Self-organized intracellular twisters

Abstract

Life in complex systems, such as cities and organisms, comes to a standstill when global coordination of mass, energy, and information flows is disrupted. Global coordination is no less important in single cells, especially in large oocytes and newly formed embryos, which commonly use fast fluid flows for dynamic reorganization of their cytoplasm. Here, we combine theory, computing, and imaging to investigate such flows in the Drosophila oocyte, where streaming has been proposed to spontaneously arise from hydrodynamic interactions among cortically anchored microtubules loaded with cargo-carrying molecular motors. We use a fast, accurate, and scalable numerical approach to investigate fluid-structure interactions of 1000s of flexible fibers and demonstrate the robust emergence and evolution of cell-spanning vortices, or twisters. Dominated by a rigid body rotation and secondary toroidal components, these flows are likely involved in rapid mixing and transport of ooplasmic components.

Figure 1:. Hydrodynamic interactions of motor-loaded microtubules generate intracellular flows.

(A) A schematic illustrating a microtubule (blue) anchored normal to the cell surface (green) subject to active forcing and immersed in a viscous Newtonian fluid (i.e. cytoplasm). Large red arrows show compressive stress on the microtubule, small orange arrows represent stress on the fluid, and black lines indicate flows in the fluid. (B) Three regimes of microtubule behavior: stable regime with little microtubule deformation (left), beating regime with microtubules oscillating (middle; case I), and the streaming regime with microtubules (collectively) bending (right; case II). Microtubule colors represent time evolution. (C) A phase diagram of microtubule behaviors as a function of adimensional microtubule areal density $\bar{\rho }$ and motor forces $\bar{\sigma }$. The regions in yellow, red, and blue represent stable, streaming, and oscillation phases, respectively. The color of the red circles represents the characteristic streaming speed (flow speed at a distance 0.8R from the center). The Case I data point has $\bar{\rho }=5$ and $\bar{\sigma }=90$, while case II has $\bar{\rho }=15$ and $\bar{\sigma }=45$. (D,E) For cases I and II, cut-away view of instantaneous microtubule configurations in the spherical cell; (i) 2D projection of velocity field in the sectioning equatorial plane, at four time points; (ii) the adimensional azimuthal velocity component at the three points in the equatorial plane, as a function of $\bar{t}$; (iii) and 3D streamlines integrated from the 3D velocity field (iv). Curve colors correspond to the colored point in preceding velocity plots.

Figure 2:. Twisters are a combination of a strong vortical flow and a weak bitoroidal flow.

(A) Streamlines from a simulation of case II in the streaming region. Streamlines starting near the equatorial plane remain there, showing nearly circular paths (yellow). Streamlines starting near the poles move inwards on a spiral path towards the equatorial plane and then expand toward the periphery (red and blue). Streamlines starting above (or below) the equatorial plane near the cell periphery show the return flow back towards the pole (green). (B) Streamlines from two simple solutions to the Stokes equations inside a sphere: a strong 2D constant vorticity flow (thick streamlines) and a weaker bitoroidal flow (thin streamlines). Both velocity fields are tangent to the confining sphere. (C) Streamlines from best-fit combination of these two flows in approximating the flow in (A). (D) A schematic defining the microtubule polarity vector $\mathbf{p}$. The blue fiber represents the configuration of a microtubule anchored to the surface. The black vectors represent the end-to-end vector in 3D and its projection $\mathbf{p}$ on the surface. Note $\left|\mathbf{p}\right|\le 1$. (E) Microtubules configuration near the defect for simulation in (A). Microtubules with $p<0.7$ (less bent) are colored in red, and those with $p\ge 0.7$ are colored blue. (F) View of the microtubule polarity vector-field $p$ at the defect center shown in (E) with low microtubule alignment. (G,H,I) 2D projection of velocity fields from simulation in (A) in the sectioning equatorial planes when the plane normal is aligned with the z-axis (G), has angle $\pi /4$ relative to the z-axis (H), and is perpendicular to the z-axis (I).

Figure 3:. Surface microtubule orientation and cytoplasmic velocity fields.

(A) Microtubule configurations at steady state from a simulation on a sphere from the streaming region (left) and selected regions from the same snapshot with corresponding velocity fields in the vicinity of the surface (right). (B) Adimensional speed as a function of normalized distance $\bar{r}=r/R$ from the vortex center in a cross-section (shown as inset) for simulation in (A). (C) 2D velocity field in an oocyte measured by overset grid particle image velocimetry. Arrows show the direction and relative magnitude of the velocity. Measurements were done at consecutive z-sections from the oocyte surface. Scale bar, $25\mu \mathrm{m}$. (D) Computed 3D streamlines near the oocyte surface from the reconstructed velocity field. The color indicates depth. Red arrows show the flow direction. Scale bar, $25\mu m$. (E) Flow speed as a function of normalized distance $\bar{r}$ from the vortex center (shown as inset) measured in oocytes. Here, $\bar{r}=r/r_0$, where $r_0$ is the shortest distance from the vortex center to the periphery. Gray lines indicate measurements from four different oocytes, and the black line shows their average. (F,G) Near-surface microtubules in oocytes imaged in maternally derived $\text{GFP-}\alpha \text{tub}$ Drosophila. Orange lines represent microtubules’ local orientation with length representing the local degree of microtubules alignment measured by Gabor filter response. The cytoplasmic velocity field (black arrows) measured by PIV in the same oocyte in a plane $15\mu \mathrm{m}$ from the surface for corresponding regions are shown. The colored arrows are average microtubules orientation in the corresponding regions. Scale bars, $25\mu m$.

Figure 4:. The structure of the streaming flow is robust.

(A) Microtubule surface polarity vectors ${\mathbf{p}}_i$ (shown as arrows) and the scalar polar order parameter $P$ from multiple views. Views (i) and (ii) are at early time $\left(t=0.025{\tau }_r\right.,\left.t=0.035{\tau }_r\right)$ and (iii) at long time $\left(t=0.5{\tau }_r\right).P=1$ (bright) represents highest level of local alignment and $P=0$ (dark) represent lack of local alignment. (B-C) Angular power $s_l$ of spherical harmonic coefficients of the $P$ field, $s_l(t)={\sum }_{m=-l}^{m=l} {\left|{\tilde{P}}_{lm}\left(t\right)\right|}^2/(2l+1)$, for two simulations with microtubules positioned at the same place but with different initial configuration. The right and left axes represent $l=0$ and $l\ne 0$, respectively. (D) Bar graphs representing the mean steady-state value of $s_l$ ($l=0$ shown in inset) for $n=19$ such simulations. Error bars show standard deviation. Similar statistics are found by sampling different microtubule anchoring distributions (supplementary Fig. S5). (E) Microtubule configurations from a simulation with cell geometry similar to Drosophila oocyte from an early (i) and a late (iii) time point, using control parameters $\bar{\sigma }$ and $\bar{\rho }$ of case II. Integrated streamlines of the instantaneous flow field at the corresponding times (ii, iv). The disks represent the point of origin of the streamline. The streamlines in red and blue has point of origin near the regions with the lowest microtubule polarity $P$, denoted by black spheres. (F) Normalized distance between the two defect centers as a function of time for 13 simulations in the oocyte geometry, as indicated in the inset (gray, individual simulations, black, average). (G) Distance between the defect centers (blue) and total elastic energy $ℰ$ of microtubules (red) as a function of time for simulation in (E).