Noninvasive rheological inference from stable flows in confined tissues

TL;DR

A self-driven, rheometer-like assay in which collective migration generates stationary shear flows, allowing rheological parameters to be inferred directly from image sequences, is introduced, proposing a noninvasive route to rheological inference in migrating epithelial tissues and, more generally, in actively flowing granular materials.

Abstract

Quantifying the in-plane rheology of epithelial monolayers remains challenging due to the difficulty of imposing controlled shear. We introduce a self-driven, rheometer-like assay in which collective migration generates stationary shear flows, allowing rheological parameters to be inferred directly from image sequences. The assay relies on two sets of ring-shaped fibronectin patches, micropatterned in arrays for high-throughput imaging. Within isolated rings, the epithelial tissue exhibits persistent rotation, from which we infer active migration stresses and substrate friction. Within partially overlapping rings, the tissue exhibits sustained shear, from which we infer the elastic and viscous responses of the cells. The emergence of a Maxwell-like viscoelastic relation -- characterized by a linear relationship between mean cell deformation and neighbor-exchange rate -- is specifically recapitulated within a wet vertex-model framework, which reproduces experimental measurements only when intercellular viscous dissipation is included alongside substrate friction. We apply our method to discriminate the respective roles of two myosin~II isoforms in tissue mechanics. Overall, by harnessing self-generated stresses instead of externally imposed ones, we propose a noninvasive route to rheological inference in migrating epithelial tissues and, more generally, in actively flowing granular materials.

Figures (4)

• Figure 1: Spontaneous shear at double-rings overlap. (A-B) Madin-Darby Canine Kidney (MDCK) cells are deposited on confining fibronectin patterns that are shaped as two rings in close contact. Upon reaching confluency, the directions of rotations (DR) within each ring are either (A) opposite (ODR) or (B) same (SDR) patterns (scale bars: $100 \, \mu\mathrm{m}$. (C) Average angular velocity within each ring in ODR (light green) and SDR (dark green) experiments. We report the timing of the ring-ring contact time ($t_{\mathrm{c}}$) and the confluence time ($t_{\mathrm{f}}$) in the ODR experiment. (D) Observation frequency of the ODR, SDR, and unstable rotation patterns, (E) Difference ($t_{\mathrm{c}}-t_{\mathrm{f}}$) between the ring-ring contact time ($t_{\mathrm{c}}$) and the confluence time ($t_{\mathrm{f}}$) in the ODR (light green) and SDR (dark green) cases. (F) Mean velocities at steady state in the ODR (light green) and SDR (dark green) cases (averaged over $n=23$ rings).
• Figure 2: Cell deformability estimation (A) Cell shape: connectivity graph between cell barycenters; time-averaging defines the local strain tensor $\bm{\varepsilon}_{\mathrm{cell}}$, represented as ellipses with axes proportional to eigenvalues (“coffee bean” representation Graner2008). (B) Topological changes: junction appearance (green) and disappearance (red); time-averaging yields the rearrangement rate tensor $\dot{\bm{\varepsilon}}_{r}$, see Eq. (\ref{['eq:Tfield']}) . (C–H) Same Direction of Rotation (SDR) experiment. (C) Instantaneous connectivity graph. (D) Elongation axis, $\bm{\varepsilon}^{\mathrm{dev}}_{\mathrm{cell}}$: eigenvector orientation and eigenvalue amplitude of the deviatoric tensor, see (A). (E) Instantaneous appearance (green) and disappearance (red) axes. (F) Rearrangement rate tensor, $\dot{\bm{\varepsilon}}^{\mathrm{dev}}_{r}$ as defined in (B). (G) Component-to-component plot of $\bm{\varepsilon}^{\mathrm{dev}}_{\mathrm{cell}}$ vs. $\dot{\bm{\varepsilon}}^{\mathrm{dev}}_{r}$, expressed in terms of their diagonal (XX, YY, upward triangles) and off-diagonal (XY, downward) components. The viscoelastic time $\tau$ is the slope (dashed regression line). The symbol color codes for the position at which the tensors are sampled within the ring–ring contact region (inset). (H) Distribution of viscoelastic times in ODR (light green) and SDR (dark green) modes ($n=23$ rings).
• Figure 3: Myosin perturbations. (A–G) Myosin-IIA silencing (ShIIA). (A–B) Single-ring. (A) Brightfield image: concentric rings with increasing radii migrate at increasing speeds (scale bar: $50 \, \mu\mathrm{m}$). (B) Left: Angular velocity vs. the rms ring radius $r= \sqrt{(R_{\mathrm{in}}^2+R_{\mathrm{out}}^2)/2}$ (red, $n=3$) fitted with Eq. (\ref{['eq:vfinal']}); WT (green). Right: Model sketch with $V_1$ (ring velocity), $v_a$ (active traction), and $1/\mu_a$ (active coupling time) reduced in ShIIA and ShIIB (red and blue curves, respectively) as compared to WT (green) (shaded areas: model $80\%$ confidence interval). (C–G) Double-rings. (C–D) Brightfield image of ShIIA-treated tissues displaying (C) stable opposite (ODR) and (D) same (SDR) direction of rotation modes (scale bar: $100 \, \mu\mathrm{m}$). (E–G) Statistics of elongation, velocity, and viscoelastic time: experiments (WT, dark green; ShIIA, dark red) and simulations (light red: best ShIIA fit). Red arrows highlight deviations from WT simulations. (B; H–N) Myosin-IIB silencing (ShIIB). (H) Brightfield image (scale bar: $100 \, \mu\mathrm{m}$). (I) Angular velocity, showing no net sign. (K–M) The oscillation phase dominates in (K) experiments and (L–N) simulations with (L) weaker alignment $\mu_a$, (M) lower traction $v_a$, or (N) smaller preferred perimeter $P_0$ (higher junctional tension) than in the control simulations (Table S1, SM).
• Figure 4: Emergent Maxwell model interpretation. (A) Time evolution of simulations with a highlight on cells undergoing a rearrangement (colored in blue) in an SDR case with either (top) low cell-cell viscosity or low rotation speed and (bottom) high cell-cell viscosity or high rotation speeds (parameters in Supplemental Material SM). (B-C) Component-to-component relationship in the deviator of the cell rearrangement rate ($\dot{\varepsilon}^{\mathrm{dev}}_{r}$) and strain ($\varepsilon^{\mathrm{dev}}_{\mathrm{cell}}$) tensors, in (B) the low and (C) the high-velocity cases. Triangles oriented upward for the diagonal components (XX, YY) and downward for the off-diagonal ones (XY). (D) Viscoelastic times for simulations with increased active traction forces, $v_a$; each dot represents the mean over simulations at a fixed polar-velocity coupling $1/\mu_a$ value. (E) Viscoelastic time as a function of (axis 1, yellow to red lines) the polar-velocity coupling $1/\mu_a$, or (axis 2, magenta line) the cell-cell junction viscosity $\eta$.