Fig 1.
A schematic of cell center model depicting the arrangement of cells and the forces acting on them.
(a) A 2-D monolayer of epithelial cells, confined inside a circular geometry is considered with cells represented as points at their center. (b) Delaunay triangulation (blue) has been used to model cell—cell connectivity, which finds the nearest neighbors of each point and form the connectivity array accordingly. Because of the greater clarity it affords and better connection with the experimental geometry, Voronoi tessellation (topological dual of Delaunay triangulation) is used for visualization of cells. (c) When two originally connected cells move apart and form new neighbors, the connectivity of the system is updated using Delaunay triangulation. This connectivity update automatically takes T1 transitions into account. (d) Enlarged view of a representative cell i, along with its connection to neighboring cells. The position vector of this cell center is denoted by ri and position vector of its jth neighbor is denoted by rj. The blue arrow indicates the force, Fij acting between cells i and j. The total force acting on ith cell is the sum of the contributions from all the connecting neighbors. (e) The interaction between two adjacent cells is either compressive or tensile, depending upon the relative deformation of connecting spring with respect to its undeformed length, a0. Here compressive and tensile stiffness of each spring is represented by kc and kt, respectively. While kc mimics the bulk cell stiffness, kt mimics cell-cell cohesivity. It is assumed that if the deformation of any spring is greater than dmax, the cell-cell connection is broken and there is no force transfer between these two cells. (f) Force acting on each cell is resolved along anti-parallel (Fll) and perpendicular(F⊥) to the direction of the cell’s polarization(). Here v denotes the velocity vector on each particle. (g) Velocity profile in the direction of polarization as a function of Fll.
Fig 2.
Coherent rotation of cells on circular geometry.
(a) The time evolution of polarization vector, and velocity vector
is shown for ξ = 0.1. The evolution rule for polarization is chosen in such a way that, from an initial random orientation,
will try to orient along velocity vector with time. (b) The coordination between
and
is decided by the parameter ξ. The higher the value of ξ, higher is the tendency of
to orient along
. The orientation of
and
at steady state for ξ = 0.5 and ξ = 1 are also shown. (c) Mean vorticity for systems with different ξ is plotted as a function of time. (d) The tendency of polarization vector to orient with velocity vector is shown by the plot between
and time. As the value of ξ increases, value of
approaches 1, indicating perfect alignment between two vectors. (e) A plot of velocity correlation length for varying system size shows that correlation length equal to the confinement size. (f) A plot of correlation function with time shows that the velocity correlation length increases with time, till the coherent rotation sets in.
Fig 3.
Cell crowding leads to fluidisation of tissue.
(a) The relationship between velocity and radial distance is examined for varying number density. Keeping the values of other parameters same as in previous simulations, the absolute velocity, |v| averaged over time, after the system reaches steady state, is plotted as a function of radial distance for varying number of cells N = {140, 150, 160, 170}. As the number density of system increases, the velocity-radial distance curve become less linear, indicating the presence of shear in the system. (b) Variation of principal shear strain rate along the radial distance plotted as a function of number density. Increase in shear rate with number density illustrates the fluidisation of tissue induced by cell density. (c) Vorticity of system decreases with increase in cell density. (d) Without considering the effect of contact inhibition, mean velocity of the system increases with number density.
Fig 4.
Cell motility dictates the fluidized behavior of tissue.
(a) Mean velocity for varying values of cell motility (v0). (b) Normalized velocity-radial distance plot for varying values of v0 for N = 170.
Fig 5.
Tissue size, cell stiffness and cell cohesivity influence the fluid-like behavior of tissue.
(a) The relationship between velocity and radial distance is examined for three systems with varying radius, while keeping the number density approximately same for all. The number of cells in the systems are taken as N = {1170, 520, 130} for R = {15, 10, 5}, respectively. The values of other parameters are chosen as that of the previous simulations. It is observed that, while keeping the number density constant, with increase in system size, the velocity versus radial-distance profile become less linear as more number of cells tend to move with a velocity comparable to v0; this shows the presence of shear strain rate in the system (see S19 Video). (b) Increase in cell stiffness by increasing the value of compressive stiffness (kc) of a system will make the system stiff and resulting rotational behavior will be more like a solid. (c) Reduction in cell cohesivity (kt) leads to fluid-like tissue behavior.
Fig 6.
Coherent rotation of cells confined in annular geometry.
As observed for circular substrates, cells confined inside annular geometries also exhibit coherent rotation. Simulations done on an annular shaped geometry with outer radius, R and thickness, t show that mean vorticity of system decreases with increase in number density as in the case of a circle. Furthermore, simulations done with two sets of cell numbers (N = 100, represented by green curve and N = 140, represented by blue curve) show that at lower densities, system behaves like an elastic solid, roughly matching their values with analytical results (red points). For higher number of cells, cell behavior is more like an elastic solid for thicker sections. As the thickness of annulus reduced, cell state transitioned from an elastic solid to viscous fluid. The red and magenta curves showing the analytical values of elastic solid and viscous fluid respectively, are derived as explained in main text.
Fig 7.
Synchronous cell division changes the sense of coherent rotation.
(a) Cells are allowed to undergo division, where each mother cell on attaining a mature cell cycle number will divide and become two daughter cells with equal and opposite polarization (as indicated by black arrows inside the daughter cells). Two cases are analysed where cells are allowed to divide either synchronously or asynchronously. For both cases, initially starting with 40 number of cells, division is continued till number of cells become 80. (b) Even though incapable of change the direction of overall rotation, asynchronized cell division causes local perturbations in the pattern of rotation, which shortly dies down and system continues the steady rotational mode, indicated by the same sign of mean vorticity before and after cell division. (c) In the case of synchronous division of cells, the large perturbations introduced into the system in a short time is capable of inducing the change in the direction of rotation indicated by the opposite signs of mean vorticity before and after the division. Even though the reversal in the direction of rotation after synchronous division has not happened in all the cases analysed, we observed a preferential bias in the change in direction tendency.
Fig 8.
Cell stiffness and cohesivity dictate invasion pattern from coherent motion.
Three different systems of cells are taken with different stiffness of cell-cell connections. Simulations for (a) a soft system with kc = 1 and kt = 1 (b) a medium stiff system with kc = 10 and kt = 1 (c) stiff system with kc = 10 and kt = 10. The number of cells in all the three cases are same and equal to 100. After reaching a steady state of rotation, confinement was removed at time, t = 50. The snapshots of cell migratory patterns at t = 55 and t = 60 are also shown. For the case of intermediate stiff system, cells migrate in clusters compared to softer system where cell invasion pattern is more scattered. At the highest stiffness, cells continue to rotate even after removal of boundary. The length scale for each set of figure is shown below them.