Balance of microtubule stiffness and cortical tension determines the size of blood cells with marginal band across species
Serge Dmitrieff, Adolfo Alsina, Aastha Mathur, François Nédélec
BBalance of microtubule stiffness and cortical tension determines the size of blood cellswith marginal band across species
Serge Dmitrieff, Adolfo Alsina, Aastha Mathur, and Fran¸cois N´ed´elec Cell Biology and Biophysics Unit, European Molecular Biology Laboratory, Meyerhofstrasse 1, 69117 Heidelberg, Germany. (Dated: This manuscript was compiled on November 10, 2018)The fast blood stream of animals is associated with large shear stresses. Consequently, bloodcells have evolved a special morphology and a specific internal architecture allowing them to main-tain their integrity over several weeks. For instance, non-mammalian red blood cells, mammalianerythroblasts and platelets have a peripheral ring of microtubules, called the marginal band, thatflattens the overall cell morphology by pushing on the cell cortex. In this article, we model how theshape of these cells stems from the balance between marginal band elasticity and cortical tension.We predict that the diameter of the cell scales with the total microtubule polymer, and verify thepredicted law across a wide range of species. Our analysis also shows that the combination of themarginal band rigidity and cortical tension increases the ability of the cell to withstand forces with-out deformation. Finally, we model the marginal band coiling that occurs during the disc-to-spheretransition observed for instance at the onset of blood platelet activation. We show that when cor-tical tension increases faster than crosslinkers can unbind, the marginal band will coil, whereas ifthe tension increases slower, the marginal band may shorten as microtubules slide relative to eachother.
The shape of animal cells is determined by the cy-toskeleton, including microtubules (MTs), contractilenetworks of actin filaments, intermediate filaments andother mechanical elements. The 3D geometry of mostcells in a multi-cellular organism is also largely deter-mined by their adhesion to neighbouring cells or to theextra-cellular matrix [1]. This is not however the case forblood cells as they circulate freely within the fluid envi-ronment of the blood plasma. Red blood cells (RBC) andthrombocytes in non-mammalian animals [2, 3], plateletsand erythroblasts in mammals [4, 5] adopt a simple ellip-soidal shape (Fig. 1A). This shape is determined by twocomponents: a ring of MTs, called the marginal band(MB), and a protein cortex at the cell periphery.In the case of platelets and non-mammalian red bloodcells, both components are relatively well characterized(Fig. 1). The cortex is a composite structure made ofspectrin, actin and intermediate filaments (Fig. 1B), andits complex architecture is likely to be dynamic [11–13].It is a thin network under tension [14], that on its ownwould lead to a spherical morphology [15]. This effectis counterbalanced by the MB, a ring made of multipledynamic MTs, held together by crosslinkers and molecu-lar motors into a closed circular bundle [4, 16] (Fig. 1C).The MB is essential to maintain the flat morphology, andtreatment with a MT destabilising agent causes plateletsto round up [17]. It was also reported that when the cellis activated, the MB is often seen to buckle [3]. Thisphenomenon is reminiscent of the buckling of a closedelastic ring [18], but the MB is not a continuous struc-ture of constant length.Indeed, an important feature of the MB is that is itmade of multiple MTs, connected by dynamic crosslink-ers. The rearrangement of connectors could allow MTsto slide relative to one another, and thus would allow thelength of the MB to change. Secondly, MT growth or de-polymerisation would also induce reorganisation. How- ever, in the absence of sliding, elongation or shorteningof single MTs would principally affect the thickness ofthe MB ( i.e. the number of MT in the cross-section)rather than its length. It was also suggested that molec-ular motors may drive the elongation of the MB [19], butthis possibility remains mechanistically unclear. Theseaspects have received little attention so far, and muchremains to be done before we can understand how theoriginal architecture of these cells is adapted to their un-usual environment, and to the mechanical constraints as-sociated with it [7].We argue here that despite the potential complexity ofthe system, the equilibrium between MB elasticity andcortical tension can be understood in simple mechanicalterms. We first predict that the main cell radius shouldscale with the total length of MT polymer and inverselywith the cortical tension, and test the predicted relation-ship using data from a wide range of species. We thensimulate the shape changes observed during platelets ac-tivation [20], discussing that a rapid increase of tensionleads to MB coiling accompanied by a shortening of thering, while a slow increase of tension leads to a short-ening of the ring without coiling. Finally, by computingthe buckling force of a ring confined within an ellipsoid,we find that the resistance of the cell to external forces isdramatically increased compared to the resistance of thering alone.
RESULTSCell size is controlled by total microtubule polymerand cortical tension
We first apply scaling arguments to explore how cellshape is determined by the mechanical equilibrium be-tween MB elasticity and cortical tension. In their rest- a r X i v : . [ q - b i o . S C ] N ov FIG. 1. A) Scanning electron micrographs of Platelets and Erythrocytes shown at the same scale [6–8], scale bar 1 µm . B)The actin/spectrin cortex of platelets, EM from [9], scale bar 0 . µm . C) The MB of platelets is made of multiple MTs bundledby motors and crosslinkers [10], EM from [6]. D) In our model, the shape of the cell is determined by the balance of two forces.Because of microtubule stiffness κ , the MB pushes against the tense cortex, which resists by virtue of its surface tension σ . ing state, the cells are flat ellipsoids and the MB is con-tained in a plane that is orthogonal to the minor cellaxis. Assuming that the cell is discoid for simplicity( R = R = R ) the major radius R is also approxi-mately the radius of the MB (Fig 1D), and thus the MTsbundled together in the MB have a curvature ∼ /R .We first consider timescales larger than the dynamics ofMT crosslinker binding and unbinding (about 10 seconds[21]), for which we can ignore the mechanical contribu-tion of crosslinkers [10]. Using the measured flexuralrigidity κ = 22 pN µm of MTs [22], and defining L asthe sum of all MTs length, the elastic energy of the MBis E B = κ L /R . At time scales larger than a few seconds,the cortex can reorganize and therefore we do not haveto consider its rigidity [23]. Its effect can then be mod-eled by a surface energy associated with a surface tension σ (Fig 1D). The surface area is S = 2 πR [1 + O ( rR )], inwhich 2 r is the thickness of the cell. Assuming the cell tobe flat enough, its surface area is therefore approximately2 πR and the energy is E T ∼ πσR . The equilibriumof the system corresponds to ∂ R ( E B + E T ) = 0, leadingto : R = κ L πσ . (1)All other things constant, we thus expect R ∝ L / . Toverify this relationship, we compiled data from 25 speciesavailable from the literature [2], computing L by multi-plying the number of microtubules in a cross-section bythe length of the marginal band. The scaling is remark-ably respected, over more than two orders of magnitudes(Fig. 2A). Using equation 1, the fit provides an estimate of the tension of σ ∼ . pN/µm , which is low comparedto the tension σ ∼ pN/µm of the actomyosin cortex ofblebbing cell [24]. However, RBC have a cortex made ofspectrin rather than actomyosin, and thus have a muchlower tension, that compensates a negative membranetension [25]. In Human RBC, membrane tension wasshown to be negative with a magnitude of 0 . pN/µm [26], close to the magnitude derived from our theory. Incontrast to RBC, we predict σ = 40 pN/µm for Humanblood platelets, given that R ≈ µm and L ≈ µm [27], which is close to the value reported for blood gran-ulocytes (35 pN/µm ) [14].The precise scaling observed in the experimental dataconfirms our mechanically driven hypothesis where theMB pushes on the cell cortex, and in which at long timescales (on the order of a minute), only the bending rigid-ity of the MTs and the cortical tension need to be consid-ered (Fig 1D). To verify that this result is still valid fora ring of multiple dynamically crosslinked MTs, we de-veloped a numerical model of cells with MBs in Cytosim ,a cytoskeleton simulation engine [28].
Cytosim simulatesstochastic binding/unbinding of connectors, and repre-sents them by a Hookean spring between two MTs. Forthis work, we extended
Cytosim to be able to model acontractile surface under tension that can be deformedby the MTs. Cell shape is restricted to remain ellip-soidal, and is described by six parameters: three axeslength R , R , r and a rotation matrix, i.e. three angles.The three lengths are constrained such that the volumeof the ellipsoid remains constant. To implement confine-ment, any MT model-point located outside the cell issubject to inward-directed force f = kδ , in which δ is the FIG. 2. A) Cell radius as a function of total MT length L . Dots: data from 25 species ([2]). L was estimated fromthe number of microtubules in a cross-section, measured inelectron microscopy, and the cell radius. B) Cell radius as afunction of L κ/σ in simulations with 0 (gray points) or 10000(black points) crosslinkers. On both graphs, the dashed lineindicates the theory 4 πR = κ L /σ . shortest vector between the point and the surface and k the confining stiffness. Here for each force f applied ona MT, an opposite force − f is applied to the surface, inagreement to Newton’s third law. The rates of change ofthe ellipsoid parameters are then given by the net forceon each axis, divided by µ , an effective viscosity param-eter (see Suppl. 1.I.A). The value of µ affects the rateof cell shape change but not the stationary cell shape.This approach is much simpler than using a tessellatedsurface to represent the cell, and still general enough tocapture the shape of blood platelets [3, 29] and severalRBCs [8, 30], see Fig. 1A.To model resting platelets, we simulated a ring madeof 10–20 MTs of length 9–16 µm [4] with 0 or 10000crosslinkers, confined in a cell of volume 8 . µm witha tension σ ∼ . pN/µm , for over six minutes, untilequilibration. We find that the numerical results agreewith the scaling law, over a very large range of parame-ter values as illustrated in Fig. 2B. Interestingly, we findthat simulated cells are slightly larger than predicted an- FIG. 3. A) MB of a live platelet labeled with SIR-tubulindye. Fluorescence images were segmented at the specifiedtime after the addition of ADP, a platelet activator, to obtainthe MB size L . B) Simulation of a platelet at different times.A limited increase of the tension (90 pN/µm ) causes the MBto shorten while a large increase of the tension (220 pN/µm )causes the buckling of the MB. C) Simulations show that if cellrounding is fast enough, the MB buckles because crosslinkerscannot reorganize. This represent an elastic behavior, butat longer times, the MB rearranges, leading to a viscoelasticresponse. alytically. This is because MTs of finite length do not ex-actly follow the cell radius, and their ends are less curved,thus exerting more force on the cell. This means thatthe value of the tension we computed from the biologicaldata ( σ ∼ . pN/µm ) is slightly under-estimated. Moreimportantly, the simulation shows that with or withoutcrosslinkers, the cell has the same size at equilibrium(compare black and gray dots on Fig. 2B), confirm-ing that, because they can freely reorganize, crosslink-ers should not affect the long-term elasticity of the MB.To understand the mechanics of blood cells with MBs atshort time scale, however, it is necessary to consider thecrosslinkers. The marginal band behaves like a viscoelastic system
During activation, mammalian platelets round up be-fore spreading, and their MB coils during this processwhich occurs within a few seconds [19]. Similar reportswere made for thrombocytes [3]. To observe these re-sults experimentally, we extracted mice platelets, and ac-tivated them by exposing them to adenosine diphosphateADP, causing an often reversible response. By monitor-ing the MB with SiR-tubulin, a bright docytaxe-basedMT dye, we could capture the MB coiling live, Fig. 3A.As it coils, the MB adopts the shape of the baseball seamcurve, which is the shape that an incompressible elasticring would adopt when constrained into a sphere smallerthan its natural radius [31]. Thus, at short time scale, theMB seems to behave as an incompressible ring, and wereasoned that this must be because crosslinkers preventMTs from sliding relative to each other. To analyse thisprocess further, we returned to
Cytosim . After an initiali-sation time, in which the MB assembles as a ring of MTsconnected by crosslinkers, cortical tension is increasedstepwise. The cell as a consequence becomes spherical,and, because its volume is conserved, the largest radiusis reduced compared to that of the discoid resting state.As a result, the MB adopts a baseball seam shape (Fig.3B). Over a longer period, however, the MB regained aflat shape, as MTs rearranged into a new, smaller, ring(Fig. 3A). In conclusion, the simulated MB is viscoelas-tic (Fig. 3B). At short time scales, MTs do not havetime to slide, and the MB behaves as an incompressibleelastic ring. At long time scales, the MB behaves as ifcrosslinkers were not present, with an overall elastic en-ergy that is the sum of individual MT energies. Thusoverall, the ring seems to transition from a purely elasticat short time scales, to a viscoelastic Kelvin-Voigt lawat long time scales (Fig. 3C). The transition betweenthe two regimes is determined by the timescale at whichcrosslinkers permit MTs to slide.
The cell is unexpectedly robust
The MB in blood cells is necessary to establish a flatmorphology, but also to maintain this morphology in faceof transient mechanical challenges, for example as the cellpasses through a narrow capillary [7]. In this section, wecalculate the response of a cell to a fast mechanical stim-uli during which crosslinkers do not reorganize. There-fore, we can assume that the ring is uniform and of con-stant length, to investigate how cortical tension affectsthe resistance of the cell to coiling. Firstly, we examinethe mechanics of a closed ring of length L and rigidity κ r within a sphere, and then extend these results to a non-deformable ellipsoid. The shape of a ring in a sphere waspreviously calculated numerically [31], and we extendedthese result by deriving analytically the force f B requiredto buckle a confined ring (see Suppl. 1.II.B). If E B is theenergy of a buckled MB, the force is : f B = − lim L → πR ∂ R E B = 8 π κ r R (2)We verified this relation in simulations, with L =2 πR (1+ (cid:15) ), where 1 (cid:29) (cid:15) >
0, which made the ring slightlyoversized compared to its confinement. Given the con-fining stiffness k , the force applied to each model-point FIG. 4. A) Degree of coiling as a function of normalized con-finement stiffness k/k c and isotropy r/R of the fixed oblateellipsoid in which the ring is confined. The light shade in-dicates regions of uncoiled states and the red area indicatescoiled states, as determined by simulations. The dashed linerepresents the empirical function k ∗ = k c ( rR ) e α (1 − Rr ) , where α = 2 .
587 is a phenomenological parameter that depends on (cid:15) , itself defined from the MB length as L = 2 πR (1 + (cid:15) ). B)Illustrations of MB shapes in different regimes, as indicatedby the circled numbers. of the ring is kR(cid:15) . If n is the number of model-points inthe rings ( i.e. n = L/s where s is the segmentation), thetotal centripetal force is nkR(cid:15) . Hence, we expect thatthe ring will buckle if k exceeds k c = nR(cid:15) f B . Uponsystematically varying k in the simulation (see Meth-ods), we indeed found that the ring coils for k > k c ,Fig. 4A. We next simulated oblate ellipsoidal cells, with R = R = R and r < R , and we varied the flatness ofthe cell by changing r/R . We found that the measuredcritical confinement k ∗ is indeed k c for r = R , but in-creases exponentially with 1 − R/r , Fig. 4. The bucklingforce of a MB is thus much higher when the MB is con-fined. This is important mechanically, as it implies thatthe flat state of the MB should be metastable, and thiscould make a blood platelet 50 times more resilient tobuckling (assuming an isotropy ratio r/R = 0 . Coiling stems from cortical tension overcoming MBrigidity
We can now consider the case of a ring inside a de-formable ellipsoid of constant volume V = 4 / πR , gov- FIG. 5. A) Configuration of the MB as a function of renor-malized tension σR /κ r and renormalized MB length L/R ,in which the volume of the cell is πR . The state of the MBis indicated by colours: gray: flat MB ; red : always buckledMB ; pink : bistable region, in which the MB can be eitherbuckled or flat. B) A cut through the phase diagram, for aMB of length L = 7 . R . The degree of coiling (see methodsfor definition) as a function of tension, in a cell initially flat(black dots) or buckled (gray dots), shows the metastabilityof the flat state. Arrows illustrate the hysteresis. erned by a surface tension σ . The length of the ring L is set with L > πR , such that we expect the ringto remain flat, at low tension, and to be coiled, at hightension, because it does not fit in the sphere of radius R . In simulations, starting from a flat ring, we ob-serve as predicted the existence of a critical tension σ ∗ f leading to buckling, Fig. 5A. This shows that increas-ing σR /κ r , i.e. increasing the ratio of cortical tensionover ring rigidity, leads to cell rounding. Thus, eitherincreasing the cortical tension or weakening the ring willlead to coiling. Starting from a buckled ring, decreasingthe tension below a critical tension σ ∗ b also leads to thecell flattening, as predicted. However, our simulationsshow that σ ∗ b < σ ∗ f : a cell initially flat will remain flatfor σ ∗ b < σ < σ ∗ f , while a cell initially round will remainround for σ ∗ b < σ < σ ∗ f , Fig. 5A,B. Hysteresis is the hall-mark of bistability, and we had predicted this bistabilityin the previous section by showing that the flat config-uration is metastable. This metastability, i.e. the factthat a MB in a flat cell has a higher buckling thresholdthan in a spherical cell, allows the cell to withstand verylarge mechanical constraints such as shear stresses. DISCUSSION
We have examined how the forces determining the mor-phology of blood cells balance each other. In particular,we predicted a scaling law 4 πR = κ L /σ , if the elastic-ity of MTs is compensated by cortical tension, in which L is the sum of the lengths of the MTs inside the cell, κ the bending rigidity of MTs and σ the cortical ten-sion. Remarkably, this scaling law is well respected byvalues of R and L measured for 25 species. We cautionthat these observations were made for non-discoidal RBC(where the two major axes differ), indicating that otherfactors not considered here must be at work [7]. In humanRBC, perturbation of the spectrin meshwork can lead toelliptical RBC [32], showing that the cortex can imposeanisotropic tensions, while another study suggests thatMB-associated actin can sequester the MB into an ellip-tical shape [33]. Cortical anisotropy would be an excitingtopic for future studies, but this may not be needed tounderstand wild-type mammalian platelets.Using analytical theory and numerical simulations, weanalyzed the mechanical response of cells with MB, andshowed a complex viscoelastic behavior characterized bya timescale τ c that is determined by crosslinker reorga-nization. At long time scales ( t (cid:29) τ c ), the MB behaveselastically, and its elasticity is the sum of all MTs rigid-ity. At short time scales ( t < τ c ), the MB behaves asan incompressible elastic ring of fixed length becausecrosslinkers do not yield. At this time scale, the stiff-ness of the ring exceeds the sum of the individual MTstiffness as long as the crosslinkers connect neighboringMT tightly [34]. Buckling leads to the baseball seamcurve, which is a configuration of minimum elastic en-ergy. This explains the coiled shape of the MB observedin mouse platelets, as well human platelets [19] and wellas dogfish thrombocytes [3]. Thus an increase of corti-cal tension over bundle rigidity can cause coiling, if thecell deforms faster than the MB can reorganize. A fastincrease of tension is a likely mechanism supported byseveral experimental evidence [35–37]. In dogfish throm-bocytes and platelets, blebs are concomitant with MBcoiling, suggesting a strong increase of cortical tension[3]. We note however that a recent study suggests thatMB destabilization could be due to ring extension [19].Finally, calculating the buckling force of a cell con-taining an elastic MB and a contractile cortex led toa surprising result. We found that the buckling forceincreased exponentially with the cell flatness, becausethe cortex reinforces the ring laterally. This makes themarginal band a particularly efficient system to main-tain the structural integrity of blood cells. For transientmechanical constraints, the MB behaves elastically andthe flat shape is metastable, allowing the cell to over-come large forces without deformation. However, as weobserved, the viscoelasticity of the MB allows the cell toadapt its shape when constraints are applied over longtimescales, exceeding the time necessary for MB remod-eling by crosslinker binding and unbinding. This studysuggest that it will be particularly interesting to comparethe time-scale at which blood cells experience mechanicalstimulations in vivo , with the time scale determined bythe dynamics of the MT crosslinkers. METHODS
MTs of persistence length l p are described as bendablefilaments of rigidity κ = k B T l p , in which k B T is the ther-mal energy. We can write the energy of such a filamentof length L as the integral of its curvature squared : E = κ (cid:90) L (cid:18) d r ds (cid:19) ds (3)Where r ( s ) is the position as a function of the arclength s along the filament. The dynamics of such a system wassimulated in Cytosim, an Open Source simulation soft-ware [28]. In Cytosim, a filament is represented by modelpoints distributed regularly defining segments of length s . Fibers are confined inside a convex region of space Ωby adding a force to every model points that is outsideΩ. The force is f = k ( p − r ), where p is the projectionof the model point r on the edge of Ω. For this work, weimplemented a deformable elliptical surface confining theMTs, parametrized by six parameters. The evolution ofthese parameters is implemented using an effective vis-cosity (see Suppl. 1.I.C). To verify the accuracy of ourapproach, we first simulated a straight elastic filament,which would buckle when submitted to a force exceed-ing π κ/L , as shown by Euler. Cytosim recovered thisresult numerically. For a closed circular ring, we alsofind that the critical tension necessary for buckling cor-responds very precisely to the theoretical prediction [38].This is also true for an elastic ring confined inside a pro-late ellipsoid of tension σ (see Suppl. 1.I.D).To simulate cell radius as a function of L κ/σ , we useda volume of 8 π/ µm (close to the volume of a platelet),with a tension σ ∼ . pN/µm , consistent with phys-iological values. The MTs have a rigidity 22 pN µm asmeasured experimentally [22]. We simulate 10 −
20 MTsof length 9 − µm , and with a segmentation of 125 nm ,we used more than 70 points per MTs. The crosslinkershave a resting length of 40 nm , a stiffness of 91 pN/µm , abinding rate of 10 s − , a binding range of 50 nm , and anunbinding rate of 6 s − . An example of simulation con-figuration file is provided in Suppl. 2. When consideringan incompressible elastic ring, we used a cell of volume4 / πR , where R is the radius of the resting (spherical)cell. For simplicity, we can renormalize all lengths by R and thus all energies by κ R /R . We simulate a cellwith a tension σ = 5 − κ R /R , and a ring of length1 − . × πR . To test the effect of confinement, weplace an elastic ring of rigidity κ in an ellipsoid space ofradii R , R , r R , in which r <
1. The elastic ring hasa length (1 + (cid:15) )2 πR , in which (cid:15) = 0 .
05. To describehow coiled is a MB, we first perform a principal compo-nent analysis using all the MTs model points. The vector u z is then set in the direction of the smallest eigenvaluewhile u x , u y are set orthogonally. We can then define thedegree of coiling C as the deviation in Z divided by thedeviations in XY : C = (cid:115) (cid:80) z (cid:80) x + y (4)Thus, C is independent of the size of the cell and onlymeasures the deformation of the MB. To measure thecritical value of a parameter µ (e.g. tension or confine-ment) leading to coiling, we computed the derivative ofthe degree of coiling C with respect to this parameter.Because buckling is analogous to a first-order transition,the critical value µ ∗ can be defined by : ∂ µ C (cid:12)(cid:12) µ ∗ = max (cid:107) ∂ µ C (cid:107) (5)Platelets were extracted using a previously publishedprotocol [39], and labeled by SiR-tubulin [40] purchasedfrom Spirochrome. I. SI I : SIMULATION OFMICROTUBULES/CORTEX INTERACTION
To understand cell shape maintenance, one needs tomodel the interaction between the cellular cortex andthe microtubule marginal band. The structure of themarginal band is well known, compared to the organi-zation of the cortex which is not caracterized. We thusdecided to represent the microtubules individually, andthe cortex effectively as a contractile surface. The inter-actions between a discretized ( e.g. triangulated) surfaceand discrete filaments can be a demanding problem com-putationally, since such a surface would have a very largenumber of degrees of freedom. In contrast, we describehere how the problem remains relatively simple for a con-tinuous shape that is described by a limited number ofparameters.
A. General formulation
1. Forces and parametrization
Let S ( p k ) be the surface defined by the set of pa-rameters { p k } k 2. Constraints In many cases, constraints can be introduced using La-grange multipliers, by inserting them into the energy E .For instance, to maintain the volume, we can define anenergy E (cid:48) = E + P V where V is the volume and P isthe pression ; here P is also a Lagrange multiplier andwe have to calculate its value appropriately to obtain V = V . The pseudo-forces φ kP associated to pressure are: φ kP = − P ∂V∂p k (10) B. Contractile Ellipsoid In this section, we describe a more complex, 3D sur-face. We model an ellipsoid centered around 0 that has afixed volume V and a a surface tension σ , which meansan energy E n = σS , where S is the surface area ofthe ellipsoid. The ellipsoid is described by its eigenvec-tors u , , and their eigenvalues (i.e. the radii of theellipse) a , , . We will also use the orientation matrix M = [ u , u , u ]. By construction, M is a rotation ma-trix of determinant 1. 1. Surface Tension We can compute the pseudo-forces associated to sur-face tension as: φ kσ = − σ ∂S∂a k (11)The surface area of an ellipsoid is a complex special func-tion that is not a combination of the usual functions. Forconvenience, we used an analytical approximation of thearea : S ( a , a , a ) (cid:39) π (cid:18) ( a a ) p + ( a a ) p + ( a a ) p (cid:19) p (12) For which p = 1 . 2. Point Forces To add the contribution of the external forces exertedby the microtubules on the surface, we need to determine ∂ r /∂p k . The position of a point on the surface of theellipsoid is defined by two angles θ , φ as : r = u r cos θ sin φ + u r sin θ sin φ + u r cos φ (13)Therefore, we have : ∂ r ∂r k = r . u k (14)It is clear that only the component of the force normal tothe surface is providing work upon changing r k , therefore,we can discard the tangential component when comput-ing the radial forces. The contribution of a force f at apoint r with a local normal n to the pseudo force φ kf istherefore : φ kf = f n n . u k (cid:107) u k (cid:107) (15)With : f n = f . n (16)We can now compute the torque generated by f . In 2D,it would be convenient to describe the ellipse orientationby an angle θ , and the result is that the ”angular force” φ θ is the torque r × f . We will assume that this is generaland stays true in 3D ; thus we can write φ ang directly asa vector : φ ang = r × f (17) a. Volume conservation To implement volume con-servation we only need to find a pressure P such that( V − V ) /V < (cid:15) where (cid:15) is the tolerance. Many tech-niques allow the convergence to a suitable value of P -and the choice of method has no physical implication.Here we used a gradient descent method known as theshooting method. For this, we start with an initial valueof P = 0 and we compute V ( P ). If V ( P ) − V /V > (cid:15) ,we compute V ( P + δP ) to get the gradient of the volumewith respect to pressure. We then follow this gradientuntil we reach the desired aim for V . This method worksvery well if V ( P ) is monotonous, which is always the casehere.The volume of the ellipse is V = πa a a and there-fore, using the Lagrange multiplier P to conserve thevolume we can write : φ kP = 43 πP a a a a k (18) . . 510 2 . . . . . . t e n s i o n σ / ( κ / R ) length L/R FIG. 6. Phase diagram of the degree of buckling as a functionof the length and the tension. Red means that the filamentis buckled, gray that it is flat. The dashed line represents thecritical tension calculated in Eq. 22. C. Time Evolution We can now define the time evolution of the ellipse.We assume a unique viscosity µ associated to the changeof size of the ellipse, and a rotational viscosity η ang .˙ a k = 1 µ (cid:16) φ kP + φ kσ + (cid:88) φ kf (cid:17) (19)˙ M = R ( u ) with u = 1 η ang (cid:88) φ angf (20)In which R ( u ) is the rotation matrix generated from themoment vector u . D. Simulations Validation To validate out numerical method and its implemen-tation, we first simulated a microtubule bundle confinedinside an ellipsoid cell of tension σ and volume πR . Aclassical result of analytical mechanics is that a filamentshould buckle under a force tangential force f is this forceis larger than a critical force : f ∗ = κπ L (21)Assuming the microtubules to be sliding freely, the criti-cal buckling force of a microtubules is thus f ∗ n = nf . Weconfined the microtubule in a contractile ellipsoid, whichthus takes the shape of a prolate ellipsoid. Let us call a = R the longer axis of this ellipsoid, and the shorteraxis are a , a = (cid:112) R /R . The force exerted on the mi-crotubule is f σ = 2 σ∂ R S ( R, (cid:112) R /R, (cid:112) R /R ), with S defined in equation 12. Starting with a microtubule oflength L , buckling will occur for a critical tension : σ ∗ = κπ R (cid:18) ∂ R S ( R, (cid:113) R /R, (cid:113) R /R ) (cid:19) − (22) II. SI II : MECHANICS OF A CONFINED RINGA. Formulation Let us consider a rod of length L lying on a sphereof radius R . We can describe this rod by its position r , parametrized by its arclength s , such that the energyreads : E ( R, L ) = κ (cid:90) L ¨ r ds (23)Because the rod lies on the unit sphere, and because s is the arclength, we have the constraints : (cid:107) r (cid:107) = R and (cid:107) ˙ r (cid:107) = 1 (24)We can introduce this as constraints in the energy usingtwo Lagrange multipliers α and β , to define : E = κ (cid:90) L (cid:2) ¨ r + α ( R − (cid:107) r (cid:107) ) + β ( (cid:107) ˙ r (cid:107) − (cid:3) ds (25)Minimizing this energy yields the Euler-Lagrange equa-tion : r (4) = α r + ˙ β ˙ r + β ¨ r (26)Since the curve is lying on a sphere, we can use theidentity : ¨ r ( s ) = k i ( s ) [ r ( s ) × ˙ r ( s )] − R r ( s ) (27)In which k ( i ) is the intrinsic (geodesic) curvature. Even-tually, we find : ¨ k i = k i R (cid:18) γ − R k i (cid:19) (28)In which γ is a constant [31]. To find the shape of aclosed ring, one needs to find the value of α and k i (0)such that the curve is of length L is a closed ring, i.e. : r ( L ) = r (0) and ˙ r ( L ) = ˙ r (0) (29)Numerically, we determined γ and k i (0) using a shootingmethod. B. Weakly deformed ring For a weakly deformed ring, equation 28 can be sim-plified to ¨ k i = γR k i (30)Periodicity imposes √− γ → m when L → πR , inwhich m ∈ N . Since the lowest energy curve has a pe-riod L/ 2, we can conclude that m = 2, i.e. γ → − . . 510 0 . . 85 0 . . 95 1 E × R / κ R/R E (1) − πκ ( R − R ) /R Euler-Lagrange FIG. 7. Bending energy of an incompressible elastic ring oflength 2 πR (the marginal band) in a sphere of radius R 0, one finds : k i ( s ) = 6 √ b cos 2 s/R + O (cid:16) b (cid:17) (32)¨ k i ( s ) = γ × √ b cos 2 s/R + O (cid:16) b (cid:17) , (33)with γ = − E ( R, b ) = (cid:90) π (cid:18) R + k i (cid:19) (cid:107) ∂ t R (cid:107) dt (34)For small deformations b → 0, we have : E ( R, b ) = κ R (cid:0) π + 36 πb + O (cid:0) b (cid:1)(cid:1) (35)We can also compute the length of the marginal band,and the energy : L ( R, b ) = 2 πR (cid:0) b + O (cid:0) b (cid:1)(cid:1) , (36) E ( R, L ) → κ R (cid:18) π + 3 L − πRR (cid:19) (37) We then find the force exerted by a nearly flat ring onthe sphere L = 2 πR : f B = lim L → πR ∂ R E ( R, L ) = 8 πκR (38)This result is in agreement with solving the full shapeequation (Eq. 28), as illustrated in figure 7. f b is the forceexerted by a nearly flat ring on a sphere ; by constructionit is also the critical force at which a ring will buckle.Numerically, we can study ring buckling in two cases :when the ring is undergoing an elastic confinement, andwhen the ring is confined by a contractile surface. C. Ring under elastic confinement Let us consider a ring of length L confined in a a sphereof radius R such that L = 2 π ( R + (cid:15) ) by an elastic confine-ment k (see [28] for the implementation of confinement).The confinement force is here f c = kn(cid:15) , in which n is thenumber of points describing the discrete ring. The ringwill buckle if f c > f B ; using Eq. 38, we find that thering will buckle above a critical confinement : k c = 8 πκn(cid:15)R , (39)in which n = L/ds , where ds is the segmentation. D. Ring in a contractile ellipsoid Using our computed value of f B , we can compute ana-lytically the critical value of the tension that will bucklea ring in a contractile space, assuming that space to beellipsoid and near-spherical. For this, we take the verysame approach as we did for the bundle in a prolate ellip-soid, although now the ellipsoid is oblate, and the buck-ling force is that of a ring rather than an open bundle. σ ∗ = f B ∂ R S ( R, R, R /R ) (40) [1] T. 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