Spectral approaches to stress relaxation in epithelial monolayers

TL;DR

This work addresses how epithelial monolayers relax toward equilibrium after perturbations by analyzing the vertex model through a spectral lens. It combines the gradient-flow nature of the energy with singular-value decomposition to derive model-dependent Laplacians $\mathcal{L}_c^{\mathsf{G}}$ and $\boldsymbol{\mathcal{L}}_v^{\mathsf{G}}$ that connect vertex motions to cell-area/perimeter changes, linking relaxation times to Laplacian spectra modified by evolving prestress. A key finding is that geometric stiffness arising from prestress typically dominates the relaxation dynamics, while cell-dilation (diffusive-like) modes can control the fastest relaxation in certain limits; the full spectrum reflects both global geometry and local cell stresses, especially under jammed conditions. The approach provides a framework to interpret tissue mechanics across scales and under perturbations such as stretching, with implications for understanding mechano-responses in development and disease.

Abstract

We investigate the viscoelastic relaxation to equilibrium of a disordered planar epithelium described using the cell vertex model. In its standard form, the model is formulated as coupled evolution equations for the locations of vertices of confluent polygonal cells. Exploiting the model's gradient-flow structure, we use singular-value decomposition to project modes of deformation of vertices onto modes of deformation of cells. We show how eigenmodes of discrete Laplacian operators (specified by constitutive assumptions related to dissipation and mechanical energy) provide a spatial basis for evolving fields, and demonstrate how the operators can incorporate approximations of conventional spatial derivatives. We relate the spectrum of relaxation times to the eigenvalues of the Laplacians, modified by corrections that account for the fact that the cell network (and therefore the Laplacians) evolve during relaxation to an equilibrium prestressed state, providing the monolayer with geometric stiffness. While dilational modes of the Laplacians capture rapid relaxation in some circumstances, showing diffusive dynamics, geometric stiffness is typically a dominant source of monolayer rigidity, as we illustrate for monolayers exposed to unsteady stretching deformations.

Figures (8)

• Figure 1: Left: a diagram summarising the action of matrix operators $\mathsfbfit{M}$, $\mathsf{G}$ and $\mathsf{E}$. ${\mathsfbfit{M}}$ has components that are vectors describing how cell areas and perimeters change under small vertex displacements. 'Vertex displacements' describes the tangent space $\mathcal{T}\mathcal{Z}$ of the state space $\mathcal{Z}$. 'Forces at vertices' describes the dual space $\mathcal{T}\mathcal{W}$; 'Pressures, tensions' sit in a space $\mathcal{TX}$ that is dual to the tangent space $\mathcal{TY}$ denoted 'Areas, perimeters.' The operators $\mathsfbfit{M}\cdot$ and $\mathsfbfit{M}^\top$ are singular with rank $q<2N_c$ and therefore have nontrivial kernels mapping to $\varnothing$. Laplacians $\mathcal{L}_c^{\mathsf{G}}$ and $\boldsymbol{\mathcal{L}}_v^{\mathsf{G}}$ are defined in (\ref{['eq:lapcv']}); dual operators are given in (\ref{['eq:lapcvdual']}). Right: kernels in $\mathcal{TZ}$ and $\mathcal{TY}$ are shown as red boxes. Perturbations are mapped between these spaces by divergence $\langle\mathsfbfit{N}^\top, \cdot \rangle_{\mathsf{E}}$ and gradient $\langle\mathsfbfit{N}, \cdot \rangle_{\mathsf{G}}$ matrix operators. Elements of $\mathrm{ker}(\bar{\mathsfbfit{M}}^\top)$ are denoted States of Self-Stress (SSS); elements of $\mathrm{ker}(\bar{\mathsfbfit{M}}\cdot)$ are denoted Zero Modes (ZM).
• Figure 2: (a,b) Spectra for a symmetric configuration of $N_c=127$ hexagonal cells, as depicted in the inset; (c,d) spectra for a disordered monolayer of $N_c=100$ cells; $\Gamma=0.5$, $L_0=1$, $\gamma_A=\gamma_L=0$ in both cases. (e) shows the corresponding pressure, tension and isotropic and deviatoric cell stress components $P_{\mathrm{eff},i}$ and $\zeta_i$ in (\ref{['eq:peffshear']}) of the equilibrium disordered monolayer. (a,c) compare the eigenvalues $\lambda^{(n)}$, $n=1,\dots, 2N_v$ of the Hessian operator (\ref{['eq:lin']}a, black) to the $2N_v$ eigenvalues of the vertex Laplacian (\ref{['eq:lapcv']}, blue) and the $q$ non-zero eigenvalues of the cell Laplacian (\ref{['eq:lapcv']}, red); $N_v=294$ and $q=N_c$ in (a) and $N_v=228$ and $q=2N_c-1=199$ in (c); zero modes of $\bar{\boldsymbol{\mathcal{L}}}_v^\mathsf{G}$ have $\lambda^{(n)}<10^{-10}$; translation and rotation modes of the Hessian are indicated; the $N_c$ zero modes in (a) and single zero mode in (c) of $\bar{\mathcal{L}}_c^{\mathsf{G}}$ are indicated as states of self-stress (SSS); cell shear and cell dilation modes are indicated in (c). (b, d) show the Hessian spectrum on an enlarged scale, excluding the translation and rotation zero modes, decomposed using (\ref{['eq:buildspectrum']}) into contributions from material stiffness (green, derived from $\mathcal{L}_c^{\mathsf{G}}$) and geometric stiffness (orange, derived from prestress).
• Figure 3: A selection of eigenmodes of the Hessian of (a) the hexagonal monolayer and (b) the disordered monolayer shown in Fig. \ref{['fig:2']}. The corresponding eigenvalues are ranked by index $n$; arrows show vertex displacements, colours show corresponding cell area variations, with red and blue indicating variations of opposite signs.
• Figure 4: Insets show eigenmodes of the cell Laplacian $\bar{\mathcal{L}}_c^{\mathsf{G}}$, for the 100-cell disordered monolayer shown in Fig. \ref{['fig:2']}, with mode numbers as indicated; colours indicate patterns of relative area and perimeter variation (red and blue show opposite signs). Shear [dilation] modes (left [right]) are defined to have area and perimeter values of opposite [the same] parity; the categorisation is clear for the majority of modes, as demonstrated in the scatterplot which counts the number of cells which have the same sign for their area and perimeter values. Mode $n=1$ is the SSS.
• Figure 5: (a) Spectra for $L_0=1$ and $\Gamma=0.01, 0.1, 1, 10$ for realisations of disordered monolayers with $N_c=100, N_v=228$, showing contributions of geometric (orange) and material (green) stiffness to the full Hessian (black; translation and rotation modes are not shown). (b) Cell dilation modes collapse for small $\Gamma$ (blue, purple); the remainder of the spectrum collapses for large $\Gamma$ (red, orange). The inset shows how the spectra of $\mathsf{A}_c^{-1}\mathcal{L}_A$ for $\Gamma=0.001$ (purple) and $\Gamma L_0 \mathsf{L}_{c}^{-1} \mathcal{L}_{L}$ for $\Gamma=10$ (red) predict the most rapidly decaying component of the full spectrum (grey). (c) The same as (b) but for a monolayer of hexagons with $N_c=127, N_v=294$.