Multiscale Polar Theory of Microtubule and Motor-Protein Assemblies

Microtubules and motor proteins are building blocks of self-organized subcellular biological structures such as the mitotic spindle and the centrosomal microtubule array. These same ingredients can form new “bioactive” liquid-crystalline fluids that are intrinsically out of equilibrium and which display complex flows and defect dynamics. It is not yet well understood how microscopic activity, which involves polarity-dependent interactions between motor proteins and microtubules, yields such larger-scale dynamical structures. In our multiscale theory, Brownian dynamics simulations of polar microtubule ensembles driven by cross-linking motors allow us to study microscopic organization and stresses. Polarity sorting and cross-link relaxation emerge as two polar-specific sources of active destabilizing stress. On larger length scales, our continuum Doi-Onsager theory captures the hydrodynamic flows generated by polarity-dependent active stresses. The results connect local polar structure to flow structures and defect dynamics.

Figure 1

(a) Schematic of a cluster of polar-aligned and antialigned MTs, with their plus ends marked by red rings. Cross-linking motors walk on neighboring MTs at speed $v$ and (b) exert springlike forces, with a linear force-velocity relation. (c) An antialigned MT pair being polarity sorted by active cross-links. The left- (right-)pointing MT moves right (left) with velocity $v^L$ ($v^R$). (d) A polar-aligned MT pair upon which cross-link forces are relaxing due to the force-velocity relation. In both, the gray arrows characterize the magnitude of an induced extensile stress.

Figure 2

Results of the BD MC particle simulations. (a) Histogram of cross-link occupancy as a function of the particle pair longitudinal displacement $s_{ij}$. (b) Histogram of cross-link extension $r_c$. (c) Variation of extensile pair stresslet $S$ (unit of force $\times$ length) with motor run length $\ell$. (d) Typical variation of extensile pair stresslet with local polarity $m_i$.

Figure 3

Snapshots of streaming MT nematics on an immersed interface. (a) The nematic director field $\mathbf{n}$ superimposed on the color map of the scalar order parameter (twice the positive eigenvalue of $\mathbf{Q}$); ${\lambda }_{\text{cr}}$ is a calculated characteristic length between the cracks. (b) The polarity vector field $\mathbf{q}$ superimposed upon its magnitude. (c),(d) Polarity-dependent active stress magnitudes, showing principal eigenvalues of the active stresses [${\mathbf{\Sigma }}_{aa}$ in (c) and ${\mathbf{\Sigma }}_{pa}$ in (d)]. (e) The vector field of the active force ${\mathbf{f}}^a=\mathbf{\nabla }·{\mathbf{\Sigma }}^a$ superimposed upon its magnitude. In (a)–(e), circular areas labeled A and B mark regions of high and low polarity, respectively. Positions of $+1/2$-order defects are marked by circles. (f)–(h) Predicted results of a photobleaching experiment of fluorescent MTs for a bleached spot in a region of high polar order [(g), area A], and in a region of low polar order [(h), area B]. Arrows represent MTs with arrowheads denoting plus ends. A dimensionless time is used with scale $b/\nu v_w$, where $\nu$ is the effective volume fraction.

Figure 4

Linear stability analysis (a) and nonlinear simulation (b),(c) for strongly antialigned MTs. (a) The real part of the growth rate as a function of wave number $k$ for several values of $\alpha$ ($\alpha ={\alpha }_{aa}+{\alpha }_{pa}$). Here $k_{\text{cr}}$ corresponds to a maximum growth rate. Inset: Real part of the growth rate as a function of wave angle $\theta$ when fixing $k=k_{\text{cr}}$. (b) Crack formation. Inset: The fluid velocity vector field (blue) and the eigenmode (red line) associated with $k_{\text{cr}}$. (c) Genesis of defects at late times.