Cell-induced wrinkling patterns on soft substrates

Abstract

Cells exert traction forces on compliant substrates and can induce surface instabilities that appear as characteristic wrinkling patterns. Here, we develop a mechanical description of cell-induced wrinkling on soft substrates using a thin film elastic framework based on the Föppl-von Kármán equations coupled to a phase-field model of a single cell. We model in-plane contractile stresses driven by cellular activity and study how their magnitude, spatial distribution, and symmetry determine the onset of wrinkling and the resulting pattern selection. The theory predicts transitions between distinct morphologies, such as radial, circumferential, and anisotropic wrinkle arrangements, and provides scaling relations for wrinkle wavelength and amplitude as functions of elastic parameters and imposed cellular forcing. We compare these predictions with available experimental observations of cell-driven wrinkling on compliant gels and find good agreement for both qualitative pattern classes and quantitative wavelength trends. Our results offer a minimal modelling framework to interpret wrinkling assays and connect observed surface patterns to underlying cellular forces.

Figures (6)

• Figure 1: Schematic representation of the modelling framework. (a) Schematic representation of the experimental setup. PDMS is deposited on a glass surface. The top part is then plasma irradiated such that a thin layer of higher elasticity (Young modulus $E_1 > E_2$) is formed. The cell is placed on top of a stiffer layer, leading to a wrinkling pattern formation. (b) Simulation algorithm: we use a 2D phase-field model of a single cell to obtain the in-plane components of stresses that the cell exerts onto the substrate. The stress tensor is then used in the Föppl-von Kármán setup to calculate the in- and out-of-plane deformation of this plate. (c) Schematic illustration of how the cell contractility is modelled.
• Figure 2: Wrinkling pattern of a spherical cell. (a) Snapshots of stress tensor components, $\sigma_{xx}^{cell}$, $\sigma_{yy}^{cell}$, $\sigma_{xy}^{cell}$, at the end of simulation. (b) Snapshot of a substrate height, $w_z$, obtained from the simulations. The solid black line represents the boundary of the cell. The green dotted line indicates a cross-section along which the height profile is extracted in panel (c). (c) Substrate height profile along the x-axis at $y=0$. (d) Normalised extent of wrinkles, i.e. distance the deformations propagate from the cell body, as a function of time, demonstrating that the system reaches values close to equilibrium.
• Figure 3: Effect of material properties on the propagation of wrinkles from the cell. Phase diagrams for $\mu_f$ vs $\nu_f$ (a), $\nu_s$ vs $\nu_f$ (b) characterise the propagation against experimentally-tunable parameters. The profile for the wrinkling extent as a function of $\mu_f$ keeping $\nu_f =0.01$ and $nu_s = 0.001$ (c) and against $\nu_f$ keeping $\mu_f = 0.1$ and $nu_s = 0.001$ (d).
• Figure 4: Effect of cell aspect ratio on wrinkle patterns. Plots of the wrinkle pattern as the aspect ratio is changed from an elongated ellipsoid shape to a circle (a--d). The colour map depicts the height of the wrinkle with the director field plotted as black lines. The contour of the phase field, $\phi = 0.5$, is drawn to show the outline of the cell. The activity in all is set to $\zeta = 0.04$.
• Figure 5: Impact of cell contractility on wrinkling patterns. Wrinkle properties, such as normalised extent (a), mean amplitude within 10 units from the cell boundary (c), and normalised wavelength (e) plotted against activity. Representative snapshots on the right display the director and wrinkle fields for different regimes in the nematic, corresponding to the marked vertical lines. These regimes correspond to a stationary nematic field (b), bend waves (d), and active turbulence (f). Both extent and wavelength are normalised by the major axis.