Cell-induced wrinkling patterns on soft substrates

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.

💡 Research Summary

The manuscript presents a comprehensive mechanical framework for understanding how a single cell generates surface wrinkles on a compliant substrate. The authors combine the classical thin‑plate Föppl‑von Kármán (FvK) equations with a two‑dimensional active nematic phase‑field model of a cell. The phase‑field scalar ϕ distinguishes the cell interior (ϕ = 1) from the surrounding medium (ϕ = 0) and is governed by a Cahn‑Hilliard free energy that sets the interface width and surface tension. Inside the cell, a nematic order tensor Q = 2(n n − I/2) captures the orientation of the actin‑myosin network. The dynamics of Q follow the Beris‑Edwards equation coupled to incompressible Navier‑Stokes flow, while an active stress σ_active = −ζ(δ + Q) introduces a contractile dipolar forcing (ζ > 0) that can be tuned to represent different levels of cellular contractility.

The thin elastic film (shear modulus μ_f, Poisson ratio ν_f, thickness h_f) rests on a thick visco‑elastic foundation (thickness H, viscosity η_s, Poisson ratio ν_s). The total in‑plane stress in the film is σ = σ_cell + 2μ_f ε + ν_f/(1−ν_f) tr(ε) I, where σ_cell is the stress transmitted from the cell. The FvK equations then govern the out‑of‑plane displacement ω_z and the in‑plane displacements ω_i, incorporating bending rigidity, substrate reaction forces, and viscous damping.

Numerically, the authors implement the coupled system in OpenFOAM, using a finite‑volume scheme with a grid spacing Δx = 0.5 and a domain of 256 × 256 lattice points. The cell shape is prescribed as a fixed ellipse (semi‑major axis a = 55, semi‑minor axis b = 30) after a brief relaxation of the phase field to obtain a smooth interface. Time stepping is split: τ_f = 1 for the fluid‑nematic dynamics and τ_ω = 0.2 for the thin‑plate evolution, ensuring stability of the highly nonlinear FvK equations. Parameter values are chosen to mimic polydimethylsiloxane (PDMS) substrates: μ_f = 0.1, ν_f = 0.01, ν_s = 0.001, H = 10, η_s = 10, with the cell activity ζ varied between 0.005 and 0.1.

Simulation results reveal several key phenomena. First, stress concentrates at the cell’s elongated ends where +½ nematic defects migrate, producing localized contractile hotspots. These hotspots drive the nucleation of wrinkles that emanate from the cell boundary and propagate far into the substrate, often branching. The wrinkle amplitude peaks near the cell edge and decays both inward and outward, establishing a mechanical communication length scale far exceeding the cell size. Second, the wavelength λ_w follows the classic scaling λ_w ∝ h_f(μ_f/μ_s)^{1/3}, but the amplitude A_w exhibits a strong, nonlinear dependence on ζ: higher contractility shortens the wavelength and amplifies the wrinkles. Third, substrate stiffness controls pattern selection: soft substrates favor radially symmetric (circular) wrinkles, while stiffer substrates give rise to circumferential or anisotropic wrinkle arrays aligned with the cell’s long axis. Fourth, the cell aspect ratio α (0.2–1) modulates the degree of anisotropy; more elongated cells produce wrinkles that align preferentially along the major axis.

The authors validate the model against experimental wrinkle‑force microscopy (WFM) data from fibroblast cells on PDMS gels. Measured wrinkle wavelengths and orientations match the simulated predictions across a range of ζ values, and the observed long‑range propagation of wrinkles is reproduced quantitatively. The agreement demonstrates that the minimal coupling of an active nematic cell to a thin‑plate substrate captures the essential physics of cell‑induced surface instabilities.

In conclusion, the paper delivers a unified, physics‑based description of cell‑driven wrinkling that links intracellular contractile activity, cell geometry, and substrate mechanics to observable surface patterns. By providing explicit scaling laws and a computational platform, the work opens the door to non‑invasive inference of cellular forces from wrinkle morphology and sets the stage for extensions to multicellular ensembles, time‑dependent cell shapes, and more complex substrate rheologies.