Balance of Microtubule Stiffness and Cortical Tension Determines the Size of Blood Cells with Marginal Band Across Species

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

Serge Dmitrieffa, Adolfo Alsinaa, Aastha Mathura, and Franc¸ois J. Ned´ elec´ a,1

aCell Biology and Biophysics Unit, European Molecular Biology Laboratory, 69117 Heidelberg, Germany

Edited by Timothy J. Mitchison, Harvard Medical School, Boston, MA, and approved February 14, 2017 (received for review November 1, 2016) The fast bloodstream of animals is associated with large shear not static, but instead bind and unbind, MTs could slide rela- stresses. To withstand these conditions, blood cells have evolved tive to one another, allowing the length of the MB to change. It a special morphology and a specific internal architecture to main- was suggested in particular that molecular motors may drive the tain their integrity over several weeks. For instance, nonmam- elongation of the MB (19), but this possibility remains mechanis- malian red blood cells, mammalian erythroblasts, and platelets tically unclear. Moreover, the MB changes as MTs assemble and have a peripheral ring of microtubules, called the marginal band, disassemble. However, in the absence of sliding, elongation or that flattens the overall cell morphology by pushing on the cell shortening of single MTs would principally affect the thickness cortex. In this work, we model how the shape of these cells stems of the MB (i.e., the number of MTs in the cross-section) rather from the balance between marginal band rigidity and cortical ten- than its length. These aspects have received little attention so sion. We predict that the diameter of the cell scales with the total far, and much remains to be done before we can understand how microtubule polymer and verify the predicted law across a wide the original architecture of these cells is adapted to their unusual range of species. Our analysis also shows that the combination of environment and to the mechanical constraints associated with the marginal band rigidity and cortical tension increases the abil- it (7). ity of the cell to withstand forces without deformation. Finally, We argue here that, despite the potential complexity of the we model the marginal band coiling that occurs during the disk- system, the equilibrium between MB elasticity and cortical ten- to-sphere transition observed, for instance, at the onset of blood sion can be understood in simple mechanical terms. We first pre- platelet activation. We show that when cortical tension increases dict that the main cell radius should scale with the total length faster than cross-linkers can unbind, the marginal band will coil, of MT polymer and inversely with the cortical tension, and test whereas if the tension increases more slowly, the marginal band the predicted relationship by using data from a wide range of may shorten as microtubules slide relative to each other. 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 cytoskeleton | scaling | mechanics | blood platelet | theory ring, whereas a slow increase of tension leads to a shortening of the ring without coiling. Finally, by computing the buckling force he shape of animal cells is determined by the cytoskele- of a ring confined within an ellipsoid, we find that the resistance Tton, including microtubules (MTs), contractile networks of of the cell to external forces is dramatically increased compared actin filaments, intermediate filaments, and other mechanical with the resistance of the ring alone. elements. The 3D geometry of most cells in a multicellular organ- Results ism is also largely determined by their adhesion to neighboring cells or to the extracellular matrix (1). This is, however, not Cell Size Is Controlled by Total MT Polymer and Cortical Tension. the case for blood cells because they circulate freely within the We first apply scaling arguments to explore how cell shape is fluid environment of the blood plasma. Red blood cells (RBCs) determined by the mechanical equilibrium between MB elasticity and thrombocytes in nonmammalian animals (2, 3), as well as platelets and erythroblasts in mammals (4, 5), adopt a simple Significance ellipsoidal shape (Fig. 1A). This shape is determined by two components: a ring of MTs, called the marginal band (MB), and a The discoidal shape of many blood cells is essential to their protein cortex at the cell periphery. proper function within the organism. For blood platelets and In the case of platelets and nonmammalian RBCs, both com- other cells, this shape is maintained by the marginal band, ponents are relatively well characterized (Fig. 1). The cortex is which is a closed ring of filaments called microtubules. This a composite structure made of spectrin, actin, and intermediate ring is elastic and pushes on the cell cortex, a tense polymer filaments (Fig. 1B), and its complex architecture is likely to be scaffold associated with the plasma membrane. Dmitrieff et dynamic (11–13). It is a thin network under tension (14), that on al. examined how the mechanical balance between these two its own would lead to a spherical morphology (15). This effect components determine cell size, uncovering a scaling law that is counterbalanced by the MB, a ring made of multiple dynamic is observed in data collected from 25 species. The analysis also MTs, held together by cross-linkers and molecular motors into indicated that the cell can resist much higher mechanical chal- a closed circular bundle (4, 16) (Fig. 1C). The MB is essen- lenges than the microtubule ring alone, in the same way as a tial to maintain the flat morphology, and treatment with a MT- tent with its cloth is stronger than the poles alone. destabilizing agent causes platelets to round up (17). Platelets also respond to biochemical signals indicating a damage of the Author contributions: S.D. and F.J.N. designed research; S.D., A.A., and A.M. performed blood vessels, and during this activation, the MB is often seen to research; S.D. and A.A. analyzed data; and S.D., A.M., and F.J.N. wrote the paper. buckle (3). This phenomenon is reminiscent of the buckling of The authors declare no conflict of interest. a closed elastic ring (18), but an important difference is that the This article is a PNAS Direct Submission. MB is not a continuous structure of constant length. 1To whom correspondence should be addressed. Email: [email protected]. Indeed, the MB is made of multiple dynamic MTs that are This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10. linked by MT-associated proteins. Because these connectors are 1073/pnas.1618041114/-/DCSupplemental.

4418–4423 | PNAS | April 25, 2017 | vol. 114 | no. 17 www.pnas.org/cgi/doi/10.1073/pnas.1618041114 Downloaded by guest on October 1, 2021 Downloaded by guest on October 1, 2021 T’lnts n rt h oa lsi nryas energy elastic total the write and lengths, MTs’ define can We igand ring ii,teMsaemcaial needn,adw a assume can we can this and κ independent, In we mechanically (10). which are MTs cross-linkers for bind- the of limit, (22)], cross-linker contribution s mechanical MT 10 the of ignore [approximately dynamics unbinding the and than ing larger scales time and material as the written often bundle, the of mtif tal. et Dmitrieff expect thus we constant, things other All to: to finally corresponds leading system as the energy of the equilibrium The approximate can we flat, sufficiently thickness of cell energy a surface tension a surface by a modeled be with from then associated reorga- contributions can effect include can Its to cortex (24). have rigidity the not its seconds, do we few therefore a and than nize, larger scales time At simplic- to for discoid orthogonal is is cell that the plane (R that a Assuming ity in axis. ellip- cell contained flat minor is are the cells MB the the state, and resting soids, their In tension. cortical and stiffness MT of Because forces. two of balance the tension by surface determined its is cell 6; the ref. of from permission shape with the reprinted EM (10); cross-linkers and motors by bundled MTs 9, ref. from EM 1. 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I π σ ≈ h ubro T nacosscino the of cross-section a in MTs of number the R 2r t oeto nri)(1.W rtconsider first We (21). inertia) of moment its 2π ∼ 27 µm 2 1 100 ≈ L Rn κ is L o ua lo ltlt,given platelets, blood human for pN/µm σ r 2 BC A S C ymliligtenme fMsin MTs of number the multiplying by c ∼ h iiiyo nidvda T(23). MT individual an of rigidity the ≈0. hc orsod otesmo the of sum the to corresponds which , R 100 fteatmoi otxof cortex actomyosin the of pN/µm 2 = 2 actin/spectrin cortex Human 0.1 platelet 4 = C 65 = π h tpoie netmt of estimate an provides fit the 1, YI 2) n hsi ls othe to close is this and (28), µm πκ ,wihi o compared low is which pN/µm, R ∼1/R 2) eas h spec- the because (26), pN/µm 4πσ σ κL 2 )(14). pN/µm) r (with 1+ [1 .Tesraeae of area surface The 1D). 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BIOPHYSICS AND COMPUTATIONAL BIOLOGY in Cytosim, a cytoskeleton simulation engine (29). Cytosim solves A the Langevin equation (viscosity × velocity = forces + Brownian noise), describing the motion of bendable filaments that are dis- cretized 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 B is restricted to remain ellipsoidal and is described by six parameters: the lengths of three axes R1, R2, and r and a rotation matrix (i.e., three angles describing the cell orientation in the space). Because RBCs have active mechanisms to maintain their volume (30), we also constrained the three lengths to keep the volume of the ellipsoid constant. To implement confinement, any MT model point located outside the cell is subject to inward-directed force f = kδ, in which δ is the shortest vector between the point and the surface and k the confining stiffness. Here, for each force f applied on a MT, an opposite force −f is applied to the surface, in agreement with Newton’s third law. The rates of change of the ellipsoid parameters are then given by the net force on each C axis, divided by µ, an effective viscosity parameter (SI Text, section 1.3 and Fig. S1). The value of µ affects the rate at which the cell shape can change, but not the shape that will eventually be reached. This approach is much simpler than using a tessellated surface to represent the cell, and still general enough to cap- ture the shape of blood platelets (3, 6) and several RBCs (8, 31) (Fig. 1A). To model resting platelets, we simulated marginal bands made of 10–20 MTs of fixed length 9–16 µm (4) with 0 or 10,000 Fig. 3. (A) MB of a live platelet labeled with SIR-tubulin dye. Fluorescence 3 cross-linkers, confined in a cell of volume 8.4 µm with a ten- images were segmented at the specified time after the addition of ADP, a sion σ ∼ 0.45–45 pN/µm. Initially the filaments had random ori- platelet activator, to obtain the MB size L. (B) Simulation of a platelet at entations, and we simulated the system for >6 min, which was different times. A limited increase of the tension (90 pN/µm) causes the enough time for them to align at the periphery and balance the MB to shorten, whereas a large increase of the tension (220 pN/µm) causes cortical tension given the viscosity. This also allowed for multi- the MB to buckle. (C) In simulations, the MB buckles if cell rounding is fast ple rounds of cross-linker binding/unbinding events. We found enough because cross-linkers cannot reorganize. This represents an elastic behavior, but at longer times, the MB rearranges, leading to a viscoelastic that the numerical results agree with the scaling law over a very response. large range of parameter values, as illustrated in Fig. 2B. Inter- estingly, simulated cells were slightly larger than predicted analytically. This is because MTs of finite length do not exactly fol- the MB seems to behave as an incompressible ring, and we rea- low the cell radius, and their ends are less curved, thus exerting soned that this must be because cross-linkers prevent MTs from more force on the cell. This means that the value of the tension sliding relative to each other. To analyze this process further, computed from the biological data (σ ∼ 0.1 pN/µm) is slightly we returned to Cytosim. After an initialization time, in which underestimated. More importantly, a simulated cell has the same the MB assembles as a ring of MTs connected by cross-linkers, size at equilibrium with or without cross-linkers (compare black cortical tension is increased stepwise. The cell as a consequence and gray dots on Fig. 2B). This shows that, if they are given time becomes nearly spherical, and, because we assumed that the vol- to freely reorganize, cross-linkers do not affect the mechanical ume should be constant, the radius of this sphere was smaller equilibrium of the system. To understand the response of the than the largest radius that the cell had at low tension. As a system that occurs at short time scale, it is, however, necessary result, the MB adopted a baseball-seam shape (Fig. 3B). Over to consider the cross-linkers. a longer period, however, the MB regained a flat shape, as MTs The MB Behaves Like a Viscoelastic System. During activation, rearranged into a new, smaller ring (Fig. 3A). In conclusion, the mammalian platelets round up before spreading, and within a simulated MB is viscoelastic (Fig. 3B). At short time scales, MTs few seconds, their MB coils (19). A similar response is seen do not have time to slide, and the MB behaves as an incompress- also in thrombocytes (3). This process can be triggered by ible elastic ring. At long time scales, the MB behaves as if cross- several activators, including thrombin and ADP, that bind to linkers were not present, with an overall elastic energy that is the G-protein-coupled receptors (32) and activate several down- sum of individual MT energies. Thus, overall, the ring behavior stream events. Among them, RhoA may induce actin contrac- seems to transition from purely elastic at short time scales, to tion (33), possibly through its role in myosin light-chain phos- viscoelastic Kelvin–Voigt law at long time scales (Fig. 3C). The phorylation (34). To observe platelet activation experimentally, transition between the two regimes is determined by the time we extracted mice platelets and exposed them to ADP, causing scale at which cross-linkers permit MTs to slide. an often-reversible activation. By monitoring the MB with SiR- tubulin, a bright docetaxel-based MT dye, we could record the The Cell Is Unexpectedly Robust. The MB in blood cells is nec- MB coiling live, Fig. 3A (SI Text, section 4). The MB coils accord- essary to establish a flat morphology, but also to maintain ing to the baseball-seam curve, which is the shape that an incom- this morphology in the face of transient mechanical challenges, pressible elastic ring would adopt when constrained into a sphere for example as the cell passes through a narrow capillary (7). smaller than its natural radius (35). Thus, at short time scale, We thus investigated how the cortex effectively reinforces the

4420 | www.pnas.org/cgi/doi/10.1073/pnas.1618041114 Dmitrieff et al. 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BIOPHYSICS AND COMPUTATIONAL BIOLOGY A 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 lp are described as bendable fila- B ments of rigidity κ = kBTlp, in which kBT is the thermal energy. The asso- 1 R L 2 2 2 ciated bending energy is 2 κ 0(d r/ds ) ds, where r(s) is the position as a function of the arclength s along the filament. The dynamics of such a system was simulated in Cytosim, an Open Source simulation software (29). In Cytosim, a filament is represented by model points distributed regularly defining segments of length s. Fibers are confined inside a convex region of space Ω by adding a force to every model point that is outside Ω. The force is f = k(p − r), where p is the projection of the model point r on the edge of Ω, and k is a stiffness constant. For this work, we implemented a deformable elliptical surface confining the MTs, parametrized by six parameters. The evolution of these parameters is implemented using an effective Fig. 5. (A) The equilibrium configuration of the MB is calculated as a func- viscosity (SI Text, section 1.3). To verify the accuracy of our approach, we 3 tion of renormalized tension σR0/κr and renormalized MB length L/R0, in first simulated a straight elastic filament, which would buckle when sub- 4 3 2 2 which the volume of the cell is 3 πR0. The gray dots indicate the simulations mitted to a force exceeding π κ/L , as shown by Euler. Cytosim recovered performed to calculate the diagram. The configuration of the MB is indi- this result numerically. We then calculated the critical tension necessary cated by colors: white, flat; red, buckled; and pink, bistable (i.e., buckled or for the buckling of MTs in a prolate ellipsoid. The simulated critical ten- flat). The topmost scale indicates the shape parameter of the cell (isotropy sion corresponds very precisely to the theoretical prediction (43) (SI Text, r/R), at equilibrium in the case where the MB is flat and has a length equal section 1.4 and Fig. S1). to the cell perimeter. (B) A cut through the phase diagram, for a MB of To calculate the cell radius as a function of Lκ/σ, we used a volume of 3 length L = 7.5R0. The degree of coiling (see Methods for definition) is indi- 8π/3 µm (close to the volume of a platelet), with a tension σ ∼ 0.45 to cated as a function of tension, for a cell that is initially flat (black dots) or 45 pN/µm, consistent with physiological values. We simulate 10 − 20 MTs buckled (gray dots). In the metastable region, the two trajectories are sepa- with a rigidity 22 pN µm2 as measured experimentally (23). MTs have lengths rated, and the arrows illustrate hysteresis in the system. in 9 − 16 µm and are finely represented with a segmentation of 125nm. The cross-linkers have a resting length of 40 nm, a stiffness of 91 pN/µm, a binding rate of 10 s−1, a binding range of 50 nm, and an unbinding rate Using analytical theory and numerical simulations, we ana- of 6 s−1. An example of simulation configuration file is provided in SI Text, lyzed the mechanical response of cells with MB and uncovered section 2. When considering an incompressible elastic ring, we used a cell of / π 3 a complex viscoelastic behavior characterized by a time scale τc volume 4 3 R0, where R0 is the radius of the spherical cell. For simplicity, we can renormalize all lengths by R0 and thus all energies by κr /R0, where that is determined by cross-linker reorganization. At long time 3 κr is the ring rigidity. We simulate a cell with a tension σ = 5 − 18κr /R , scales (t τc ), the MB behaves elastically, and its elasticity is 0 and a ring of length 1 − 1.6 × 2πR0. To test the effect of confinement, we the sum of all MTs’ rigidity. At short time scales (t < τc ), the place an elastic ring of rigidity κr in an ellipsoid space of principal radii MB behaves as an incompressible elastic ring of fixed length R, R, r R, in which r < 1. The elastic ring has a length (1 + )2πR, in which because cross-linkers do not yield. At this time scale, the stiff- = 0.05. An extensive list of parameters and their values are given in SI ness of the ring exceeds the sum of the individual MT stiffness, Text, section 1.2. To estimate the coiling level of a MB, we first perform a as long as the cross-linkers connect neighboring MT tightly (39). principal component analysis using all of the MTs’ model points. The coordi- Buckling leads to the baseball-seam curve, which is a configura- nate system is then rotated to bring the vector uz in the direction of the smallest eigenvalue. We then define the degree of coiling as the devia- tion of minimum elastic energy. This explains the coiled shape of pP P the MB observed in mouse platelets (Fig. 3A), human platelets tion in Z divided by the deviations in XY: C = z2/ x2 + y2. Thus, C is (19), and dogfish thrombocytes (3). Thus, an increase of cortical independent of the size of the cell and only measures the deformation of the MB. tension over bundle rigidity can cause coiling, if the cell deforms To measure the critical value of a parameter θ (e.g., tension or con- faster than the MB can reorganize. A fast increase of tension is a finement) leading to coiling, we computed the derivative of the degree likely mechanism supported by evidence of several experiments of coiling C, with respect to this parameter. Because buckling is analo- (40–42). In dogfish thrombocytes and platelets, blebs are con- gous to a first-order transition, the critical value θ∗ can be defined by: ∂ | = k∂ k comitant with MB coiling, suggesting a strong increase of cortical θ C θ∗ max θ C . We calculated the Fourier modes of deformation by tension (3). A recent study concluded that MB coiling could be transforming the z-coordinates of the MT model points. triggered by the extension of the MB, leading to coiling without an increase of cortical tension (19). However, the fact that the ACKNOWLEDGMENTS. We thank S. Correia for technical assistance; MB elongates during activation was inferred there by averaging A. Diz-munoz, R. Prevedel, and N. Minc for critical reading; and European Molecular Biology Laboratory (EMBL) IT support for performance comput- over populations of fixed platelets, rather than observed at the ing. This work was supported by the EMBL and the Center for Modeling in single cell level. the Biosciences (S.D.).

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