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

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Edited by Timothy J. Mitchison, Harvard Medical School, Boston, MA, and approved February 14, 2017 (received for review November 1, 2016)

## Significance

The discoidal shape of many blood cells is essential to their proper function within the organism. For blood platelets and other cells, this shape is maintained by the marginal band, which is a closed ring of filaments called microtubules. This ring is elastic and pushes on the cell cortex, a tense polymer scaffold associated with the plasma membrane. Dmitrieff et al. examined how the mechanical balance between these two components determine cell size, uncovering a scaling law that is observed in data collected from 25 species. The analysis also indicated that the cell can resist much higher mechanical challenges than the microtubule ring alone, in the same way as a tent with its cloth is stronger than the poles alone.

## 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.

The shape of animal cells is determined by the cytoskeleton, including microtubules (MTs), contractile networks of actin filaments, intermediate filaments, and other mechanical elements. The 3D geometry of most cells in a multicellular organism is also largely determined by their adhesion to neighboring cells or to the extracellular matrix (1). This is, however, not the case for blood cells because they circulate freely within the fluid environment of the blood plasma. Red blood cells (RBCs) and thrombocytes in nonmammalian animals (2, 3), as well as platelets and erythroblasts in mammals (4, 5), adopt a simple ellipsoidal shape (Fig. 1*A*). This shape is determined by two components: a ring of MTs, called the marginal band (MB), and a protein cortex at the cell periphery.

In the case of platelets and nonmammalian RBCs, both components are relatively well characterized (Fig. 1). The cortex is a composite structure made of spectrin, actin, and intermediate filaments (Fig. 1*B*), and its complex architecture is likely to be dynamic (11⇓–13). It is a thin network under tension (14), that on its own would lead to a spherical morphology (15). This effect is counterbalanced by the MB, a ring made of multiple dynamic MTs, held together by cross-linkers and molecular motors into a closed circular bundle (4, 16) (Fig. 1*C*). The MB is essential to maintain the flat morphology, and treatment with a MT-destabilizing agent causes platelets to round up (17). Platelets also respond to biochemical signals indicating a damage of the blood vessels, and during this activation, the MB is often seen to buckle (3). This phenomenon is reminiscent of the buckling of a closed elastic ring (18), but an important difference is that the MB is not a continuous structure of constant length.

Indeed, the MB is made of multiple dynamic MTs that are linked by MT-associated proteins. Because these connectors are not static, but instead bind and unbind, MTs could slide relative to one another, allowing the length of the MB to change. It was suggested in particular that molecular motors may drive the elongation of the MB (19), but this possibility remains mechanistically unclear. Moreover, the MB changes as MTs assemble and disassemble. However, in the absence of sliding, elongation or shortening of single MTs would principally affect the thickness of the MB (i.e., the number of MTs in the cross-section) rather than its length. These aspects have received little attention so far, and much remains to be done before we can understand how the original architecture of these cells is adapted to their unusual environment and to the mechanical constraints associated with it (7).

We argue here that, despite the potential complexity of the system, the equilibrium between MB elasticity and cortical tension can be understood in simple mechanical terms. We first predict that the main cell radius should scale with the total length of MT polymer and inversely with the cortical tension, and test the predicted relationship by using data from a wide range of species. We then simulate the shape changes observed during platelet activation (20), discussing that a rapid increase of tension leads to MB coiling, accompanied by a shortening of the ring, whereas a slow increase of tension leads to a shortening of the ring without coiling. Finally, by computing the buckling force of a ring confined within an ellipsoid, we find that the resistance of the cell to external forces is dramatically increased compared with the resistance of the ring alone.

## Results

### Cell Size Is Controlled by Total MT Polymer and Cortical Tension.

We first apply scaling arguments to explore how cell shape is determined by the mechanical equilibrium between MB elasticity and cortical tension. In their resting state, the cells are flat ellipsoids, and the MB is contained in a plane that is orthogonal to the minor cell axis. Assuming that the cell is discoid for simplicity (*D*), and thus the MTs bundled together in the MB have a curvature *D*). The surface area of a cell of thickness *A*). By using Eq. **1**, the fit provides an estimate of the cortical tension of

The scaling observed across 25 species seems to confirm that, at long time scales, the mechanical balance between bending rigidity of the MTs and cortical tension define cell size (Fig. 1*D*). To verify the validity of this result for a ring made of multiple dynamically cross-linked MTs, we developed a numerical model in Cytosim, a cytoskeleton simulation engine (29). Cytosim solves the Langevin equation (*viscosity* *velocity = forces + Brownian noise*), describing the motion of bendable filaments that are discretized into model points. The forces stem from the rigidity of the filaments (tending to minimize bending energy), links between filaments (modeled as Hookean springs between filaments), and confinement within the cell. The Brownian noise is a stochastic force calibrated from temperature. For this work, we extended Cytosim to be able to model a contractile surface under tension that can be deformed by the MTs. The cell shape is restricted to remain ellipsoidal and is described by six parameters: the lengths of three axes *SI Text*, section 1.3 and Fig. S1). The value of *A*).

To model resting platelets, we simulated marginal bands made of *B*. Interestingly, simulated cells were slightly larger than predicted analytically. This is because MTs of finite length do not exactly follow the cell radius, and their ends are less curved, thus exerting more force on the cell. This means that the value of the tension computed from the biological data (*B*). This shows that, if they are given time to freely reorganize, cross-linkers do not affect the mechanical equilibrium of the system. To understand the response of the system that occurs at short time scale, it is, however, necessary to consider the cross-linkers.

### The MB Behaves Like a Viscoelastic System.

During activation, mammalian platelets round up before spreading, and within a few seconds, their MB coils (19). A similar response is seen also in thrombocytes (3). This process can be triggered by several activators, including thrombin and ADP, that bind to G-protein-coupled receptors (32) and activate several downstream events. Among them, RhoA may induce actin contraction (33), possibly through its role in myosin light-chain phosphorylation (34). To observe platelet activation experimentally, we extracted mice platelets and exposed them to ADP, causing an often-reversible activation. By monitoring the MB with SiR-tubulin, a bright docetaxel-based MT dye, we could record the MB coiling live, Fig. 3*A* (*SI Text*, section 4). The MB coils according to the baseball-seam curve, which is the shape that an incompressible elastic ring would adopt when constrained into a sphere smaller than its natural radius (35). Thus, at short time scale, the MB seems to behave as an incompressible ring, and we reasoned that this must be because cross-linkers prevent MTs from sliding relative to each other. To analyze this process further, we returned to Cytosim. After an initialization time, in which the MB assembles as a ring of MTs connected by cross-linkers, cortical tension is increased stepwise. The cell as a consequence becomes nearly spherical, and, because we assumed that the volume should be constant, the radius of this sphere was smaller than the largest radius that the cell had at low tension. As a result, the MB adopted a baseball-seam shape (Fig. 3*B*). Over a longer period, however, the MB regained a flat shape, as MTs rearranged into a new, smaller ring (Fig. 3*A*). In conclusion, the simulated MB is viscoelastic (Fig. 3*B*). At short time scales, MTs do not have time to slide, and the MB behaves as an incompressible elastic ring. At long time scales, the MB behaves as if cross-linkers were not present, with an overall elastic energy that is the sum of individual MT energies. Thus, overall, the ring behavior seems to transition from purely elastic at short time scales, to viscoelastic Kelvin–Voigt law at long time scales (Fig. 3*C*). The transition between the two regimes is determined by the time scale at which cross-linkers permit MTs to slide.

### The Cell Is Unexpectedly Robust.

The MB in blood cells is necessary to establish a flat morphology, but also to maintain this morphology in the face of transient mechanical challenges, for example as the cell passes through a narrow capillary (7). We thus investigated how the cortex effectively reinforces the MB, making the cell a stronger object than the MB alone. Specifically, we calculated the resistance of the cell to deformations that would require its marginal band to coil, on a short time scale, during which cross-linkers do not reorganize. We therefore considered the MB as a closed ring of constant length *SI Text*, section 1.2.2 and Fig. S2). If

We verified this relation in simulations, with *A*). We next simulated oblate ellipsoidal cells, with *A* and *B*). We also found that the mode of deformation increases with the cell flatness (Fig. 4*A*, shades of red). This is because, as the cell gets flatter, large deformations along the short axis are increasingly penalized, and higher modes of deformation (such as the chair shape; Fig. 4*C*, *c*) become more favorable than the baseball-seam curve (Fig. 4*C*, *d*), because the magnitude of their out-of-plane deviations is smaller. This increase of the critical buckling force with cell flatness implies that an uncoiled marginal band in a flat cell could be metastable.

Platelets and nonmammalian RBCs have an isotropy ratio *C*, *e*), which makes them >10 times more resilient than a spherical cell with similar characteristics. Direct micropipette aspiration showed that destabilizing MTs or actin lead in both cases to an increased deformability, confirming that actin and MT systems determine the rigidity of the cell together (36).

### Coiling Stems from Cortical Tension Overcoming MB Rigidity.

We then considered the case of a ring inside a deformable ellipsoid of constant volume *A*). This shows that increasing

## Discussion

We have examined how MB elasticity and cortical tension determine the morphology of blood cells. Equilibrium between these forces predicts a scaling law,

Using analytical theory and numerical simulations, we analyzed the mechanical response of cells with MB and uncovered a complex viscoelastic behavior characterized by a time scale *A*), human platelets (19), and dogfish thrombocytes (3). Thus, an increase of cortical tension over bundle rigidity can cause coiling, if the cell deforms faster than the MB can reorganize. A fast increase of tension is a likely mechanism supported by evidence of several experiments (40⇓–42). In dogfish thrombocytes and platelets, blebs are concomitant with MB coiling, suggesting a strong increase of cortical tension (3). A recent study concluded that MB coiling could be triggered by the extension of the MB, leading to coiling without an increase of cortical tension (19). However, the fact that the MB elongates during activation was inferred there by averaging over populations of fixed platelets, rather than observed at the single cell level.

Finally, calculating the buckling force of a cell containing an elastic ring and a contractile cortex led to a surprising result. We found that the buckling force increased exponentially with the cell flatness, because the cortex reinforces the ring laterally. This makes the MB a particularly efficient system to maintain the structural integrity of blood cells. For transient mechanical constraints, the MB behaves elastically, and the flat shape is metastable, allowing the cell to overcome large forces without deformation. However, as we observed, the viscoelasticity of the MB allows the cell to adapt its shape when constraints are applied over long time scales, exceeding the time necessary for MB remodeling by cross-linker binding and unbinding. It will thus be particularly interesting to compare the time scale at which blood cells experience mechanical stimulations in vivo with the time scale determined by the dynamics of the MT cross-linkers.

## Methods

### Simulations.

MTs of persistence length *SI Text*, section 1.3). To verify the accuracy of our approach, we first simulated a straight elastic filament, which would buckle when submitted to a force exceeding *SI Text*, section 1.4 and Fig. S1).

To calculate the cell radius as a function of *SI Text*, section 2. When considering an incompressible elastic ring, we used a cell of volume *SI Text*, section 1.2. To estimate the coiling level of a MB, we first perform a principal component analysis using all of the MTs’ model points. The coordinate system is then rotated to bring the vector

To measure the critical value of a parameter **-**coordinates of the MT model points.

## SI Text

## 1. Simulation of MTs/Cortex Interaction

To understand cell shape maintenance, one needs to model the interaction between the cellular cortex and the MB. The structure of the MB is well known, compared with the organization of the cortex, which is less well characterized. We thus decided to represent the MTs individually, and the cortex effectively as a continuous deformable surface. Treating the interactions between a discretized (e.g., triangulated) surface and discrete filaments can be demanding computationally, because such a surface would have a very large number of degrees of freedom. In contrast, we describe here how the problem remains relatively simple for a continuous shape that is defined by a small set of parameters.

### 1.1. General Formulation.

#### 1.1.1. Forces and parametrization.

Let

We can define

To write Eq. **S3**, we had to assume that any displacement of the surface (allowed by the constraints) can be described in terms of

#### 1.1.2. Constraints.

In many cases, constraints can be added by modifying the energy functional following the method of Lagrange. For instance, to maintain the volume constant, we define an energy

### 1.2. Deformable Ellipsoid.

In this section, we describe a surface in 3D. We model an ellipsoid centered around the origin, with a fixed volume

#### 1.2.1. Surface tension.

We can compute the pseudoforces associated to surface tension as:

The surface area of an ellipsoid is a complex special function that is not a combination of the usual functions. For convenience, we used an analytical approximation of the area:

#### 1.2.2. Point forces.

To add the contribution of the external forces exerted by the MTs on the surface, we need to determine

Therefore, we have:

Because we assumed that the boundaries offer no friction, all forces are normal to the surface. The contribution

We can now compute the torque generated by

#### 1.2.3. Volume conservation.

To implement volume conservation, we need to find a pressure

The volume of the ellipse is

### 1.3. Time Evolution.

We can now define the time evolution of the ellipse. We assume a unique viscosity

### 1.4. Validations.

To validate our numerical method and its implementation, we first simulated a MT bundle confined inside an ellipsoid cell of tension

We confined the MT in a deformable ellipsoid, which thus takes the shape of a prolate ellipsoid. Let us call **S7**. Starting with a MT of length

In simulations, we find that the MT buckles for

## 2. Mechanics of a Confined Elastic Ring

### 2.1. Formulation.

Let us consider a rod of length

Because the rod lies on the unit sphere, and because

We can introduce this as constraints in the energy using two Lagrange multipliers

Minimizing this energy yields the Euler–Lagrange equation:

Because the curve is lying on a sphere, we can use the identity:

Numerically, we determined

### 2.2. Case of a Weakly Deformed Ring.

For a weakly deformed ring, Eq. **S22** can be simplified to

Periodicity imposes **S24** for small deformation as follows:**S25**, we can compute the bending energy of the MB:

We can also compute the length of the MB, and the energy:

We then find the force exerted by a nearly flat ring on the sphere

This result is in agreement with solving the full shape equation (Eq. **S22**), as illustrated in Fig. S2.

### 2.3. Ring Under Elastic Confinement.

Let us consider a ring of length **S32**, we find that the ring will buckle above a critical confinement:

### 2.4. Ring Confined in a Deformable Ellipsoid.

Using our computed value of

## 3. Simulation Parameters

The parameters used for Figs. 4 and 5 were:

For the incompressible elastic ring in a fixed shape ellipsoid (Fig. 4), we built a closed ring by linking the first and lastpoint of the filament with a zero-resting length link of rigidity

For the simulations of the incompressible elastic ring in a deformable ellipsoid (Fig. 5), we used:

## 4. Experimental Methods

The experiments were performed on platelets extracted from the blood of the common inbred laboratory mouse strain (C57BL/6) and prepared according to standard protocols (44). Blood was collected by cardiac puncture and mixed with *g*. The upper phase (platelet-rich Plasma), was carefully removed and centrifuged at 2,000 × *g* for 2 min. The plasma was discarded, and the platelet pellet was resuspended in Tyrode’s albumin buffer. We labeled MTs with

## Acknowledgments

We thank S. Correia for technical assistance; A. Diz-munoz, R. Prevedel, and N. Minc for critical reading; and European Molecular Biology Laboratory (EMBL) IT support for performance computing. This work was supported by the EMBL and the Center for Modeling in the Biosciences (S.D.).

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. Email: nedelec{at}embl.de.

Author contributions: S.D. and F.J.N. designed research; S.D., A.A., and A.M. performed research; S.D. and A.A. analyzed data; and S.D., A.M., and F.J.N. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1618041114/-/DCSupplemental.

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