Multiscale Microtubule Dynamics in Active Nematics

In microtubule-based active nematics, motor-driven extensile motion of microtubule bundles powers chaotic large-scale dynamics. We quantify the interfilament sliding motion both in isolated bundles and in a dense active nematic. The extension speed of an isolated microtubule pair is comparable to the molecular motor stepping speed. In contrast, the net extension in dense 2D active nematics is significantly slower; the interfilament sliding speeds are widely distributed about the average and the filaments exhibit both contractile and extensile relative motion. These measurements highlight the challenge of connecting the extension rate of isolated bundles to the multimotor and multifilament interactions present in a dense 2D active nematic. They also provide quantitative data that is essential for building multiscale models.

Figure 1

Dynamics of extensile bundles. (a) Time series of two oppositely polarized microtubules cross-linked by kinesin motor clusters. Depletion induces bundle formation at $t=0\mathrm{s}$. Thereafter, kinesin clusters generate bundle extension. Scale bar, $10\mu \mathrm{m}$. (b) Bundle length over time for four representative events. The red line indicates the average velocity of eight events, with shaded standard deviation. Average extension velocity $v=540\pm 100\mathrm{nm}/\mathrm{s}$.

Figure 2

Analysis of photobleached active nematics. (a) Time evolution of a photobleached region within a nematic assembled at $10\mu \mathrm{M}$ ATP. Scale bar, $20\mu \mathrm{m}$. (b) Intensity profiles $I_{\perp }$ along the dotted lines in (a) and averaged across the length of the photobleached region. (c) The width of the photobleached region plotted over time, fitted to $w=a{e}^{-t/\tau }$. (d) Initially sharp, the edge of the bleached square along the director roughens over time. (e) Intensity profiles $I_{||}(x)$ along the director averaged across the edge of the bleached region (points) and the prediction from the model of diffusion in Eq. (1) (lines). (f) The effective diffusion coefficient $D_{\text{bleach}}$ extracted for Eq. (1) versus ATP concentration. (g) The strain rate as a function of ATP concentration. Strain rates extracted from the exponential fit ${\gamma }_{\perp }=1/\tau$ (blue) and from a fit to the diffusion-convection model Eq. (1) (red). The gray lines are fits to the Michaelis-Menten equation $\gamma ={\gamma }_{\max }[\mathrm{ATP}]/(K_{\mathrm{M}}-[\mathrm{ATP}])$ with ${\gamma }_{\perp ,\max }=0.026{\mathrm{s}}^{-1}$ and $K_{\perp ,\mathrm{M}}=54\mu \mathrm{M}$ (blue, dashed) and ${\gamma }_{||,\max }=0.027{\mathrm{s}}^{-1}$ and $K_{||,\mathrm{M}}=46\mu \mathrm{M}$ (red, solid). Error bars in (f) and (g) are the standard errors of multiple measurements.

Figure 3

Tracking microtubules in active nematics. (a) An active nematic doped with fluorescent microtubules. (b) An image of dilute fluorescent microtubules. The position (dot) and orientation (bar) of an example reference microtubule is overlaid in blue, and the aligned microtubules for which the relative velocities are calculated are plotted in red. One out of twenty-thousand microtubules is labeled. Scale bar, $25\mu \mathrm{m}$. (c) The average Lagrangian flow field in the reference frame of a microtubule located at the origin and oriented along $x$. The flow field was obtained from binning the relative velocities by distance and averaging over time and filament pairs ($1000\mu \mathrm{M}$ ATP). (d) The average flow field with relative microtubule positions and velocities collapsed into the first quadrant, so that $V_{||}>0$ indicates an extending pair.

Figure 4

Analysis of filament tracking data. (a) The average velocity of microtubule pairs along the director $V_{||}(x,0)$ as a function of interfilament distance along the director for all ATP concentrations [legend in panel (b)]. Inset: $5\mu \mathrm{M}$ velocity profile with a $y$-axis re-scaled to $\mathrm{nm}/\mathrm{s}$. (b) The average relative velocity perpendicular to the director $V_{\perp }(0,y)$ of microtubules as a function of the interfilament distance perpendicular to the director. (c) The strain rate calculated from the slope of the velocity profiles along the director (red) and perpendicular to the director (blue) plotted versus ATP concentration. The gray lines are fits of the data to the Michaelis-Menten equation with ${\gamma }_{||,\max }=0.037{\mathrm{s}}^{-1}$ and $K_{||,\mathrm{M}}=57\mu \mathrm{M}$ (red, solid), and ${\gamma }_{\perp ,\max }=0.035{\mathrm{s}}^{-1}$ and $K_{\perp ,\mathrm{M}}=63\mu \mathrm{M}$ (blue, dashed). Error bars are standard errors of multiple measurements. (d) The distribution of the microtubule sliding velocities for interfilament distance less than $6\mu \mathrm{m}$ ($50\mu \mathrm{M}$ ATP). Lines are fits of the data to Gaussians $P=A*e^{[-(v-\overline{v}{)}^2/2{\sigma }^2]}$. Along the director (red), ${\overline{v}}_{||}=0.04$ and ${\sigma }_{||}=0.12\mu \mathrm{m}/\mathrm{s}$. Perpendicular to the director (blue), ${\overline{v}}_{\perp }=-0.04$ and ${\sigma }_{\perp }=0.07\mu \mathrm{m}/\mathrm{s}$.

Figure 5

Comparison of photobleaching and filament tracking results. (a) Strain rate along the director extracted from the slope of velocity profiles (blue dots) and Eq. (1) (gold squares). Inset: Strain rate perpendicular to the director from slope of velocity profiles (blue dots) and exponential fit of photobleached region width (gold squares). Gray lines are fits to the Michaelis Menten equation. (b) Width of velocity distribution along the director ${\sigma }_{||}$ (blue dots, left axis) and effective diffusion coefficient $D_{\text{bleach}}$ (gold squares, right axis) versus ATP concentration.