Spiral-eyes: A soft active matter model of in vivo corneal epithelial cell migration
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eLife Assessment
This useful study describes a physical mechanism for the emergence of spiral patterns in the outer epithelial layer of the mammalian cornea independent of pre-patterning or guidance cues, using an agent-based model of self-propelled particles with alignment. The model is well constructed, however the central premise of the manuscript, that the spiral patterning of epithelial corneal cells occurs without guidance cues, is incomplete and not fully supported. Several significant questions remain unanswered, such as the role of the corneal curvature or the importance of topological defects. Furthermore, comparison between the model and data are qualitative at best for the moment.
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Abstract
Abstract
The mammalian cornea constantly regenerates its outer epithelial layer. Cells lost by abrasion are replaced by division of both corneal epithelial cells and populations of stem cells around the corneal periphery, the limbus. Limbal-derived epithelial cells migrate into the cornea, retaining equal cell loss and replacement rates (the ‘XYZ hypothesis’). This process leads to a striking stable spiral cell motion pattern across the corneal surface, with a central vortex. Here, we show that the emergence of the spiral pattern can be explained by the interplay of the position of the limbus, cell division, extrusion, and collective cell migration along the curved surface of the cornea. Using dissected LacZ mosaic murine corneas, we inferred the flow field on the curved surface by following stripe edges, revealing a tightening spiral. To explain the flow fields, we developed a cell-level, in silico model of the mouse cornea. Corneal epithelial cells were treated as mechanically soft, self-propelled particles with density-dependent proliferation and extrusion rates, and noisy alignment of the direction of migration. Even without any global guidance cues, the model predicted that migration patterns form stripes and spirals closely recapitulating those in the experiment. To understand the origin and properties of this flow field, we complemented the particle-based model with a continuum description of cell flux on the cornea that generalises the XYZ hypothesis of steady-state balance between cell divisions and extrusions. The work, therefore, demonstrates how the physics of swarms on curved surfaces can provide quantitative explanations of biological processes at the tissue scale.
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Author response:
We thank the referees for finding our work well written and systematic. We are planning a revision of the manuscript based on the public review and the confidential recommendations of the referees.
The role of axons:
Indeed, radial axon projections appear before mature epithelial stripes in the cornea (Iannaccone et al., 2012). Our claim is, however, not that guidance cues are absent, but that global cues are unnecessary. The alignment term in our model, together with evidence that corneal epithelial cells follow contact-mediated substrate cues (Walczysko et al., 2016), show that corneal cells migration is responsive to external forces, and the underlying patterns of axonal projections could be one of those cues.
Experiments (Collinson et al., 2002) and simulations in this work show that a rapid spiral epithelial flow …
Author response:
We thank the referees for finding our work well written and systematic. We are planning a revision of the manuscript based on the public review and the confidential recommendations of the referees.
The role of axons:
Indeed, radial axon projections appear before mature epithelial stripes in the cornea (Iannaccone et al., 2012). Our claim is, however, not that guidance cues are absent, but that global cues are unnecessary. The alignment term in our model, together with evidence that corneal epithelial cells follow contact-mediated substrate cues (Walczysko et al., 2016), show that corneal cells migration is responsive to external forces, and the underlying patterns of axonal projections could be one of those cues.
Experiments (Collinson et al., 2002) and simulations in this work show that a rapid spiral epithelial flow forms first, with cells migrating radially for ~2 weeks before stripes become visible. Axons seeking the path of least resistance within this moving basal layer would therefore appear radial early on. By contrast, establishing visible stripes requires an entire cohort of epithelial cells to travel from the limbus to the central cornea (Fig. 7). Extensive in-vivo studies (Song et al., 2004; Leiper et al., 2009) find no evidence that axons direct epithelial migration; if anything, epithelial flow dictates axonal trajectories.
Geometry and boundaries:
The spiral also forms on a flat disc, but its exact shape changes with curvature and cap angle; this variation is seen across mammals, including humans (Dua et al., 1993) and in diseases such as keratoconus. On a spherical cap the boundary winding number fixes the interior index, so ongoing limbal influx keeps the total index = 1.
In the revised version, we will therefore simulate a range of curvatures, cap angles, a prolate ellipsoid, and cases without limbal division, then compare with published data and disease states.
In-vitro data and parameter fits:
Although our dataset is limited, the inferred parameters match three independent invitro estimates (Kostanjevec et al. 2020; Saraswathibhatla et al. 2021; Kammeraat et al. in prep.). Spatial correlations exceed those expected from persistence alone, implying some polar alignment - consistent with Saraswathibhatla et al. 2021. Slide-scanner images that we will include in the revision show cells are neither elongated nor nematically ordered. In the revision we will detail our parameter extraction, highlight evidence for alignment, stress the substrate-based activity mechanism, and draw attention to the supplementary videos.
Topological clarification:
Stagnation points can be seen as topological defects because classification depends only on vector directions. Boundary conditions can remove such defects in fluids, yet two sources/sinks still interact via the same logarithmic Green’s function that governs disclinations, despite di^erent physics. The Euler characteristic is a property of the surface; while the boundary winding number fixes the field index, it does not alter the surface’s Euler characteristic.
In the revision, we will add a concise primer on the di^erential-geometric concepts to make these points explicit.
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eLife Assessment
This useful study describes a physical mechanism for the emergence of spiral patterns in the outer epithelial layer of the mammalian cornea independent of pre-patterning or guidance cues, using an agent-based model of self-propelled particles with alignment. The model is well constructed, however the central premise of the manuscript, that the spiral patterning of epithelial corneal cells occurs without guidance cues, is incomplete and not fully supported. Several significant questions remain unanswered, such as the role of the corneal curvature or the importance of topological defects. Furthermore, comparison between the model and data are qualitative at best for the moment.
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Reviewer #1 (Public review):
Summary:
The manuscript by Kostanjevec et al. investigates the mechanism behind spiral pattern formation in the cornea. The authors demonstrate that the spiral motion pattern on the mammalian corneal surface emerges from the interaction between the limbus position, cell division, extrusion, and collective cell migration. Using LacZ mosaic murine corneas, they reveal a tightening spiral flow pattern and show that their cell-based, in silico model accurately reproduces these patterns without global guidance cues. Additionally, they present a continuum model that extends the XYZ hypothesis to describe cell flux on the cornea, offering a quantitative explanation for tissue-scale processes on curved surfaces.
Strengths:
The manuscript is well-written, with a systematic approach that clearly explains experimental …
Reviewer #1 (Public review):
Summary:
The manuscript by Kostanjevec et al. investigates the mechanism behind spiral pattern formation in the cornea. The authors demonstrate that the spiral motion pattern on the mammalian corneal surface emerges from the interaction between the limbus position, cell division, extrusion, and collective cell migration. Using LacZ mosaic murine corneas, they reveal a tightening spiral flow pattern and show that their cell-based, in silico model accurately reproduces these patterns without global guidance cues. Additionally, they present a continuum model that extends the XYZ hypothesis to describe cell flux on the cornea, offering a quantitative explanation for tissue-scale processes on curved surfaces.
Strengths:
The manuscript is well-written, with a systematic approach that clearly explains experimental setups, model construction, assumptions, parameter selection, and predictions. The discussion also provides insightful perspectives on the broader implications of the results for both physics and biology.
Weaknesses:
The central premise of the manuscript, that the spiral patterning of epithelial corneal cells occurs without guidance cues, is not fully supported. The authors overlook the potential role of axons in guiding epithelial cells, despite clear evidence of spiral axon patterns in their own Fig. 1b. Previous literature indicates that axon patterning precedes epithelial cell patterning, suggesting that epithelial migration might be influenced by pre-existing neural structures (e.g., Leiper et al. 2002, IOVS 2013). The authors need to address this point, possibly by exploring whether axonal patterns serve as a template for epithelial cell migration, or by providing experimental evidence to rule out axon-based guidance.
While the model is well-constructed, it currently falls short of its stated goal of elucidating the mechanisms of spiral formation. Key questions remain unanswered:
Is the curvature of the cornea necessary for spiral formation, or would a simpler disk geometry suffice?
What role do boundary conditions play?
How well do the model's predictions quantitatively match experimental data?
The current comparisons in Fig. 4c-f lack quantitative agreement, and this discrepancy should be discussed with possible explanations.The authors emphasize polar alignment as a key feature of the spiral pattern based on simulation results. However, they do not provide experimental evidence for this polar alignment. The manuscript includes discussions of polar and nematic symmetries that, without supporting data, feel somewhat distracting. If direct experimental evidence for polar alignment is not available, the authors could instead quantify nematic alignment as the spiral forms. This would also allow them to explore potential crosstalk between nematic cell orientation and the polar alignment of self-propulsion, especially considering recent studies showing alternative mechanisms for vortex formation in similar systems.
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Reviewer #2 (Public review):
In K. Kostanjevec et.al, the authors study a possible mechanism for the formation of spiral patterns in the cornea. First the authors analyze an inferred velocity field, which is deduced from images of fixed corneas, and then determine the position-dependent spiral angle of this velocity fields. Next, the authors analysed two possible markers of cell polarity: the direction of the centrosome-nuclei and the axis of mitosis. Then the authors introduce a stochastic agent-based model of self-propelled particles with over-damped dynamics and with aligning interactions to the orientation of the nearest neighbors and to the particle's velocity. The authors claim to be able to reproduce the equal-time autocorrelation function and the velocity Fourier spectrum. Then the authors introduce the geometry of the cornea by …
Reviewer #2 (Public review):
In K. Kostanjevec et.al, the authors study a possible mechanism for the formation of spiral patterns in the cornea. First the authors analyze an inferred velocity field, which is deduced from images of fixed corneas, and then determine the position-dependent spiral angle of this velocity fields. Next, the authors analysed two possible markers of cell polarity: the direction of the centrosome-nuclei and the axis of mitosis. Then the authors introduce a stochastic agent-based model of self-propelled particles with over-damped dynamics and with aligning interactions to the orientation of the nearest neighbors and to the particle's velocity. The authors claim to be able to reproduce the equal-time autocorrelation function and the velocity Fourier spectrum. Then the authors introduce the geometry of the cornea by constraining the dynamics on a spherical cap and show that their model can reproduce a typical trajectory in experiments. Finally, the authors produce a phase diagram of the states at a fixed time point as a function of the spherical cap radius and the strength of the coupling aligning constant. Finally, the authors propose an interpretation of the cell fluxes based on the equation of mass conservation.
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Excerpt
Unraveling the mysteries of corneal cell migration with active matter physics.
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