Why is mitotic spindle important

Menin has a bookmarking role during mitosis, but its expression is reduced during mitosis compared with interphase Interestingly, a nontranscriptional role for MLL in mitosis was recently described 20 , 21 and we hypothesize that menin will also have a nontranscriptional activity during cell division. To better understand how menin misregulation can promote tumorigenesis, we studied menin specifically during cell division. Depletion of menin in HeLa cells and HCT cells led to defects in spindle assembly, chromosome congression, lagging chromosomes during anaphase, cytokinetic defects, multinucleated interphase cells, and cell death.

In addition, pharmacological inhibition of the menin-MLL1 interaction revealed similar cell division defects. These data define a function for menin in ensuring proper mitotic spindle assembly and cell division. Furthermore, they advance our understanding of how mutation of MEN1 that is likely to lead to misregulation of cell division promotes the downstream disease pathology associated with endocrine tumors that harbor MEN1 mutations. Immunofluorescence microscopy was performed as described previously 24 with the following modifications.

The Leica Application Suite 3D Deconvolution software was then used to deconvolve the images and they were subsequently exported as tagged image file format TIFF files.

Images were then converted to Audio Video Interleave movies. Each frame represents a minute interval. To create the green fluorescent protein GFP -menin expression plasmid, the full-length open reading frame of human wild-type menin from pCR2. Western blotting was performed as previously reported Supplemental figures and video can be found in an online repository To better understand the function of menin and its misregulation in tumorigenesis, we began by asking if menin localized to microtubule structures similar to MLL1 during cell division.

Subcellular localization of both subunits was determined in this study. During mitosis, menin localized robustly to the spindle poles during prophase and prometaphase and to a lesser extent during metaphase and postmetaphase, similar to MLL1-N and MLL1-C [ Fig. Cell cycle subcellular localization of menin and MLL1-N. Note that menin localizes to the mitotic spindle poles and mitotic spindle during early mitosis and to intercellular bridge microtubules during cytokinesis, similar to MLL1-N.

That previous studies could not demonstrate menin localization to the mitotic apparatus could be due to different immunofluorescence microscopy conditions or different lots of the antibody As expected, the siMEN-treated cells showed a decrease in menin protein levels by immunoblot analysis and menin was not observed at the mitotic apparatus Fig. To further address this issue, we visualized overexpressed GFP-tagged menin. Together, these data indicated that menin was localizing to microtubule-based structures during mitosis, spindle poles in early mitosis, and intercellular bridge microtubules during cytokinesis, similar to MLL1.

Importantly, to our knowledge, MLL1 had not been previously shown to localize to intercellular bridge microtubules. Next, we asked if menin had a role in cell division by characterizing mitotic defects in HeLa cells deficient in menin expression.

HeLa cells were transfected with siCont or siMEN, validated to decrease menin protein levels by immunofluorescence and immunoblot analyses Fig. Although siMEN cells showed no apparent perturbation in the localization of MLL1-N and MLL1-C to the mitotic spindle and intercellular bridge microtubules, they did show an increase in the percentage of mitotic cells with defective spindles multipolar and unfocused; siMEN: These results indicated that depletion of menin led to defects in early mitosis and cytokinesis.

Depletion of menin leads to cell division defects. Note that siMEN cells show multiple aberrancies, including multipolar spindles and unaligned chromosomes in metaphase, lagging chromosomes in anaphase, multipolar cytokinesis, and multinucleated interphase cells. Arrows point to uncongressed chromosomes in a metaphase cell panel with four arrows and lagging chromosomes in a telophase cell panel with one arrow. B Quantification of the percentage of mitotic cells with defective spindles, uncongressed chromosomes, and cytokinetic defects and interphase cells with more than one nucleus multinucleated.

Representative cell division defects are shown, including cytokinetic arrest, multipolar cell division with cell death, and regression of a dividing cell into a binucleated cell. Time is in minutes. D The percentage of cells undergoing normal cell division, dying during cell division, undergoing aberrant cytokinesis and failing cytokinesis, and regressing to a binucleated state were quantified for siCont- or siMEN-treated cells.

Consistently, live-cell time-lapse fluorescent microscopy with HCTGFP-H2B cells showed that siMEN cells exhibited cell division defects, including cytokinetic arrest with the two cells linked by a cytokinetic bridge, a failure to divide with regression into one binucleated cell, and cell division followed by death of one or both cells [ Fig. Together, these data indicated that depletion of menin led to defects in spindle assembly, chromosome congression, chromosome segregation, and cytokinesis, which resulted in multinucleated interphase cells and an increase in cell death.

Next, we asked if the menin-MLL1 interaction was important for cell division by analyzing the consequences of inhibiting the menin-MLL1 interaction pharmacologically with the small molecule inhibitor MI-2 Although MI-2—treated cells showed no apparent perturbation in the localization of MLL1-N and MLL1-C to the mitotic spindle and intercellular bridge microtubules, they did show an increase in the percentage of mitotic cells with defective spindles multipolar and unfocused; MI MI-2—treated cells also had a pronounced increase in the percentage of cytokinetic cells undergoing a defective cytokinesis MI These results indicated that pharmacological inhibition of the menin-MLL1 interaction led to defects in early mitosis and cytokinesis.

Note that MI-2—treated cells show multiple aberrancies, including multipolar spindles and unaligned chromosomes in metaphase, lagging chromosomes in anaphase, multipolar cytokinesis, and multinucleated interphase cells. The arrow points to lagging chromosomes in a telophase cell. Representative cell division defects are shown, including multipolar cytokinesis and regression of dividing cells into binucleated cells.

D The percentage of cells undergoing normal cell division, dying during cell division, undergoing defective divisions and failing cytokinesis, and regressing to a binucleated state were quantified for DMSO- or MI-2—treated cells.

Consistent with these results, live-cell time-lapse microscopy showed that MI-2—treated HCTGFP-H2B cells exhibited cell division defects, including cytokinetic arrest with the two cells linked by a cytokinetic bridge, a failure to divide with regression into one binucleated cell, and cell division followed by death of one or both cells [ Fig.

Together, these data indicated that disruption of the menin-MLL1 interaction led to defects in spindle assembly, chromosome congression, chromosome segregation, and cytokinesis, which resulted in multinucleated interphase cells and an increase in cell death. Because MLL1 specifically the C-terminal subunit regulates Kif2A localization to the mitotic spindle to ensure proper chromosome alignment during mitosis 20 , we analyzed whether menin similarly affected Kif2A localization during mitosis.

However, the ability of menin to localize to the mitotic spindle during mitosis was reduced in the presence of MI-2 treatment Fig. Together, these results indicated that in contrast to depletion of MLL1, depletion of menin or inhibition of the MLL1-menin interaction did not affect the localization of Kif2A during mitosis.

In addition, these results indicated that the MLL1-menin interaction is important for the localization of menin to the mitotic spindle. Note that Kif2A remains localized to the mitotic spindle in MI-2—treated cells, whereas menin localization to the mitotic spindle decreases. Menin is mutated in patients with MEN1 syndrome and the related sporadic endocrine tumors It was previously reported that menin remains bound to chromatin with MLL1 during mitosis albeit at reduced levels compared with interphase, suggesting it shares a mitotic bookmarking role with MLL1 We demonstrate that menin also has a nontranscriptional role during mitosis.

Menin localized to the mitotic apparatus specifically to the spindle poles in early mitosis and intercellular bridge microtubules during cytokinesis, similar to the subcellular localization of MLL1. In addition, live-cell time-lapse video microscopy showed that cell division in menin-depleted HCT cells frequently resulted in either cell death or multinucleated cells.

On the basis of these observations, we propose that, like MLL1, menin plays a dual role during mitosis. Interestingly, the results of our studies suggest there is a direct role for the MLL1 N -terminal subunit with menin during mitosis.

Long protein fibers called microtubules extend from the centrioles in all possible directions, forming what is called a spindle. Some of the microtubules attach the poles to the chromosomes by connecting to protein complexes called kinetochores. Kinetochores are protein formations that develop on each chromosome around the centromere, which is a region located near the middle of a chromosome.

Other microtubules bind to the chromosome arms or extend to the opposite end of the cell. During the cell division phase called metaphase, the microtubules pull the chromosomes back and forth until they align in a plane along the equator of the cell, which is called the equatorial plane.

The cell goes through an important checkpoint to ensure that all of the chromosomes are attached to the spindle and ready to be divided before it proceeds with division. Next, during anaphase, the chromosomes are simultaneously separated and pulled by the spindle to opposite poles of the cell.

The correspondence between fluorescent tubulin intensity and the mass of tubulin polymerised in the spindle can also be used to calculate the abundance and critical concentration of tubulin subunits within the fission yeast cell Table 1. This calculation uses the well-defined structure of microtubules Howard, to estimate the total number of tubulin subunits within the spindle.

The intensity ratio between GFP-labelled tubulin in the spindle and cytoplasm can then be used to extrapolate this quantity to the abundance of tubulin subunits in the cytoplasm.

Finally, the regular spherocylindrical shape of fission yeast cells, with a uniform radius and a well-defined length upon entry into mitosis, enables accurate estimates of the intracellular volume and thus concentration to be obtained. The estimates of tubulin abundance are in good agreement with mass-spectrometry studies Marguerat et al. Yeast samples were prepared as in Roque et al.

Tilt-series were reconstructed, joined in the cases where spindles spanned several adjacent subsections, and tracked manually using the software package IMOD Kremer et al. In the electron tomograms, microtubules appear as cylindrical tubes with a diameter of around 18 nm compared with the 25 nm expected for protofilament microtubules. The microtubule shrinkage is caused by the freeze-substitution methods used to fix the cells, but other aspects of the microtubule structure appeared unperturbed.

The granularity of the isotropic fibre tracking analysis IFTA solution is controlled by a single parameter, R s , which specifies the separation of neighbouring coordinates that define the axis of the spindle in three-dimensions.

This parameter is set to nm for tracking anaphase B spindles in fission yeast and at nm for computing the histograms of nearest-neighbour distances and microtubule packing angles.

The IFTA algorithm proceeds by first detecting the pole with largest number of microtubule minus-ends that lie within the threshold distance, R s. The centre-of-mass of these ends is used as the first point defining the spindle. A sphere with a radius, R s , is then centred at the pole, and used to detect the positions where microtubules intersect the ball's surface.

A constrained optimisation calculation is then used to determine the point lying on the sphere's surface that minimises the mean-squared distance from the microtubule crossings. This point is selected as the second position in the three-dimensional interpolation of the spindle axis whilst microtubule coordinates that lie within the sphere are masked. This procedure is repeated until the spindle coordinates reach the opposite spindle pole.

The transverse organisation of the spindle is then determined by detecting the positions where microtubules intersect a plane halfway between and perpendicular to the points that define the spindle axis.

The calculation of the transverse stiffness is made by assuming that microtubules are uniform, hollow cylinders and that the cross-linkers are rectangular support elements with identical material properties.

The parameters are taken from the known dimensions of microtubules and the microtubule numbers and organisation determined in this study. Alternative boundary conditions lead the critical force to be altered by a multiplicative constant. The third property of the beam that determines the critical force is the area moment of inertia Figure 2.

The contribution that a single microtubule with its centre a distance, y, from the neutral axis makes to the bundle's area moment of inertia in the y-direction, I xx , is given by the following equation. The term A MT is the microtubule cross-sectional area and I MT is the isotropic moment of inertia of a single microtubule about its axis.

A symmetric expression exists for the moment of inertia in the x-direction, and the product moment of inertia I xy , is given by. The components of the moment of inertia tensor can be obtained by summing the contributions of individual microtubules. The formula for the area moment of inertia tensor, J , can thus be written compactly in matrix notation as.

A similar expression can be obtained for a composite beam containing microtubules and cross-linkers, if it is assumed that the cross-linkers are rectangular support elements with a width, w, and a height, h and the same material properties as microtubules.

The effect that polymer mass conservation has on the spindle's stiffness can be investigated by treating the spindle as a solid, homogeneous cylinder. If the volume of the cylinder, V , is held constant, to model the conservation of polymer mass, then the cylindrical beam becomes thinner as it elongates. The response of the spindle to compressive forces was investigated using the cytoskeletal modelling software Cytosim, which solves the over-damped Langevin equations of cytoskeletal filaments using an implicit numerical integration scheme Nedelec and Foethke, Simulation results were analysed using Matlab The MathWorks inc.

The simulations were designed to reproduce the morphology and biophysical characteristics of each spindle sufficiently closely to estimate the spindle's critical force.

The SPBs were modelled as cylindrical elastic solids, associated with a scalar drag coefficient. Each microtubule was connected to the SPB by a pair of Hookean springs. The first of these was coupled to the minus-end of the microtubule, and was given a large elastic constant to model the high resistance of wild-type SPBs to pushing forces Toya et al. The steric exclusion between microtubules was implemented using a one-sided quadratic potential with a minimum at the steric radius of 30 nm Loughlin et al.

The number and length of microtubules in the models of spindles from wild-type fission yeast cells were determined directly from ET reconstructions, whilst the SPB separation or spindle length was set to the IFTA-derived contour length between the poles of the ET spindle.

The lengths of microtubules in budding yeast and cdc The budding yeast spindles contain short microtubules with lengths less than 0. These fibres are unlikely to contribute to the structural integrity of the spindle, and account for a small proportion 7. These microtubules were therefore neglected in the spindle models. Two of the budding yeast spindles also contain pole-to-pole microtubules that cannot be unambiguously assigned to a specific spindle pole. In the first spindle numbered 12 in Winey et al.

The second spindle numbered 14 in Winey et al. Having determined the spindle length, the number of microtubules and their lengths, the positions of the microtubule minus-ends at each SPB are set by sampling a random position on the circular face of each SPB using a Monte Carlo method.

Rejection sampling was first used to sample the area on the disk's surface with uniform probability. The Euclidean distance between a candidate point and the other microtubules was determined, and the point was only accepted if its separation was greater than the twice the steric radius of each microtubule.

This process was repeated until the SPB was populated with the correct number of microtubules. This procedure set the position of the microtubule minus-ends with respect to the SPB, with the position of plus-ends and the transverse organisation of microtubules at the midzone determined by simulating cross-linker attachment and detachment from the microtubule lattice.

The SPBs were confined to the x-axis throughout the simulation to aid visualisation. The simulation of microtubule organisation at the spindle midzone was performed using cross-linkers that only bind to pairs of anti-parallel microtubules Janson et al. The cross-linkers were also confined to cylindrical region with a total length of 2. When bound to a pair of microtubules, cross-linkers behave as elastic bridging elements that set the centre-to-centre between pairs of microtubules at 50 nm.

The stiffness of the cross-linkers is consistent with the known Young's modulus of the alpha-helical class of proteins to which the dimeric kinesins and Map65 proteins belong.

The microtubules were also subjected to weak 20 Pa centring forces to prevent rotational diffusion of the microtubules away from the spindle axis and thus ensure that cross-links were formed between the two halves of the bipolar spindle. Throughout the initialisation of spindle architecture, the SPBs were connected to each other by a stiff spring with the same resting length as the spindle to prevent the SPB separation being altered by diffusion.

After simulating the spindle for fifty seconds, almost all of the cross-linkers are bound to the midzone and the two halves of the spindle are strongly connected. Under these conditions, the cross-linkers are capable of forming the idealised square-packed arrays observed in yeast spindles, albeit with lower efficiency than we observe in electron tomogram reconstructions of wild-type spindles data not shown.

Upon completion of the initialisation step, the rate with which cross-linkers detach from the microtubule lattice was set to zero in order to probe the spindles' elastic response to increasing forces.

The stiffness of the elastic element was maintained at the maximal value for a further 50 s, during which time the critical force on the spindle was determined. This was carried out by averaging the forces borne by the elastic element connecting the pair of SPBs.

In simulations of spindles with elastic reinforcement, each microtubule model point was confined by an elastic potential with a given degree of stiffness Figure 6A—D. Simulation parameters are provided in Table 2.

The simulations of the null models of spindle architecture were identical to those used to determine the critical force of wild-type spindles, except that microtubule length and number were sampled from probability distributions. The objective of this procedure was to sample a large number of alternative spindle morphologies to investigate the degree to which wild-type spindles are mechanically optimal.

In constructing the null statistical models, the number of microtubules emanating from each SPB was sampled from a Poisson distribution with a mean equal to that observed in the wild-type spindle with the same length. The lengths of the microtubules in each random model were determined by randomly partitioning the total polymer present in the wild-type spindle between the N MT microtubules.

In cases where the length of a sampled microtubule exceeded the spindle length, the microtubule was truncated to the length of the spindle with the remaining polymer used to set the length of one or more additional microtubules that were assigned to one of the two SPBs at random.

This procedure increased slightly the average number of microtubules in the random spindles but ensured that the overall polymer mass was conserved. At SPBs that contained in excess of six microtubules, the radius of the circular face of the SPB was increased so that its area increased linearly with microtubule number, and that the density of microtubules on the surface of the SPB was constant.

The viscous drag on the fission yeast nucleus is substantially larger than is predicted by Stokes' law due to the narrow separation between the nuclear envelope and the enclosing cell wall Foethke et al. The equation for translational motility of the nucleus in the cell geometry can be approximated as. A description of the other variables is shown in Table 3.

After ablation of the spindle midzone, the forces acting on the daughter nuclei are. An exponential fit to the relaxation data provides values for these variables, which can then also be combined to give a rough estimate of the total force resisting spindle elongation Table 3. An edited version of the letter sent to the authors after peer review is shown, indicating the substantive concerns or comments; minor concerns are not usually shown.

Reviewers have the opportunity to discuss the decision before the letter is sent see review process. Similarly, the author response typically shows only responses to the major concerns raised by the reviewers. Your article has been favorably evaluated by Richard Losick Senior editor and 3 reviewers, one of whom is a member of our Board of Reviewing Editors.

The Reviewing editor and the other reviewers discussed their comments before we reached this decision, and the Reviewing editor has assembled the following comments to help you prepare a revised submission.

This interesting paper combines computer simulations and electron tomography to understand spindle mechanics. This is a very important topic, and the approach that the authors use is novel and exciting.

While EM structures of yeast spindles have been discussed before Ding et al. Also, the combination of mechanical models with EM data is novel and interesting.

However, in the manuscript, it is not carefully explained what was known before and what the new contribution of the present work is. This needs to be better clarified. Another more serious problem with the present manuscript is that it is, in many points, unclear and very difficult to read, in particular when it comes to the analysis of compressive strength.

The main text is insufficient to understand key arguments. When following the referrals to figure captions and methods, it is often difficult to find some of the information needed. The figure captions in general are not fully satisfactory.

All of these weaknesses of the presentation make the paper hard to read. Below is a list of points that the authors should address in order to improve the manuscript. I do not understand how Figure 1 supports those claims. Figure 1D shows that spindles with similar lengths have very different cross-sections, different degrees of microtubule overlap, and even different microtubule lengths. The cross-sections also look quite disordered, and Figure 1E shows very broad distribution of packing angles, thus, it seems to me that the spindles are not well ordered.

It would be helpful if the authors tried to quantify the degree of order and the extent to which the spindles are stereotyped, perhaps by measuring a 2D crystalline order parameter. This is a very interesting methodology. An alternative null model would be to generate random bundles of microtubules that would be expected to form with unregulated, non-specific crosslinkers.

It would be very interesting for the authors to perform such simulations to test if there is really something special about the arrangement of microtubules in the spindle, or if the same resistance to buckling would result from unregulated crosslinked arrays of microtubules.

Since the spindles are already bent to some extent, then forces on the poles cannot cause a buckling which would only occur from a starting straight configuration , but additional forces could cause further bending. Given this, the relevance of studying how a simulated straight spindle would buckle is unclear.

It would be helpful for the authors to explain their reasoning in studying buckling. It would also be helpful for the authors to comment on the observed bending of the spindle in the tomograms and explain how it relates to their studies of the mechanics of bending of spindles. However, it does not become clear whether an optimal strength is needed to withstand the compressive forces during anaphase B.

What is the compressive strength needed for the task? How many individual microtubules would be needed to withstand the compressive forces that arise? There is a discussion of this subject in the Discussion section, but it is unclear to me. Also, the text following this sentence is unclear. I also do not see how Figure 2—figure supplement 1 should help to provide clarification. Where is cross-linker density defined and discussed?

It would be useful to have clear definitions of these terms and to clarify their relationship. Switching from one term to another in the text is confusing as the reader is not sure whether a different spindle property is meant. However, I did not find a discussion of this simplification and whether, in the case of microtubule bundles, this is a good simplification.

How is overlap maximized? Does it mean overlap along the whole length? What does optimal mean? What is optimized under what conditions? It sounds like an exaggeration and the reader is left in the dark as to what exactly is meant. This should be clarified. This point could be used to motivate the numerical work discussed in the Results.

As the reviewers mention in several of their comments, it is correct that any demonstration of spindle optimality is contingent on the model that is used, which, for a system as complex as the spindle, is always necessarily a simplification. We have therefore altered the tone of the article, so that we instead focus on the mechanical properties of the spindle that are likely to enhance its ability to generate force rather than on its optimality per se.

We have also revised the text extensively to improve both its clarity and precision, using a more precise terminology throughout. Our revised manuscript includes two additional figures that respond to specific comments made by the reviewers. The first of these presents a more detailed description of the calculations of spindle stiffness.

This is intended to provide the reader with a more intuitive picture of how the architecture of the spindle influences its mechanics.

We would be happy to receive guidance from the editorial team on this matter, considering that the necessary changes can easily be implemented by us. This figure explores the effect of external elastic reinforcement on the forces that the wild-type spindle can support during its elongation.

As mentioned in the letter above, we have made extensive changes to the manuscript to improve its clarity and justify its major findings more carefully. The contribution from Ding et al. The description of our wild-type electron tomograms now includes a comparison with the earlier serial-section reconstructions of cdc The later theoretical sections of the paper explore the mechanical properties of both wild-type and cdc Comparisons of these two architectures provide insight into how the fission yeast spindle adapts to changes in cytoplasmic volume.

The nearest-neighbor approach has been used extensively for measuring distances and packing angles Ding et al. The most notable of these is associated with the angle measurements, which are defined with respect to the two nearest MTs.

These triplets of MTs are not necessarily all from the same crystalline unit cell, which leads to a less uniform distribution of packing angles and distances being measured for the spindle. This effect partly explains the broad distribution of packing angles that are present, particularly in the earlier spindles.

We agree that the transverse organization of cytoskeletal arrays is an area worthy of future study that would certainly benefit from the application of techniques from solid-state physics, but these techniques are relatively involved.

We also feel that this quantitation would not provide a great deal of additional insight beyond the simple and intuitive nearest-neighbor method, and is therefore beyond the scope of this paper. This is an interesting suggestion, as the impact of cross-linker specificity on the organization of the spindle is currently unknown. It is also correct that the number of potential spindle organizations that could possibly exist is far larger than has been probed in the current study.

However, these architectures are likely to be subject to other constraints on how they are formed and mediate elongation that cannot be probed using the simulation methodology that is described in the paper. The simulations were designed to approximate the organization of the spindle at a particular stage of its elongation without needing to simulate its progression through the earlier stages of mitosis.

It is unclear whether the alternative architectures could ever be realized in vivo. We have responded to this comment by qualifying the discussion of this result more carefully. The bends in the spindle are an interesting feature of the electron tomographic reconstructions, and are indeed present in all of the anaphase B spindles. These deflections are more pronounced in the early anaphase B spindles, where they appear to be inconsistent with the linear spindle morphology that can be observed in living cells via light microscopy.

As the reviewer also rightly suggests, it is unclear how a buckled spindle could elongate at such a uniform rate in the presence of variable resistive forces. These two lines of evidence suggest that the deflections are caused partially or entirely by the standard preparation of the cell for electron tomography and that the native spindles have a straighter morphology. Live-cell imaging studies suggest that the spindles in living cells are close to straight but that pathologically large compressive forces in mutant cells can lead to buckling and the eventual breakage of anaphase B spindles.

These observations provide the rationale for studying the critical force of a spindle that is initially in a straight configuration. It is likely that curved microtubules were also present in reconstructions of the cdc Electron tomography remains the best technique that is available for studying cellular volumes of the size of the yeast spindle at high resolution.

The advantage of ET over serial-section EM is that the full three-dimensional organization of the spindle can be recovered, whilst any distortions of the spindle can be corrected using the IFTA algorithm. This approach thus allows us to obtain spindle reconstructions across the full range of lengths that characterize anaphase B spindle elongation.

These estimates suggest that the critical force of the metaphase spindle is greater than the force it is likely to bear during this phase of mitosis. Our calculations also suggest that the later anaphase B spindles can support the drag forces from the elongating nuclei. However, as the reviewer points out, this force seems intuitively relatively small, for example compared with the combined forces that the kinetochores could produce.

In the original manuscript, this information was split between the Methods section and the main text, and admittedly was difficult to follow. We modeled increases in cross-linker density by increasing the width of the rectangular support elements between the microtubules. We have amended the Results and Methods sections, in addition to the content and caption of the revised Figure 3 to explain this point more clearly.

The new Figure 2 explains the concept of transverse stiffness as a 2D tensor in greater detail. We have amended the text to refer to the relevant Figure 3E. The results of the numerical simulations, which are presented in subsequent figures, are not dependent on this assumption, as our algorithm considers each spindle as a composite structure that is not homogeneous.

We, however, also provide arguments that the properties of the yeast spindle, in particular its slow dynamics, crystalline architecture and high abundance of crosslinkers, implies that the elastic assumption is a reasonable first approximation for this system. The rationale for this model of spindle organization is now explained in greater detail in the manuscript the overlap length is maximized, under two constraints: the total polymer mass L T is given, and the pole-to-pole distance L s is fixed.

We have also added a diagram to Figure 5 panels A and B that explains how the overlap is maximised, given the constraints on polymer mass and the width of the midzone. In this case, we describe the spindle architecture in detail and refer to it as a maximal midzone-overlap MMO model, which is both precise and does not exaggerate its significance. The expanded discussion of the stiffness tensor now explains how the anisotropy of a beam leads to deviations from simple, 1-dimensional buckling.

We also note that the behavior of beams that do not have a constant cross-section is complex, which provides a valid justification for using simulations. These two potential sources of confusion have largely been resolved by the changes made in response to major comment 8. We also have substantially expanded the caption of Figure 5 and the discussion of theoretical models of a maximally overlapping spindle to explain these results and their consequences more clearly.

The funders had no role in study design, data collection and interpretation, or the decision to submit the work for publication. We thank Phong Tran for help in initiating the project. We thank Ken Sawin and Fred Chang for providing yeast strains, and Damian Brunner and members of his lab for yeast strains and technical help.

This article is distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use and redistribution provided that the original author and source are credited. Article citation count generated by polling the highest count across the following sources: Scopus , Crossref , PubMed Central. Phagocytosis requires rapid actin reorganization and spatially controlled force generation to ingest targets ranging from pathogens to apoptotic cells.

How actomyosin activity directs membrane extensions to engulf such diverse targets remains unclear. Here, we combine lattice light-sheet microscopy LLSM with microparticle traction force microscopy MP-TFM to quantify actin dynamics and subcellular forces during macrophage phagocytosis. We show that spatially localized forces leading to target constriction are prominent during phagocytosis of antibody-opsonized targets. Contractile myosin-II activity contributes to late-stage phagocytic force generation and progression, supporting a specific role in phagocytic cup closure.

Observations of partial target eating attempts and sudden target release via a popping mechanism suggest that constriction may be critical for resolving complex in vivo target encounters.

Overall, our findings present a phagocytic cup shaping mechanism that is distinct from cytoskeletal remodeling in 2D cell motility and may contribute to mechanosensing and phagocytic plasticity. Key processes of biological condensates are diffusion and material exchange with their environment. Experimentally, diffusive dynamics are typically probed via fluorescent labels.

However, to date, a physics-based, quantitative framework for the dynamics of labeled condensate components is lacking. Here, we derive the corresponding dynamic equations, building on the physics of phase separation, and quantitatively validate the related framework via experiments. We show that by using our framework, we can precisely determine diffusion coefficients inside liquid condensates via a spatio-temporal analysis of fluorescence recovery after photobleaching FRAP experiments.

We showcase the accuracy and precision of our approach by considering space- and time-resolved data of protein condensates and two different polyelectrolyte-coacervate systems.