Balance of microtubule stiffness and cortical tension determines the size of blood cells with marginal band across species

Abstract

The fast bloodstream of animals is associated with large shear stresses. To withstand these conditions, blood cells have evolved a special morphology and a specific internal architecture to maintain their integrity over several weeks. For instance, nonmammalian red blood cells, mammalian erythroblasts, and platelets have a peripheral ring of microtubules, called the marginal band, that flattens the overall cell morphology by pushing on the cell cortex. In this work, we model how the shape of these cells stems from the balance between marginal band rigidity and cortical tension. We predict that the diameter of the cell scales with the total microtubule polymer and verify the predicted law across a wide range of species. Our analysis also shows that the combination of the marginal band rigidity and cortical tension increases the ability of the cell to withstand forces without deformation. Finally, we model the marginal band coiling that occurs during the disk-to-sphere transition observed, for instance, at the onset of blood platelet activation. We show that when cortical tension increases faster than cross-linkers can unbind, the marginal band will coil, whereas if the tension increases more slowly, the marginal band may shorten as microtubules slide relative to each other.

Keywords: blood platelet; cytoskeleton; mechanics; scaling; theory.

Fig. 1.

(A) Scanning electron micrographs of platelets and erythrocytes shown at the same scale (–8). (Scale bar, $1\mu$m). (B) The actin/spectrin cortex of platelets; EM from ref. , $\mathrm{\copyright }$ Hartwig and DeSisto, 1991. Journal of Cell Biology, DOI:10.1083/jcb.112.3.407. (Scale bar, $0.5\mu$m). (C) The MB of platelets is made of multiple MTs bundled by motors and cross-linkers (10); EM reprinted with permission from ref. ; www.sciencedirect.com/science/book/9780123878373. (D) In our model, the shape of the cell is determined by the balance of two forces. Because of MT stiffness $\kappa$, the MB pushes against the tense cortex, which resists by virtue of its surface tension $\sigma$.

Fig. 2.

(A) Cell radius as a function of total MT length $\mathcal{L}$. Dots represent data from 25 species (2). $\mathcal{L}$ was estimated from the number of MTs in a cross-section, measured in EM, and the cell radius. (B) Cell radius as a function of $\mathcal{L}\kappa /\sigma$ in simulations with $0$ (gray dots) or 10,000 (black dots) cross-linkers. On both graphs, the dashed line indicates the scaling law $4\pi R^4=\kappa \mathcal{L}/\sigma$.

Fig. S1.

Phase diagram of the degree of buckling as a function of the length and the tension. Red means that the filament is buckled and gray that it is flat. The dashed line represents the critical tension calculated in Eq. S16.

Fig. 3.

(A) MB of a live platelet labeled with SIR-tubulin dye. Fluorescence images were segmented at the specified time after the addition of ADP, a platelet activator, to obtain the MB size $L$. (B) Simulation of a platelet at different times. A limited increase of the tension ($90\text{pN}/\mu$m) causes the MB to shorten, whereas a large increase of the tension ($220\text{pN}/\mu$m) causes the MB to buckle. (C) In simulations, the MB buckles if cell rounding is fast enough because cross-linkers cannot reorganize. This represents an elastic behavior, but at longer times, the MB rearranges, leading to a viscoelastic response.

Fig. S2.

Bending energy of an incompressible elastic ring of length $2\pi R_0$ (the MB) in a sphere of radius $R<R_0$. The solid line represents the numerical solution to the Euler–Lagrange equations (Eq. S22), while the dashed line represents the small deformation approximation, Eq. S32.

Fig. 4.

(A) Coiling diagram of an elastic ring confined inside a fixed oblate ellipsoid. The configuration of the ring is indicated by the color (white, uncoiled; colored, coiled), as a function of the isotropy $r/R$ of the confining ellipsoid and the normalized confinement stiffness $k/k_c$. The gray dots indicate the simulations performed to calculate the diagram. The dashed line indicates the predicted critical buckling confinement in a spherical cell (i.e., $r=R$). The color indicates the main Fourier mode, from 2 (pink) to 5 (darker red). (B) A closeup reveals that the critical confinement is exponential for mode 2: $k^{\ast }=k_c{(\frac{r}{R})}^2{e}^{\alpha (1-\frac{R}{r})}$ (red line), where $\alpha =2.587$ is a phenomenological parameter that depends on the excess length $ϵ$, defined from the MB length as $L=2\pi R(1+ϵ)$. (C) Illustrations of MB shapes from the phase diagram, as indicated by the letters. Flatter cells (C, a and b) are deformed in higher modes than rounder cells (C, c and d). The normal physiology of a resting platelet corresponds to condition (C, e).

Fig. 5.

(A) The equilibrium configuration of the MB is calculated as a function of renormalized tension $\sigma R_0^3/{\kappa }_r$ and renormalized MB length $L/R_0$, in which the volume of the cell is $\frac{4}{3}\pi R_0^3$. The gray dots indicate the simulations performed to calculate the diagram. The configuration of the MB is indicated by colors: white, flat; red, buckled; and pink, bistable (i.e., buckled or flat). The topmost scale indicates the shape parameter of the cell (isotropy $r/R$), at equilibrium in the case where the MB is flat and has a length equal to the cell perimeter. (B) A cut through the phase diagram, for a MB of length $L=7.5R_0$. The degree of coiling (see Methods for definition) is indicated as a function of tension, for a cell that is initially flat (black dots) or buckled (gray dots). In the metastable region, the two trajectories are separated, and the arrows illustrate hysteresis in the system.