Swirling Instability of the Microtubule Cytoskeleton

In the cellular phenomena of cytoplasmic streaming, molecular motors carrying cargo along a network of microtubules entrain the surrounding fluid. The piconewton forces produced by individual motors are sufficient to deform long microtubules, as are the collective fluid flows generated by many moving motors. Studies of streaming during oocyte development in the fruit fly Drosophila melanogaster have shown a transition from a spatially disordered cytoskeleton, supporting flows with only short-ranged correlations, to an ordered state with a cell-spanning vortical flow. To test the hypothesis that this transition is driven by fluid-structure interactions, we study a discrete-filament model and a coarse-grained continuum theory for motors moving on a deformable cytoskeleton, both of which are shown to exhibit a swirling instability to spontaneous large-scale rotational motion, as observed.

Figure 1

Cytoplasmic streaming in the Drosophila oocyte. The three-dimensional oocyte shape is approximately given by rotating the cross section about its anterior-posterior axis. (a) Experimental flow field [15] and schematic of the disordered swirling flows and microtubule organization in early stages of development. (b) Later flows organize into a single vortex as MTs lie parallel to the cell periphery.

Figure 2

Discrete-filament computations. (a) $N$ equally spaced filaments clamped at their attachment points, reach inward from a no-slip spherical shell. Each has a continuous distribution of tangential point forces (green) that (b) exert a force $\mathbf{\Gamma }$ on the fluid and an equal and opposite compressive force on the filament. Synchronous oscillations ($N=7$, $\sigma =1700$), (d) steady, bent configuration ($N=11$, $\sigma =1100$) and swirling flow field.

Figure 3

Continuum model in planar geometry. (a) Bi-infinite array of MTs, clamped vertically at a no-slip boundary. Results of computations at $\rho =4.65$: (b) $\sigma =70$ and (c) $\sigma =39$. Colors denote time, from cyan (early) to pink (late).

Figure 4

Continuum model in cylindrical geometry. (a) Results of linear stability analysis about the radially aligned state, with $R=5L$. (b) Steady-state fiber deformations and velocity field for $\sigma =150$ and $\rho =80$. Density of visualized fibers corresponds to the physical density. The top inset shows deformed MTs and the dynamics of a representative one (see also the Supplemental Video [34]). The bottom inset shows the azimuthal velocity field as a function of $r$. (c) Dimensional streaming velocities in parameter space; hatched region is consistent with in vivo estimates of $100–400\mathrm{nm}/\mathrm{s}$. Yellow dot denotes simulation shown in (b).

Figure 5

Self-consistent model. An upstream point force $\mathbf{F}$ parallel to the distal filament end produces shear flow that deflects the filament. Two variants: (i) rigid rod with a torsional spring at its base and (ii) a clamped elastic filament.