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. b Azimuthally (ϕ) averaged spiral angle α as a function of polar cornea angle θ for n = 8 wild-type eyes. 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 from the central defect. 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.

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.

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 = 0.0018 (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.

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 coe?cient ζ are not shown since only their ratio k/ζ matters in our model.

Matching experimental and simulated cell flows, 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 each case, we included nsim = 3 simulations with all parameters fixed but with three different alignment strengths, J. 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 a. c and d Normalised Fourier space velocity correlation ϕ = | ⟨ v⟩ | /νrms, with shading and symbols same as in function ⟨v(q)2| ⟩ / ⟨ν2⟩ for the ‘explant’ and ‘plastic’ experiments. e and f Normalised velocity autocorrelation function ⟨v(t) ⋅ v(0) ⟩/ ⟨ν2⟩ for ‘explant’ and ‘plastic’ experiments.

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, Fact 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 ν 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).

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.

Flux on the 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.

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 use 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 obtain (1) the outline of the dissected tissue (2) the location of the cornea and (3) the stripe locations. We place 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 relax 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 are able to infer the velocity direction on points coloured red (red arrows). We then relax 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.