Fluid-Solid Pattern Formation and Strain Localisation via Shear Banding Instability in Model Biological Tissues

TL;DR

It is shown that spontaneous symmetry breaking may also arise via a mechanical instability in the strain field of a deformed tissue, leading to a patterned coexistence of fluid and solid regions, with a strong localisation of the strain into shear bands.

Abstract

The rheological properties of biological tissues are core to processes such as cancer metastasis, wound healing and embryo development. The emergence of tissue and organ structures during morphogenesis requires the precise formation of spatial patterns. Dating back to Turing, pattern formation has been suggested to arise in tissues via spontaneous symmetry breaking instabilities in the concentration field of chemical morphogens. Within the vertex model of tissue mechanics, we show that spontaneous symmetry breaking may also arise via a mechanical instability in the strain field of a deformed tissue, leading to a patterned coexistence of fluid and solid regions, with a strong localisation of the strain into shear bands. The nature of the bands differs between tissues in which internal cell-cell dissipation dominates external drag against a substrate, and vice versa.

Figures (4)

• Figure 1: Yielding phenomenology of the vertex model with internal dissipation (left) and external dissipation (right). Shown as a function of strain $\gamma$ are the stress $\Sigma$(a+b), standard deviation of plasticity $\delta$(c+d), and correlation length $\xi$(e+f), with $\delta$ and $\xi$ defined in the main text. System sizes $C = 4096, 8192, 16384$ in red, blue and black curves respectively. Data in (c-f) are averaged over strain bins of size $\Delta \gamma=0.2$. Distribution of T1 events across the flow-gradient direction $y$ versus strain $\gamma$ (in a single realisation, binned over $\Delta\gamma = 0.1$ and $\Delta y= 1$) for $C = 4096$(g+h), $C=8192$(i+j) and $C=16384$(k+l). No T1 events arise in black regions. $p_{0} = 3.5$, $\dot{\gamma} = 10^{-4}$.
• Figure 2: Correlation length with internal dissipation (a+c) and external dissipation (b+d), as a function of shear rate for system sizes $C=1024, 2048, .. 16384$ increasing in curves from light to dark (a+b), and as a function of system size for shear rates $\dot{\gamma}=10^{-n}$ with $n=0.0, -0.5, \cdots -4.0$, the shear rate decreasing in curves from light to dark (c+d). Data averaged over the strain interval $2.0<\gamma<6.0$ and over $10$ realisations. Dashed lines show power laws indicated, as a guide to the eye. Shape parameter $p_0 = 3.5$.
• Figure 3: Amplitude of shear banding in the vertex model with (a) internal dissipation and (b) external dissipation. The amplitude is quantified by the standard deviation $\delta_{\rm max}$ defined in the main text, plotted as a function of shear rate for values of the shape parameter $p_0=3.50, 3.55, \cdots 4.00$ in curves red, orange $\cdots$ purple. Data averaged over $10$ realisations. System size $C=8192$. In snapshots of the kind shown in Fig. \ref{['fig:pheno']}g-l) and \ref{['fig:longTimes']}c-f), shear bands are visually apparent for values of $\delta_{\rm max}$ above the dashed line.
• Figure 4: Curves as in Fig. \ref{['fig:pheno']} (a+b) and (g-j), but now over the larger strain interval $0<\gamma<45$ to show the evolution of the shear bands at long times and large strains.