Fig 1.
Schematic of the interacting active surface framework.
A tissue is described by a collection of triangular meshes representing each cell. The dynamics of the tissue is described by the dynamics of the vertices making the cellular meshes, similar to how the movement of the vertices of a vertex model describe the deformation of a tissue. The motion of the cell mesh is obtained by coarse-graining continuum mechanics equations of a theory of active surfaces via the finite element method. In this theory, cortical flows, cortical tensions, intracellular pressures, bending moments and forces arising from cell-cell interactions are taken into account.
Fig 2.
A smooth surface representing a cell is obtained based on a triangular control mesh with vertex positions Xa (A) and a set of basis functions per vertex a (B).
(A) To define the mapping between the control mesh and the cell surface, the barycentric coordinates of points in a triangular element e in the control mesh , which span a reference triangle, are used to define a point on the cell surface
,
(Eq (9)). Points
are obtained by summing basis functions
, weighted by Xa, over vertices a whose basis functions have a non-zero contribution to this element, an ensemble denoted 〈e〉. (B) Example of the basis function associated to a vertex in the mesh. For Loop subdivision surfaces basis functions, the basis function spans the first and second rows of elements surrounding the vertex (thicker white line). The vertices that interact with vertex a in the same cell, represented by the set 〈〈a〉〉 (green) are formed by the first, second, and third nearest neighbours in the mesh.
Fig 3.
(A) Physical picture of two interacting cells, I and J, that adhere via stretchable linkers.
The concentrations cI and cJ describe the number of free linkers in cells I and J per unit area. When free linkers from cells I and J are in close proximity, they can react to generate a linker joining the two cells. The ensemble of connected linkers is characterised by a two-point concentration field cIJ(XI, XJ) with units of inverse of an area squared. (B) Concentration of connected linkers as a function of the distance r = |XI − XJ|, for linear elastic bonds with stiffness k and preferred elongation rmin, fast equilibration of linker density and assuming that free linkers are in contact with a reservoir imposing their equilibrium concentration. (C) Green: effective potential of interaction of two points on two surfaces as a function of their distance r. Black: the Morse potential resembles the effective potential of interaction but with additional short-range repulsion.
Fig 4.
Single cell dynamics and convergence of the numerical framework.
(A) An inhomogeneous pattern of active surface tension γ is imposed on a spherical surface. (B) Analytically computed velocity field generated by the active tension profile in A. The velocity field is decomposed into its normal (colormap) and tangential (arrows) components. (C) Numerical solution for the velocity profile generated by the active tension profile in A. (D) Discretisation error, evaluated here in terms of the L2 norm of the difference between the analytical and numerical solutions for the velocity, as a function of the average mesh size h and for (blue),
(orange) and
(green). Other parameters are ξℓ2/η = 4, C0ℓ = 1.
Fig 5.
Dynamics and steady-state shape of a cell doublet.
(A) Normalised area of contact as a function of time (,
). (B) Snapshots of deforming doublet at times indicated in (A), with the cell surface velocity field superimposed. (C) Coloured lines: ratio of contact surface area
to cell surface area A, for simulations with different values of
and
. Black dotted line: theoretical approximation valid in the limit of
. (D) Snapshots of doublet equilibrium shape for increasing adhesion strength; different parameters in (D1)-(D3) correspond to points labelled in C. (E) Comparison of a slice for the different simulations, with values of
and
indicated in C, and
. The value of
affects the shape smoothness of the edge of the adhesion patch. (F) Convergence of the method evaluated by computing the L2-norm of the error in the initial velocity field for two different
(
(orange) and
(blue)), and different average mesh sizes h, and comparing the results with a simulation with h/ℓ ≈ 2 ⋅ 10−2 (finer). (G) Schematic of adhering doublet, with different active tensions γ1 and γ2 for each cell. (H) Position of the cell centre of mass X1 and X2, as a function of active tension asymmetry between the two adhering cells, α = (γ1 − γ2)/(γ1 + γ2), where γ1 and γ2 are the surface tensions of the two cells. Beyond α = 0.69, the cell with lowest tension completely engulfs the one with highest tension. (I) Snapshots of doublet equilibrium shape, clipped by a plane passing by the line joining the cell centres, for increasing difference of active surface tension; corresponding to points labelled in H. In H, I:
,
,
.
Fig 6.
(A) Simulation results (equilibrium shapes) for a planar sheet for different values of the adhesion parameter (
,
).
(B) Reduced volume v as a function of . (C) Schematic for the measured apical and basal cell surface area Aab, lateral surface area Al, side length a and cell thickness hc. (D) Results are compared to a simple 3D vertex model with a lateral surface tension γl and apical and basal surface tension γab. (E) Box plots: ratio of apico-basal to lateral surface area 2Aab/Al for the center cells, as a function of the adhesion parameter
. Dashed black, blue and green lines: prediction of simplified theories describing the cell shape as an hexagonal prism, a cylinder with two spherical caps, and the union of a hexagonal prism with two spherical caps. Inset: cellular aspect ratio a/hc, with a measuring the side of the hexagonal face and hc the thickness of the sheet, as a function of
. Here
and hc = Al/(6a). (F) Schematic for the polar vector P characterising the asymmetry of the cell shape. (G) Box plot for Pz (blue) and |Pz| (red). As adhesion increases, cells deform asymmetrically in the direction orthogonal to the planar sheet.
Fig 7.
Growth of small cell aggregates, driven by synchronous cell divisions and cell volume increase (,
,
).
(A) Each cell doubles its volume between birth and division, over a cell cycle time tD. Cell division is introduced by splitting the mother cell with a plane passing through the cell centre, and generating two daughter cells separated by a distance d* (d*/2 from the division plane). (B-D) Simulation results for two values of the ratio τ/tD. (B): Largest centre-to-centre cell distance max〈I,J〉 RIJ, as a function of time. Jumps correspond to cell division events. (C) Average reduced volume 〈v〉 as a function of time. (D) Snapshots of simulations of two growing aggregates.