Integer topological defects of cell monolayers: Mechanics and flows

Monolayers of anisotropic cells exhibit long-ranged orientational order and topological defects. During the development of organisms, orientational order often influences morphogenetic events. However, the linkage between the mechanics of cell monolayers and topological defects remains largely unexplored. This holds specifically at the timescales relevant for tissue morphogenesis. Here, we build on the physics of liquid crystals to determine material parameters of cell monolayers. In particular, we use a hydrodynamical description of an active polar fluid to study the steady-state mechanical patterns at integer topological defects. Our description includes three distinct sources of activity: traction forces accounting for cell-substrate interactions as well as anisotropic and isotropic active nematic stresses accounting for cell-cell interactions. We apply our approach to C2C12 cell monolayers in small circular confinements, which form isolated aster or spiral topological defects. By analyzing the velocity and orientational order fields in spirals as well as the forces and cell number density fields in asters, we determine mechanical parameters of C2C12 cell monolayers. Our work shows how topological defects can be used to fully characterize the mechanical properties of biological active matter.

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 $S$ 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 panels (b) and (d): $50\mu \mathrm{m}$.

Figure 2

Active forces associated with integer topological defects: asters (a, c, e) and spirals (b, d, f). Active forces generated only by traction forces $T_0\mathbf{p}$ (a, b), by anisotropic active stresses proportional to $\zeta \mathrm{\Delta }\mu$ (c, d), and by isotropic active stresses proportional to ${\zeta }^{\prime \prime }\mathrm{\Delta }\mu$ (e, f). Gray lines indicate the polarization field, which points outwards. The angle of the spiral is ${\psi }_0=\pi /3$ (b, d, f). Magenta arrows: surface active force density at $r/R=\{1/3,2/3,1\}, {\mathbf{f}}^{a,s}$ in Eq. (27). Green arrows: line active force density, ${\mathbf{f}}^{a,l}$ in Eq. (28). Black circle: boundary at $r=R$. The shafts of the magenta arrows are scaled by ${\mathrm{f}}^{a,s}(r=R)$ and of the green arrows by $R{\mathrm{f}}^{a,s}(r=R)$, where $f^{a,s}=\left|{\mathbf{f}}^{a,s}\right|$. Scale bars indicate ${\mathrm{f}}^{a,s}(r=R)=R{\mathrm{f}}^{a,s}(r=R)=1$. We assumed $T_0,\zeta \mathrm{\Delta }\mu ,{\zeta }^{\prime \prime }\mathrm{\Delta }\mu >0$.

Figure 3

Steady-state profiles for asters. (a) Cell number density $B(n-n^{\mathrm{tot}})/n_0$, Eq. (32), and (b) radial force density ${\mathbf{f}}_i·\widehat{\mathbf{r}}$, Eq. (39), as a function of the radial distance $r$ for varying values of the dimensionless ratio $T_{0}R/\zeta \mathrm{\Delta }\mu$ as indicated in the legend. We consider ${\zeta }^{\prime \prime }\mathrm{\Delta }\mu =0$ (a) and $-\frac{{\zeta }^{\prime \prime }\mathrm{\Delta }\mu }{2}-B\frac{n^{\mathrm{tot}}-n_0}{n_0}=0$ (b). Units are set by $\zeta \mathrm{\Delta }\mu =R=1$.

Figure 4

Steady-state profiles of the orientational order in spirals and with $R^2\ll \mathcal{K}/\chi$. (a) Polarization angle $\psi$ and (b) polar order parameter $S$. Purple lines: $\psi ={\psi }_0$ and $S=r/R$, respectively. Green dots: numerical solution of the dynamic equations. Parameter values are $\chi =0.1, \nu =-1.4, \zeta ={10}^{-2}, T_0=0, \eta ={10}^2$, and $\xi =1$ with the units being set by $R=\mathcal{K}=\gamma =1$. For these parameter values $|\gamma \nu v_{r\theta }sin\left(2{\psi }_0\right)|<2\times {10}^{-5}\ll \chi$.

Figure 5

Steady-state azimuthal velocity for flows driven by traction forces and with $R^2\ll \mathcal{K}/\chi$. (a) Regime I with $\eta =50$, 100, 200 and $\xi =1$, (b) Regime II with $100\eta =0.5$, 1, 2 and $\xi ={10}^{-2}$, (c) Regime III with $\eta =100$ and ${10}^{-5}\xi =0.5$, 1, 2, and (d) Regime III with $\eta =0.01$ and ${10}^{-3}\xi =0.5$, 1, 2. Purple lines: Eq. (43). Green dots: numerical solutions of the dynamic equations. Other parameter values are $\chi ={10}^{-1}, \nu =-1.4, T_0={10}^{-2}$, and $\zeta \mathrm{\Delta }\mu =0$ with the units being set by $R=\mathcal{K}=\gamma =1$.

Figure 6

Steady-state azimuthal velocity for flows driven by gradients in active stresses and with $R^2\ll \mathcal{K}/\chi$. (a) Regime I with $\eta =50$, 100, 200 and $\xi =1$, (b) Regime II with ${10}^4\eta =0.5$, 1, 2 and $\xi ={10}^{-2}$, (c) Regime III with $\eta =100$ and ${10}^{-5}\xi =0.5$, 1, 2, and (d) Regime III with $\eta =0.01$ and ${10}^{-3}\xi =0.5$, 1. Purple lines: (a) Eq. (48), (b) Eq. (49), (c, d) Eqs. (50) and (52). Green dots: numerical solution of the dynamic equations. Other parameter values are $\chi ={10}^{-1}, \nu =-1.4, T_0=0$, and $\zeta \mathrm{\Delta }\mu ={10}^{-2}$ with the units being set by $R=\mathcal{K}=\gamma =1$.

Figure 7

Half-integer topological defects in C2C12 myoblast monolayers. (a) Schematic representation of the director field for a $+1/2$ topological defect. (b) Theoretical profile $\varphi \left(\theta \right)$, Eq. (67), with $s=+1/2$ for varying $\epsilon$ as indicated in the legend. The ratio of Frank constants is ${\mathcal{K}}_1/{\mathcal{K}}_3=\{0.25,0.54,1,1.86,4.\}$ for $\epsilon =\{-0.6,-0.3,0,0.3,0.6\}$. (c) Representative experimental curves ${\varphi }^e\left(\theta \right)$ for varying radial distance $r$ as indicated in the legend. (d) Fitted ratio ${\mathcal{K}}_1/{\mathcal{K}}_3$ as a function of the radial coordinate $r$. Error bars correspond to the standard deviation of all values of $\epsilon$ that lead to $\mathcal{E}<1.1{\mathcal{E}}_{\min }$.

Figure 8

Probability density of the polarization angle with respect to the radial direction $\psi$. The data were obtained from C2C12 monolayers in spiral configurations that were confined to an island of $50\mu \mathrm{m}$ radius ($N=11$). The dashed line indicates $\psi ={90}^{\circ }$.

Figure 9

Parameter values leading to an error $\mathcal{E}<1.1{\mathcal{E}}^{\min }$ for the error function (70). The cuts of the parameter space are (a) $T_0$ vs $\zeta \mathrm{\Delta }\mu$, (b) $\eta$ vs $\xi$, (c) $\chi$ vs $\zeta \mathrm{\Delta }\mu$, and (d) $\zeta \mathrm{\Delta }\mu \xi /T_0\eta$ vs $\xi$. The units are fixed by $\mathcal{K}=\gamma =R=1$, and $\nu =-1.2$. Gray areas indicate parameter regions that were not analyzed. Green squares: active stress predominant region, dark green star: local minimum. Magenta circles: traction force dominant region, dark magenta star: global minimum.

Figure 10

Theoretical fits to experimental data. (a) Polar order parameters $S$ and (b) azimuthal velocity $v_{\theta }$ as a function of the radial distance $r$. Mean theoretical profiles for the active stress predominant parameter region in solid magenta and for the traction force dominant parameter region in dashed green; see Fig. 9 and Table 1. Blue: experimental profiles of spirals that were treated with $10\mu \mathrm{M}$ mitomycin-C ($N=(11,12,5)$ for confining domain radius $(50,100,150)\mu \text{m}$). Error bars in theoretical fits correspond to the standard deviation of parameter values that lead to $\mathcal{E}<1.1{\mathcal{E}}_{\min }$ and in experimental curves to the standard error of the mean. Profiles for three different confinement radii $R=50$, 100, and $150\mu \mathrm{m}$ are shown. The theoretical curves are endowed with physical units such that $S\left(R\right)=1$ and $v_{\theta }\left(R\right)=21.4\mu \mathrm{m}$/h for $R=50\mu \text{m}$.

Figure 11

Theoretical fits of steady-state profiles for asters. (a) Cell number density $n$ and (b) radial force density as a function of the radial distance $r$. Averaged experimental profiles (orange filled circles in (a): $N=10$ spirals treated with $10\mu \mathrm{M}$ mitomycin-C 1 day after seeding, purple empty circles in (a): $N=9$ asters 2 days 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) of the cell number density of asters and the radial force density, respectively. The theoretical solutions are Eq. (32) in (a) and Eq. (39) in (b). Parameters are given in Table 2. We used ${\zeta }^{\prime \prime }\mathrm{\Delta }\mu =0$. Error bars in theoretical fits correspond to the standard deviation of all parameter values with $\mathcal{E}<1.1{\mathcal{E}}_{\min }$ and in experimental curves to the standard error of the mean.

Figure 12

Polarization field of C2C12 monolayers. (a) Phase-contrast image for a spiral defect. Yellow segments represent the polarization field obtained from the structure tensor method in Ref. [44]. The yellow segments in the inset correspond to the actual experimental resolution. (b) Fluorescence image of SiR-actin for a spiral defect. In both panels (a) and (b) the circular domain has a radius of $50\mu \text{m}$. (c) Histogram of ${\psi }_{BF}$ defined as the angle between the radial direction and the polarization field from phase-contrast images. The average angle is ${\psi }_{BF}={76}^{\circ }\pm {22}^{\circ }$ ($\text{mean}\pm \text{std}, N=11$). Panel (c) is the same as Fig. 8. (d) Histogram of ${\psi }_{SA}$ defined as the angle between the radial direction and the polarization field from fluorescence images of SiR-actin. The average angle is ${\psi }_{SA}={82}^{\circ }\pm {34}^{\circ }$ ($\text{mean}\pm \text{std}, N=7$). For panels (c) and (d), values of the angles positioned at a radial distance from the circular domain center below $48\mu \text{m}$ as well as with a polar order parameter $S>0.4$ were considered. Dashed lines in panels (c) and (d) indicate $\psi ={90}^{\circ }$.