A constitutive model for discontinuous shear thickening in epithelial tissues

Abstract

The rheological properties of biological tissues, though fundamental to many physiological and pathological processes such as embryonic development, wound healing, and tumor progression, remain poorly understood. A recent study showed that the active vertex model of biological tissues exhibits discontinuous shear thickening (DST), where stress and viscosity suddenly increase at a critical shear rate. What is the mechanism of DST here? Is it another nontrivial feature of activity or an inherent property of the system? To address this, we show that the thermal vertex model also exhibits DST at a small but non-zero temperature $T$. Solid-like and liquid-like cells coexist at the stress jump, and the stress-controlled flow curves exhibit the characteristic S-shape. We then introduce a constitutive model for DST in epithelial tissues. As $p_0$ increases, the theory predicts DST, followed by continuous shear thickening (CST), and finally Newtonian behavior, consistent with simulations. DST begins at the jamming point, $p_0^m$, and the Newtonian behavior starts at $p_0^*$, where the yield stress vanishes. Both $p_0^*$ and the liquid-to-solid transition stress, $σ^*$, govern the DST-CST boundary. Furthermore, $p_0^*$ and $σ^*$ also depend on $T$. Increasing $T$ reduces $p_0^*$, narrows the shear-thickening regime, and eventually destroys DST when $p_0^* \leq p_0^m$. Thus, the primary ingredients of DST in tissue models are a finite yield stress in the unjammed regime and non-zero fluctuations, whose specific form is not important. The theory agrees well with our simulation data and also provides further testable predictions.

Figures (6)

• Figure 1: Rheological properties at $T=0$. (a) Stress-strain curves for two representative target shape indices, $p_0 = 3.7$ and $p_0 = 3.9$. When the strain $\gamma$ is small, the stress, $\sigma$, increases with $\gamma$ and then reaches a steady state at large $\gamma$. (b) Representative snapshots at three selected strain values for $p_0 = 3.9$ [marked in (a)]; the colors encode the local aspect ratio. (c) Histogram of the orientation of the principal axes with respect to the flow direction; the color scale represents the magnitude of the average aspect ratio within each angular bin. When $\gamma$ is very small, the cells are less elongated and randomly oriented. As $\gamma$ increases, the cells elongate and orient along the shear direction. (d) Steady state flow curves, $\sigma$ as a function of $\dot{\gamma}$, for different $p_0$. We can fit each curve to the Herschel-Bulkley form to extract the yield stress, $\sigma_y$, which decreases with increasing $p_0$ (inset). (e) Shear viscosity, $\eta=\sigma/\dot{\gamma}$, for the same flow curves as in panel (d). The system only shows shear thinning at $T=0$. We have used $N=100$ for these simulations.
• Figure 2: Discontinuous shear thickening at finite $T$. (a) Flow curves for different values of $p_0$ at $T = 2 \times 10^{-4}$ for a system size $N = 100$. (b) Shear viscosity $\eta$ corresponding to the flow curves shown in panel (a). (c) S-shaped flow curves obtained under stress-controlled conditions at $T = 2 \times 10^{-4}$ for $N = 400$. (d) Shear viscosity corresponding to the flow curves shown in panel (c). The sudden jump in both $\sigma$ and $\eta$ and the S-shaped flow curves are key characteristics of DST.
• Figure 3: Fluid-to-solid transition and the relaxation dynamics. (a) Stress-controlled flow curve for $p_0 = 3.9$ at temperature $T = 2 \times 10^{-4}$. We show the physical states of the cells based on their perimeter, $p_i$, at the three points marked by the arrows as the stress transitions from the liquid-like to the solid-like regime. We mark the cell blue if $|p_i-p_0|\leq 0.02$, otherwise, yellow. (b) In the liquid-like branch, most cells have $p_i\simeq p_0$. (c) In the solid-like branch, the cells are deformed, with most having $p_i$ much higher than $p_0$. (d) In the DST regime, blue and yellow cells coexist. (e) We study the dynamics via the overlap function, $Q(t)$. At fixed $T = 2 \times 10^{-4}$, $Q(t)$ decays becomes faster as $p_0$ increases. (f) The relaxation time $\tau$ varies nearly linearly with $\dot{\gamma}_c^{-1}$, the characteristic shear rate at which $\sigma$ jumps from the liquid to solid branch. The solid line represents $\tau=1+0.47 \dot{\gamma}^{-1}_c$. Inset: $\tau$ as a function of $p_0$; symbols represent the data, and the dashed line represents $\tau = 247.34\,(p_0 - 3.823)^{-3/2}$.
• Figure 4: Distribution of local stress $\sigma_i$. (a) Distribution of local cell stresses, $P(\sigma_i)$, at various strain rates $\dot{\gamma}$ in the liquid-like branch. (b) $P(\sigma_i)$ shows a broader distribution at larger $\dot{\gamma}$ when the system goes to the solid-like branch. We also show $P(\sigma_i)$ in the liquid-like branch for comparison. (c)--(e) Representative snapshots illustrating three distinct rheological regimes: liquid-like, solid-like, and coexistence of liquid-like and solid-like cells. We mark the liquid-like cells blue when $|\sigma_i|\leq 0.02$, otherwise, yellow. These results are for $p_0=3.88$ and $N=100$.
• Figure 5: Comparison of theoretical predictions at $T=2\times10^{-4}$ and varying $p_0$. (a) Theoretical flow curves, Eq. (\ref{['DSTmodel']}), for various values of $p_0$, with $\delta = 1.0$ and a constant $\sigma^* = 0.008$. The curves show the characteristic S-shape of DST. (b) Fitting the parameters of $f(\sigma)$ with the simulation data, we find $\delta\simeq 0.6$, but $\sigma^*$ varies with $p_0$. The symbols represent the fitting values, and the line is $\sigma^*(p_0) = 0.107\,(p_0 - 3.823)^{3/2}$. (c) Comparison of the theoretical flow curves (lines) with stress-controlled (filled symbols) and rate-controlled (open symbols) simulation data for different $p_0$. (d) Theoretical flow curves using $\delta = 0.6$ and the $p_0$-dependent $\sigma^*$ from panel (b) in the interpolating function $f(\sigma)$. The quantitative nature of the curves is different than the constant $\sigma^*$ case. (e) Comparison of the DST phase diagram between theory (dashed line) and simulations (symbols). We have indicated the different regimes: DST, then CST, and the Newtonian liquid. (f) The comparison of the stress drop, $\Delta \sigma$, plotted as a function of $p_0$ from the theory (dashed line) and simulation (symbols). The theoretical predictions agree well with the simulation data.