Quantifying Material Properties of Cell Monolayers by Analyzing Integer Topological Defects

In developing organisms, internal cellular processes generate mechanical stresses at the tissue scale. The resulting deformations depend on the material properties of the tissue, which can exhibit long-ranged orientational order and topological defects. It remains a challenge to determine these properties on the time scales relevant for developmental processes. Here, we build on the physics of liquid crystals to determine material parameters of cell monolayers. Specifically, we use a hydrodynamic description to characterize the stationary states of compressible active polar fluids around defects. We illustrate our approach by analyzing monolayers of C2C12 cells in small circular confinements, where they form a single topological defect with integer charge. We find that such monolayers exert compressive stresses at the defect centers, where localized cell differentiation and formation of three-dimensional shapes is observed.

Figure 1

Confined C2C12 monolayers. (a) Schematic of the experimental setup. (b) Phase-contract image of a spiral in a circular domain of $100\mu \mathrm{m}$ radius. (c) Orientational order (left) and velocity fields (right) averaged over $N=12$ spirals. Colors correspond to the polar order parameter $S$ and speeds, see legend, the vertical line separates the two fields. Gray lines: velocity stream lines. (d) Phase-contrast image of an aster in a circular domain of $100\mu \mathrm{m}$ radius. Scale bar in (b),(d): $50\mu \mathrm{m}$.

Figure 2

Spiral steady-state patterns. (a) Profiles of the polar order parameter $S$ and the velocity components $v_r$ and $v_{\theta }$ for myoblast monolayers ($\mathrm{mean}\pm$, standard error of the mean (sem), $N=12$). (b) Theoretical profiles of $S$ for varying values of $\chi$. (c),(d) Theoretical profiles of $v_{\theta }$ for varying $\zeta \mathrm{\Delta }\mu$ (c) and varying $\xi$ (d). Unless stated otherwise parameters are $\nu =-1.2$, $\zeta \mathrm{\Delta }\mu ={10}^{-2}$, $\eta =\xi =\chi =1$, $T_0={10}^{-3}$ in (b),(c) and $T_0=0$ in (d). The units are set by $\mathcal{K}=\gamma =R=1$.

Figure 3

Theoretical fits to experimental data. (a) Polar order parameter $S$ and (b) azimuthal velocity $v_{\theta }$ as a function of the radial distance $r$. Averaged experimental profiles for spirals that were treated with $10\mu \mathrm{M}$ Mitomycin-C (blue, $N=11$, 12, 5 for confining domain radius 50,100, $150\mu \mathrm{m}$), mean fit in the active-stress predominant (magenta, full lines) and in the traction force dominant parameter region (green, dashed lines). The theoretical curves are endowed with physical units such that $S(R)=1$ and $v_{\theta }(R)=21.4\mu \mathrm{m}/\mathrm{h}$ for $R=50\mu \mathrm{m}$. Error bars in theoretical fits correspond to std of all values of parameters with $\mathcal{E}<1.1{\mathcal{E}}_{\min }$, and experimental curves to sem.

Figure 4

Theoretical fits of steady-state profiles for asters. (a) Cell number density $n$, (b) radial force density as a function of the radial distance $r$. Averaged experimental profiles (purple empty circles in (a): $N=9$ asters at 2 day after seeding, blue cross in (b): $N=3$ experiments), mean fit in the active stress predominant (magenta, full lines) and in the traction force dominant parameter region (green, dashed lines). Parameters are given in Table 1. We used ${\zeta }^{\prime \prime }\mathrm{\Delta }\mu =0$. Error bars in theoretical fits correspond to std of all values of parameters with $\mathcal{E}<1.1{\mathcal{E}}_{\min }$, and experimental curves to sem.