Nonlinear Rheology in a Model Biological Tissue

The rheological response of dense active matter is a topic of fundamental importance for many processes in nature such as the mechanics of biological tissues. One prominent way to probe mechanical properties of tissues is to study their response to externally applied forces. Using a particle-based model featuring random apoptosis and environment-dependent division rates, we evidence a crossover from linear flow to a shear-thinning regime with an increasing shear rate. To rationalize this nonlinear flow we derive a theoretical mean-field scenario that accounts for the interplay of mechanical and active noise in local stresses. These noises are, respectively, generated by the elastic response of the cell matrix to cell rearrangements and by the internal activity.

Figure 1

(a) Microscopic events in a system with cell division and apoptosis under shear. From left to right, $T1$ event (passive rearrangement), cell division, and apoptosis (active rearrangements). (b) Typical macroscopic stress-strain curve; the average slope of the increasing elastic parts on the curve corresponds to the elastic modulus $G_0$.

Figure 2

Flow curves for passive and active systems. (a) Steady-state average shear stress $⟨{\sigma }_{xy}⟩$ versus the applied external shear rate $\dot{\gamma }$ for different apoptosis rates $a$. The symbols indicate the microscopic simulations results for the different apoptosis rates. The dashed line corresponds to a Herschel-Bulkley fit of the passive system with packing fraction $\mathrm{\Phi }\approx 0.94$. The dashed-dotted lines are the mean-field model fits with fitting parameter $D_0$. (b) Shear-rate dependence of the corresponding packing fraction $\mathrm{\Phi }$ for the same values of the apoptosis rates.

Figure 3

(a) Fitted value of the diffusion coefficient $D_0$ of the active noise as a function of the apoptosis rate $a$ from the mean-field fit on the simulation data. The dashed line is a power-law fit with an exponent 0.94. (b) Activity-dependent crossover shear rate ${\dot{\gamma }}^{*}(a)$ measured from the flow curves (see text for details). Crosses are determined values from the pure simulation data, whereas stars come from the mean-field fit of the simulations.

Figure 4

Viscosity $\eta =⟨{\sigma }_{xy}⟩/\dot{\gamma }$ as a function of the apoptosis rate $a$ for different small values of the shear rate (symbols for the numerical data and dash-dotted lines for the mean-field prediction).