The self-organization of plant microtubules inside the cell volume yields their cortical localization, stable alignment, and sensitivity to external cues

Abstract

Many cell functions rely on the ability of microtubules to self-organize as complex networks. In plants, cortical microtubules are essential to determine cell shape as they guide the deposition of cellulose microfibrils, and thus control mechanical anisotropy of the cell wall. Here we analyze how, in turn, cell shape may influence microtubule behavior. Building upon previous models that confined microtubules to the cell surface, we introduce an agent model of microtubules enclosed in a three-dimensional volume. We show that the microtubule network has spontaneous aligned configurations that could explain many experimental observations without resorting to specific regulation. In particular, we find that the preferred cortical localization of microtubules emerges from directional persistence of the microtubules, and their interactions with each other and with the stiff wall. We also identify microtubule parameters that seem relatively insensitive to cell shape, such as length or number. In contrast, microtubule array anisotropy depends on local curvature of the cell surface and global orientation follows robustly the longest axis of the cell. Lastly, we find that geometric cues may be overcome, as the network is capable of reorienting toward weak external directional cues. Altogether our simulations show that the microtubule network is a good transducer of weak external polarity, while at the same time, easily reaching stable global configurations.

Fig 1. Shapes of simulated cells.

(A) Top left: Two time steps of a simulation when the microtubule grows freely inside the cell and simultaneously shrinks from the minus end; at the plus side (arrowhead), the new vector is calculated by adding a small random deviation to the previous vector. Bottom left: Example of a zippering process—When the new microtubule encounters an existing one at a shallow angle (if the angle a is smaller than threshold α), the new direction of the vectors follows that of the leading microtubule; a steep angle would lead to plus end shrinkage. Top right: Strong anchoring to the plasma membrane—The microtubule grows tangentially to the local plane. Bottom right: Weak anchoring to the membrane—When the microtubule encounters the membrane at a shallow angle (a < α), it continues tangentially until the random deviation leads it to leave the membrane. (B) Shapes of the envelope of the cell: cube, “square”, and “elongated”. First row: smooth shapes. Second row: sharp shapes. Scales are not respected. (C,D) Example of a simulation in a sharp square with weak anchoring and default parameters (see Methods); the length of the simulation is 50000 timesteps, corresponding to approximately 100 minutes, also see S2 Video. (C) Snapshot showing all microtubules (left), microtubules located in a central slice that is ∼2.4μm-thick (center), and the projection of this slice in 2D (right). (D) Snapshots taken after 10000 timesteps intervals (∼20 min); the pictures show a confocal microscopy-like z-projection (see Methods).

Fig 2. Influence of the anchoring to the membrane.

(A) Mean distance of microtubules to the membrane according to cell shape and to weak/strong anchoring to the membrane. (B) Effect of anchoring on three properties of the microtubule network (data for all shapes pooled together). Left: total number of microtubules. Middle: mean microtubule length. Right: proportion of tubulin in bundles. Model parameters have the default values (see Methods).

Fig 3. Influence of global shape on three properties of the network.

Data for sharp shapes: cube, square, and long (Fig 1). From left to right: number of microtubules, mean microtubule length, and proportion of tubulin in bundles. Top row: weak anchoring; bottom row: strong anchoring. Model parameters have the default values (see Methods).

Fig 4. Anisotropy of the network.

(A) Effect of global shape: Anisotropy for the three shapes (sharp/smooth and weak/strong anchoring were pooled together). (B) Effect of curvature: Anisotropy for sharp shapes vs. smooth shapes with strong anchoring. (C) Effect of curvature: Anisotropy for sharp shapes vs. smooth shapes with strong anchoring. (D) Effect of anchoring strength: Anisotropy for all shapes (pooled). In all panels, the data are pooled except for the parameter that is varied in the subfigure; model parameters have the default values (see Methods).

Fig 5. Orientation of the network.

Distribution of the orientation of unit vectors (local orientation of microtubules) located close to the top and bottom faces. Top and bottom faces refer to the largest faces of the square shape represented in C, and to arbitrary opposite large faces of the cube or the long shapes. The reference angle 0 corresponds to the longest axis of the long cell and is parallel to one of the edges of the top face in the two other cases. (A,B) Distribution of orientations for sharp cube, sharp square, and sharp long shapes, as labeled in Fig 1. (A) Weak anchoring and (B) strong anchoring. (C) Arrow showing the reference for angle measurement for the long shape (top) and for the square and cube shapes (bottom). The shapes are not at the same scale. Model parameters have the default values (see Methods).

Fig 6. Influence of directional cues on the global orientation of the network.

Simulations in an ellipsoidal cell with a circumferential cue. (A) Histogram of the scalar product between the unit vectors (defining microtubules locally) and the long axis of the ellipsoid. Black: no directional cue. Colors: directional cue with influence on the direction. The strength, b_d, of the signal is: 0.1% (yellow), 0.2% (green), 1% (red), and 2% (blue). (B) Simulation result for a weight of 2%, also see S2 Video. (C) Simulation result with no directional cue. Model parameters: n_p = 2.4 ⋅ 10⁻⁷, n_s = 10⁻³, α = 40°, weak anchoring.