a polar angle of 70°.

Each cell is self-propelled by an internal active force, F act, in addition to being subject to a short-range elastic force Fel due to mechanical interactions with neighbouring cells. Due to slow migration speeds, cell motion is overdamped, and forces are balanced by friction with the substrate, giving rise to cell velocity v. Inwards from a layer of immobile barrier cells (grey), the stem cells of the limbal niche (green and pink) divide with rate dL to give rise to transient amplifying (TA) cells (blue and white). TA cells divide with the density-dependent rate dz into cells with the same label and are extruded with rate a.

Emergence of the spiral flow in experiment and in a simulated R = 700 µm cornea.

a X-Gal-stained beta-galatosidase mosaic corneas from ‘H253’ (XLacZ) female mice, hemizygous for X-linked LacZ transgene (Collinson et al., 2002), showing the gradual appearance of the spiral over 3 - 4 weeks. Scale bar is 500 µm. b Simulated time sequence of the emerging spiral pattern with labelled ‘blue’ and ‘white’ TA cells. The spiral is completely formed by simulated day 20. c Corresponding hydrodynamic flow fields vH of the cells, coloured according to the x (left-right) component of the flow direction in 3D space from blue to red. The flow is determined by topological defects (white +1, black −1) that merge to give rise to a central +1 topological defect by day 8. d Velocity (top) and director (bottom) topological defect polar angle as a function of time. +1 and −1 defect pairs merge progressively, the snapshots in b and c correspond to the blue vertical lines. e Central +1 defect track for both velocity and director overlaid on the velocity field.

Properties of the spiral as a function of the corneal radius R and alignment strength J.

a Schematic phase diagram with lineage tracing and select flow fields after 20 days. Spirals appear at all R, but need J ≈ 0.075 h−1 to emerge. We include our best fit for R = 1500 µm life-size cornea. b and c Spiral angle α with respect to the inward −êθ direction as a function of polar angle θ as a function of J for R = 500 µm (b) and as a function of R for J = 0.1 h−1 (c). d and e Coarse-grained flow velocity magnitude υ as a function of polar angle θ as a function of J for R = 500 µm (d) and as a function of R for J = 0.1 h−1 (e). Figure 7–Figure supplement 1. Emergence of the corneal spiral as a function of J for R = 500 µm simulated corneas, snapshots are at 20 days. Figure 7–Figure supplement 2. Emergence of the corneal spiral as a function of R for J = 0.1 h−1 simulated corneas, snapshots are at 20 days. Note that simulations for R = 200, 400 and 1000 µm have patterned 48 instead of 12 corneal stripes into the limbus.

Flux on the simulated corneal surface, and XYZ hypothesis.

a Radial flux F as a function of polar angle θ for different simulated corneal radii R at J = 0.1 h−1. Dashed lines: predicted flux from eqn. (5) using b the measured difference between cell division and extrusion rates A as a function of θ. c Radial flux emerges as a function of alignment strength J at R = 500 µm, and is consistent with (dashed lines) the prediction from net loss rate A in d, where the elevated loss rate near the limbus for the non-aligning system is clearly visible.

Importance of the limbal stem cells for spiral formation.

Result of simulations without limbal stem cells, at R = 700 and different J, with the TA cells on top and the hydrodynamic flow fields vH of the cells at the bottom. a-b J = 0.1, the direct equivalent of Fig. 6, c-d J = 0.15 and e-f J = 0.2. g Defect traces for velocity (top) and director field (bottom) for J = 0.15. None of these systems forms a central spiral.

Migration pattern in other geometries, with the ‘blue’ and ‘white’ TA cells marked on the left, and on the right hydrodynamic flow fields vH of the cells, representation and other parameters as in the main model and Figure 6.

a Flat cornea, R = 700 µm. b Spherical cornea with a cap angle of 30°, R = 500 µm. c Spherical cornea with a cap angle of 120°, R = 300 µm. d Defect traces for velocity (top) and director field (bottom) for the system in panel e. e Prolate ellipsoidal cornea (radii a = b = 300 µm and c = 500 µm). f Oblate ellipsoidal cornea (radii a = b = 500 µm and c = 300 µm). We observe stripe and spiral formation in all systems.

Extracting digitised flow fields from fixed eyes.

a Side (top) and top (bottom) view of a fixed mouse eye, stained for LacZ. The line of the limbus is clearly visible. b Dissected, flattened cornea, note radial cuts and the clearly visible stripes. c Using Inkscape, we used different colours to draw (1) the cut boundaries (2) a circle at the position of the limbus and most importantly (3) all visible stripe edges as new layers on the image from (b). d After processing this drawing with MATLAB, we obtained (1) the outline of the dissected tissue (2) the location of the cornea and (3) the stripe locations. We placed 40 concentric rings of evenly spaced points on the cornea, omitting the cuts. e ‘Moulding’ the cornea: Starting from the centre and moving outwards, we relaxed each ring by equilibrating the springs indicated in green; a cut location is outlined in blue. f Location of the points defined in (d) on the cornea after ‘moulding’. g Using the local tangent to the stripe edges, we inferred the velocity direction on points coloured red (red arrows). We then relaxed the director at points without stripe edge (blue) assuming director alignment between neighbours (black lines). h Final estimate for the flow direction of the cornea, including central spiral defect.

Emergence of the hydrodynamic velocity.

a Instantaneous velocity arrows on the R = 700µm cornea shown in Figure 6, colored by x (left/right) component of the velocity. b-d Hydrodynamic velocity arrows defined as for increasing time intervals Δt, showing convergence to smooth field after Δt = 2.5h. e Distribution of the angle ϕ of the velocity vectors (as defined in Figure 3), weighted by velocity magnitude, showing increasingly sharply peaked inward (pink) motion as Δt increases.

Packing fraction ϕ = πr2ρ for the simulations as a function of angle θ from the centre, for different alignment strengths J and radii R.

Note that for J above the alignment transition and all R the apparent regions of lower ϕ at θ > 50° are due to corneas with asymmetric defect patterns clipping the edge of the cornea at those angles away from the defect.

Fixed, stained and dissected LacZ-mosaic corneas (n = 8) used for flow inference.

The striped corneal surface (white and blue) is flattened by making radial cuts, opening wedge-shaped spaces. The limbal boundary is located at the transition to the brown outer tissue.

Inferred flow fields for corneas in experiments 1-8.

Emergence of the corneal spiral as a function of J for R = 500 µm simulated corneas, snapshots are at 20 days.

Emergence of the corneal spiral as a function of R for J = 0.1 h−1 simulated corneas, snapshots are at 20 days.

Note that simulations for R = 200, 400 and 1000 µm have patterned 48 instead of 12 corneal stripes into the limbus.