Self-organized centripetal movement of corneal epithelium in the absence of external cues

Maintaining the structure of the cornea is essential for high-quality vision. In adult mammals, corneal epithelial cells emanate from stem cells in the limbus, driven by an unknown mechanism towards the centre of the cornea as cohesive clonal groups. Here we use complementary mathematical and biological models to show that corneal epithelial cells can self-organize into a cohesive, centripetal growth pattern in the absence of external physiological cues. Three conditions are required: a circumferential location of stem cells, a limited number of cell divisions and mobility in response to population pressure. We have used these complementary models to provide explanations for the increased rate of centripetal migration caused by wounding and the potential for stem cell leakage to account for stable transplants derived from central corneal tissue, despite the predominantly limbal location of stem cells.

, and the keratin-14 positivity of real mouse eyes (a-d), whose identification numbers are given above the graphs, or density of low generation number cells in simulations (e) were determined by intensity analysis using Fiji software. The real eyes (a-d) were from 4 independent mice injected with tamoxifen at 6 weeks of age and imaged 2 weeks later. The images analysed in a-c were obtained from the EYFP channel of living mice; those of d and e are the same as those in Fig. 5, b and d, respectively, and are included within the same figure here also for ease of comparison. Figure 8. Robustness of centripetal migration to variations in the pressure equilibration time and TAC lifespan in the simulation model. To determine whether centripetal migration in the simulation model is sensitive to variations in the pressure equilibration time in the movement rule or lifespan of TACs, the effects of increasing or decreasing these parameters from their usual levels were tested. The mean LD ± SD are shown for 25 simulations of each case. (a) During routine use of the simulation program, a maximum of 100 iterations of the simulation algorithm was allowed to occur at each time point to enable pressure equilibration to occur within the epithelial sheet. To determine whether the duration of pressure equilibration had a major effect on corneal pattern maintenance, the LD was also calculated for each zone depicted in Fig. 3 using 75 or 150 iterations per time point. (b) The lifespan of TACs was varied from the usual 75 time-steps to either 60 or 100 time-steps, as indicated, and the effect on centripetal migration was calculated and plotted as in (a).  Figure 1e. For each of the 10 clones plotted in Fig. 1e, the number of cells, the mean direction of the path from the LESC that gave rise to the clone and the mean linear displacement were calculated.

Parameter
In vivo parameter value Simulation parameter value and comments Shape of cornea Circular dome Flat circle. The ratio of central cells to peripheral cells will be somewhat lower in simulated corneas than in real corneas. Size of cornea and basal corneal epithelial cell The radius of a flat-mounted mouse cornea is ~100 cells and 1,500µm, giving the basal epithelial cells ~7.5µm radius 1 The radius of the cornea is usually 30 cells, enabling 227 LESCs. This is large enough to maintain the complexity of the cornea and is the largest radius for computional tractability for routine simulations. Predominant location of epithelial stem cells The limbus 2 The limbus Replicative potential of TAC

Purpose
The model is used to investigate the key mechanisms of corneal epithelial maintenance. Specifically, it can be used to investigate the roles of stem cell location, population pressure, cell longevity, replicative capacity, growth rate and migration on the clonal lineage structure of the cornea.

Entities, state variables, and scales
The model space is a circular region representing the cornea and of adjacent limbus consisting of approximately 4,000 cells within the basal layer. Each cell has the following attributes, or state variables: 1. Position: The position of a cell is given determined by a single pair of co-ordinates, as we ignore the curvature of the basal layer of the cornea. Collectively, the set of cell positions is used to determine the edges and vertices of cells, using geometric structures known as Voronoi diagrams. We assume that the limbus consists exclusively of stem cells, which are all within a single cell rim a fixed region away from the center of the corneal region. 2. Lineage identification: Each stem cell is endowed with a distinct lineage identification code, and all future progeny of that stem cell in the cornea share the same cell lineage identification code. 3. Age: Cells are attributed with a current age. Processes such as time to cell death, symmetric cell division and differentiation off the basal layer are assumed to occur approximately periodically. 4. Type: Each cell is characterized by a type, reflecting either the phenotypic behavior of cells in the cornea, or they are used to maintain the model. Cell types are: Ghosts, whose position is exterior to the circular region that represents the cornea, and are designed to play no active part in corneal dynamics, but are necessary to maintain the spatial structure of the model; epithelial stem cell (ESC) types that represent the phenotypic properties of stem cells found either in the limbus (LESC) or in the cornea (CESC); and transit amplifying cell (TAC) types within the cornea. CESCs behave like LESCs with respect to lifespan and replication, but like TACs with respect to motility and spatial assignment of daughter cells following cell division. Cell types ESC and TAC differ in the following ways: -Lifespan: The lifespan of ESCs are normally distributed with a mean and standard deviation of 750 and 25 time steps, respectively. The lifespan of TACs are normally distributed with a mean and standard deviation of 75 and 25 time steps, respectively. -Replicative capacity: ESCs are assumed to have a replicative capacity that is much greater than the time scale represented by the model, and are therefore characterized by limitless replicative potential for both symmetric and asymmetric proliferation in the model. ESCs are assumed to replicate symmetrically, producing two ESCs, or asymmetrically, producing an ESC and a TAC. TACs can divide asymmetrically to produce another TAC and a terminally differentiated cell (TDC). TDCs do not contribute to the basal layer and are not depicted in simulations. TACs have a limited number of rounds of symmetric cell division, known as TAC(max), after which they divide symmetrically to produce two TDCs, both of which do not contribute to the basal layer. -Replicative frequency: Replication rates for ESCs and TACs are chosen to maintain corneal equilibrium over time.

Process overview and scheduling
At each time step, the following processes occur in this order: The other source of stochasticity in the model occurs when a neighboring LESC is selected to undergo symmetric cell division in response to the removal (death) of a LESC. 10. Collectives: there is no pre-determined collective behavior in the model. 11. Observation: position, lineage identification code, age and proliferative capacity is recorded for all cells in the model at every time step.

Initialization:
Cell positions are initially chosen so that the Voronoi diagram generated is a regular hexagonal grid (the distance between the positions of neighboring cells is a constant, s, known as the idealized cell diameter). All ESCs are given a random age (uniformly distributed from 0 to their lifespan), and a unique cell lineage identification code. Initially, all TACs are not assigned a lineage identification code. TACs derived from ESCs after initialization inherit the lineage identification code of their parent cell. Cell types are chosen such that LESCs have only two adjacent LESC neighbors, and that TACs that have less proliferative capacity are placed towards the center of the cornea. This placement has been validated through simulations where the proliferative capacity of TACs had no spatial relationship, and the simulations were run until homeostasis was reached. At that point, TACs near the center had less proliferative capacity than cells near the limbus.

Input data:
There are no external input data that alter the model during the simulations.

Sub-models:
Cell : where j represent the cells that neighbor cell, i, and w i,j is the length of the shared cell wall between cells i and j, and leaves the position of LESCs unperturbed. Each time the cell movement rule is applied, the set of Voronoi neighbors (found through Delaunay Triangulation) is recalculated. 2. Render and record the state of the simulation: All data and figures are collected during this point in the process. Cells with a particular replicative capacity remaining are color-coded in generational maps, and cells that share the same lineage identification code are assigned to a specific color, unique to that identification code in the clonal lineage maps. 3. Age cells: All cells are assumed to age synchronously. 4. Symmetric and asymmetric cell division of ESCs: • LESCs that have reached their lifespan are removed, and are synchronously replaced by symmetric proliferation of a neighboring LESC, which is chosen at random. The cell lineage identification of the new cell matches that of the mother cell, and the lifespan of the new cell is chosen randomly from N(750, 25). • In the representations where LESCs can give rise to CESCs, the other neighboring LESC gives rise to a CESC, with probability p. In this case, the CESC shares the lineage identification of the mother cell, and its position is 0.5s closer to the center of the cornea than the mother cell, and the lifespan of the CESC is chosen randomly from N(750, 25). • LESCs give rise to TAC(max)s at a rate designed to maintain corneal equilibrium. Newly proliferated TACs share the lineage identification of the mother cell, its position is 0.5s closer to the center of the cornea than the mother cell, and the lifespan of the TAC(max) is chosen randomly from N(75, 25). The rate proliferation rate r lt of LESCs to TACs is given by Equation 2: !" = 3679 227 * 75 * (2 !"#!! − 1) as 227 is the number of LESCs, which enclose an average 3769 interior cells, and 75 is the average age of TACs. 5. Cells differentiation off the basal layer: TACs that have reached their lifespan and exhausted their replicative potential die by becoming TDCs in the suprabasal layers of the epithelium, and are removed from the model. 6. Symmetric division of a TAC with x rounds of cell division remaining produces two daughter TACs, both having x-1 potential rounds of cell division remaining. Both these daughter cells share the lineage identification code as the mother cell, and are placed within 0.5s of the position of the mother cell, with no other predetermined spatial assignment.