SUPPLEMENTARY NOTES Pivoting of microtubules around the spindle pole accelerates kinetochore capture

Size: px

Start display at page:

Download "SUPPLEMENTARY NOTES Pivoting of microtubules around the spindle pole accelerates kinetochore capture"

Kerrie Mosley
5 years ago
Views:

1 SUPPLEMENTARY NOTES Pivoting of microtubules around the spindle pole accelerates kinetochore capture Iana Kalinina 1, Amitabha Nandi 2, Petrina Delivani 1, Mariola R. Chacón 1, Anna Klemm 1, Damien Ramunno-Johnson 1, Alexander Krull 1, Benjamin Lindner 2, Nenad Pavin 1,2,3, and Iva M. Tolić-Nørrelykke 1 1 Max Planck Institute of Molecular Cell Biology and Genetics, Dresden 01307, Germany 2 Max Planck Institute for the Physics of Complex Systems, Dresden 01187, Germany 3 Department of Physics, Faculty of Science, University of Zagreb, Zagreb 10002, Croatia Supplementary Note 1: The movement of polar s is most likely driven by thermal fluctuations Thermally driven motion of an object depends on its size and shape, as well as on the viscosity of the medium. A polar can be described as a thin stiff rod with one end freely jointed to a fixed point, while the other end is free to move, allowing the rod to perform angular movement (scheme in Supplementary Fig. S3b). The angular diffusion coefficient D scales approximately with ln(l/d)/l 3, where L and d denote the rod length and diameter, respectively 1-3. Therefore, to test if motion is thermally driven, we determined the relationship between the length and the angular diffusion coefficient, which was calculated from the mean squared angular displacement of the s. Here we used cells expressing Mal3-GFP (Refs. 4, 5 ), a homolog of EB1, which decorated the plus end of the, and Sid4-GFP to label the SPBs (Supplementary Fig. S3a; Supplementary Movie S3). To obtain precise measurements of length and orientation, we developed software similar to that for tracking KCs (see Methods). The spindle and polar s were assumed to be straight lines. The intensity of the image stack was assumed to be a mixture of multiple three-dimensional Gaussian distributions, which represent Mal3-GFP spots along the spindle and s, and a uniform distribution representing the background. In a second step, the positions of the spindle and s in three dimensions were estimated by fitting lines to the Gaussian centers. Finally, two-dimensional projections of the positions were used for further analysis. The diffusion coefficient showed a strong dependence on the length of the, consistent with the theoretical prediction for diffusion of a thin rod (Supplementary Fig. S3b). The corresponding fit with a single free parameter being the effective viscosity for the nucleoplasm suggests that the nucleoplasm is 2600 times more viscous than water (Supplementary Fig. S3b). A similar value of times the viscosity of water was previously measured for the cytoplasmic viscosity of these cells, by following the movement of lipid granules 6. For comparison, the viscosity of vertebrate cells in the vicinity of chromosomes, measured on large particles, was on average 282 times larger than the viscosity of water 7. Thus, the viscosity of vertebrate cells is roughly 10 times smaller than the viscosity of the S. pombe nucleoplasm.

2 For small particles, however, the effective viscosity is smaller than for large particles. The diffusion coefficient of GFP in the cytoplasm of mammalian cells was measured to be 3.2 times smaller than in water (the corresponding value in water was 87 µm 2 /s) 8, and the diffusion coefficient of kda dextrans in the nucleus of mammalian cells was 6-20 times smaller than in water 9. We measured the diffusion coefficient of GFP in the cytoplasm and in the nucleus of S. pombe to be 10 and 16 times smaller than in water, respectively (Supplementary Fig. S5d). In sum, S. pombe cells are 3-10 times more viscous than higher eukaryotic cells, for both large and small particles. Thus, the dependence of effective viscosity on the particle size may be similar in S. pombe and in higher eukaryotic cells. Supplementary Note 2: Theory The model for KC capture based on random angular movement of the and random movement of the KC. In our three-dimensional description, which is depicted in Fig. 3b, s are thin stiff rods of fixed length L performing angular diffusion around the SPB. The SPB is on the nuclear envelope, which is described as a sphere of radius R. Our model includes n s, but we first describe the behavior of a single. In a spherical coordinate system with the origin at the SPB, the θ, φ obeys the stochastic differential equations: orientation ( ) dθ cosθ = D + 2D ξ (1) θ sinθ dφ 2D = ξφ (2) sinθ Here, D denotes the angular diffusion coefficient of the and t is the time. The r, θ, φ diffuses, where its radial distance from the origin, KC with coordinates ( ) r KC, is governed by KC KC KC dr KC = 2 D KC + 2D KC " r. (3) r KC The angular coordinates of the KC, θ KC and φ KC, are determined by similar dynamics 2 as for the, when in Eqs. (1) and (2) D is replaced by D / r and by KC. Here ξ i is a Gaussian white noise obeying ξi() t ξ j() t = δi, jδ( t t), where i,j = θ, φ, KCθ, KCφ, r (for derivation of Eqs. (1)-(3) see below). The movement of the and KC is constrained by the nuclear envelope. The constraint for the is described by the reflecting boundary condition 0 θ θ max and the periodic boundary condition 0 φ 2π, where θ max = acos(l/2r). For the parameters given in Fig. 3c, θ max = π/3. The boundary conditions for the KC are obtained by replacing the subscript by KC and L by r KC. Note that φ and α from Fig. 1e describe the same orientation of the in the x-y plane for the case in which the SPB is at the lower pole of the nuclear sphere. Finally, we define KC capture as the first encounter of the with the KC, which in our model occurs when rkc < L and a a θ θkc < and φ φkc <. Here, a corresponds to the size of the 2rKC 2rKC sinθkc KC. Because of our definition of KC capture, a is approximately the arc length of the edge of the latitude-longitude square that represents the KC, and this approximation is KC KC

3 good for. The model outlined above describes the case with a single. However, we generalized the model to n s that move independently by replacing with k, where k = 1,,n. We numerically solved Eqs. (1)-(3) starting with random initial orientations of both and KC, and with a fixed distance of the KC from the origin. Brownian motion of a particle in spherical coordinates described by Langevin equations. The dynamics of a Brownian particle in 3D in the over-damped limit is given by the following Langevin equation " x% "( x % d $ y $ = 2D $ ( y $, (4) # z& #( z & where " x," y," z are independent white Gaussian noise sources with " m (t)" n (t) = # m,n #(t $ t), m,n = x, y,z. In spherical coordinates, defined by the velocity components are given as, x = rsin" cos#, y = rsin" sin#, z = rcos", (5) x = r sin" cos# + r " cos" cos# $ r# sin" sin#, y = r sin" sin# + r" cos" sin# + r# sin" cos#, z = r cos" $ r" sin", (6) With respect to the time derivatives, this is a linear system of equations that can be inverted, yielding in matrix form the equation where $ r $ d & ) " & ) = B & & %#( % B = * x * y * z ) ), (7) ( % ( sin" cos# sin" sin# cos" * 1 2D r cos" cos# 1 r cos" sin# $ 1 r sin" * * $ 1 * sin# 1 cos# 0 * & r sin" r sin" ) (8) Although we started with Eq. (4) that contained only additive noise, by the transformation, we obtained Eqs. (7) with a multiplicative noise, i.e. the prefactors of the noise terms depend on the state variables. Because we used the standard calculus for the coordinate transformation, we have to interpret Eqs. (7) in the sense of Stratonovich 10. We can, however, simplify the equations as follows. First, we convert the equations to an Ito form (see Ref. 10, pp , for a discussion)

4 $ r $ * x d & ) " & ) = A + B & ) * y & ), (9) %#( %* z ( where the spurious drift term is given as A i = 1 " B kj # k B ij. (10) 2 j,k From this definition, we obtain by means of Eq. (8) A r = 2D r, A " = D r 2 cos" sin", A # = 0 (11) Second, we rewrite the noise terms in Eqs. (9) into new and simpler ones. Because the three noise sources " x," y," z are independent, in each of the equations their weighted sum can be lumped into one single noise " r," #, or " $, respectively. In each equation, the three noise intensities (prefactors under the square roots) add up to the intensity of the respective new noise source. The noise in the equation for r, for instance, can be written as follows 2Dcos 2 " cos 2 #$ x + 2Dcos 2 " sin 2 #$ y + 2Dsin 2 "$ z = 2Dcos 2 " cos 2 # + 2Dcos 2 " sin 2 # + 2Dsin 2 "$ r = 2D$ r (12) In a similar way the noise terms appearing in the equations of " and " can be simplified using trigonometric relations. Furthermore, one can show that the three new noise terms " r," #, " $ are uncorrelated (due to the fact that the scalar product of two distinct rows of the matrix B vanishes). Putting everything together, we arrive at the Eqs. (1)-(3). Supplementary Note 3: Estimation of the systematic error resulting from twodimensional measurements of length The measurement of length in the maximum-intensity projections introduces a systematic error, resulting in underestimated length. If L denotes the real length in three dimensions, then the measured length in the projection is, where is the inclination angle of the, such that is the orientation of the with respect to the imaging plane. If one assumes that s extend from the SPB isotropically, the measured length averaged over all orientations in three dimensions is, implying that the real length is 27% larger than the average measured length. Here, averaging over the azimuth angle is omitted because the integrated function does not depend on. The average measured length was even closer to the real length because the measured length shorter than 0.75 µm, corresponding to 4.3 pixels, was not observed (Supplementary Fig. S2g), because the signal of such short s overlapped with the signal of the SPB. Thus, s extending at an angle and were not included in the analysis, where. For

5 the minimal angle is rad (or 25 ). Consequently, the real length was 20% larger than the average measured length. Furthermore, we calculate the angular diffusion coefficient of the from the measured angle in the maximum-intensity projections, which corresponds to in the model, as. This calculation introduces a systematic error, resulting in overestimated angular diffusion coefficient because the real angular diffusion coefficient,, should be calculated as, where. To estimate the relation between the averaged over all orientations in three dimensions,, and, we assume that s extend from the SPB isotropically. We also take into account that in our calculation of, we used only s with the projected length larger than 1 µm, thus only s extending at an angle were included in this calculation, where rad. Consequently,, implying that the real is 30% smaller than the measured. Here, averaging over is omitted because the integrated function does not depend on. The real corrections are most likely smaller than those calculated above for the following reason. Assuming that the SPB has equal probability to be at any point of the sphere of the nucleus, the SPB has higher probability to be found close to the equatorial plane of the nucleus than close to the poles. Because length is similar to the radius of the nucleus and because they are constrained by the nuclear envelope, s do not extend from the SPB isotropically, but in a conical region centered at the equatorial plane, for SPBs in this plane. Therefore, the correction of 20% on length and 30% on are most likely upper limits and the real corrections are not known. Nevertheless, we tested whether including the upper-limit corrections affects the behavior of the model. We calculated the fraction of lost KCs with L = 1.8 µm and D = rad 2 /s (other parameters as in Fig. 3c), and found that a simultaneous change in these two parameters results in a negligible difference in the fraction of lost KCs (the fraction of lost KCs at 4 minutes showed less than 3% difference). Supplementary References 1. Broersma, S. Rotational diffusion constant of a cylindrical particle. J Chem Phys 32, (1960). 2. Hunt, A.J., Gittes, F. & Howard, J. The force exerted by a single kinesin molecule against a viscous load. Biophys J 67, (1994). 3. Tirado, M.M. & de la Torre, J.G. Translational friction coefficients of rigid, symmetric top macromolecules. Application to circular cylinders. J Chem Phys 71, (1979). 4. Beinhauer, J.D., Hagan, I.M., Hegemann, J.H. & Fleig, U. Mal3, the fission yeast homologue of the human APC-interacting protein EB-1 is required for

6 microtubule integrity and the maintenance of cell form. J Cell Biol 139, (1997). 5. Busch, K.E. & Brunner, D. The microtubule plus end-tracking proteins mal3p and tip1p cooperate for cell-end targeting of interphase microtubules. Curr Biol 14, (2004). 6. Tolic-Norrelykke, I.M., Munteanu, E.L., Thon, G., Oddershede, L. & Berg- Sorensen, K. Anomalous diffusion in living yeast cells. Phys Rev Lett 93, (2004). 7. Alexander, S.P. & Rieder, C.L. Chromosome motion during attachment to the vertebrate spindle: initial saltatory-like behavior of chromosomes and quantitative analysis of force production by nascent kinetochore fibers. J Cell Biol 113, (1991). 8. Swaminathan, R., Hoang, C.P. & Verkman, A.S. Photobleaching recovery and anisotropy decay of green fluorescent protein GFP-S65T in solution and cells: cytoplasmic viscosity probed by green fluorescent protein translational and rotational diffusion. Biophys J 72, (1997). 9. Braga, J., Desterro, J.M. & Carmo-Fonseca, M. Intracellular macromolecular mobility measured by fluorescence recovery after photobleaching with confocal laser scanning microscopes. Mol Biol Cell 15, (2004). 10. Gardiner, C.W. Handbook of Stochastic Processes. (Springer-Verlag, Berlin, 1985). 11. Paul, R. et al. Computer simulations predict that chromosome movements and rotations accelerate mitotic spindle assembly without compromising accuracy. Proc Natl Acad Sci USA 106, (2009). 12. Elowitz, M.B., Surette, M.G., Wolf, P.E., Stock, J.B. & Leibler, S. Protein mobility in the cytoplasm of Escherichia coli. J Bact 181, (1999).

Pulling forces in Cell Division

Pulling forces in Cell Division Frank Jülicher Max Planck Institute for the Physics of Complex Systems Dresden, Germany Max Planck Institute for the Physics of Complex Systems A. Zumdieck A. J.-Dalmaroni