Cell strain-stiffening drives cell breakout from embedded spheroids

TL;DR

The paper develops a three-dimensional mechanical framework that links intracellular stresses, cell–cell cohesion, and cell–ECM interactions to predict invasion from embedded spheroids. It extends a 3D vertex model to a fibrous ECM and derives a 3D Cauchy stress tensor to quantify per-cell stresses, revealing distinct stress patterns for solid-like versus fluid-like spheroids and showing strain-stiffening enables boundary cells to transiently generate high local forces. The authors demonstrate that single-cell breakout requires both strain stiffening and reduced adhesion, while multi-cell streaming invasion emerges only with anisotropic adhesion aligned with the elongation axis, identifying two distinct mechanical pathways to invasion. The work provides a multiscale, testable framework linking spheroid rheology, cell-scale mechanics, and adhesion organization to invasion modes, with implications for understanding tumor invasion and ECM remodeling.

Abstract

Understanding how cells escape from embedded spheroids requires a mechanical framework linking stress generation within cells, across cells, and between cells and the surrounding extracellular matrix (ECM). We develop such a framework by coupling a 3D vertex model of a spheroid to a fibrous ECM network and deriving a 3D Cauchy stress tensor for deformable polyhedral cells, enabling direct cell-level stress quantification in three dimensions. We analyze maximum shear stress in solid-like and fluid-like spheroids: solid-like spheroids exhibit broader stress distributions and radial stress gradients, while fluid-like spheroids show lower stresses with weak spatial organization. Cell shape anisotropy is not generically aligned with principal stress directions, indicating that morphology alone is an unreliable proxy for mechanical state. We further demonstrate strain stiffening at the single-cell level, where elongation produces nonlinear increases in maximum shear stress, allowing boundary cells in otherwise low-stress, fluid-like spheroids to transiently generate forces sufficient to remodel the matrix. To connect strain-induced stress amplification to invasion modes, we introduce an extended 3D vertex model with explicit, tunable cell-cell adhesion springs. In this minimal mechanical framework, single-cell breakout results from strain stiffening combined with reduced adhesion, whereas multi-cell streaming additionally requires anisotropic adhesion strengthened along the elongation axis and weakened orthogonally. Together, these results identify distinct mechanical pathways coupling cell strain, stress amplification, and adhesion organization to spheroid invasion.

Figures (9)

• Figure 1: Integrating a 3D vertex model of a cell spheroid with a fiber-network model to study cell--ECM interactions. Left: Full system at the final simulation time, i.e., time $t = t_f$ and Right: Zoom in of the spheroid. The black denote active linker springs coupling the cells to the fibers.
• Figure 2: Distributions of cellular shear stress, cell shape, and stress--shape correlation for solid-like and fluid-like spheroids at fiber-network occupation probability $p=0.8$. (a) Histogram of the maximum shear stress for the cells for two different target $s_0$s. Gamma fit parameters: $s_0 = 5.2$: $\alpha = 3.06, \theta = 9.34\times10^{-4}$; $s_0 = 5.8$: $\alpha = 1.19, \theta = 7.70\times10^{-4}$. (b) Histogram of the cell shape anisotropy for the cells for two different target $s_0$s. Gamma fit parameters: $s_0 = 5.2$: $\alpha = 3.20, \theta = 1.66\times10^{-2}$; $s_0 = 5.8$: $\alpha = 2.78, \theta = 6.21\times10^{-2}$. (c) The overlap, or dot product, between the eigenvector associated with the largest maximum shear stress and the eigenvector associated with the largest gyration eigenvalue.
• Figure 3: Spatial distribution of cellular stresses and cellular shape in solid-like spheroids. (a) Sample cross-section of the cellular maximum shear stress. (b) Sample cross-section of the cellular shape anisotropy. (c) Maximum shear stress for both fluid-like and solid-like spheroids as a function of radial distance from the center of mass of the spheroid. The inset shows the result for $p=0$, i.e., the non-embedded spheroid with no surrounding fiber network.
• Figure 4: Cellular strain stiffening for volume-preserving deformations. (a) Initial cell configuration and a strained cell configuration for cells from a fluid-like spheroid. (b) Distribution of maximum bulk shear stress for the initial cells (dark blue) and for the strained cells (light blue) for $s_0=5.8$. (c) The average maximum shear stress versus strain demonstrating a nonlinear relationship for both types of the spheroids and, thus, exhibiting strain stiffening phenomena. The inset shows a deviation from linear behavior around 0.1 strain, yielding an estimate of the onset of strain stiffening. By linearly fitting the low and high strain parts of the stress-strain curve, the intersection of the two fits yields a second estimate for the crossover between low and high strain behavior at approximately 0.4 (dashed vertical line). (d) The average cell shape anisotropy versus strain.
• Figure 5: Different spheroid cell breakout modes. (a) Experiment with MEF spheroid in 1.5 mg/ml collagen I. matrix. (b) Single-cell breakout as the target spring length of the four fibers attached to the deep yellow cell decreases. The thicker purple cell-cell adhesions are weaker than the thinner red ones. (c) Two-cell breakout as the target spring length of the four fibers attached to the deep yellow cell decreases. The cyan denotes strong anisotropic cell-cell adhesion springs between the leader boundary dark yellow cell and the center blue cell that is becoming elongated in the dark yellow cell extension to ultimately follow it.

Theorems & Definitions (7)

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• Corollary 1.1
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• Corollary 2.1
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