Non-linear visco-elasto-plastic rheology of a viscous vertex model

Abstract

Morphogenesis involves complex shape changes of biological tissues. Yet, tissue shape changes depend on tissue rheology, which in turn arises from the interplay of large numbers of cells. Here, we link cell- and tissue-scale mechanics by constructing mean-field rheological relations for the vertex model. In contrast to past work in the field, we study a vertex model with an explicit viscous friction. We also include two different cellular mechanisms creating active, anisotropic stresses. Our mean-field model accounts for cell shape and the non-linear elastic and visco-plastic regimes. We validate our results by predicting the response to large-amplitude oscillatory shear. There are several vertex model variants, and comparing to results from the literature, we show that their rheology depends on a number of model details. Our approach should be sufficiently general to construct non-linear mean-field constitutive relations for any cell-based tissue model.

Figures (9)

• Figure 1: A schematic showing the implementation of the two active cellular mechanisms: (a) anisotropic interface tension and (b) active crawling forces.
• Figure 2: Linear shear rheology. (a) Shear protocol, $\gamma(t)$, and corresponding cell shape $Q(t)$ and shear stress $\sigma(t)$. Parameter values: $\gamma_0=10^{-4}$ and $\omega=10^{-1}$. (b) Storage and loss moduli, $G'$ and $G"$, as a function of $\omega$ for strain amplitude $\gamma_0=10^{-7}$. Error bars, which are mostly smaller than symbols size, show the standard error of the mean.
• Figure 3: Non-linear elastic regime. (a) Shear protocol $\gamma(t)$ and measured $Q$ and $\sigma$. (b) Average steady-state stress $\tilde{\sigma}$ as a function of the average steady-state cell shape $Q$. The dashed line shows a fit to Eq. \ref{['eq:sigma Q']}. Fixing the value $G_0=0.82$ from the linear rheology (\ref{['fig:osc-shear']}b), we obtain $G_2=0.86$. Error bars show the standard error of the mean.
• Figure 4: Non-linear elasticity with linear viscosity. (a) Shear protocol $\gamma(t)$ and measured $Q$ and $\sigma$. (b) Average steady-state stress $\sigma$ and $Q$ as a function of shear rate $\tilde{v}$. (c) The part of the shear stress $\tilde{\sigma}$ not captured by the elastic stress $\tilde{\sigma}_\mathrm{el}$ from \ref{['fig:stress-Q']}b increases linearly with the shear rate (black circles). Yet, the associated viscosity $\eta_s$ (blue line) is substantially larger than the viscosity $\eta_0$ obtained from the linear rheology (orange dashed line, cf. \ref{['fig:osc-shear']}b). (d) Conversely, the part of the shear stress $\tilde{\sigma}$ not captured by the viscous stress expected from the linear rheology, i.e. $\tilde{\sigma}-2\eta_0\tilde{v}$, is larger than the elastic stress function extrapolated from \ref{['fig:stress-Q']}b (blue dashed line). The blue data points are the same as in \ref{['fig:stress-Q']}b and the black data points are those from the constant shear protocol, panel b. We take the union of both data sets and fit the 5th-order polynomial $\tilde{\sigma}_\mathrm{el}=2G_0Q + 4G_2Q^3 + 6G_4Q^5$ (black solid line). We obtain $G_2=-0.85$ and $G_4=5.4$ when imposing $G_0=0.82$ from the linear rheology. (e) Comparison of the time-dependent shear stress $\tilde{\sigma}$ measured in the simulations (blue and orange solid lines) with predictions for two different shear rates $\tilde{v}$ (black solid and dashed lines). Solid lines show predictions assuming the scenario in panel c: the elastic stress from Eq. \ref{['eq:sigma Q']} and \ref{['fig:stress-Q']}b with an increased viscosity when there are T1 transitions, Eq. \ref{['eq:viscosity']}. To this end, $R(t)$ is computed using a binned average. Dashed lines show predictions assuming the scenario from panel d: the elastic stress from the 5th-order polynomial with the linear-rheology viscosity $\eta_0$. Shaded regions indicate the standard error of the mean.
• Figure 5: Shear stress created by activity. The part of the shear stress $\tilde{\sigma}$ not captured by the passive stress from Eq. \ref{['eq:sigma Q v']} as a function of the respective activity parameter for varying shear rate $\tilde{v}$. The dashed lines represent linear fits. (a) Active, anisotropic interface tensions and (b) active crawl forces. (insets) The respective total shear stress $\tilde{\sigma}$ as a function of $\tilde{v}$ for different activity strengths. The dashed lines are the consistency checks using Eqs. \ref{['eq:sigma with activity ait']} and \ref{['eq:sigma with activity crawl']}, respectively.