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Corrigendum: Paired arrangement of kinetochores together with microtubule pivoting and dynamics drive kinetochore capture in meiosis I

Scientific Reports 6: Article number: 25736; published online: 11 May 2016; updated: 22 September 2016 In this Article, the Acknowledgments section is incomplete. “We thank M. Sato, S. Hauf, G. Rodel and the Yeast Genetic Resource Center for strains and plasmids; B. Schroth-Diez, Bert Nitzsche and Jan Peychl from the Light Microscopy Facility of MPI-CBG; V.

However, random probing of space by MTs while searching for multiple small targets is likely to be slow. Thus, a number of models have been introduced to describe how the capturing process can be accelerated: by a biased MTs growth toward the KC 5,6 , by MTs nucleated from the KC and by KC movement 7,8 . We have recently shown that MTs explore the space by pivoting around the SPB, which accelerates KC capture in mitosis in fission yeast 9 . All these mechanisms of KC capture contribute most likely to a different extent in different spindles.
In the present study, we combine experiments and theory to explore the relative contribution of the mechanisms that lead to KC capture at the onset of meiosis I in fission yeast. Based on our experimental results, our theoretical model includes MT angular movement around the SPB and MT dynamics. In addition, we model KCs on homologous chromosomes as a single object because we have observed that they move in a correlated manner. Our simulations show that both MT dynamics and their angular movement around the SPB accelerate KC capture. Moreover, the paired configuration of KCs on homologous chromosomes significantly increases capture efficiency.

Results
Movement of KCs is spatially correlated, and their capture and retrieval show specific time dynamics. At the onset of meiosis I in the fission yeast S. pombe, the KCs are far apart from SPBs and telomeres, and SPBs are clustered together 2 . The SPBs nucleate MTs that form the spindle. At the same time other MTs grow, probe the space, bind to KCs and pull them toward SPBs.
To quantify the kinetics of KC capture, we measured the decrease in number of free KCs over time. Free KCs were defined as KCs that freely move within the nucleoplasm without being connected to any MT or to SPBs. A KC was considered captured when the signals from the KC and MT overlap and the KC starts to move toward the SPBs. The time reference was chosen as the frame when the first capture was observed. For this system, the process of capturing lasts, on average, for about 12 minutes and only for one cell, we observed that the process of capturing took more than 15 minutes. The average number of the free KCs was halved within 2.31 ± 0.09 minutes (Fig. 1A), which defines the typical capture time in this system.
By live-cell imaging of cells with KCs labeled in green (Ndc80-GFP) and MTs in magenta (α -Tub-mCherry) we observed 6 KCs that seemed to be grouped in three pairs (Fig. 1B). To address this point, we followed the position of KCs and SPBs as function of time, using a custom made tracking software 10,11 , and we computed the distances between KCs and SPBs. We found a linear correlation between KC-SPB distances of the same pair ( Fig. 1C), but no correlation between KC-SPB distances of KC's from different pairs (inset of Fig. 1C). This result suggests that KCs are grouped in three pairs, the movement of KCs from the same pair is spatially correlated and when one pair gets in close proximity of the SPB either KC in pair is close to the SPB.
Interestingly, two types of movements of the KCs were observed, a first one that seemed to be random and a second one that was directed toward the SPBs and coincided with the moment when a KC gets in the vicinity of a MT. We first focused on the movement of KCs before the moment of capture, and to address this point we looked at the mean squared displacement (MSD) of the center of mass of a single pair. The position of the center of mass was determined from the positions of the two KCs. By computing the MSD of a single pair, we found that the movement of single pairs before capture is diffusive, with a diffusion coefficient D = 3.9 ± 0.2* 10 −4 μ m/s (Fig. 1D).
To determine whether and how the capture of a KC influences the movement and capture kinetics of the other KCs, we analyzed how the distances of the KCs and SPBs are changing over time. As can be observed in kymograph Fig. 1E, KCs initially move together toward the SPBs, but after a while only one KC continues the movement and the second one remains behind and moves randomly until it is also captured. The movement of a KC that occurs after the capture is known as "KC retrieval" [12][13][14] . In Fig. 1F, we show the distances between each KC and the SPB where it was retrieved. Each pair of KCs is depicted with a different color. As can be observed for all three pairs, although the retrieval of the first KC in a pair determines the movement of the second KC from the same pair closer to the SPB, the distance between KCs from the same pair increases (Fig. 1F bottom). The density of MTs is higher closer to SPBs and this is expected to increase the chance that the second KC in a pair is captured faster. We calculated the delay of the capture time between pairs and between KCs inside of the same pair (Fig. 1G). The delay in capture between pairs was more than 2 minutes and inside each pair for about 1 minute. These findings suggest that the capture of a KC influences the movement and the capture kinetics of the KC within the pair, but not the capture kinetics of the other KCs.
between the distances KC-SPB for each pair (in green). On the x-axis is the distance between the first captured KC in a pair and the SPB, and on the y-axis the distance between the second captured KC and the corresponding SPB. In the inset, no correlation between KC-SPB distances of KC's from different pairs (black). On the x-axis is the distance between a KC and the SPB, and on y-axis is the distance between each KC from other pairs and SPB. (D) Mean squared displacement of the KC pair. A weighted fit to the equation MSD = 4D KC Δ t + offset (green line) yielded a diffusion coefficient of KC pair D KC = 3.9 ± 0.2 *10 −4 μ m/s (mean ± SD, n = 51). KCs were tracked with subpixel resolution (Methods) and the center of mass of a pair was calculated. Grey denotes the area corresponding to subpixel movement. (E) Time-lapse images representing the capture and retrieval of a KC pair. First, one KC is captured and pulled toward the SPB (white area). The second KCin this pair does not move for about 15 seconds (4 frames), but afterwards starts to move towards the SPB. Time difference between frames Δ t = 3.85 seconds. Scale bar is 1 μ m. (F) Top: distance between each KC and the SPB. Circles, squares and triangles mark different KC pairs, see legend. Bottom: distance between the KCs from the same pair. The magenta dashed lines represent the capture event of the first KC in each pair. (G) Box plots of the delay capture time between pairs (left) and delay capture time between KCs inside each pair (right).
Scientific RepoRts | 6:25736 | DOI: 10.1038/srep25736 Highly dynamic MTs perform angular movement to explore the entire nucleus. Additionally to the kinetics of the KCs we investigated the dynamics and the movement of MTs. To address the dynamics of the MTs, we performed live-cell imaging of cells with MTs labeled in green (α -Tub-GFP) and, as can be observed in the time-lapse from Fig. 2A, in only 30 seconds MTs covered most parts of the nucleus. We found a MT's growth rate v g (2.4 ± 0.7 μ m/min) consistent with 9,13,15,16 and shrinkage rate, v s , almost three times higher (6.4 ± 1.9 μ m/min). Unlike as has been reported in 9 , the life of most MTs (78%) consisted of two periods, growth and shrinkage, and only for few of them a pausing period was observed (Fig. 2B). We then computed the MT catastrophe frequency for different MT lengths. We found that MTs undergo catastrophe with a rate that increases linearly 17 with their length with a slope of α ~1.1 (μ m*min) −1 (Fig. 2D). MTs lived on average for ~1.2 min and the averaged maximum length was L = 2.22 ± 0.66 μ m. They spend L/v g ≈ 0.9 min in the growth phase and L/v s ≈ 0.3 min in the shrink phase. These calculations show that MTs have short pausing intervals and a fast turnover during meiosis I.
To understand how the fast turnover of the MTs influences the process of capture, we measured the number of MTs over time. At the end of the meiotic prophase few intra-nuclear MTs are barely visible. Their number increases in time and reaches a peak of about 5-6 MTs/cell in the first minute after the first capturing event occurs (Fig. 2C). This high number of MTs is maintained for about 2 minutes increasing the chances that a MT finds and attaches to a KC. This timing coincides with the time when the number of KCs was halved. Although the MTs shrink rapidly a high number of MTs is maintained constant by fast turnover of the MTs.
To complete the picture of the behavior of MTs, we analyzed their movement. As previously was reported in 9 , we also observed that intra-nuclear MTs change their orientation with time in meiosis I (Fig. 2E). We calculated the mean squared angular displacement (MSAD) 18 , for MTs with an average length of 1.49 ± 0.18 μ m, to distinguish whether MTs angular movement is directed or random. We found that the MSAD increases roughly linearly with time (inset of Fig. 2F) and we know that such linear relationship is associated with unbiased random movement. The corresponding angular diffusion coefficient of MTs was calculated from the slope after fitting the experimental data. Furthermore we found that the angular diffusion coefficient of the MTs decreases for longer MTs (Fig. 2F). This is in agreement with the Brownian angular motion of a thin rigid rod [19][20][21] . Taken all together, our observations regarding MT's movement suggest that MTs are freely jointed to SPBs, pivot around them and that the movement of MTs is Brownian.

A simple model can explain the observed captured times of the KCs.
To understand the mechanisms that lead to the observed KC capture times we developed a stochastic model that takes into account MT pivoting around the SPB (as was described in our previous work 9 , MT dynamics and paired movement of the KCs. The nucleus is described as a sphere of radius 1.5 μ m corresponding to the nuclear membrane 9 . We consider that the two SPBs are located together at one pole of the sphere and acting as a single MT nucleation source. This choice is based on our observation that two SPBs are close to each other during most of the capturing process. MTs are described as thin, rigid rods that can grow, shrink and perform angular diffusion while fixed with one end at the SPB (Fig. 3A). They can switch from growth to shrinkage (catastrophe) with a probability per unit time that increases linearly with MT length, as shown by our experiments (Fig. 2B) and as was observed previously in 17,22 . The model does not include the possibility of switching from shrinkage to growth (rescue), since we did not observe this in our experiments. The simulations were performed using a constant number of MTs, so if one MT shrinks until its length equals zero it immediately starts growing again, with a new random orientation. A pair of KCs was described as a single rigid sphere that has a radius twice as large as that of a single KC. To check that this description can be used for studying capture we plotted the experimental fraction of free KC pairs as a function of time and observed that it obeys a kinetics that is very similar to that of the capture of single KCs.
The motion of both KCs and MTs was described using Langevin's equations in spherical coordinates (shown in Fig. 3B). These equations describe a realization of the stochastic Brownian motion of objects immersed in a fluid by considering the effect of the fluid to be similar to that of a random force exerted on the particle. We solve these equations by numerical integration using the Euler method. When integrating the equations, we consider that the motion of the KCs and MTs is confined within the sphere (the boundary conditions are given in Fig. 3B). When a KC touches the membrane, it is reflected back towards the center of the nucleus. When the tip of a MT grows until it touches the nuclear membrane (that is, the boundary is reached due to a change in L) it slides along it until it does not touch it anymore. If a MT hits the boundary by pivoting (due to a change of one of the angular variables), it is reflected towards the center of the nucleus. The exact equations used to implement the boundary conditions in the simulations are given in the supplementary text. A KC capture event was considered to occur when the MT overlaps with the volume of the KC, according to the equations in Fig. 3B. Note that this allows for lateral capture. After a KC is captured by a MT, that MT undergoes catastrophe and starts to shrink, carrying the KC to the SPB. The parameters of the model have values that we measured experimentally, with two exceptions: the radius of the kinetochore, which is taken from 23 , and the radius of the nucleus, which is taken from 24 . The parameters of the model are given in Fig. 3C. The initial configuration was set by putting the MTs' lengths to zero and positioning the KCs close to the equator of the nucleus, as measured at the onset of meiosis I. The details of the initial conditions and the parameters of the model are given in Fig. 3C. The model was tested by computing the number of free KCs as a function of time for increasing numbers of MTs. The best agreement with the experimental data is obtained for 6 MTs (Fig. 4A, S1A), which is a MT number that is in good agreement with what we observed experimentally (Fig. 2C). One has to note that in the experimental images MTs that have a length below 0.75 μ m are not always observed, while they are always included in the simulation MT count. In figure S1A we show the average KC capture time (which is equivalent to the mean first passage time for each MT-KC encounter, averaged for all KCs) for other values of the MT number. As for the experimental data, time zero is considered when the first KC pair is captured, so the average is taken over the remaining two pairs. We have then checked that the model we used for describing MT catastrophe does not influence our results (see Supplementary text and supplementary figures S1A-F and S2A-C for a detailed discussion). The results above validate our model and show its robustness.
We then simulated the capture of six unpaired KCs instead of three KC pairs and observed that the average capture time has almost doubled. The main reason for this is that a pair has a larger surface available for interaction with MTs, thus facilitating attachment. This is also supported by the prediction that when increasing the number of KCs without changing their properties the average capture time stays the same, but the total capture time (or the lifetime of the last kinetochore that is being captured) increases from 4.81 minutes to 7.76 minutes (Fig. 4B). This result predicts a second crucial contribution to capturing, which has not been suggested before: the pairing of KCs.
The theoretical model was then used to make predictions regarding the mechanisms that contribute to KC capture and to estimate their contributions. Simulation parameters were changed in turn, as follows: (i) D MT = 0, D KC = 0 (ii) D KC = 0, and (iii) D MT = 0. All the other parameters were kept as in Fig. 3C. Case (iii) corresponds to the "search and capture" model of Mitchinson and Kirchner 4 . We denote by case (0) the simulations that were done using the measured parameters. The results of all cases are shown in Fig. 4C. When both the angular diffusion coefficient of the MTs and the diffusion coefficient of the KCs were put to zero (case (i)) the predicted average The value for a was obtained by doubling the value for a single KC that is given in 26  capture time increased with a factor of ~2.75 with respect to case (0). This could be explained by the fact that, when neither the MTs nor the KCs are diffusing, the MTs have to reach for the fixed KCs just by growing. Time is lost when a MT is not growing directly towards a KC, so the search and capture mechanism alone is the slowest search strategy. In the simulations in which only one of the two diffusion coefficients is put to zero (cases (ii) and (iii)) the average capture times were smaller than the one of case (i) but larger than in case (0). This points to the fact that at least one component of random motion (Brownian motion of either MTs or KCs) is needed for the capture process. This can be explained as follows: a MT, which does not perform Brownian motion, only captures a fixed KC by growing directly towards it. However, if the KCs move even with a small speed their motion will contribute to the capture because it increases the chances of an encounter with the MT. Differently put, the difference in capture time between the these situations is strongly related to how much space is sampled in a given time by MTs and KCs together. The presence of Brownian motion maximizes this time. Similarly, if the KC is fixed but the MT pivots, the search is accelerated because a MT that does not grow directly towards a KC can still capture it through a change in direction. Actually, in the presence of MT pivoting the effect of the KC's motion becomes negligible (case (ii) compared to case (iii)), because pivoting is faster than the diffusion of KCs, so the latter becomes just a small addition to the amount of space that is being covered.
We also performed simulations in which the MT length is kept constant at different values ranging from 1.4 to 2.5 μ m. The results of these simulations are shown in Fig. 4D. The average capture time decreases when the MT length is increased until it reaches an optimal value at ~2 μ m and then it starts increasing again. This happens because the amount of space sampled by the MT is determined by two factors: its length and its angular velocity. A long MT swipes more space to search for the KC, but it moves slower, while a short one pivots fast but samples less space. MTs shorter than 1.4 μ m (the initial distance between the KC and the SPB) will take a very long time to capture KCs. This is because they cannot reach a KC by pivoting, so they have to 'wait' until a KC diffuses towards them. Taken together, the results from Fig. 4C,D suggest that MT dynamic instability and angular motion are important contributors to KC capture.
In order to quantify the contribution of pivoting to the capturing process we computed the fraction of lateral capture events. We considered a capturing event to be lateral capture if at the moment of capture the following condition is fulfilled:

KC MT
Using δr equals the radius of the KC, we obtained that approximately half of the kinetochores are captured laterally. Under the assumption that all lateral capture events occur due to pivoting, and all the tip capture events occur due to dynamic instability, this implies that the two mechanisms have an equal contribution. This assumption could, however, underestimate the contribution of pivoting. Using δr = 8 nm (the length of a tubulin) the ratio between the number of tip capture events and lateral capture events switches to 0.32. These values show that pivoting is indeed important for KC capture.

Discussion
At the onset of mitosis and meiosis, MTs capture KCs to incorporate them into the spindle, which is required for the proper segregation of chromosomes. The mechanisms of KC capture and their relative contribution in different cell types are still not clear. We have shown that at the beginning of meiosis I in fission yeast the paired configuration of KCs on homologous chromosomes significantly increases capture efficiency (Fig. 5A). This is a novel mechanism that accelerates capture, which is specific to meiosis I. Our experiments have shown that the KCs in each of the 3 pairs, which are presumably located on a pair of homologous chromosomes, move in a correlated manner. This is most likely a consequence of the fact that homologous chromosomes are connected by a reciprocal recombination (chiasmata) at the end of meiotic prophase and the onset of meiosis I 25 . The paired configuration of KCs on homologous chromosomes may contribute to the acceleration of the capture process in two ways. First, when one KC in a pair is captured by a MT and starts to move towards the SPB, this movement brings the other KC in the same pair closer to the SPB (Fig. 1G), where the density of the MTs is higher and hence the probability of capture is also higher. Second, two KCs on a pair of homologous chromosomes provide a larger surface that has to be targeted by the MTs than a single KC, resulting in a shorter total capture time, as predicted by our model (Fig. 4B). Even though in our model a pair of KCs is described as a single large KC, the model could be extended to describe a pair of KCs as 2 KCs connected by a viscoelastic element. The prediction that the paired configuration of KCs accelerates KC capture could be tested experimentally by measuring the capture kinetics in cells lacking meiosis-specific proteins responsible for the formation of double-strand breaks and chiasmata, such as Rec12 26 . In these cells, homologous chromosomes are not organized as 3 bivalents, but as 6 univalents that are not connected with each other 27,28 . Thus, we expect that all 6 KCs would move independently in these cells, and our model predicts a roughly 3 times longer total capture time for this scenario.
The second mechanism that contributes to KC capture in meiosis I is MT dynamics (Fig. 5B). We observed that at the onset of meiosis I several MTs grow from the SPB, similarly as in 3 . These MTs spend most of their lifetime either growing or shrinking (Fig. 2B), thus they are highly dynamic. The dynamics of MTs was the first suggested mechanism of KC capture 4 . Indeed, our results show that the highest number of MTs is reached when the capture starts and it is maintained constant for about 3 minutes (Fig. 2C). This is similar to the time in which half of the KCs are captured. MTs in meiosis I are more numerous than in mitosis (4-5 in meiosis versus 2-3 in mitosis, on average) and more dynamic than MTs in mitosis, which spend ~70% of their lifetime at a constant length 9 . Future work will identify the molecular mechanisms that differentially regulate MT number and dynamics during meiosis and mitosis.
Finally, angular movement of MTs around the SPB accelerates KC capture (Fig. 5C). We have recently shown that this is the major mechanism underlying KC capture in mitosis in fission yeast, during spindle re-assembly after cold-induced disassembly 9 . Similarly, angular movement of astral MTs accelerates their search for cortical anchor sites during mitosis in budding yeast 29 . In general, angular movement allows MTs to explore space laterally rather than by the tip only, thereby accelerating their search for various targets such as KCs 1 . As in mitosis 9 , MT angular movement in meiosis I is random and characterized by a similar angular diffusion coefficient. Thus, MT movement in meiosis I is likely thermally driven as the movement in mitosis 9 . Our model predicts that the contribution of the angular movement of MTs to KC capture is the largest if MTs are roughly 2 μ m long, which matches the experimentally measured values of MT length. It would be interesting to identify the mechanisms of MT length determination at the onset of meiosis I and to experimentally test the effect of altered MT length on the kinetics of KC capture. Ultimately, it would be important to understand what makes a particular combination of capture mechanisms suitable for the settings of meiosis or mitosis in fission yeast, as well as to what extent these mechanisms contribute to KC capture in other cell types.

Methods
Cells culture. S. pombe strains used in this work are listed in Table 2 Generation of S. pombe strains. GFP provides a higher signal-to-noise ratio than the available red fluorescence proteins (e.g. tdTomato and mCherry) and a lower bleaching rate during live cell imaging, allowing for more accurate tracking of tagged structures through a larger number of time-lapse images. Thus, we used strains expressing Ndc80-GFP to measure the kinetics of the KCs and strains expressing GFP-alpha2-tubulin (GFP-atb2) to measure the dynamics and kinetics of the MTs. We cannot exclude small differences in MT dynamics between the strains we used due to different tagging. However, here we measured MT dynamics in a strain expressing GFP-atb2, and these data are in agreement with the previously reported 16 MT dynamics in strains expressing mCherry-atb2. Strain KI011 was obtained by crossing SV54 with strain KI006. C-terminus tagging of Ndc80 with GFP was performed by using the polymerase chain reaction (PCR) gene-targeting method 30 . The open reading frame of GFP and a KanMX resistance-cassette for selection were integrated at the C-terminus of the original gene locus of Ndc80 by homologous recombination. The primers were designed using the web tool http://www.bahlerlab.info/resources/ 31 . The tagging-plasmid pFA6a-GFP-KanMX was used as a template (gift from K. Sawin, University of Edinburgh, UK). The product of the PCR reaction was purified using the QIAquick PCR Purification Kit (QIAGEN, Hilden, Germany), then concentrated by isopropanol precipitation with the pellet being resolved in 10 mM Tris-Cl, pH 8.5. To create GCAK1, the strain MJ95 (mCherry-atb2-hphMX leu1 ura4-D18, h-, gift of M.Sato) was transformed with 15 μ g of the purified PCR product by electroporation at 1.5 kV using an Eppendorf Electroporator 2510 (Eppendorf) 32 . Ndc80-GFP-KanMX expressing strains were selected by growth on YE + supplements + hygromycin + kanamycin medium. GC03 and GC04 were obtained by crossing SI661 (Sid4-mCherry-NatMX, gift of Silke Hauf, Virginia Tech, USA) with GCAK1.
Time-lapse live cell imaging. Cells were also imaged and manipulated using an Olympus IX71 inverted microscope with a Yokogawa CSU10 spinning-disc scan head (Yokogawa Electric Corporation) equipped with a fast piezo objective z-positioner (PIFOC, Physik Instrumente GmbH & K.G.) and an Olympus UPlanSApo × 100/1.4 NA oil objective (Olympus). For cells expressing GFP and tdTomato/mCherry, we performed sequential imaging and images were acquired at a 2.1-4.7 s time intervals. For excitation, a sapphire 488 nm solid-state laser (75 mW; Coherent) and a Jive 561 nm solid-state laser (75 mW; Cobolt) were used for GFP and tdTomato/mCherry, respectively. The laser intensity was controlled using the acousto-optic tunable filter inside the Andor Revolution Laser Combiner (ALC, Andor Technology). The emission wavelength was selected using respective emission filters BL 525/30 (Semrock) and ET 605/70 (Chroma) mounted in a fast, motorized filter wheel Scientific RepoRts | 6:25736 | DOI: 10.1038/srep25736 (Lambda 10B, Sutter Instrument Company). The images have a xy-pixel size of 168 nm and the z-distance between optical sections was 500 nm. The system was controlled by Andor iQ software version 1.9.1 (Andor Technology).
Image processing and data analysis. Measurements of MT and KC positions were performed in the maximum-intensity projections of the z-stacks. Measurement of positions along the z axis were not performed because the corresponding point-spread function of the microscope is about 800 nm, which is roughly half of the length of a typical MT. An estimate of the systematic error resulting from two-dimensional measurements was presented in 9 . Maximum-intensity projections were calculated with ImageJ (National Institutes of Health) using the plug-in Stacks-Z-function-Grouped ZProjector. The colour-merge images were obtained by overlay of projections in green and magenta channels using the plug-in Colour functions-Colour merge. MT tips were tracked manually in the maximum-intensity projections using the plug-in Particle analysis-Manual tracking. Custom software 10 was used to determine the SPBs and KCs position in the maximum-intensity projections. Data analysis and plots were performed using scripts written in MATLAB (The Mathworks).