Harmonic fields and the mechanical response of a cellular monolayer to ablation

Abstract

Multicellular tissues, such as the epithelium coating a developing embryo, often combine complex tissue shapes with heterogeneity in the spatial arrangement of individual cells. Discrete approximations, such as the cell vertex model, can accommodate these geometric features, but techniques for analysis of such models are underdeveloped. Here, we express differential operators defined on a network representing a monolayer of confluent cells in a framework inspired by discrete exterior calculus, considering scalar fields defined over cell vertices and centres and vector fields defined over cell edges. We achieve this by defining Hodge stars, wedge products and musical isomorphisms that are appropriate for a disordered monolayer for which cell edges and links between cell centres are not orthogonal, as is generic for epithelia. We use this framework to evaluate the harmonic vector field arising in an ablated planar monolayer, demonstrating an approximate 1/\textit{r} scaling of the upper bound of the field's amplitude, where \textit{r} is the distance from the ablation. Using a vertex model that incorporates osmotic effects, we then calculate the mechanical response of a monolayer in a jammed state to ablation. Perturbation displacements exhibit long-range coherence, monopolar and quadrupolar features, and an approximate 1/\textit{r} near-hole upper-bound scaling, implicating the harmonic field. The upper bounds on perturbation stress amplitudes scale approximately like 1/\textit{r}$^2$, a feature relevant to long-range mechanical signalling.

Figures (13)

• Figure 1: (a) Schematic diagram illustrating the primal network $\mathcal{N}$ (green cell edges $\mathbf{t}_{j'}$) and the dual network $\mathcal{N}^\rhd$ (purple links $\mathbf{T}_{j'}$). Cell vertices ($\mathbf{r}_k$ and $\mathbf{r}_{k'}$, blue dots) are associated with triangle areas ($E_k$, shaded blue). Edge centroids ($\mathbf{c}_j$ and $\mathbf{c}_{j'}$, green dots), and edge-link intersections ($\mathbf{b}_j$, purple dots) are associated with quadrilateral areas ($\tfrac{1}{2}F_j$, shaded green). Cell centres ($\mathbf{R}_i$, red dots) are associated with cell areas ($A_i$, shaded grey). Cell orientation $\boldsymbol{\epsilon}_i$ and the opposite triangle orientation $\boldsymbol{\epsilon}_k$ are indicated. (b) The tangent plane $T\mathcal{M}\vert_{\mathbf{b}_j}$, showing basis vectors $\mathbf{e}_j^\parallel\equiv \mathbf{t}_j$, $\mathbf{E}_j^\parallel\equiv \mathbf{T}_j$, rotated vectors $\mathbf{e}_j^\perp$, $\mathbf{E}_j^\perp$ and the projection $\{v_j^\parallel,v_j^\perp\}^\top$ of a vector $\mathbf{v}_j$ onto $\mathbf{e}_j^\parallel$ and $\mathbf{e}_j^\perp$.
• Figure 2: An ablated monolayer showing network construction and representations of geometric quantities. (a) Primal network $\mathcal{N}$ of cell edges (green lines) and dual network $\mathcal{N}^\rhd$ of cell links (purple lines), as shown in detail in Fig. \ref{['fig:Figure1schematic']}(a). Links are bounded by cell centres (red dots) or, at the monolayer periphery, edge centroids (green dots). (b) Randomly coloured polygons show cell areas $A_i$, overlaid on $\mathcal{N}$ and $\mathcal{N}^\rhd$. (c) The reduced dual network $\hat{\mathcal{N}}^\rhd$ (purple lines) lacks links to peripheral edge midpoints; randomly coloured polygons show internal triangle areas $E_k$, overlaid on $\mathcal{N}$. (d) The reduced primal network $\hat{\mathcal{N}}$ (green lines) lacks peripheral edges and peripheral vertices; blue dots show the vertices of $\hat{\mathcal{N}}$; randomly coloured polygons show quadrilateral areas $\tfrac{1}{2}F_j$ at non-peripheral edges. In $\hat{\mathcal{N}}$, edges normal to the periphery do not terminate in a vertex.
• Figure 3: DEC-inspired operators constructed from the maps shown in (\ref{['eq:dr']}). Operators acting on vectors parallel to edges and links are paired with those perpendicular to edges and links, to form grad (\ref{['eq:grad']}), curl (\ref{['eq:curl']}), $-$div (\ref{['eq:div']}) and rot (\ref{['eq:rot']}) defined over the primal and dual networks. Coloured arrows show the corresponding maps between spaces. Labels of the same colour show the compact DEC representation plus the related operator components, listed below in Table \ref{['tab:diffOpDef']}. Operators on the left-hand-side [right-hand-side] of the diagram have superscript $v$ [$c$], with $\mathrm{d}$ being $\mathsf{A}_1^*$ or $\mathsf{A}_1^{*\top}$ [$\mathsf{B}_1^*$ or $\mathsf{B}_1^{*\top}$]. $\mathrm{grad}$ is magenta; $\mathrm{curl}$ is brown. The corresponding codifferentials $-{\mathrm{div}}$ and ${\mathrm{rot}}$ are green and blue (dashed) respectively.
• Figure 4: Two illustrations of the rotated force potential. (a,d) show cell monolayers, at mechanical equilibrium, with clusters of central cells highlighted, with cells coloured arbitrarily. (b,e) show the clusters with the network $N^\Diamond$ connecting adjacent edge centroids $\mathbf{c}_j$ superimposed. An edge of $N^\Diamond$ in cell $i$ adjacent to vertex $k$ is given by $\mathbf{s}_{ik}$, as defined in (\ref{['eq:sik']}). The corresponding forces $\mathbf{f}_{ik}$ are rotated by $\pi/2$ and assembled to form networks (e,f), with vertices $\mathbf{h}_j$, preserving the colours assigned in (a,d) and (b,e). The monolayers are in equilibrium, so that rotated forces form closed triangles around vertices ($\sum_i \mathbf{f}_{ik}=\mathbf{0}$) and closed polygons around cells ($\sum_k \mathbf{f}_{ik}=\mathbf{0}$), so sharing the topology of $N^\Diamond$. The network in (c) is non-overlapping because the monolayer simulated in (a) is subject to isotropic compression at its periphery. The monolayer in (d) is stress-free at its periphery, showing that the rotated force network is typically a non-planar graph.
• Figure 5: Harmonic fields on edges $\mathsfbf{x}^{(m)}$ in (\ref{['eq:edgex']}) are shown in the two left-hand columns; harmonic fields on links $\mathsfbf{X}^{(m)}$ in (\ref{['eq:edgeX']}) are shown in the two right-hand columns. Top row: the 1st eigenmode ($m=1$) of a system with 1 hole, for $\{z^{\parallel(1)},z^{\perp(1)}\}=\{1,0\}$ or $\{0,1\}$. Rows 2 and 3: the 1st ($m=1$) and 2nd ($m=2$) eigenmodes of a system with 2 holes.