SUPPLEMENTARY INFORMATION

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1 doi: /nature10286 Supplementary Model 1 Introduction, summary of previous descriptions of pole mechanics during cytokinesis We describe here a theoretical model for the mechanics of the poles of a cell undergoing cytokinesis. Many studies of cytokinetic mechanics focus on force generation at the equator, where the contractile ring drives furrow ingression 1 3. Although the acto-myosin cortex preferentially accumulates at the cell equator during cytokinesis 4, a clear layer of actin and myosin remains present at the cell poles throughout division 5 7. Previous studies have provided contradictory evidence as to the role of this polar cortex. While disruption of the polar cortex by local treatment with cytochalasin D (CD), an inhibitor of actin polymerization, inhibited cytokinesis in normal rat kidney cells 8, a similar treatment did not interfere with the division of sand dollar blastomeres 9. Overall, the mechanical contribution of the poles to cytokinetic mechanics has been subject to debate and often neglected in physical models of cytokinesis 10. However, to understand the shape of the entire dividing cell, a physical description of the cell poles is necessary 11, 12. An early model describing cytokinesis in sea urchin eggs explicitly including the mechanical contribution of the poles, described the poles as portions of spheres with a surface tension that resists the ingression forces exerted by the contractile ring 13. In the simple geometry of a symmetrically dividing cell with two spherical poles of equal sizes, the authors could estimate the minimal force required for cleavage. In recent years, this model has been expanded and extensively tested experimentally using Dictyostelium as a model system 5, 14. It has been shown that polar resistive forces depend not only on myosin II, but also on globally distributed cross-linkers such as dynacortin, fimbrin and enlazin Moreover, microtubules have been involved in the spatial regulation of the mechanical properties of the polar cortex via the RacE/ pathway 18, 19. The model we propose here differs from the previous models mentioned above in that it does not assume that the poles are of equal size. Similarly to the model of Yoneda and Dan 13, 14, we describe the dividing cell as two connected portions of spheres with a cortex under tension underlying the membrane, but we allow the size of the poles to vary. We can thus investigate the influence of cortical tension on the stability of the shape of the dividing cell (see Fig. 1a and d in the main text). We first consider the case where the two polar cortices have a constant tension and we show that a symmetry-breaking instability is expected to occur for a sufficiently large polar tension, leading to the contraction of one pole at the expense of the other (Section 2). Motivated by the oscillating behaviour of the actin cortex during cellular oscillations, we then incorporate a description of the dynamics of cortical actin in the model (Section 3). We show that coupling actin turnover dynamics and cell mechanics results in the apparition of an oscillating region in parameter space, where the symmetry between the two poles of the cell is broken. This symmetry-breaking transition occurs at a smaller cortical tension than the constant-tension symmetry breaking described in Section 2. By relating myosin fluorescence intensity to cortical tension, we then show that the experimental measurements of the volumes of the poles and actin and myosin fluorescence intensities can be related using the driving equations of the model (Section 4). This allows the extraction of the main parameters of the model. Using these parameters, we position the oscillating cells in the predicted state diagram and show that they mostly fall into the oscillatory region. Simulations of the equations with the extracted parameters very accurately reproduce the experimental oscillations (Fig. 3 and Supplementary Fig. 8). 1

2 2 Constant polar tension model 2.1 General equations We describe a cell undergoing cytokinesis as two portions of spheres (the polar regions) with tensions T 1 and T 2 connected by a short cylinder of wih a and radius R c (the equatorial contractile ring) under tension T c (Supplementary Fig. 4a). The two polar regions can have two differents radii of curvature R 1 and R 2, and we assume that the total volume of the mother cell V 1 + V 2 =2V 0 is conserved (see Section 5.1). We assume that the cytoplasm of each half of the cell can be modelled as a poro-elastic medium 20. Indeed, the cellular cytoplasm is filled with fluid cytosol, and also with cytoskeletal and membrane structures, the two sets of chromosomes, the astral microtubules and the central spindle, which are likely to exhibit solid-like properties and oppose cell shape changes. In this description each daughter cell has a bulk elasticity, taken to be uniform and linear for simplicity, with a reference volume V 0 and a bulk elastic modulus K (Supplementary Fig. 4a). The elasticity of the cortex itself, as well as substrate adhesions, could also contribute to cell elasticity (see Sections 5.2 and 5.3). As shown below, the elastic response of the cell is essential in this description to account for the stability of the successfully dividing control cell. With these hypotheses, the force balance in the dividing cell reads: dr c α c α dv 1 V 0 = [at c R c (T 1 cos θ 1 + T 2 cos θ 2 )] (1) = α [ dv 2 V 0 = 2T1 2T 2 + K ] (V 1 V 2 ) (2) R 1 R 2 V 0 where Eq. (1) corresponds to the equilibrium of tensions between the contractile ring and the two polar cortices (Supplementary Fig. 4a) and Eq. (2) corresponds to the equilibrium of pressures between the two halves of the cell. The angles θ 1, θ 2 (defined in Supplementary Fig. 4a) and the radii of curvature R i are geometrically related to the volumes V i and to the contractile ring radius R c by cos θ i = g(3v i /(πrc)) 3 and R i = R c [1 [ g(3v i /(πrc)) 3 ] ] 2 1/2, where g is defined such that x = g(y) is the real solution of the polynomial equation (1 + x)(2 x) 2 = y 2 (1 x) 3. In the left-hand side of Eqs.(1) and (2) we have introduced two generic friction terms limiting the contraction of the ring and the volume exchange between the two cellular halves, with effective friction coefficients α c and α/v 0, respectively. The permeation of the fluid cytosol through the poroelastic cytoplasm, described by Darcy s law, is a possible source of dissipation giving rise to the friction term in Eq.(2). The coefficient α c could be related to the fluid viscosity of the actin network constituting the contractile ring and to the resistance of the cytoplasm to the contraction of the ring. 2.2 Spontaneous symmetry breaking of a cell with constant polar tension We now consider the case of a symmetric cytokinetic cell whose two polar cortices have the same cortical tension T 0. If cell shape remains symmetric during cytokinesis, the volumes of the two poles are constant due to volume conservation (V 1 = V 2 = V 0 ) and Eq.(2) is automatically satisfied. We do not make any particular assumption about the process controlling cleavage furrow dynamics, so that T c may in general depend on R c and on time 2

3 t. The dynamics of contraction is given by: α c dr s c = [ at c (R s c,t) 2T 0 R s cg ( )] 4R 3 0 Rc s3 (3) where R s c(t) is the solution giving the ingression of the furrow in the symmetric case, and where we have introduced the length R 0, the radius of one spherical daughter cell at the end of symmetric cytokinesis, defined by V 0 = 4 3 πr3 0. To study the stability of this solution, we expand the dynamic equations (1) and (2) to linear order around the symmetric solution and for a symmetry-breaking perturbation: V 1 = V 0 (1 + v), V 2 = V 0 (1 v) and R c = R s c + δr c. This yields α c dδr c α dv = = [ a dt c 2T 0 [2 2g(4R dr 0/R 3 c) 3 2g(4R0/R 3 c) 3 2 c ] +g(4r0/r 3 c) 3 3 ] δr c (4) [ T0 16R0 4 R 0 3Rc 4 g(4r0/r 3 c) 3 [ 1 g(4r0/r 3 c) 3 ] ] 2 2K v (5) Since these two equations are decoupled, they can be analysed separately. Eq.(5) gives the linear stability of the cell with respect to symmetry-breaking in the volumes of the two poles. The cell is unstable when the coefficient in front of δv 1 in the right-hand side of Eq.(5) is positive, as in that case any perturbation in the difference of volume between the poles will be amplified exponentially. This happens for: T 0 R 0 K > 3 8 ( Rc R 0 ) 4 1 g(4r 3 0 /R3 c)[1 g(4r 3 0 /R3 c)] 2 (6) The corresponding stability diagram is plotted on Supplementary Fig. 4b. In the limit T R c 0, inequality (6) simply becomes 0 R 0 K > 3 2. This result suggests that cell elasticity is essential to ensure symmetric division, as for large polar tension compared to cell elasticity a symmetry-breaking instability should occur. In Section 3, we show that taking into account the dynamic behaviour of the actin cortex leads to a symmetry breaking instability at even smaller values of the critical parameter T 0 R 0 K. 2.3 Bifurcation induced by a difference of tension between the two poles We now consider the effect of a difference of tension T = T 1 T 2 on the equilibrium of the cell in the stable region of the diagram of Supplementary Fig. 4b. We assume here that the dynamics of contraction of the ring is slow compared to the rate of volume exchange and R c is taken constant. If the symmetric solution is stable, for small enough tension differences T, a new mechanical equilibrium is reached where the side of the cell with the larger tension slighty contracts at the expense of the other side. However, for larger values of T it is not guaranteed that an equilibrium state exists. Indeed, developing Eq.(5) close to the spontaneous symmetry breaking transition T T (R c /R 0 ) where T is the critical value of tension given by Eq.(6), yields the following expansion to third order in δv 1 : dδv 1 = A ( Rc R 0 ) T +2 T T KR 0 T δv 1 + B ( Rc R 0 ) T R 0 K δv C ( Rc R 0 ) δv 3 1 (7) 3

4 where A(R c )= 2R 0 R c 1 [ g(4r 3 0 /R3 c) ] 2 and the functions B and C have more lengthy expressions. The function C is positive for R c <R c 0.93R 0 and negative for R c >R c. The general form of Eq.(7) is well-documented in bifurcation theory 21, and one can show that for R c <R c: for T = 0 the pitchfork symmetry-breaking bifurcation occuring at T = T described previously is subcritical, so that for T > T a discontinuous jump occurs from the symmetric to an asymmetric solution; in the region T < T and for a sufficiently small value of T T a saddle-node bifurcation occurs at increasing T, so that the stable solution is lost and the system jumps to another equilibrium solution. This suggests that for a value of average tension close enough to the symmetry-breaking instability and for a sufficient ingression of the cleavage furrow R c <R c, introducing a difference of tension between the two poles will result in a bifurcation to a strongly asymmetric solution. To obtain the corresponding phase diagram, we first write that any equilibrium solution has to be a solution of the pressure balance given by Eq.(2) 2T 1 R 1 + K V 1 V 0 = 2T 2 R 2 + K V 2 V 0 (8) The bifurcation is then found by performing a linear stability analysis of this solution. This can be done by expanding the dynamic equation Eq.(2) around the solution of Eq.(8) and studying the sign of the linear term in the dynamics. One finds that the linear term cancels when T 1 R1 4 cos θ 1 (1 + cos θ 1 ) 2 + T 2 R 4 2 cos θ 2 (1 + cos θ 2 ) 2 = 3K 4R 3 0 Solving these two equations gives the phase diagram shown in Supplementary Fig. 4c, plotted here for a cell constricted to a furrow radius R c =0.6R 0. The diagram shows that as the average tension increases, a smaller difference of tension is required to undergo the T m R 0 K bifurcation. For 1.5 the solution for T 1 = T 2 is unstable, which corresponds to the symmetry-breaking transition shown on Supplementary Fig. 4b. In the diagram, a bipolar shaped solution is defined as an equilibrium shape where both angles θ i are smaller than π/2. One can verify that after the bifurcation, in the region of high tension in the diagram, no bipolar shaped solution can be found. 3 Model coupling cortex turnover dynamics and cell mechanics: shape oscillations The model described in the above section predicts that the shape of the dividing cell is unstable for a large cortical tension, however it does not predict oscillations of the shape of the cell. In this section, we describe a more complete model of the poles of a dividing cell where we take into account the dynamic behaviour of the actin cortex. This model can account for the shape oscillations we observe. The model is based on the observation that during oscillations, both cortical actin and myosin undergo cycles of accumulation and disassembly, suggesting that the tensions of the poles are not constant. Here we show that with simple assumptions on the polymerization and depolymerization of the actin cortex, the coupling between cell shape and actin dynamics leads to an oscillating behaviour. (9) 4

5 3.1 Accumulation and turnover of the actin cortex Actin dynamics The polar actin cortex is an active layer with a turnover resulting from actin polymerization and depolymerization 22. The dynamics of such a layer on a surface with varying area can be described by the following equation for the actin density at the pole i, c i a: d(s i c i a) = S i τ (c0 a c i a),i =1, 2 (10) where τ is the turnover time of the cortex, c 0 a is the reference actin density, and S i is the surface area of the pole i. This equation takes into account both matter conservation of actin and the polymerization and depolymerization kinetics of the actin layer, as becomes apparent when considering two simple limits: In the absence of surface variation, ds i = 0, Eq. (10) reduces to dci a = 1 τ (c0 a c i a), which describes the exponential relaxation upon perturbation of the actin density to a reference density c 0 a. Such a process could for instance result from cortex polymerization at the membrane 3 with a rate of monomer addition k p and depolymerization in bulk with a rate k d. In that case, in the absence of surface variation, the actin density dynamics reads dci a = k p k d c i a, which gives Eq. (10) with τ = 1 k d and c 0 a = kp k d. For slow turn-over of the layer (τ ), Eq. (10) reduces to d(s ic i a ) = 0, which simply expresses that in the absence of polymerization and depolymerization, the total quantity of actin at one pole S i c i a is conserved when the surface area of the pole varies Myosin dynamics The analysis of experimental data shows that the intensities of actin and myosin are proportional at all times during oscillations induced by anillin depletion or laser ablation (Supplementary Figure 9b, see Section 4 below). Therefore, we restrict our analysis to the description of only one of these two cortical components. We note that such a proportionality relationship could result from fast first-order binding kinetics of myosin to the actin cortex: dc i m = k m onc i a koff m ci m (11) where c i m is the density of myosin in the cortex of the pole i, kon m is the binding rate of myosin to the cortex and koff m the unbinding rate. When the unbinding timescale 1 koff m is small compared to the characteristic times of the oscillation, the previous equation can be approximated by c i m km on koff m c i a Relation between cortex tension and the densities of actin and myosin In general, the tension of the cortex could be a function of both actin and myosin densities T = T (c a,c m ). Following Ref 23, we assume that the cortical tension T can be simply written as a saturating function of myosin intensity: T = T 0 (c c m /c 0 m + 1) c + c m /c 0 m = T 0 (c c a /c 0 a + 1) c + c a /c 0 a (12) 5

6 where c 0 m = km on koff m c 0 a, and c is a non-dimensional factor setting the strength of the saturation: c corresponds to a linear dependency of tension with myosin density, whereas c = 0 corresponds to a tension independent of the myosin concentration. Ref. 23 shows that this choice of a saturating function for the tension exerted by myosins in the cortex explains cortical flow profiles during polarization of the C. Elegans embryo. 3.2 Stability analysis We then used the expression of tension introduced above, Eq. (12), to rewrite the mechanical equation for the difference of volume between the two poles (Eq. (2)): α dv [ 2T (c 1 = m ) 2T ] (c2 m) +2Kv +2K 3 v 3 (13) R 1 R 2 where v is the volume difference ratio introduced in Section 2 above, v = v 1 v 2 = V 1 V 2 V 1 +V 2. v is 0 for a symmetric cell and is equal to 1 or 1 when the volume of one pole vanishes. We have also introduced an additional non-linear elastic term (term proportional to K 3 ). This term is necessary in our description to limit the amplitude of the oscillation, and we also found as explained in Section 4.2 that this non-linear elasticity can be observed in our data. Besides, we note that this equation involves a characteristic time τ c = α 2K, which sets the timescale of cell volume variation around the symmetric state as a response to an instantaneous difference of pressure applied at the poles. Eqs. (10) and (13) form a dynamical system for the three variables v, c 1 a and c 2 a. The symmetric state v = 0, c 1 a = c 0 a and c 2 a = c 0 a is a stationary state of this system. We now proceed to a linear stability analysis around this symmetric state. To linear order, we find: α dv d(δt 1 δt 2 ) d(δt 1 + δt 2 ) = 2 δt [ ] 1 δt 2 2T0 j(r c /R 0 )+ h(r c /R 0 ) 2K v (14) R 0 R 0 = 1 τ (δt 1 δt 2 ) 2T 0 c 1+c s(r c/r 0 ) dv (15) = 1 τ (δt 1 + δt 2 ) (16) where we have introduced for convenience the tension variations δt 1 and δt 2 and the functions of ratio of the equatorial ring radius to the radius of a daughter cell r c = R c /R 0 : h(r c ) = 8 g(4/r 3 3rc 4 c ) [ 1 g(4/rc 3 ) ] ( ) ) 2 2 1/2, j(rc ) = 1 r c (1 g 4 and s(r rc 3 c ) = 4/rc 3 g (4/rc 3 )/(1 g(4/rc 3 )). Eq. (16) is independent of Eqs. (14) and (15) and describes the exponential relaxation of the sum δt 1 + δt 2 to its reference value as a response to a perturbation, so that we can restrict the analysis to the variables v and δt 1 δt 2. The dynamics of the system at linear order then reads: ( d v δt 1 δt 2 ) = 2K α ( T 0 R 0 K h(r c) 1 T 0 R 0 K j(r ) [ ] c) 2c 1+c s(r c ) T0 R 0 K h(r c) 1 α 2Kτ + T 0 R 0 K s(r c)j(r c ) 2c 1+c ( ) v (17) δt 1 δt 2 The linear stability of the symmetric state can be obtained by studying the eigenvalues of the matrix in Eq. (17): eigenvalues with a positive real part indicate an unstable state, 6

7 and eigenvalues with a non zero imaginary part indicate that any perturbation of the symmetric state will result in oscillations of the dynamic variables, which will be damped or amplified according to the real part of the eigenvalue. The stability of the symmetric state depends on 4 parameters. Two of them were introduced in the constant tension model (Section 2): the ratio of polar cortex tension to cell elasticity T 0 R 0 K the ratio of the radius of the contractile ring to the radius of one daughter cell Rc R 0. The two other parameters are: the ratio of the timescale of the mechanical response of the cell to the turnover of the cortex α 2τK = τc τ the saturation parameter c introduced in Eq. (12). On Fig. 3d in the main text, we represent the resulting linear stability phase diagram in the parameter space ( τc τ, T 0 Rc R 0 K ), calculated for R 0 =0.8 and c =0.8, and on Supplementary Fig. 9a in the parameter space ( Rc T R 0, 0 τc R 0 K ), calculated for τ =0.3 and c =0.8. In both diagrams we have set the saturation parameter to c =0.8, following the choice of Ref 23, and other constant values were set to their average values obtained from data extraction of anillin-depleted and laser-ablated oscillating cells (see Section 4). In the phase diagrams, the yellow region corresponds to a stable symmetric state, the white region to an oscillatory unstable state and the grey region to a non-oscillatory unstable state. Note that depending on the non-linearities, the system could exhibit non-linear oscillations in the linearly unstable region (grey regions in Fig. 3d and Supplementary Fig. 9a). For fast turnover of the cortex relative to the timescale of cell mechanical response, τ <<τ c (right region of the diagram of Fig. 3d), the tension of the cortex is constant and the linear stability analysis reduces to the constant-tension case (Supplementary Fig. T 4b where for R c = 0, the symmetric state is stable for 0 R 0 K < 3 2 and unstable otherwise). When the turnover timescale is comparable to or larger than the cell mechanical timescale τ τ c, a region of the phase diagram appears where for intermediate values of the stability parameter T 0 R 0 K, the eigenvalues are imaginary, corresponding to unstable oscillatory states. In this regime, any fluctuation in the volume or tension difference between the poles will result in a spontaneous oscillation of the system. Near the bifurcation to an unstable oscillatory state, the spontaneous oscillation grows with the period T = 2π ττ c 1 T 0 R 0 K h(r c) (18) which increases both with the turnover of the actin layer τ and the cellular mechanical time scale τ c. In the unstable oscillatory region, the stability diagram predicts an exponential growth of the oscillation following a perturbation. Addition of the non-linear elastic term introduced in Eq. (13) limits the amplitude of the oscillation, so that a small perturbation to the symmetric state results in a transient growth of the oscillation followed by a stationary oscillating regime. To get an insight into the exact profiles of oscillating variables in this stationary regime, we then simulated Eqs. (10) and (13) after a small initial perturbation of the symmetric state, and for parameters extracted from a fit of the data to the driving equations of the 7

8 model (Fig. 3c, τ c = 20 s, τ = 64 s, T 0 /R 0 K =1.15, K 3 /K =0.5). The waveforms of cortex actin density and difference of volume between the poles are very similar to the experimental measurements: the volume difference shows a relatively smooth oscillation, whereas the actin density oscillates with successive sharp peaks with a faster decreasing than increasing part (Supplementary Fig. 8e). The simulation also predicts an asymmetric variation of the amplitude of the actin density around its average, with larger relative maxima of density than relative minima, which is also observed experimentally. This asymmetry arises from the non linear coupling between the variation of the size of a pole and the actin density at this pole in Eq. (10). We have also verified that for every analyzed cell, the periods given by the simulations in the stationary regime are close to the mean periods experimentally measured. We could only compare orders of magnitude at this point as the period of individual cells displayed significant variability during oscillations (Supplementary Fig. 12). In the representative examples shown on Fig. 3 (anillin-depleted L929 cell) and Supplementary Fig. 8 (laser-ablated L929 cell), we found the simulated periods to be T = 315 s and T = 368 s respectively, of the same order of magnitude as the mean periods of the 4 first large-amplitude oscillation cycles measured experimentally, T = 379 ± 85 s and T = 340 ± 36 s respectively (note that the measured period slows down after the first oscillation cycles). 4 Data analysis and extraction of the critical parameters 4.1 Data fitting We then tested whether the driving equations of the model, Eqs. (10) and (13) could accurately fit the experimental data. To obtain the volume difference v from the experimental measurements, we extracted the cross-sectional area of the two poles from the segmentation of confocal sections (see Supplementary Fig. 8a and Methods). The two poles have roughly spherical shapes, consistent with the description of the dividing cell as two connected portions of spheres (see Supplementary Fig. 4). To estimate the volume V i and cortex surface area S i of the two poles from this measurement, we assumed that the cell had a rotational symmetry around an axis joining the center of the two poles, and that the confocal sections were taken in a plane of maximal surface area. Actin and myosin densities in the cortex c a and c m were taken proportional to fluorescence intensity of Lifeact-mCherry and myosin regulatory light-chain-gfp, and scaled to their average values c a and c m. The large number of sampling points available during the course of the oscillation ( 10 3 sampling points for an oscillation lasting 1.5 hours) allowed for a precise verification of the driving equations. We first verified that actin and myosin densities were proportional during oscillation (Supplementary Fig. 9b), thus justifying that our theoretical description involves only one of the two cortical components. We then computed the different terms involved in Eqs. (10) and (13) and performed the following multilinear regression fit to verify these two equations: [ ] dv [ 1 d(s i c i ] a/c a ) S i [ 2T (c 1 = C 0 + C m /c m ) 1 2T ] (c2 m/c m ) [ + C 2 [v]+c 3 v 3 ] (19) T 0 R 1 T 0 R 2 [ ] c i = C 4 + C a 5 (20) c a where the quantities in brackets were obtained from experimental data and the constants C k were extracted from the fitting procedure. The driving equations for the density at 8

9 the two poles were included into a single fit since in this description they should obey the same driving equation. We found that both driving equations could be fitted with good accuracy for nearly all the cells studied (see Fig. 3e and Supplementary Fig. 9c for two representative examples, the total number of cells analyzed were N = 5 anillin depleted, N = 6 laser ablated, one anillin-depleted cell and one laser ablated cell were rejected due to poor fitting). The linear relationship expected from Eq. (10) can be verified by directly plotting the quantity 1 d(s i c i a /ca) S i against the actin density c i a/c a (see Fig. 3e for a representative anillin-depleted cell and Supplementary Fig. 9c for a laser-ablated cell). For all cells, the constant C 0 had small values as expected from the symmetry of the problem. The constant C 4 yielded the value of the reference concentration relative to the average concentration during the oscillation, c 0 a/c a. From the other constants C k obtained from T this fitting procedure, we could extract the parameters 0 R 0 K, K 3 K, τ and τ c. As explained in Section 3, these measurements allowed us to position the cells on the theoretical stability diagram (Fig. 3d and Supplementary Fig. 9a). On the diagram, error bars indicate the T standard deviations of the fitted values for 0 τc R 0 K and τ and the standard deviation of the time variation during the oscillation for Rc R 0. Strikingly, we found from this analysis that within the estimated error in the parameters, all the cells fell into the unstable oscillatory region of the diagram, showing the self-consistency of the model and the agreement with our assumption that the oscillations arise from a spontaneous symmetry-breaking of the dividing cell shape. Besides, we could also extract the average turnover time of actin in the cortex, τ = 136 s, and average timescale of cell mechanical response, τ c = 29 s. The non-dimensional ratio T 0 R 0 K was found to be of order 1. A direct measurement of the polar surface tension of a dividing cell is difficult, in particular due to an active response of mitotic cells to micropipette aspiration 24. However, taking the order of magnitude of the cortical tension in L929 cells in interphase T pn.µm 1 (Ref. 25, presumably a lower estimate since cortical tension has been shown to increase in mitosis 26 ) and the measured value R µm, we would obtain the estimate K 70 Pa in oscillating cells, in the range of values reported for cellular elasticity Non-linear cell elasticity In Eq. (13) we have introduced a non-linear elastic term in the elastic response of the cell. The elastic response of the cell could be a general nonlinear function of v, in which case Eq. (13) would read: α dv [ 2T (c 1 = m ) 2T ] (c2 m) +2Kf(v) (21) R 1 R 2 To obtain a visual representation of the elastic response f(v), we first performed a fit α of Eq. (21) with a linear function f(v). We then used the extracted parameters 2K and T 0 α dv R 0 K determined from this fit to obtain an estimate of the function f(v) = [ ] 2K + T (c 1 m ) KR 1 T (c2 m ) KR 2 in the more general case: as can been seen on the left diagram of Fig. 3e and on Supplementary Fig. 9c, the points align on a single curve. The curve shows a central linear region around v = 0, which corresponds to the linear elastic response of the cell. If the cell was a purely linear elastic material, the points would fall on a straight line for the full range of v, whereas one can clearly see on Fig. 3e and Supplementary Fig. 9c that the tails of the curve show a non-linear behaviour for high values of v. It is necessary to introduce a non-linear elastic term in Eq. (13) to account for these tails. We included this non-linear elasticity through an expansion in powers of v, and we found that an expansion up to 3 rd order was sufficient to fit the experimental data and reproduce the 9

10 oscillations. The expansion was reduced to odd powers of v, a restriction arising from the assumption that both poles have the same physical properties. 4.3 Additional feedback of the volume on polymerization We found in our analysis that introducing a small additional term depending on volume in Eq. (10) allowed to improve the agreement between theory and data: d(sc a ) = S τ [ c 0 a (1 βv) c a ] where β characterizes the coupling between cortex assembly and cell volume. The coefficient β was found to be relatively small and positive on average (β =0.04 ± 0.01), which means that the coupling leads to an increased polymerization rate for smaller volumes of the pole. This effect could be triggered by the active recruitment of myosin and actin in response to the decreasing radius of curvature when a pole contracts. Such an effect has been observed in Dictyostelium cells, where deformation of the cell by micropipette aspiration triggers a mechanical response leading to cortex accumulation 24. We have verified that the effect of this additional coupling on the oscillation can be neglected in the range of parameters that we extract from our data analysis. In general, with a positive sign, this coupling tends to favour instabilities of the symmetric shape, as it leads to an increase of actin density, and therefore tension, as a response to the contraction of a pole. To summarize, active mechanosensing is not necessary for the oscillation mechanism we propose but may slightly enhance the instabilities. 5 Discussion of the model assumptions 5.1 Volume conservation One of the assumptions of our model is that the total volume of the cell is conserved. Previous work on Dictyostelium indicates that cell volume is indeed conserved during cell division 5. To test this assumption in our experimental system, we monitored the volume of L929 cells during division and during shape oscillations. Within the experimental error of the volume measurement (about 10%, see Methods), we did not observe any significant volume variation (Supplementary Fig. 13). 5.2 Intracellular elasticity One central feature of our model is that we describe the cell interior as a poro-elastic material. This description was motivated by previous work, where the poro-elastic properties of the cytoplasm were shown to be essential to account for the mechanics of bleb formation 25, 28. To describe the elasticity of the intracellular material, we introduced a bulk modulus K. This bulk modulus is an effective parameter accounting for all the various sources of elastic resistance present within the cell. Many components could contribute to this cellular elasticity: Intracellular components, such as the spindle, intermediate filaments, chromosomes and intracellular organelles are thought to form a dense meshwork with elastic properties 20. (22) 10

11 The cortex itself has elastic properties, which primarily depend on actin cross-linkers and which have been shown to regulate the dynamics of cytokinesis 14. The contribution of the cortex Young s modulus E to the effective bulk modulus of the cell is of the order of Eh/R where h is the thickness of the cortex and R is the radius of the cell 25. Cell substrate adhesions could also contribute to the global elastic resistance of the cell to deformation (see Section 5.3). 5.3 Influence of cell-substrate and cell-cell adhesions Even though the cells we study round up for mitosis and can be approximated as two portions of spheres during cytokinesis, they conserve substrate adhesions throughout division via retraction fibres that connect the rounded cell to the substrate (Supplementary Movie 5). The retraction fibres are also conserved during oscillations (Supplementary Fig. 5). It is thus possible that these fibres exert resistive forces that would oppose cortex contractions. However, we observe that cell shapes are mostly round and that no large deformations can be observed at the points where retraction fibres are connected to the cell (Supplementary Movie 5), in agreement with previous studies of retraction fibres 29. This suggests that the forces exerted by adhesions are relatively small and that cell shape is dominated by cortical tension (which favours a rounded shape). To further investigate this assumption, we checked whether enhancing substrate or cell-cell attachment would limit cellular oscillations. To that aim, we cultured anillin-depleted cells on dishes coated with collagen and fibronectin (to enhance cell-substrate attachment) or at a high level of confluence (to increase cell-cell adhesion). We found that neither of these treatments significantly changed the number of oscillating cells (see Supplementary Table 1). These observations indicate that adhesions do not significantly affect cortex contractions and cellular mechanics in our system. 5.4 Pressure propagation in the poro-elastic cytoplasm: local versus global pressure We modelled the cell interior as a poro-elastic material. An important property of poroelastic materials is that they can considerably slow down pressure equilibration. Studies in melanoma M2 cells provide evidence for slow pressure equilibration (compared to the timescales of cellular deformations) because of cytoplasmic poro-elasticity 28 : using the extensively blebbing filamin-deficient M2 cells, the authors show that bleb formation can be decreased or enhanced locally, and propose a model where a poro-elastic cytoplasm with a small enough mesh-size can maintain pressure gradients through the cell on timescales > 10 seconds. However, studies in other systems indicate that intracellular pressure equilibrates more rapidly. For example, a recent work on Entamoeba histolytica suggests fast (faster than 1 s) pressure equilibration in these cells: the authors instantly stop blebbing at one end of a cell by reducing intracellular pressure by electroporating the membrane at the other end 30. In a previous study on the L929 fibroblasts we use here, we have provided evidence for pressure equilibration within seconds: we investigated the size reached by two blebs successively induced by laser ablation of the cortex and could show that the pressure release due to the growth of the 1 st bleb propagated through the entire cell within seconds 25. More generally, the timescale of pressure equilibration within a cell with a poro-elastic interior is expected to strongly depend on the effective mesh-size of the cytoplasmic material 20, 31. In our description, we assume that hydrostatic pressure 11

12 propagates rapidly. This assumption is supported by our experimental observations that inducing a bleb at one pole of an oscillating cell has an effect on the other pole and leads to reversal of the oscillation direction within 15 s (Fig. 4b). 5.5 Influence of a finite size of the cleavage furrow In our analysis, the wih of the cleavage furrow was considered small compared to R 0 and therefore the possible role of flows going from the equator towards the poles was neglected. However, many observations indicate that the furrow is often a broad cylinder rather than an actual actin ring 5, 32. We checked the influence of a non-negligible wih of the furrow on our model. Here we show that this simplification is justified in the framework of the theory we propose. We assume, as before, that the dividing cell can be described as a cylinder of wih a and radius R c connecting two portions of spheres of volume V 1 and V 2, but we now consider that a is not negligible and that there is a dissipation associated with the exchange of fluid between the central equatorial part of the cell and the two poles. In that case, the mechanical balance of the cell can be written: α dv 1 = [P 1 P c ] (23) α dv 2 = [P 2 P c ] (24) where P 1, P 2 are as previously the pressures in the two poles and P c is the pressure in the equatorial part of the cell. Moreover, we assume again conservation of the total volume of the cell V 1 + V 2 + V c =2V 0 where V c = πar 2 c is the volume of the central part of the cell. Eqs. (23) and (24) can then be rewritten: α d(v 1 V 2 ) dr c 2αaπr c = [P 1 P 2 ] (25) = α d(v 1 + V 2 ) = [2P c (P 1 + P 2 )] (26) Eq. (26) is analogous to the equation derived in Ref. 14 for the case of a symmetric division where P 1 = P 2 = 2T R, and shows that the pressure exerted by the poles, P 1 + P 2, generates a force opposing the equatorial pressure P c and resisting the ingression of the furrow. Eq. (25) is similar to the equation for the difference of volume derived before (Eq. (13)) and therefore our model is essentially unchanged when we consider significant the cellular flows going from the equator towards the poles. 5.6 Effects of introducing a difference of tension between the poles of a dividing cell Our initial model (Section 2), which assumes that the tension at each pole remains constant and that the elasticity of the cell interior is purely linear, predicted that the introduction of a sufficient tension difference between the poles precluded the existence of a symmetric cell shape. This preliminary analysis motivated us to experimentally test the effect of introducing a tension difference between the poles in stably dividing cells. These experiments led to cell shape oscillations (Fig. 2 a-d). The oscillations could be understood in the context of our extended model, which considers non-linear elastic terms and where polar tension varies as cell shape changes (Section 3). In the extended model, introducing a difference of tension between the poles would trigger a relaxing oscillation with decreasing amplitude, the number of oscillation cycles increasing as parameters get closer to the 12

13 instability threshold. Experimentally testing that the observed oscillations (Fig. 2 a-d) indeed correspond to a relaxing oscillation is not straightforward and further studies will be necessary to fully understand oscillations induced by local tension release at one of the poles. Supplementary Discussion Influence of polar bleb size on cell stability An unexpected feature of our findings is that while small blebs are likely to stabilize cell shape by buffering inhomogeneities in cortical tension between poles, large blebs can have the opposite effect. In fact, as bleb formation releases tension in a size-dependent manner 25, the formation of a sufficiently large bleb at one pole can introduce a tension imbalance sufficient to destabilize the cell (Supplementary Fig. 4c). This feature reconciles seemingly contradictory observations on the influence of blebbing on cytokinesis. On the one hand, we observe that the induction of a large bleb triggers shape oscillations (Fig. 2b, d), and excessively large blebs have been reported to induce cytokinetic shape oscillations in other systems 33. On the other hand, we show that bleb induction limits cortical contractions, which drive shape oscillations, and that inhibition of blebbing also destabilizes cell shape (Fig. 4b, c). Taken together, these observations suggest that blebs act as pressure valves stabilizing cell shape during cytokinesis but that their size must be tightly controlled so that the tension released by their formation remains under the threshold, above which a tension difference between the poles precludes symmetric division. Implications for asymmetric division Our findings, which demonstrate that the symmetric shape of the dividing cell is intrinsically unstable, may have important implications for asymmetric cell division. Many mechanisms have been reported to lead to asymmetric division. In most cases, asymmetry between the daughter cells directly results from asymmetric spindle positioning. In flattened cells, that divide by crawling away from each other, such as the Dictyostelium myosin II-null cell, daughter cells of unequal sizes can result from asymmetric crawling 18, 34. Strikingly, a recent study in C. elegans QR.a neuroblasts 35 shows that asymmetric divisions can also be the direct result of myosin-driven cortex contraction at one of the poles at the expense of the other. Other studies report that during asymmetric division in Drosophila neuroblasts myosin also accumulates at the cortex of the smaller daughter cell, suggesting that a polar contraction-driven mechanism for the generation of different-sized daughter cells could also be at play there 7, 36. Finally, another recent paper reports a more extreme example, where myosin-driven contraction locally deforms the cortex for polar body formation during mouse meiosis 37. Such asymmetric contractions are similar to the initial symmetry breaking step of the shape instabilities we observe. An important open question is how such an asymmetric shape, with more cortical myosin and thus a higher tension in the smaller pole, can be stable. Our findings suggest that bleb formation or elasticity increase could stabilize the asymmetric shape. It will be interesting to investigate the mechanics of such contractility-driven asymmetric divisions. Cortical oscillations in other systems Finally, the mechanism for cortex oscillations proposed here could be relevant for a number of other systems where contractility-driven oscillations are observed. Interphase cells 13

14 and cell fragments with depolymerized microtubules have been shown to display shape oscillations morphologically similar to the cytokinetic oscillations reported here 38. Oscillating contractions have been involved in tissue morphogenesis during gastrulation 39 and dorsal closure 40 in Drosophila as well as during convergent extension in Xenopus embryos 41. A common feature of all these oscillations is that they display cycles of cortex contraction, disassembly and reassembly, suggesting they might be driven by a common underlying mechanism coupling contractility, cortex turnover and cell elasticity. Our theoretical model may provide a general framework for the study of contractile oscillations. References [1] Pollard, T. D. Mechanics of cytokinesis in eukaryotes. Curr. Opin. Cell Biol. 22, (2010). [2] Mukhina, S., Wang, Y.-L. & Murata-Hori, M. Alpha-actinin is required for tightly regulated remodeling of the actin cortical network during cytokinesis. Dev. Cell 13, (2007). [3] Salbreux, G., Prost, J. & Joanny, J. F. Hydrodynamics of cellular cortical flows and the formation of contractile rings. Phys. Rev. Lett. 103, (2009). [4] Bray, D. & White, J. G. Cortical flow in animal cells. Science 239, (1988). [5] Robinson, D. N., Cavet, G., Warrick, H. M. & Spudich, J. A. Quantitation of the distribution and flux of myosin-ii during cytokinesis. BMC Cell Biol. 3, 4 (2002). [6] Werner, M., Munro, E. & Glotzer, M. Astral signals spatially bias cortical myosin recruitment to break symmetry and promote cytokinesis. Curr. Biol. 17, (2007). [7] Barros, C. S., Phelps, C. B. & Brand, A. H. Drosophila nonmuscle myosin II promotes the asymmetric segregation of cell fate determinants by cortical exclusion rather than active transport. Dev. Cell 5, (2003). [8] O Connell, C. B., Warner, A. K. & Wang, Y. L. Distinct roles of the equatorial and polar cortices in the cleavage of adherent cells. Curr. Biol. 11, (2001). [9] Rappaport, R. Absence of furrowing activity following regional cortical tension reduction in sand dollar blastomere and fertilized egg fragment surfaces. Dev. Growth Differ. 41, (1999). [10] Rappaport, R. Cytokinesis in Animal Cells (Cambridge University Press, 1996). [11] Robinson, D. N. & Spudich, J. A. Mechanics and regulation of cytokinesis. Curr. Opin. Cell Biol. 16, (2004). [12] Wang, Y.-L. The mechanism of cortical ingression during early cytokinesis: thinking beyond the contractile ring hypothesis. Trends Cell Biol. 15, (2005). [13] Yoneda, M. & Dan, K. Tension at the surface of the dividing sea-urchin egg. J. Exp. Biol. 57, (1972). [14] Zhang, W. & Robinson, D. N. Balance of actively generated contractile and resistive forces controls cytokinesis dynamics. Proc. Natl. Acad. Sci. U. S. A. 102, (2005). 14

15 [15] Girard, K. D., Chaney, C., Delannoy, M., Kuo, S. C. & Robinson, D. N. Dynacortin contributes to cortical viscoelasticity and helps define the shape changes of cytokinesis. EMBO J. 23, (2004). [16] Octtaviani, E., Effler, J. C. & Robinson, D. N. Enlazin, a natural fusion of two classes of canonical cytoskeletal proteins, contributes to cytokinesis dynamics. Mol. Biol. Cell 17, (2006). [17] Reichl, E. M. et al. Interactions between myosin and actin crosslinkers control cytokinesis contractility dynamics and mechanics. Curr. Biol. 18, (2008). [18] Zhou, Q. et al coordinates microtubules, Rac, and myosin II to control cell mechanics and cytokinesis. Curr. Biol. 20, (2010). [19] Surcel, A., Kee, Y.-S., Luo, T. & Robinson, D. N. Cytokinesis through biochemicalmechanical feedback loops. Semin. Cell Dev. Biol. 21, (2010). [20] Mitchison, T. J., Charras, G. T. & Mahadevan, L. Implications of a poroelastic cytoplasm for the dynamics of animal cell shape. Semin. Cell Dev. Biol. 19, (2008). [21] Guckenheimer, J. & Holmes, P. Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, vol. 42 of Applied Mathematical Sciences (Springer, 1983). [22] Murthy, K. & Wadsworth, P. Myosin-II-dependent localization and dynamics of F- actin during cytokinesis. Curr. Biol. 15, (2005). [23] Mayer, M., Depken, M., Bois, J. S., Jülicher, F. & Grill, S. W. Anisotropies in cortical tension reveal the physical basis of polarizing cortical flows. Nature 467, (2010). [24] Effler, J. C. et al. Mitosis-specific mechanosensing and contractile-protein redistribution control cell shape. Curr. Biol. 16, (2006). [25] Tinevez, J.-Y. et al. Role of cortical tension in bleb growth. Proc. Natl. Acad. Sci. U. S. A. 106, (2009). [26] Kunda, P., Pelling, A. E., Liu, T. & Baum, B. Moesin controls cortical rigidity, cell rounding, and spindle morphogenesis during mitosis. Curr. Biol. 18, (2008). [27] Charras, G. T., Mitchison, T. J. & Mahadevan, L. Animal cell hydraulics. J. Cell Sci. 122, (2009). [28] Charras, G. T., Yarrow, J. C., Horton, M. A., Mahadevan, L. & Mitchison, T. J. Non-equilibration of hydrostatic pressure in blebbing cells. Nature 435, (2005). [29] Théry, M., Jiménez-Dalmaroni, A., Racine, V., Bornens, M. & Jülicher, F. Experimental and theoretical study of mitotic spindle orientation. Nature 447, (2007). [30] Maugis, B. et al. Dynamic instability of the intracellular pressure drives bleb-based motility. J. Cell Sci. 123, (2010). [31] Charras, G. & Paluch, E. Blebs lead the way: how to migrate without lamellipodia. Nat. Rev. Mol. Cell Biol. 9, (2008). 15

16 [32] Fishkind, D. J. & Wang, Y. L. Orientation and three-dimensional organization of actin filaments in dividing cultured cells. J. Cell Biol. 123, (1993). [33] Rankin, K. E. & Wordeman, L. Long astral microtubules uncouple mitotic spindles from the cytokinetic furrow. J. Cell Biol. 190, (2010). [34] Weber, I. et al. Two-step positioning of a cleavage furrow by cortexillin and myosin II. Curr. Biol. 10, (2000). [35] Ou, G., Stuurman, N., D Ambrosio, M. & Vale, R. D. Polarized myosin produces unequal-size daughters during asymmetric cell division. Science 330, (2010). [36] Cabernard, C., Prehoda, K. E. & Doe, C. Q. A spindle-independent cleavage furrow positioning pathway. Nature 467, (2010). [37] Larson, S. M. et al. Cortical mechanics and meiosis II completion in mammalian oocytes are mediated by myosin-ii and Ezrin-Radixin-Moesin (ERM) proteins. Mol. Biol. Cell 21, (2010). [38] Paluch, E., Piel, M., Prost, J., Bornens, M. & Sykes, C. Cortical actomyosin breakage triggers shape oscillations in cells and cell fragments. Biophys. J. 89, (2005). [39] Martin, A. C., Kaschube, M. & Wieschaus, E. F. Pulsed contractions of an actinmyosin network drive apical constriction. Nature 457, (2009). [40] Blanchard, G. B., Murugesu, S., Adams, R. J., Martinez-Arias, A. & Gorfinkiel, N. Cytoskeletal dynamics and supracellular organisation of cell shape fluctuations during dorsal closure. Development 137, (2010). [41] Kim, H. Y. & Davidson, L. A. Punctuated actin contractions during convergent extension and their permissive regulation by the non-canonical Wnt-signaling pathway. J. Cell Sci. 124, (2011). 16

17 Supplementary Figures and Legends #!!!! ;!! $%&'()*+,-./)0%&,12,32,.*34,31*45. 6: !6:!69!68!67!!#$%&'( '&$%) ;!! <%1*4*%3.()%3=.2,)).>,&*>?,&5 $%&'()*+,-./)0%&,12,32,.*34,31*45. 6: !6:!69!68!67!!#$%&'( *+,%) ;!! <%1*4*%3.()%3=.2,)).>,&*>?,&5 Supplementary Figure 1. An acto-myosin cortical layer is present at the cell poles during cytokinesis. Mean fluorescence intensities of F-actin (Lifeact-GFP) and myosin (MRLC-tdRFP) in the cortex (solid lines) as a function of the position along the cell contour (as indicated on the drawing). The red square marks the start and end position along the cell contour. The length of the contour was subdivided into 1000 points so that profiles from different cells with different contour lengths could be compared. The intensity values were normalized to the global mean intensity of the cortical protein of interest (see Methods). The dotted line marks the mean fluorescence intensity of the cytoplasm (normalized to the global mean intensity in the cortical region). The data points are means ± SD over n=10 L929 cells. 17

18 ! : 7*%!89!+*-*// #,!8 $&+!8 %)+!8 -./01234!567#8 9( : #% 9#% 10 µm 9#, $ % &, (< ;27.!58: 7*%!89!+*-*// ($ )!!#& & #$%&'()*+(+,-#&)%.&/-*// )!!#& ($ & #$%&'()*+(+, %.*!%*6-*// )!!#& ($ &!!#%!!#%!!#% (+ ' (+ ' (+ '!!#$!!#$!!#$ (, (, (, $( '' $( '' $( '' $% ' $% ' $% ' $* $* $* Supplementary Figure 2. Cytoplasmic flows during metaphase and cytokinesis. a, Stills from a time-lapse of a metaphase L929 cell with corresponding PIV flowfields. The scale of the arrows is the same as on Fig. 1b. b, Mean velocity of the PIV flowfield projected on the long cell axis as a function of time in a metaphase L929 cell (black circles). The solid line shows data smoothed with a moving average window of 2 points. c, Distribution of the mean velocity vectors of the PIV flowfields during time-lapses in a representative metaphase cell, control cytokinetic cell and cytokinetic cell treated with 10 µm Y27632 (L929 cells). We considered that a cell displayed small oscillations if a significant number of the PIV vectors were of amplitude at least two-fold higher than those for metaphase cells. Each vector is the mean of the flowfield at a single time-point. The axis corresponds to the pole-pole axis of the cell. The vectors are colour-coded as indicated in Methods. 18

19 '()*+,- $%&!# doi: /nature10286./012 Supplementary Figure 3. Time-lapse of shape oscillations in a dividing HeLa cell stably expressing EB3-GFP and Lifeact-mCherry and treated with 5 µm nocodazole. Following nocodazole treatment the spindle was depolymerized (see EB3 channel). Eventually the cell underwent high-amplitude shape oscillations and failed cytokinesis (9 out of 36 L929 cells and 7 out of 16 HeLa cells). White stars indicate the positions of the two sets of chromosomes. 0 s: addition of nocodazole. Scale bar: 10 µm. 19

20 !!!! & < 1 < 2!! 1 2 & & # $!$ ' ' ' # $ % & % ' ()*+,-./$0#12. # %343.)#56+$ -7.,+* #-9*):-.)#56+$- 73,*)*5-; #!-; > I I I & ' & E & ' ' & =156#9). 7? ' & H6#9). > >B' >B@ >BC >BD & &B'! $! > >BA > > H6#9). H6#9). 9+G3)#4 53)*6+31 F3-9+G3)#4 53)*6+31 >BA & &BA ' 7!-; > Supplementary Figure 4. Constant polar tension model. a, Left: geometrical parameters used in the model. V 1 and V 2 are the volumes of the two poles. Right: physical parameters used in the model. T 1 and T 2 are the cortical tensions at the two poles. T c is the tension at the cleavage furrow, a is the wih of the furrow and R c is the radius of the cell at the furrow. P 1 and P 2 are the pressures in the cytoplasm at the two poles. b, Linear stability diagram of a symmetrically dividing cell in the constant polar tension model. R 0 is the radius of each of the daughter cells. For a sufficiently large tension compared to cellular elasticity, the symmetric state is unstable and one pole contracts while the other expands. c, Phase diagram of a cleaving cell in the presence of a tension difference between the two poles for R c =0.6R 0. T m = T 1+T 2 2 is the mean cortical tension. For a sufficiently high tension difference between the poles a subcritical bifurcation occurs, precluding the existence of a stable bipolar solution. 20

21 doi: /nature10286 Control dividing cell Oscillating cell Supplementary Figure 5. Cells maintain substrate attachments during division and during shape oscillations. Maximum intensity projection time-lapses of confocal z-stacks of dividing (left) and oscillating (right) HeLa cells stably expressing GFP-actin. The cells, outlined in red dashed lines, remained attached to the substrate during both division and oscillations via retraction fibres (arrowheads), which are visible throughout the entire processes. Scale bars: 10 µm. 21

22 . / $-7#%&)0!,! 2A341!,! 2A?20!,! G-8#!;,= &'%#(')-- 0 3A000 *$+!%$))$%-- -CE IJ$K, 5#6-%)-'(!7*'8!7+%)!5-,)*-9:)-'(!;%<:<= 0 &'()*'+ $>3>!,-./(-++-( &'()*'+ #$%! 410 0<2?@ 0< !#$%!!,-./(-++-( 20!L8 2A000 0 G-8#!;,= A B',-)-'(!%+'(C! &#++!D#*-DE#*F &'%#(') *$+!%$))$% B',-)-'(!%+'(C! &#++!D#*-DE#*F?A000 3A000?A B',-)-'(!%+'(C! &#++!D#*-DE#*F 100 B',-)-'(!%+'(C! &#++!D#*-DE#*F $'H -CE $'H Supplementary Figure 6. Anillin depletion leads to a flattening of the distribution of cortical myosin and F-actin. a, Montage from a time-lapse of oscillations after anillin depletion in an L929 cell expressing Lifeact-GFP (F-actin) and MRLC-tdRFP (myosin). b, Cortical intensity profiles of myosin in a representative control (left) and anillin-depleted (right) cytokinetic L929 cell over time. c, Integrals of intensity profiles, as a measure of deviations from uniform cortical distributions for F-actin in control cells and anillin-depleted cells shortly before the onset of oscillations. Boxes extend from the 25th to the 75th percentile, with a line at the median. N=10 for each sample. P-values were determined using the Mann-Whitney U test. d, Cortical intensity profiles of F-actin in a representative control (left) and anillin-depleted (right) cytokinetic L929 cell over time. b, d, The normalized mean fluorescence intensity of actin or myosin in the region of the cortex was plotted as a function of the position along the cell contour (horizontal axis) and of time (vertical axis). The cell contour was subdivided into 1000 points, 0 being at the equator (see Supplementary Fig. 1 and Methods for details). Pseudo-colours: normalized mean fluorescence intensity of myosin or F-actin, horizontal axis: position along the cell contour, vertical axis: time. 22

23 +!-/ '-/ (!#!-/ (%!-/ (#'-/ ($E!-/ (!-P+! )*+,-./0 &' (&' -)&$.)/ O89*= A;2/*= (>!!! (>!! '!! (>!!! *6,7-8239,:-92-8;92<54/+- +,4=-*=9,=/*9;-!&' ABCD-!!!! #!! $!! %!! )*+,-./0! '!!! '!! L2/*9*2=-452=M- 8,55-<,3*<N,3; L2/*9*2=-452=M- 8,55-<,3*<N,3; #!/+$0, )*+,-./0! O89*= A;2/*= Q*MN (>!!! >!!! G>!!!, G'! G!! '!!! ('! (!! '!!! '!!! '!! L2/*9*2=-452=M- L2/*9*2=-452=M- 8,55-<,3*<N,3; 8,55-<,3*<N,3; 82=9325 #$%&--!&!!$ 4F549,7 82=9325 '()*%&-- G:(! K# 4F549,7 C2R Supplementary Figure

24 Supplementary Figure 7. Cortex ablation in cytokinetic L929 cells triggers cycles of cortex disassembly - reassembly and leads to reinforcement of the polar cortex and shape oscillations. a, Time-lapse of shape oscillations in a cleaving L929 cell where the equatorial cortex was locally ablated, 0s: onset of ablation. Red circle: ablation region, the arrowhead points to the bleb induced by cortex ablation. b, Mean fluorescent intensities of cortical actin (Lifeact-GFP, red) and myosin (MRLCtdRFP, blue) as a function of time in a laser-ablated L929 cell. Fluorescence intensities in the cortical region were normalized to the mean fluorescence intensity detected in the cytoplasm (dashed line). Laser ablation triggered a strong decrease in cortical Lifeact and MRLC intensities, suggesting partial depolymerization of the cortex, followed by several cycles of cortex reassembly and disassembly until the initiation of shape oscillations. c, Cortical intensity profiles of actin (left) and myosin (right) in a representative control and laser-ablated cytokinetic L929 cell. The normalized mean fluorescence intensities of actin or myosin in the region of the cortex was plotted as a function of the position along the cell contour (horizontal axis) and of time (vertical axis). The cell contour was subdivided into 1000 points, 0 being at the equator (see Supplementary Fig. 1 and Methods for details). Pseudo-colours: normalized mean fluorescence intensity, horizontal axis: position along the cell contour, vertical axis: time. In the control cell, the levels of actin and myosin remained mostly unchanged at the equator during cleavage. In the laser-ablated cell, after several cycles of cortex disassembly - reassembly ( s), the fluorescence intensities of cortical proteins were less peaked at the equator and enhanced at the cell poles. During cell shape oscillations ( s), the cortex was repetitively redistributed between the two poles. d, Integrals of cortex fluorescence intensity deviations from a uniform distribution for actin and myosin in control cells and oscillating cells shortly before the onset of oscillations for L929 cells (see Fig. 2e-f and Methods for details). Boxes extend from the 25 th to the 75 th percentile, with a line at the median. N=10 for each sample. P-values were determined using the Mann-Whitney U test. 24

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upplementary Figure

26 Supplementary Figure 8. Actin and myosin dynamics during ablation-induced cytokinetic shape oscillations. a, Montage from a time-lapse of ablation-induced oscillations in a L929 cell expressing Lifeact-GFP (F-actin) and MRCL-tdRFP (myosin). b, Areas of the two parts of the oscillating cell displayed in a and mean intensities of actin and myosin in the cortex left and right of the cleavage furrow (see schematic picture in a) as a function of time. The area of the observed cell section and the mean intensities of fluorescence of cortical proteins in a thin layer under the membrane were automatically monitored during the oscillations (see Methods for details). c, Zoom on the regions delimited by dotted lines on panel b. Area, cortical actin and cortical myosin intensities as a function of time in the left and right parts of the cell b and c, Circles: datapoints, solid lines: data points smoothed with a moving average window of 2 points. Intensities are normalized to their average value during the oscillations. d, Simulation of the evolution of the difference of volumes v of the two poles (top) and of the polar cortex densities (bottom) following a small initial perturbation of the volume difference. The simulation was performed using parameters extracted from experimental measurements displayed on panel b (τ c = 23 s, τ = 79 s, T 0 /R 0 K =1.2, K 3 /K =1.8) The period of the simulated oscillation is 367 s, in good agreement with the period of the corresponding laser-ablated cell (T = 340 ± 36 s). e, Details of the oscillating curves shown in d. The waveform of the oscillation is very similar to the experimental curves on panel c, with peaks of cortex intensity with a faster decrease than increase, and the minimum volume of one pole being reached after the cortex intensity of the same pole reaches its maximum ( t). 26

27 # H! 3! I D 0./ '!,./ $! -./!! 3 2!!!'!%!#!& $! $' 3!! >:5/16,C60*6/10:D?(*A.E*,>:5/16,C60*6/10: % >:5/16,160*6/10:,(/,?2016,160*6/10:,<?61++16;9*4+*0*9= 1./ G 0./ ',./ $ -./!! -./ $,./ ' 0./ G?2016,C60*6/10:D?(*A.E*,?2016,C60*6/10: >:5/16,C60*6/10:D?(*A.E*,>:5/16,C60*6/10: 0./ ',./ $ -./ >:5/16,160*6/10:,(/,?2016,160*6/10: <@./*A;.B+.0*9=!! -./ $,./ '?2016,C60*6/10:D?(*A.E*,?2016,C60*6/10:,./ ;8<(= $ -./! +-./ +, +,./ +, +-./! -./ $ ( )*++,-+./012,3*/456/*!!G -( ,58,.2016,9*6/10: $DF,9<F,2.D2.=D90!!'!!$! +-.-, !# $ $% $& ' Supplementary Figure 9. Data fitting with the theoretical model. a, Phase diagram of the linear stability analysis of a symmetric cell with respect to volume perturbation in the parameter space (R c /R 0, T 0 /R 0 K), the third parameter is set to τ c /τ = 0.3. The yellow region corresponds to a stable symmetric state, the white region to an oscillatory unstable state and the grey region to a non-oscillatory unstable state. For intermediate values of the stability parameter T 0 /R 0 K (central region of the diagram), an oscillatory unstable region appears. Extracting the critical parameters involved in this description from experimental data allows the positioning of cells on the phase diagram (magenta points: anillin depleted cells, green points: laser-ablated cells, error bars: standard deviation of the fit parameters). b, Mean cortical myosin vs. actin fluorescence intensity during the oscillation of the representative anillin-depleted (left) and laser-ablated (right) L929 cells shown in Fig. 3 and Supplementary Fig. 8 respectively. Myosin and actin densities follow a clear proportionality relationship. c, Left: Elastic response of the cell represented in Supplementary Fig. 8, as extracted from the fit of Eq. (13). Right: Time derivative of the cortical actin density at one pole divided by the surface of the pole as a function of the actin density (right), for the representative laser-ablated cell shown in Supplementary Fig. 8b-c. The slope of the fitted linear curve yields the turnover of the cortical actin layer τ. c a is the mean cortical actin density. 27

28 doi: /nature10286!!#$ %&'()*+,-./ $001,2$3(445!#$ %&'()*+,-./ $001,2$3(445 Supplementary Figure 10. Polar bleb initiation. Time-lapses of nucleation events of blebs in an L929 cell transiently expressing Lifeact-GFP (center columns) and CAAXmCherry (membrane marker, right columns) and oscillating after anillin depletion. (Left columns: DIC). a, The arrowheads point to a local weakening of the cortex region preceding bleb formation, indicating that a rupture in the cortex may be responsible for nucleating the bleb. b, The arrows point to an intact cortex following bleb nucleation. The cortex is only disassembled during bleb retraction (arrowheads). This indicates that the bleb was likely nucleated by delamination of the membrane from the cortex, although it cannot be excluded that a rupture in the cortex occured above or below the recorded confocal slice. Scale bars: 1 µm. 28

29 doi: /nature10286! !#$%&' ()* ,-./ ,-2- ()*,-2- +,-./ Supplementary Figure 11. Calcium is not involved in polar bleb formation and shape oscillations. Time-lapses of shape oscillations in anillin-depleted L929 cells without either intracellular or extracellular Ca2+. a, Intracellular Ca2+ was chelated with 40µM BAPTA-AM. Top row: DIC, bottom row: Lifeact-GFP. Representative of 16 oscillating cells out of 30. b, DIC images of a cell oscillating in Ca2+ -free medium with additional extracellular Ca2+ -chelation with 500 µm EGTA, representative of 18 oscillating cells out of 20. Scale bars: 10 µm. 29

30 ! # -*.$/0#1 : 3:: 5:: 8:: ;:: 2<::: #*+,*!!*,!#$%&'!($) -*.$/0#1 : 2:: 3:: 4:: 5:: 2#( 3,) 4%) 5(6 7(6 8(6 9(6 -*.$/0#1 : 2:: 3:: 4:: 5:: 7:: 2#( 3,) 4%) 5(6 Supplementary Figure 12. Variability of the oscillation periods. a, Mean periods of oscillations in anillin-depleted (n=5) and laser-ablated (n=5) L929 cells. b, Graph showing the periods of consecutive oscillation cycles in a representative anillin-depleted L929 cell. c, Graph showing the periods of consecutive oscillation cycles in a representative laser-ablated L929 cell. b, c, Each period is the mean of four measured times: peak-topeak and trough-to-trough times for both the left and right sides of the cell. Data points show the mean periods and the error bars the standard errors in the mean. 30

31 Supplementary Figure 13. Volume conservation during cell division and shape oscillations. Graph of the evolution of cell volume during a control division (red) and during oscillations following anillin depletion (blue) in L929 cells. Volumes were normalized to the first time-points. The insets show 3D reconstructions of example stacks by surface rendering in Imaris. The reconstructed cells were used for volume estimation. The rendered surfaces are shown in white and the underlying fluorescent signals from the intracellular and inverted extracellular dyes (see Methods) are shown in red and blue for the control cell and the oscillating cell respectively. Cell volumes did not change significantly during division or oscillations (representative of 3 oscillating and 2 control dividing cells). 31