Reliable and robust control of nucleus centering is contingent on nonequilibrium force patterns

Summary Cell centers their division apparatus to ensure symmetric cell division, a challenging task when the governing dynamics is stochastic. Using fission yeast, we show that the patterning of nonequilibrium polymerization forces of microtubule (MT) bundles controls the precise localization of spindle pole body (SPB), and hence the division septum, at the onset of mitosis. We define two cellular objectives, reliability, the mean SPB position relative to the geometric center, and robustness, the variance of the SPB position, which are sensitive to genetic perturbations that change cell length, MT bundle number/orientation, and MT dynamics. We show that simultaneous control of reliability and robustness is required to minimize septum positioning error achieved by the wild type (WT). A stochastic model for the MT-based nucleus centering, with parameters measured directly or estimated using Bayesian inference, recapitulates the maximum fidelity of WT. Using this, we perform a sensitivity analysis of the parameters that control nuclear centering.


INTRODUCTION
The nucleus and other organelles in eukaryotic cells display precise and reproducible intracellular positioning. [1][2][3] For instance, the nucleus in sea urchin eggs and the mitotic spindle in Caenorhabditis elegans localize at the center to within 5% of the cell length. 4,5 Reliable positioning is governed by nonequilibrium intracellular forces that arise from self-assembled structures 6-10 that are cell spanning, adaptive, and simultaneously pliable and rigid, such as the active cytoskeleton. [11][12][13] However, the quantitative principles underlying the interplay between these force patterns and organelles positioning remain to be elucidated.
Indeed, the positioning of the organelles by fluctuating nonequilibrium forces from the active cytoskeleton appears to be ubiquitous across all eukaryotic life forms. This can arise as a result of pulling forces from actomyosin or motor-microtubule (MT) complexes or pushing forces from MT polymerization-depolymerization. 11,[14][15][16][17][18][19][20][21] For instance, nucleus positioning and spindle migration to the cell surface in mouse oocytes is driven by stresses exerted by actin cytoskeleton. [14][15][16][17][18] Contractile stresses arising from actomyosin are also seen to be responsible for proper nuclear positioning in mouse fibroblast cells. 18,22 In most cells, MT-cytoskeleton-based active forces operate on the cortex either via pushing forces caused by MT polymerization against the cortex or pulling forces applied by anchored, minus-end-directed motors or a combination of these. [19][20][21] Nonequilibrium stresses from both actomyosin contractility and polymerization-depolymerization of MT exhibit strong mechanical fluctuations. [23][24][25][26][27] For instance, the growth dynamics of individual MT exhibits dynamical instability, 23,24,28-30 with a well-defined growth phase before going to ''catastrophe'' leading to a rapid shrinking phase. Furthermore, the number of MT involved in active force generation is subject to fluctuations, owing to the stochastic dynamics of nucleation and growth. Evidence suggests that the mechanical fluctuations of MTs are sensitive to applied forces [31][32][33] ; this implies that MT (as indeed actomyosin) are both active force generators and force sensors, aspects that are advantageous for adaptation and robust control.
In this paper, we study the fidelity of nucleus centering in wild type (WT) and genetically perturbed fission yeast (Schizosaccharomyces pombe) cells, using a combination of live-cell microscopy and detailed statistical analysis. Fission yeast cells divide by medial fission and produce two daughter cells that are identical in size. Symmetric cell division is contingent on precise localization of the division plane apparatus. 5,34-44 The iScience Article Figure 1C. To standardize the observations between WT and the various perturbations, we used a strain with the following fluorescent tags: Sid4-mCherry (SPB) and EnvyGFP-Atb2 (MT). This background enabled the long-term observation of SPB and MTs.
To verify the validity of SPB dynamics as a readout for the nucleus dynamics, we analyze the dynamics of the centroid of the nucleus and SPB in strains with GFP-Cut11 (nuclear envelop) and Sid4-mCherry (SPB) (see Figures 1A and S1). Figures S1A-S1D show the time course of longitudinal displacement of the nucleus centroid and SPB for the several conditions. The SPB shows larger fluctuations than the nucleus, and thus directly inferring the nucleus dynamics from SPB leads to some uncertainty. However, given that the nuclear membrane is only slightly deformable ( Figure 1A), the distribution of SPB positions and the nucleus centroid should be strongly correlated (see Figure S1): Convolution of the nucleus centroid with a uniform distribution of points on the surface of a sphere fits well with the distribution of SPB positions (Figures S1A-S1D). Finally, the principal features describing the fidelity of centering, namely the reliability (d) and the robustness (s x ) of the nucleus and SPB, show significant correlations (Figures S1E and S1F) with the Pearson correlation of 0.75 (for d) and 0.78 (for s x ).
This suggests that the SPB dynamics can be used as a proxy for nucleus localization before the onset of mitosis. Analysis of the time series of the longitudinal and transverse displacements, during a 30 min period, shows them to be statistically independent ( Figure S1). Moreover, the time series is also stationary, with the mean and variance measured over a time window, being independent of time ( Figure S1). These measured quantities do not show any change as M-phase approaches, suggesting the dynamics is not subject to additional regulation. Consequently, the spatial position of SPB (and nucleus) at the onset of mitosis is determined exclusively by its prior dynamics.
From the time series of the longitudinal positions of the SPB in a given cell, we compute its stationary distribution, from which we extract d, the mean SPB displacement from the geometric center, and s x , the standard deviation of SPB displacement ( Figure 1E). d is a measure of positioning reliability; smaller d implies a more reliable centering. In $40% of WT cells, d lies significantly away from the center, implying that even WT cells have a significant variation in reliability ( Figure 1F). On the other hand, s x is a measure of robustness in the positioning, and a smaller s x corresponds to a more robustly localized nucleus. In all WT cells, s x is $3fold higher than the fluctuations observed after disintegrating MTs using MBC ( Figure 1G); thus, fluctuations arising from active forces makes a significant contribution to s x . In principle, d and s x represent two independent objectives for nucleus centering; changes in either lead to higher chances of an off-centered nucleus and, consequently, a high deviation in septum position (S) from the cell center ( Figures 1H and 1I).
In what follows, we will discuss changes upon specific genetic perturbations, keeping in mind that the perturbations inevitably change many attributes related to the organization and dynamics of MTs (quantified in Figures 1J and S2-S6, and discussed further in detail)-we will take this into account in making our inferences.

SPB dynamics is sensitive to cell length
An aspect of cell geometry that is simple to vary is cell length. Since the MT-based pushing machinery that control nucleus positioning, span the entire length of the cell, perturbations of cell length can potentially inform about adaptability and scaling. [53][54][55][56][57] We utilized the cell cycle mutant cdc25-22 in which G2-M transition is delayed leading to longer cells; and wee1-50 mutant strains, which shortens the cell cycle by inducing early G2-M transition, leading to shorter cells. 58 Together with WT, these strains provide a 4x variation in cell length (Figures 2A-2C).
The basic organization of MT cytoskeleton in these strains is similar-MTs are arranged in bundles, mostly aligned with the long axis of the cell (Figures 2A and S2). However, in the short wee1-50 cells, MTs are more bent, resulting in a dispersed local MT orientation ( Figure S2). The number of MT bundles and the net MT mass per unit length is large in cdc25-22 cells and small in wee1-50 cells compared to the WT, suggesting a systematic variation with cell length ( Figure S2). Importantly, measurement of the MT growth parameters (see STAR Methods and Figure S7) reveal that the free growth velocity, shrinkage velocity, and growth velocity during contact of MT to cell cortex are very similar to the WT ( Figure S2). iScience Article respectively ( Figure S2). Similarly, the mean dwell times are 45 s for the WT, 17 s for cdc25-22, and cells 54 s for wee1-50. Taken together, these results suggest that cell length perturbations have only a moderate effect on the basic MT organization and intrinsic growth parameters. However, the dynamic parameters that rely on the interaction between MT and cell cortex, e.g. the catastrophe and dwell times, 32 and MT mass show a dependence on cell length.
We now monitor the SPB dynamics before the onset of mitosis both in the WT and these strains. A visual inspection of the time series of the longitudinal positions immediately reveals that wee1-50 cells exhibit ). In addition, in many cdc25-22 cells, the mean SPB localization is visibly off-center. Consequently, many cells in the cdc25-22 strain have large values of d compared to the WT or wee1-50 cells, suggesting inadequate reliability of positioning. This difference is independent of Cdc25 function, since in small cdc25-22 cells, d decreases ( Figure 2D). We see that wee1-50 cells have significantly large s x in comparison to WT or cdc25-22 cells ( Figure 2E), suggesting a less robust localization of nucleus in small cells. This difference is independent of Wee1 function since small WT cells also have large s x ( Figure S2). We conclude that an increase (decrease) in cell length from the typical WT leads to less reliable (less robust) nucleus centering, respectively.
The observations of high s x in the shorter cells are reminiscent of the high fluctuations seen in in vitro assays where MT lengths are much longer than the compartment size. 59,60 The buckled MTs appear less dynamic and are associated with lower s x . 61 Together, reliability (d) and robustness (s x ) are the two independent phenotypic objectives for nucleus position at the onset of mitosis; good performance in these two objectives correspond to low values of both d and s x . We can ask how each cell in these strains fare in the realization of these joint objectives.
Owing to underlying cell-to-cell variations, we expect a cloud of points to represent each strain (the color-shaded region in Figure 2F) in an objective space. Large cdc25-22 cells occupy an expanded space where many cells have larger d than the WT and wee1-50 cells. Conversely, small wee1-50 cells occupy a region where many cells have higher s x than the WT cells. These observations suggest that collective minimization of d and s x is sensitive to the cell length.
An increase in either d or s x enhances the chances of mislocalization of the septum. We define the failurecoefficient of SPB localization (F), that quantifies the likelihood of extreme septum localization events (Figure 1I) beyond a fixed threshold, say the extreme-5% septum localization events in the WT cells. Comparing F across the three strains suggests that the chance of mislocalization is the least in the WT cell length ( Figure 2G).

Robustness in nucleus positioning increases with number of MT bundles
We now look at how one may systematically vary parameters characterizing MT-based force patterning. Here, we study how variation in MT bundle numbers affects nucleus positioning. WT cells have 3-4 MT bundles which consist of up to 12 filaments arranged in a predominantly anti-parallel manner. 7,8,47,62 We study deletion mutants where this bundle number can vary-thus rsp1D has 2-3 and mto2D has 1-2 MT bundles ( Figures 3A-3C). Rsp1 localizes to interphase-MTOCs (iMTOCs), non-SPB MTOCs which are only present during interphase, and is required for the proper functioning of iMTOCs. 63,64 Mto2 is a component of the g-tubulin activator complex, and deletion of Mto2 leads to nonfunctioning iMTOCs, which results in SPB being the only active MTOC. 47,49,65 Both these mutants have cell size similar to WT cells at the onset of mitosis ( Figure S3).
The rsp1D and mto2D cells have very similar MT mass, though significantly reduced in comparison with WT ( Figure S3). Not surprisingly, mto2D cells show a higher fraction of bent MTs ( Figure S3). We find that most of the dynamic parameters associated with MT growth are not significantly different from the WT, except for the free MT growth velocities which are elevated in the mto2D strain ( Figure S3; Table S1). High MT growth velocities and similar MT catastrophe times are consistent with the observation that mto2D cells have longer MTs than WT and rsp1D cells. Furthermore, having the same MT mass for longer MT suggests that mto2D has fewer MTs numbers than rsp1D cells. In conclusion, changes in MT bundles in these strains also changes the MT numbers and one should expect the number of MTs to follow the order WT>rsp1D>mto2D.
Next, we analyzed the SPB dynamics in these strains 30 mins before the onset of mitosis ( Figure S3). We note that in some mto2D cells, the SPB movement ceases much before the onset of mitosis. This was observed earlier 49 and attributed to an SPB-MT bundle detachment phenotype; we therefore exclude these cells from our analysis. As seen in Figure S3F, the SPB in rsp1D and mto2D cells appears much more dynamic than in WT cells. We quantified the reliability d and robustness s x of nucleus positioning for each cell ( Figures 3D and 3E Article s x appears to increase with a decrease in the number of MT bundles i.e., WT<rsp1D<mto2D. This relationship becomes more evident in the objective space of d and s x ( Figure 3F). Moreover, the transverse fluctuations of the SPB are higher in mto2D than in WT (see Figure S3). This suggests that having multiple bundles also aids in stabilizing the transverse motion of the nucleus.
In comparison to WT strain, rsp1D and mto2D strains show a much larger cell-to-cell variation in objective space ( Figure 3F). We estimate the failure-coefficient of SPB localization, F, using the values of d and s x for each condition, and find that it depends on the number of MT bundles. We find that the WT with the larger number of MT bundles has the least F. There is no appreciable gain beyond a certain number of MT bundles, suggesting a tradeoff between effectiveness and economy in cellular costs. As stated, the MT in WT cells is organized as linear bundles aligned along the long axis of the cell. In contrast, rsp1-1 mutant cells have MTs that have an aster-like organization, and MT crosslinker, Ase1, deletion strain (ase1D) has disorganized and non-bundled MTs ( Figure 4A). 63,66 While rsp1-1 cells have similar cell length as WT, in our experiments, the typical length of ase1D cells is longer than WT at the onset of mitosis; therefore we also analyze small ase1D cells (ase1D(s)) ( Figure S4). The mean scaled MT mass in rsp1-1 (ase1D) cells is slightly higher (lower) than in WT cells ( Figure S4). The orientation distribution of Despite this, most MT growth parameters in these strains are very similar to the WT ( Figure S4). The only significant difference is in catastrophe time, which is higher in both rsp1-1 (154 s) and ase1D (157 s) strains in comparison to WT (115 s) (see Table S1). Figure S4E shows typical SPB trajectories in these cells during the 30 min interval before the onset of mitosis. In some rsp1-1 cells, the SPB pauses for an extended period near the pole. These typically happen when the MTs are in an aster-like organization.
In ase1D cells, the mean position of the SPB is displaced away from the cell center. Analysis of d and s x for rsp1-1 and ase1D cells ( Figures 4D and 4E) shows that d is more widely distributed, and that the mean d is significantly higher than in the WT. On the other hand, s x in rsp1-1 and ase1D cells are not significantly different from WT cells. Since ase1D cells are typically longer than the WT cells, we also study the dynamics of SPB positioning in small ase1D cells (marked ase1D(s), also see Figure S4). In small ase1D cells, while the values of d are similar to WT cells owing to the MTs making contact with cell tips, s x is higher. Moreover, rsp1-1 and ase1D cells displayed larger fluctuations in the transverse displacements of SPB ( Figure 4F). The higher fluctuations in the transverse (and longitudinal for ase1D(s)) direction can be attributed to the large angular deviation of MTs which make prolonged contact with the cylindrical side of cells (away from the cell tips). This results in pushing forces having large angular variations, very different from the coaligned pushing forces in the WT.
In the objective space of d and s x , both rsp1-1 and ase1D strain show large cell-to-cell variation, and the failure-coefficient of SPB localization (F) increases in both of these strain ( Figure 4G). Thus, having linear bundles of MT aligned along the long axis (as in WT) leads to a more reliable and robust nucleus centering.
Optimal centering of nucleus requires favorable MT growth dynamics  S5). We find that klp5D-klp6D cells are shorter and tip1D cells are longer at mitosis compared to WT (Figure S5). While the MT mass relative to cell length is reduced in tip1D and mal3D strains as expected, 24,67,72 we also observe a significant reduction in MT mass in klp5D-klp6D strain. This may reflect a decrease in the nucleation activity consistent with the in vitro observation that Klp5-Klp6 acts as a nucleation factor 73 ( Figures 5B  and 5C). MT orientation in klp5D-klp6D and mcp1D cells is similar to the WT cells ( Figure S5). However, in tip1D and mal3D cells, the MTs are less aligned along the cell axis, highlighting the significance of long MTs. 40,74 The MT growth dynamics in these strains has the following features ( Figure S5): Both MT growth and shrinkage velocities are very similar in these strains. The growth velocities of MT during contact with cell tip are also similar in WT, klp5D-klp6D, and mcp1D cells. However, MT undergoes catastrophe much before reaching the cell tip in mal3D and tip1D cells, consequently showing higher catastrophe frequency. In our experimental sampling, we do not see much difference between MT catastrophe frequency in WT and klp5D-klp6D and mcp1D cells. However, for mcp1D cells, we do notice a wider distribution of both MT catastrophe time and dwell times.
Next, we characterized the reliability and robustness of the nucleus centering in these cells ( Figures 5D and  5E). Both, klp5D-klp6D and mcp1D cells show higher fluctuations in SPB dynamics compared to WT. In contrast, tip1D and mal3D cells, in which the MT catastrophe rate is higher thus leading to shorter MTs, show suppressed SPB dynamics away from the cell center ( Figure S5). This leads to high value of d (Figure 5D), similar to the long cdc25-22 cells. However, both klp5D-klp6D and mcp1D cells show much higher s x (Figures S5 and 5E). Earlier, high s x in klp5D-klp6D has been attributed to the lack of enhanced catastrophe at cell ends; 50 however, we propose that a reduction in MT might also contribute to this phenomenon. Cumulative analysis of d and s x in the objective space ( Figure 5F) shows that mal3D and tip1D prominently occupy a space to the right of the WT with smaller s x and klp5D-klp6D and mcp1D cells occupy space above the WT with small d. The failure-coefficient F of SPB localization ( Figure 5G) shows that the altered MT growth dynamics increases the chance of nucleus mispositioning. iScience Article Precise nucleus centering is key to medial positioning of division plane A crucial function of nuclear positioning in fission yeast at the onset of mitosis is to define the site of septum formation. We have shown that cell length, MT bundle numbers, MT architecture, and MT growth dynamics influence both reliability and robustness of nucleus centering. We next ask how do these traits impact the septum position. To this end, we observed the septum position in the studied strains at 35 C. (Note that cdc25-22 cells do not commit to mitosis at 35 C. However, we found that cdc25-22 cells growing at 35 C iScience Article immediately go to mitosis on changing the temperature to 25 C, see STAR Methods). The off-center septum is visible in many strains. We quantify the error in septum positioning by measuring the distance of the septum from the cell center (S) ( Figure 1H). Figure 6B shows the cumulative distribution of S in these strains. It is immediately clear that all mutants show a much broader distribution of septum positions than WT. Both, high d or high s x of the SPB-position get mapped onto a broad distribution of S. For example, the wee1-50 and mcp1D strains have higher s x than WT, and ase1D has higher d than WT, even though the s x is comparable-all these strains exhibit a much broader S distribution.
Based on these observations, we expect that S f , the failed septum localization (fraction S that lies beyond x f ), should be strongly correlated with F (fraction of cells with failed SPB localization), as F is a function of both d and s x ( Figure 6C). Pearson correlation statistics between the mean F and S f shows a correlation of 0.72 (p < 0.05). We note that some of the mutants explored here are also known to show some degree of  40,75,76 Given these complexities, we contend that the $72% variation in S f explained by F reflects the high influence of optimal nucleus centering on the septum position.

Stochastic model of MT-driven nucleus centering recapitulates the optimality of the WT
We next develop a stochastic model to explain how the underlying MT force patterning may lead to an optimum nucleus centering. We have seen that the position of the nucleus in the interphase cell is controlled by forces exerted by the dynamic MTs which nucleate at the nucleus and are organized in a finite number of bundles. MTs polymerizing from these bundles can reach the cell tips and exert pushing forces, undergo dynamical instability, and shrink. Consequently, the nucleus feels a stochastic force in a direction opposite to the MT growth during the contact phase and moves in response.
This is summarized in the following dynamical equations in terms of nonequilibrium forces and torques on the nucleus: where X and u are the position of the centroid of the nucleus (taken as a rigid sphere) and angle between the longitudinal axis of the cell and the vector connecting the centroid of the nucleus to SPB, respectively. The effective translational (T) and rotational (R) drag matrices zðtÞ T=R are a combination of the drag coefficients of the nucleus and MT bundles. The force f r=l and the torque T r=l are applied by N r=l MTs directed toward the right (r) and left (l) cell tips ( Figure 7B, see detailed description in STAR Methods). The nucleus experiences a pushing force only when the MTs are in contact with the right or left cell boundary, where f p is the MT polymerization-driven pushing force described by the force-velocity relationship: f s ð1 À where f s is the stall force and V + and V f + is the growth velocity of MT during the free growth and dwell phase, and f e is the critical Euler buckling force of MTs, which depends on the MTs flexural rigidity k (Table S2). Contact with the cell tips lasts only until the onset of catastrophe, wherein the MTs shrink rapidly and exert zero force until it recovers, grows, and reaches the cell tip again. The catastrophe times t cat and their recovery are stochastic, and we measure their statistical distribution. In our analysis, we include a force-dependent reduction in catastrophe time via the relation t cat ðV f + Þ = t o + tcat ðV+ Þ À to V+ V f + , where t o is the mean catastrophe time at V + = 0. 32, 77 We measure these growth (and shrinkage) velocities in the WT and the different mutant strains, and find them similar ( Figures S2-S5, Table S1).
We find that the experimentally determined t cat across strains is best modeled by a gamma distribution  where pðqÞ is the prior distribution of the model parameters, pðOjqÞ is the likelihood function of the observations O given the parameters q, and pðOÞ = R pðOjqÞpðqÞdq is the marginal likelihood distribution of the observations (see STAR Methods). To estimate an informative prior, we obtain a likelihood distribution from the rather abundant and reliable data on the Mal3 strain ( Figure S8, also STAR Methods) and combine it with a flat (uniform) prior (see STAR Methods for details). We then use this informative prior to obtain the distribution of parameters that best describes the observations t cat in each strain. As a result, we find that with the exception of mcp1D, mal3D, and tip1D mutant strains (all regulators of MT growth dynamics), all other strains studied have very similar parameters associated with the t cat distribution ( Figure 7A). In addition, a truncated distribution obtained by restricting the full distribution at experimentally observed minimum and maximum values of t cat suffices to explain the observed empirical distribution.
We now proceed to estimate the remaining MT parameters, the stall force f s , and the flexural rigidity k (the growth and shrinkage velocities have been measured directly from experiments). We do this by fitting the simulated distributions of d and s x from the model Equations 1 and 2 (see STAR Methods) to the distribution obtained from the WT data (see Figure S10). This gives a value of kz1:25pNmm 2 , which goes to set f e . This value of k falls in the lower end of the range of MT stiffness obtained in in vitro measurements, 80 and is consistent with the bent and curved MTs observed in the WT as well as the small cell strains, such as the wee1-50. The fit however does not fix f s as along as f s > 1pN. If we choose f s ranging from 4 to 6 pN, then we find that the mean catastrophe times, dwell-times of MTs growth dynamics, and auto-correlation function of nucleus also matches with the experimental observations. For this range of values of f s , we are indeed in the regime f s > f e where MT buckles before the MT growth stalls.
Having established the kinetic parameters of our model that take very similar values in the different strains (Table S2), we are now in a position to compare the nuclear trajectories and the statics of nuclear positioning, d and s x , from the model with those obtained from the experiments (Figures 7C-7E). The cellstrain-dependent features that we vary in the model include the MT number N r=l , the orientation pattern of the filaments, and the cell length ( Figures 7F and 7G).
Our stochastic model recapitulates the observation that d and s x show opposite trends upon changing MT number and orientation and cell length away from the WT. For instance, from our model, we find that independent of MT number, d increases with an increase in the cell length away from WT. This is consistent with our experimental observations. Moreover, d decreases as a function of increasing MT number, keeping everything else fixed.
On the other hand, the model shows that s x decreases with an increase in cell size away from WT and decreases with increasing MT number, again consistent with experimental observations. For small MT number, the relative fluctuations of the force bearing MTs to the left and right is large, leading to an increase in s x . 61,81 Note that the MT number affects both the MT polymerization-induced forces and the effective drag on the nucleus.
Our stochastic model recapitulates the observation that it is the combination of d and s x that is functionally relevant for the position of the nucleus at the onset of mitosis. As before, this can be expressed by the miscentered fraction of SPB localization F ( Figure 7H The anisotropy in the joint distribution of d and s x provides on the control of nucleus positioning. The bivariate distribution width is much broader in the s x direction than the d direction ( Figure 7I), implying that control on d is stiffer than s x . 82,83 We find that the reliability d is predominantly controlled by the strong correlation between the left and right-oriented MT force generators and correlations in the localization of iMTOCs on the nucleus. On the other hand, the robustness s x is controlled by cell length, MT length scale, and number of MT bundles. Figure 7J shows a comparison of the distribution of d and s x between the experimental WT and the stochastic model where the cell lengths take values over the experimental range. In addition, in our model, we can vary the left/right correlation of the MT force generators. We find that a perfect correlation reduces the width of the variation along d, while a weaker correlation increases it substantially (Figures 7K and 7L).

DISCUSSION
In this paper, we have used live-cell imaging to study the fidelity of nucleus centering (and hence of the septum) in WT and genetically perturbed fission yeast cells in terms of two phenotypic objectives for nucleus position at the onset of mitosis, namely reliability (d) and robustness (s x ). The genetic perturbations that we have studied go to change cell length, MT number, and their orientational patterning and MT dynamics. In conjunction, we have analyzed a stochastic model that describes the nucleus centering in terms of MT polymerization-induced forces. Extracting the parameters of this model from experiments, we were able to compute the corresponding d and s x , as one varied the MT number, orientation, and cell length. Both experiment and theory show that high fidelity in nucleus centering is a consequence of the collective optimization of d and s x in nucleus positioning, which is akin to finding Pareto-optimal conditions. 84-87 Our analysis has interesting implications that we list below.

Optimal MT length and implications for cell length selection
The first implication is that the length scale of MTs growth needs to be comparable with the cell length. MT length-scale is determined by the distribution of catastrophe time along with the growth and shrinkage velocities. We showed that catastrophe time distribution across many perturbations can be modeled by a gamma distribution. The surprising finding is that the parameters of the distribution do not show significant difference, suggesting that MT dynamics and consequently the MT length are unaltered by the perturbations we have imposed on the cells. This is likely because the perturbations do not affect the MT (dis)assembly machinery; specific MT regulator deletion strains do produce deviations in the MT catastrophe times distribution. While it is possible that this distribution gets altered because of force-enhanced catastrophe, 32 this appears to be small. This optimality in nucleus centering in the WT, as a consequence of the matching of the MT length (in the milieu of the yeast cell) and the WT cell length, might have implications for cell size selection in fission yeast.

Proper MT number and orientation for optimal force patterning
The proper centering of the nucleus is contingent on proper nonequilibrium force patterning that is generated by MT-polymerizing forces. Thus, the force patterning is described by the orientation of the MTs, their arrangement in bundles, and the number of MT bundles. In fission yeast, this is achieved by a noncentrosomal MT organization with MT arranged in a small number of bundles emanating from the nucleus. Our study shows that the optimal orientation of MTs is when they form bundles coaligned with the long axis of the cell. Deviation from coalignment leads to aberrant centering. Any change from this arrangement increases d and s x . Organizing MTs in coaligned 3-4 bundles distributed around the nucleus diminishes transverse fluctuations s y driven by the torque.
The performance of the two phenotypic objectives improves with an increase in coaligned MTs; 12,61,81 however beyond a certain number, there is no significant gain. Beyond approximately N = 22 MT, the improvement if any is marginal, suggesting as in any biological control system, a tradeoff between effectiveness and economy.  iScience Article We note that both in the experiments and the stochastic model of the WT, the cell-to-cell variation in robustness (s x ) is much larger than in reliability (d), suggesting that d is under stiff control by the cell, while s x is under a sloppier control. 82,83,88 This, while expected, deserves further attention. The control of s x is affected by the force-dependent catastrophe of MTs that we have included in our stochastic modelthis acts as an adaptive feedback control, and possibly operates over a certain range of cell lengths. On the other hand, cellular control of d is achieved by ensuring left (L) -right (r) symmetry in the MT-polymerizing forces. At present, we do not have a complete understanding of the molecular basis of this force correlation, which will likely involve anti-parallel crosslinkers such as Ase1 7 and motor proteins such as Kinesin-14 or Dynein, 7,89,90 an investigation of this cellular control mechanism is a task for the future.

Limitations of the study
Our research findings and theoretical model suggest that achieving a high level of fidelity through the combination of two objectives-reliability and robustness-leads to minimal errors in septum positioning. However, it is important to note that the strict mathematical sense of optimality cannot be proven through experiments that involve perturbations in only a limited number of directions. Therefore, our demonstration of the optimality of the WT method is valid only within the scope of the perturbations studied.

STAR+METHODS
Detailed methods are provided in the online version of this paper and include the following:

Lead contact
Requests for reagents should be directed to and will be fulfilled by the lead contact, Phong T. Tran (phong. tran@curie.fr).

Materials availability
Strains used in this study are available upon request.
Data and code availability d Microscopy data reported in this paper will be shared by the lead contact upon request.
d All original code has been deposited at Github and is publicly available as of the date of publication. The link is listed in the key resources table.
d Any additional information required to reanalyze the data reported in this paper is available from the lead contact upon request.

EXPERIMENTAL MODEL AND SUBJECT DETAILS
Genetics, cell culture, and strains Standard yeast genetics methods are followed to insert fluorescent markers and to create strains. 93 The EnvyGFP plasmid was a gift from the Bi Lab at the University of Pennsylvania. We use Gibson assembly to construct a pFa6a based plasmid to endogenously tag Atb2 with EnvyGFP inserted at the N-terminal and transformed it to the fission yeast cell. The list of primers used in this construct is given in the key resources table. A list of the strains used in the work can be found in the key resources table. In general, cells were maintained at 25 C on agar plates containing YE5S media. For microscopy based experiments, a small number of cells were inoculated in liquid YE5S medium a night before the experiment and incubated at 25 C under shaking for 16-18 hours, and then harvested when they reached an optical density of $ 0.5 G 0.2 a.u.

Microscopy and image analysis
Imaging was performed using a spinning disk confocal microscope. Specifically, we used a Nikon Eclipse Ti2 inverted microscope, equipped with a Nikon CFI Plan Apochromat 100x/1.4 NA objective lens, a Nikon Perfect Focus System (PFS), a Mad City Labs integrated Nano-View XYZ micro-and nano-positioner, a Yokogawa Spinning Disk CSU-X1 unit, and a Photometrics Evolve EM-CCD camera controlled by Molecular Devices MetaMorph 8.0 software. For GFP and mCherry imaging, we used solid-state lasers of 488,nm (100,mW) and 561,nm (100,mW).
To ensure experimental consistency, we used the CherryTemp microfluidics-based thermostat to precisely control the temperature during imaging. We aimed to analyze 20-50 cells in each case. To mount the samples, we immobilized them directly on a ready-made microfluidics chip-based setup (CherryTemp, CherryBiotech) using YE5S in 2% agar. After setting the temperature using the CherryTemp system, we waited for approximately 30 minutes for the cells to acclimate to the new conditions before proceeding with imaging, with the exception of the microscopy of temperature-sensitive mutants wee1-50 and cdc25-22, for which we waited approximately 2 hours before imaging. The microscopy was performed at 35 C. The specific conditions for each type of microscopy experiment are provided below.
Live cell microscopy and segmentation of SPB: A z-stack comprising a total of 7 focal planes spaced iScience Article the segmented cell images, removed background via thresholding, and found a 3x3 pixel window with the highest integrated intensity.
Live cell microscopy and segmentation of Nucleus: We followed a similar procedure as described above, with the exception that the stack was acquired over a period of approximately 1 hour with a time interval of 30 s. We used the same procedure to segment the cell and SPB as described earlier.
To segment the Cut11-GFP signal, we first constructed an image with maximum projection in the GFP channel of the segmented cell images, then binarized the images by selecting the pixels with the top 5% intensities and selecting the largest object.
Live cell microscopy and segmentation of MT in EnvyGFP-Atb2 strains: A z-stack comprising a total of 13 focal planes spaced 0.5 mm apart was acquired in the GFP (Exposure time of 100 ms, EM Gain 400) and mCherry (Exposure time of 100,ms, EM Gain 400) channels. The stack was acquired for approximately 10 minutes with a time interval of 6 seconds. The cell was segmented by setting a threshold in the GFP channel images after background subtraction and taking a maximum projection. Following this, we constructed a color-combined image stack using the GFP and mCherry channels. The MTs that emanate from the SPB were manually segmented using the ImageJ segmented line tool (see Figure S7) and analyzed using custom script written in MATLAB.
The number of MT bundles, total MT mass, and orientation were measured using static z-stacks (using the microscopy procedure described above). The number of MT bundles was counted manually. The quantification of the total MT mass and orientation was done using the following steps. First, we segmented the MT cytoskeleton using the ''Trainable Weka Segmentation'' plugin in ImageJ. 95 A manually curated dataset was used to train the classifier to distinguish between the MT cytoskeleton and cytosol. The segmented probability histogram was bimodal, and we used the value corresponding to the minima to segment the MT cytoskeleton and create a segmented mask. The MT mass was estimated by directly integrating the total intensity in a sum projection under the segmented mask. The orientations were calculated using the OrientationJ plugin in ImageJ 96 under the segmented mask.
Live cell microscopy and segmentation of MT in Mal3 strains: A z-stack comprising a total of 13 focal planes spaced 0.5 mm apart was acquired in the GFP (Exposure time 100,ms, EM Gain 400) and mCherry (Exposure time 100,ms, EM Gain 400) channels. The stack was acquired over a period of approximately 10 minutes with a time interval of 6 s. For each cell, we constructed two types of kymographs (see Figure S8): 1) a cell-wide kymograph was used to score catastrophe events associated with MTs that grow longer than 5 mm in length, and 2) a kymograph using a 2x50 pixel window centered around the SPB was used to score events that occurred on a short time scale (length < 5 mm). In both cases, we manually traced the trajectory of the MT growth and analyzed using custom script written in MATLAB.
Microscopy and segmentation of septum: We incubated the cells at 35 C for 3 hours and then imaged them to quantify the septum positioning. As known, the cdc25-22 cell does not initiate the mitotic phase at the restrictive 35 C. Thus, to quantify the septum position in cdc25-22 cells, we incubated the cells at 35 C for 3 hours and then quenched the temperature to 25 C. Most of the cells undergo mitosis immediately and form a septum after 40 À 50 mins. To segment the septum position, we manually segmented the cell end and septum using the point-tool in ImageJ.

Stochastic model for MT-dependent nucleus centering
Here we discuss a model for microtubule (MT) pushing-force dependent nucleus centering to help us understand the diversity of phenomena observed in our experiments. The framework takes into account the motion of the nucleus because of stochastic forces arising due to MT polymerization against the cell wall. The model also accounts for the non-linear growth dynamics of MT, number, and orientation of MT, etc., inside the cell. Since the imaging plane is an x-y projection, we will take the cell to be a 2-dimensional rectangle of length 2L and width 2R (see Figure 7B). The nucleus is modeled as a rigid circle, and MT nucleation sites are situated at MTOCs located at the periphery which nucleates MTs with swivel points.
As a consequence, the nucleus undergoes translational dynamics according to, where l i ðtÞ is the length of the longest MT emanating from the nucleus perimeter from angle j i and r MT radius of the MT. 24 The expressions Equations 10 and 11 are valid in the infinite rod limit, i.e. when rMT ClD ( 1: The above forms of the hydrodynamic drag are approximations to the real system. Both the finite geometry of the yeast cell that confines the cytoplasmic fluid and the permeability of the nucleus as it moves through the incompressible cytoplasmic fluid, affect the form of the drag. Denoting the permeability of the nucleus by k, the drag coefficient reduces by z nuc = 6phrnuc . 101 On the other hand, the drag on the nucleus increases because of geometric confinement. Treating the yeast cell as a cylinder of radius R, with the assumption of axial symmetry, the drag coeffcient follows z T nuc = 9p 2 ffi ffi 2 p hrnuc 4ð R À rnuc R Þ 5=2 , evaluated to the lowest order in R À rnuc R . 102 Finally, we have treated the nucleus as a rigid sphere. The deformability of the nucleus can also change the coefficient of drag. We have experimentally estimated the translational diffusion of the nucleus (using MBC treated cells) and found the translational diffusion length scale to be of order 10 À2 mm which is significantly smaller than the variation of the nucleus position because of active forces (which are of order 10 0 mm). Consequently, we neglect the effect of thermal forces in Equations 5 and 6.
The microtubules apply forces only when they are in contact with the cell boundary and this is given by Where f p is the pushing force applied by MTs because of polymerization when MTs are in contact with the cell cortex and f e is the critical buckling force. f p is given by: assuming that the applied force prominently affects the 'on' rate (the probability of a tubulin subunit to get intercalate at the tip of microtubule) of tubulin subunits (see Peskin 1993, 103 Dogterom 1997 31 ). Here f s is the stall force (i.e. force at which the MT growth halts), V o + and V f + are the mean growth rate of MTs during the free-growth phase and during the contact-phase with cell boundary (while MTs applying pushing force), l i ðtÞ is the length of the MT. b n = ½cosðq i Þ; sinðq i Þ is the unit orientation vector of the MT. The Heaviside theta function QðxðtÞ + r cosðu + jÞ À l i ðtÞcosðu i + j i + q i Þ À LÞ enforces the condition that microtubules only apply force when they are in contact with the cell boundary. f e is the critical buckling force require to buckle MT with flexural rigidity k and is given by: where l io is the length of MT which can be constrained inside the geometry of the cell after buckling. The inclusion of l io emulate constraints imposed by cell geometry on the configurations of buckled MTs.
The growth dynamics of microtubule is stochastic where microtubules show dynamic instability. During the growth phase, MT polymerize with rate V o + and during the shrinking phase, MT depolymerize with rate V À .
The microtubule will catastrophe (i.e. jump from the growing to shrinking phase) such that the catastrophe time t cat has a distribution given by Pðt cat Þ. We also model force dependent reduction of catastrophe time.
Here, if there is no force applied on MT then the catastrophe time t cat has a distribution given by Pðt cat Þ. In the presence of force, the catastrophe time is given by t cat ðV f + Þ = t o + tcat ðV+ Þ À to V+ V f + , where t o is the mean catastrophe time at V + = 0. 32,77 The value of t o is not known in vivo, but it is found to be $ 25 sec in the reconstitution experiments (see Janson 2003 32  is the mean catastrophe time of MT with age T, 78 and Dt is the time-step. The switch from shrinking state to growing state occurs instantaneously if the l i % 0. In the Numerical integration step, we update the position of the nucleus using the forward Euler method for explicit integration. We initiate the calculation by assigning a fixed number of MT-anchoring sites which are distributed on the nucleus randomly. The number of nucleation sites (i.e. number of bundles) is equal to ððN r + N l Þ O4Þ (i.e. quotient of the fraction). The MTs are then randomly assigned to these sites and the MT-anchoring site with the maximum number of MT is considered the SPB to defining j SPB . MTs orientation q follow the experimentally determined orientation statistics of MTs and each time MTs switch from the shrinkage to growing state a new q is assigned to an MT. The q of the longest MT originating from the each anchoring site is considered in evaluation of the drag forces. Throughout this study, we assumed N r = N l (unless stated otherwise). We started each simulation with the nucleus located on the tip of the cell and performed a simulation for z 7200s (a generation time of fission yeast cell), unless noted otherwise.
We utilize Pðt cat Þ obtained from Mal3 strain (cdc25-22 GFP-mal3 sid4-mCherry) and anticipated that the difference in catastrophe times seen in experiments can be attributed entirely to the force dependent change in velocities. Thus the model has only two free parametersk and f s . We systematically vary these parameters to search for a set of conditions which matches the experimental observations (see Figure S10). We see that that for k z 1.25pNmm 2 the value of s x matches with experimental observations, attributed to the dominance of buckling and f s only becomes a limiting parameters. For f s , ranging from 4À6 pN, the Ct cat D and dwell-times also matches with the experimental observations.

Segmentation and quantification of microscopy data
The segmentation and analysis were done using semi-automated scripts written in ImageJ macros and MATLAB. 91,92 The specifics for segmenting each type of microscopy experiment are described along with the microscopy condition above in the method details section of STAR Methods.

Determining MT-catastrophe time distribution using bayesian inference
Arriving at the correct statistics of the MT growth dynamics is crucial for a description of the MT-driven processes. However, estimating MT growth dynamics by observing MTs (EnvyGFP-Atb2) has limitations -we can only reliably follow the growth dynamics of the longest MT emanating from a MT-bundle. This limits the length (and time) window to observe the MT's catastrophe events, and consequently, the measured catastrophe length or time distributions only represents a truncated subset of the entire distribution. Moreover, the lengths (and time) window for observing the MTs depend on the MT number, orientation and cell length, which may introduce systematic biases in the analysis, or may result in sparse statistics. Instead of using conventional approaches, such as maximum likelihood estimate (MLE), an alternative method is to ask how well the data explains a set of model parameters. This is achieved by the Bayesian inference method. 48 Bayes rule assigns the posterior probability on a set of distribution parameters (q) given the ob- where pðqÞ is the prior distribution of the model parameters, pðOjqÞ is the likelihood function of the observations O given the parameters q and pðOÞ = R pðOjqÞpðqÞ dq is the marginal likelihood distribution of the observations. Apart from the dataset of observations O, we also require the following to evaluate the above expression: 1. a probability distribution function (e.g. exponential, gamma, etc.) which is parametrized by the set q and, 2. a distribution of prior probabilities of pðqÞ.

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