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# Mitotic cells generate protrusive extracellular forces to divide in three-dimensional microenvironments

## Abstract

During mitosis, or cell division, mammalian cells undergo extensive morphological changes, including elongation along the mitotic axis, which is perpendicular to the plane that bisects the two divided cells. Although much is known about the intracellular dynamics of mitosis, it is unclear how cells are able to divide in tissues, where the changes required for mitosis are mechanically constrained by surrounding cells and extracellular matrix. Here, by confining cells three dimensionally in hydrogels, we show that dividing cells generate substantial protrusive forces that deform their surroundings along the mitotic axis, clearing space for mitotic elongation. When forces are insufficient to create space for mitotic elongation, mitosis fails. We identify one source of protrusive force as the elongation of the interpolar spindle, an assembly of microtubules aligned with the mitotic axis. Another source of protrusive force is shown to be contraction of the cytokinetic ring, the polymeric structure that cleaves a dividing cell at its equator, which drives expansion along the mitotic axis. These findings reveal key functions for the interpolar spindle and cytokinetic ring in protrusive extracellular force generation, and explain how dividing cells overcome mechanical constraints in confining microenvironments, including some types of tumour.

## Main

In many physiological contexts, cells divide in mechanically confining microenvironments. For example, the growth of solid tumours, driven by sustained cell division, is mechanically restricted by surrounding tissues, and tumours are often under mechanical compression (Fig. 1a)1,2. It is known that cell division requires substantial changes in morphology3,4. In two-dimensional (2D) culture, where most cell division studies have been conducted, cells simply release from the substrate and divide unrestricted (Fig. 1b). Substantial insight into the intracellular forces generated during mitosis has been gleaned from these studies5,6,7. However, how cells are able to undergo these morphological changes and generate space for daughter cells in mechanically confining 3D microenvironments is unknown. It would be expected that dividing cells must generate extracellular forces in a spatially and temporally coordinated manner during cell division to facilitate these changes in morphology and create additional space for daughter cells. Indeed, it is known that cells generate substantial outward forces during swelling and rounding immediately before mitosis8,9,10,11,12,13. Yet, the magnitude, nature and sources of such extracellular force generation during mitosis remain unexamined. Here, we investigated the biophysical mechanisms underlying extracellular force generation during mitosis in three-dimensionally confining microenvironments.

## Extracellular protrusive force generation during mitosis

To elucidate the forces generated by cells dividing in a mechanically confining microenvironment, we studied the division of cells cultured in viscoelastic alginate hydrogels (Fig. 1b). Alginate is a polymer that is inert to cells, not subject to degradation by mammalian enzymes, and can be crosslinked into nanoporous hydrogel with calcium14. Soft tissues exhibit elastic moduli, or stiffness, ranging from ~100 Pa to 5 kPa, and are typically viscoelastic, exhibiting stress relaxation in response to a deformation15,16. Therefore, alginate hydrogels with an initial elastic modulus of ~3 kPa and exhibiting fast stress relaxation and high creep were primarily used for this study (Supplementary Fig. 1a–d). To remove any ligand-dependent effects, no cell-adhesion ligands were coupled to the inert alginate hydrogels, and a tumour cell line was chosen for study, as such cells exhibit anchorage-independent proliferation. We encapsulated MDA-MB-231 breast cancer cells as single cells within the hydrogels and observed that they formed multicellular spheroids over time (Supplementary Fig. 1e). As the alginate hydrogels are nanoporous and exhibit minimal degradation, the growing tumour spheroids must deform the surrounding hydrogel to expand, and therefore must generate force.

To assess how single cells deform the surrounding matrices during mitosis, fluorescent microbeads were embedded in the hydrogels. These beads displaced as the hydrogels was deformed, allowing visualization of hydrogel deformation. Cells dividing along a plane parallel to the microscope slide were selected for study, and bead displacements were traced during mitosis, using a computational bead-tracking algorithm17. Strikingly, single dividing cells substantially deformed the surrounding matrices in a protrusive manner along the mitotic axis, with the greatest matrix deformation occurring during anaphase B (AnaB) and telophase/cytokinesis (T/C) (Fig. 1c,d and Supplementary Movies 1 and 2). Kymographs of single beads along the mitotic axis confirmed the outward displacement of beads along the mitotic axis (Fig. 1e, middle). In contrast, kymographs of beads along the axis perpendicular to the mitotic axis, extending from the midplane, showed that beads moved inwardly, probably due to recovery of the hydrogel from previously accumulated deformation (Fig. 1e, right). While traditional traction force microscopy18,19 cannot be applied to measure forces in viscoelastic hydrogels, due to the time-dependent creep of viscoelastic materials, an upper bound of mechanical stress generated by the cells on the hydrogels can be estimated by assuming hydrogels to be elastic (see Supplementary Note 1 and Supplementary Fig. 2a). A lower bound of the stress on the hydrogels can be estimated by assuming that cells generate a constant minimum stress and considering hydrogel creep (see Supplementary Note 1 and Supplementary Fig. 1d). From this analysis, the lower and upper bound of protrusive stresses generated by cells on the hydrogels during AnaB and T/C were estimated to be ~1.0 kPa and ~2.0 kPa, respectively (Supplementary Fig. 2b–d). These results demonstrate that cells generate substantial protrusive forces and deform hydrogels during cell division in confining microenvironments.

We next investigated matrix deformations associated with mitotic rounding and swelling, before metaphase, to compare with those generated from metaphase to cytokinesis. Studies investigating mitotic rounding and swelling have revealed that dividing cells generate protrusive, or outward, forces8,9,10,12,13. To isolate the contribution of matrix deformation due to mitotic swelling, cells were encapsulated in hydrogels but arrested at prometaphase with S-trityl-l-cysteine (STLC), a kinesin inhibitor20. The cells arrested at division expanded their size through mitotic swelling, and thereby deformed the surrounding matrices (Fig. 1f and Supplementary Movie 3). However, the magnitude of matrix deformation associated with mitotic swelling was smaller than that found to occur during AnaB and T/C (Fig. 1g,h). The upper bound of protrusive stresses generated by cells undergoing mitotic swelling on the hydrogels were estimated to be ~800 Pa (Supplementary Fig. 2b,c). This value is consistent with the results of previous studies that measured stress generated by mitotic swelling or rounding to be 150–500 Pa (refs 8,9,10) and smaller than the values associated with AnaB and T/C.

## Biological significance of protrusive force generation

Given the observation that cells generate protrusive forces during division, we next examined the biological importance of these forces. Specifically, we tested whether cells would continue through mitosis in stiffer and more elastic hydrogels, where forces were not sufficient to deform the hydrogel for mitotic elongation. Hydrogels with an initial elastic modulus of ~16 kPa and exhibiting slower stress relaxation and low creep were used as stiff, elastic hydrogels (Supplementary Fig. 1a–d). Finite-element modelling indicated that a protrusive stress of ~2.0 kPa, the upper bound of the estimated maximum stress, was sufficient to create space for mitotic elongation in soft, viscoelastic gels, but insufficient to create space for mitotic elongation in stiff, elastic gels (Supplementary Fig. 3a–e). Cells at prometaphase were encapsulated within these hydrogels. Mitotic progression was then monitored for cells identified to be at metaphase and compared to progression in the soft and viscoelastic gels or 2D culture. Strikingly, for cells at metaphase in the stiffer and more elastic gels, most cells (88%) did not progress through mitosis (Fig. 2a,b; Supplementary Fig. 4a–c and Supplementary Movies 46). Mitotic spindle disassembly and chromosome reintegration were observed in these cells that failed to progress through mitosis. Following failure of cell division, some of the cells underwent apoptosis (Supplementary Fig. 4c). This is in contrast to cells encapsulated in the soft and viscoelastic gels or cells in 2D culture, where most cells at metaphase completed mitosis successfully (Fig. 2b and Supplementary Fig. 4d). These results highlight the importance of matrix deformation along the mitotic axis, and therefore extracellular protrusive force generation, for enabling mitotic elongation and mitosis in confining microenvironments.

To gain insight into the mechanism underlying mitosis failure in stiff and elastic hydrogels, we examined whether the spindle assembly checkpoint (SAC) prevented progression to anaphase. The SAC blocks progression to anaphase when chromosomes are improperly attached or misaligned, or the kinetochores are not under sufficient tension21. Activity of the SAC was inhibited using M2I-1, a small-molecule inhibitor of Mad2, a key component of SAC22. Cells at prometaphase were encapsulated within stiff and elastic gels and cultured in the presence of M2I-1 or a control. Cells treated with the Mad2 inhibitor were found to exhibit lower rates of mitosis failure than untreated cells (Fig. 2c and Supplementary Fig. 5). This result indicates that SAC activity causes mitosis failure in stiff and elastic gels, where mitotic elongation is unsuccessful.

## Mechanisms underlying protrusive force generation

We then sought to elucidate the mechanisms underlying protrusive extracellular force generation during AnaB and T/C. One possible source of force generation could be volume change during mitosis. Indeed, several studies have reported decreases in volume associated with mitotic exit for cells in suspension20 or on 2D substrates23, although other studies have found volume to be conserved throughout mitosis for cells on 2D substrates8,10. To test this possibility, the volumes of cells undergoing division from metaphase (Meta) to T/C were measured using confocal fluorescence microscopy and found to be conserved through these stages (Fig. 3a,b and Supplementary Movie 7). This indicates that changes in volume, driven by osmotic pressure, are not responsible for force generation beyond mitotic swelling. However, conservation of volume from Meta to T/C suggests another possible source of force generation. In this mechanism, lateral contraction in a mitotic cell due to contraction of the cytokinetic actomyosin ring at the mid-plane, driven by myosin II motor activity24,25,26, must lead to outward expansion along the mitotic axis as intracellular materials are displaced from the centre axis towards the poles, generating protrusive extracellular force (Fig. 3c). A macroscopic analogy to this phenomenon would be squeezing of a spherical water balloon at its equator, which leads to longitudinal expansion at the poles (Supplementary Fig. 6 and Supplementary Movie 8).

Examination of mitotic spindle morphologies revealed an additional source of extracellular force generation. Mitotic spindles contain kinetochore microtubules, which connect the kinetochores on chromosomes to the spindle poles, interpolar microtubules, referring to the non-kinetochore microtubules that together span the region between the two spindle poles, and astral microtubules, which extend from the centrosomes to the cell cortex5. Interestingly, very curved morphologies of mitotic spindles between the poles, suggestive of mechanical buckling of the interpolar spindles, were observed in some dividing cells (Fig. 3d). To assess whether the interpolar spindles were buckled, we measured the curvatures of interpolar spindles and obtained a distribution of curvatures that suggested two populations of interpolar spindles (Fig. 3e and Supplementary Movies 9 and 10). Buckling occurs when compression force carried by a strut reaches a critical value, and as such is a threshold phenomenon27. Thus, the two distinct populations found in the distributions of the curvatures indicate that some of the interpolar spindles were buckled under compression. As hydrogels do not generate forces on cells by themselves, the compressional force leading to interpolar spindle buckling must be a reaction force in response to the outward forces generated by interpolar spindle elongation and transmitted through astral microtubules5,28,29,30,31. As the buckling of a strut depends on both the structure of the supporting strut and the compressional force that it carries, the observation that some interpolar spindles buckle while others do not probably arises from cell-to-cell variation in forces generated and interpolar spindle structure. To rule out the possibility that asymmetric ingression of the cleavage furrow induced bending of the spindles, without buckling, analysis of cleavage furrow asymmetry and spindle curvature was carried out (Supplementary Fig. 7). Spindle curvature was not correlated with asymmetry of the cleavage furrow. Further, in all cases, a straight conformation of the interpolar spindle was geometrically accessible, and not blocked by asymmetric ingression of the cytokinetic ring. These findings indicate that the observed spindle morphologies represent buckling and not bending. Taken together, these results suggest that elongation of interpolar spindles may be directly coupled to hydrogel deformation and contribute to extracellular force generation (Fig. 3f).

After finding these two possible sources of force generation, interpolar spindle elongation and cytokinetic ring contraction, we conducted perturbation experiments to validate their role in contributing to extracellular force generation. To directly test whether the interpolar spindle elongation is mechanically coupled to the hydrogel, laser ablation experiments were performed. Laser ablation was used to sever the interpolar spindles for cells entering anaphase, and the displacement of beads embedded in the hydrogels was traced before and after ablation (Fig. 4a). The position of beads along the mitotic axis immediately retracted in the direction of cells after the interpolar spindles were severed, indicating relaxation of some portion of the protrusive forces generated during mitosis along the mitotic axis (Fig. 4b–d; Supplementary Fig. 8 and Supplementary Movies 11 and 12). This represents a direct confirmation that the interpolar spindles are mechanically coupled to the hydrogel and are under compression, indicating that their elongation must deform the hydrogel. In addition, STLC was used at low concentrations to partially inhibit kinesin-5 activity, while not perturbing spindle assembly or blocking mitotic progression32,33. Partial inhibition of kinesin-5 activity decreased matrix deformation during early stages of mitosis and the curvature of spindles at later stages (Fig. 4e,f and Supplementary Fig. 9). Together, these results demonstrate the role of interpolar spindle elongation in extracellular force generation.

Next, to validate the role of lateral contraction of the cytokinetic ring in mitotic expansion, we inhibited cytokinetic ring contraction by using small-molecule inhibitors. First, blebbistatin, a potent inhibitor of myosin II, was introduced to inhibit the lateral contraction of dividing cells without disrupting interpolar spindle elongation34. As in previous studies, cells treated with the myosin inhibitor did not undergo lateral contraction and progress beyond anaphase, but interpolar spindles of the treated cells still elongated in a manner similar to untreated cells (Fig. 4g,h and Supplementary Fig. 10). Importantly, the treated cells still deformed matrices during mitosis, but did so to a significantly lesser extent than untreated cells (Fig. 4i,j). In addition to the blebbistatin experiments, lateral contraction of the cytokinetic ring was inhibited using cytochalasin D, a potent inhibitor of actin polymerization that also leads to disassembly of the actin cortex35. In agreement with the results from the blebbistatin experiments, treatment with cytochalasin D led to a decrease in matrix deformation during mitosis (Supplementary Fig. 11). Therefore, these results confirm that contraction of the cytokinetic ring generates protrusive forces along the mitotic axis.

With these two sources of extracellular force generation during cell division now identified, we determined the extent to which each mechanism contributes to force generation using mathematical analysis. If extracellular force generation arose from interpolar spindle elongation alone, matrix deformation would be expected to be directly proportional to the change in spindle length (Fig. 5a,b, left). Alternatively, if extracellular protrusive forces arose solely from cytokinetic ring contraction, matrix deformation would be directly synced with lateral contraction along the mid-plane of the mitotic cells (Fig. 5a,b, right). Image analysis of spindle length and lateral contraction revealed that neither was linearly correlated with matrix deformation, suggesting forces to be generated from some combination of the two mechanisms (Fig. 5c,d, Supplementary Fig. 12 and Supplementary Table 1). To estimate the extent to which each mechanism contributes to extracellular force generation, mathematical analysis was applied (see Supplementary Notes 2 and 3). Briefly, it was assumed that over any given stage of mitosis, change in matrix deformation was induced by some linear combination of changes in spindle elongation and lateral contraction, and that each mechanism contributed independently (Fig. 5e). For each stage of mitosis, normalized contribution factors were defined on the basis of the ratio of matrix deformation induced by either spindle elongation or lateral contraction to the overall matrix deformation. On the basis of this analysis, interpolar spindle elongation was found to primarily drive matrix deformation during the early stages of cell division, contributing ~80% of matrix deformation from Meta to AnaA, while mitotic-axis expansion due to lateral contraction governs matrix deformation from AnaB to T/C, contributing ~88% to matrix deformation (Fig. 5f). From AnaA to AnaB, both mechanisms contribute similarly.

## Outlook

Taken together, our studies reveal that dividing cells generate substantial protrusive extracellular forces during mitosis in confining microenvironments to drive mitotic elongation, a morphological change required for cell division (Fig. 6). Force generation was found to be greatest during AnaB and T/C, and the magnitude of forces exceeded those generated during mitotic swelling. This relation between forces could be specific to cases where cells are more rounded before mitosis, as a recent study found that mitotic rounding of cells that are spread generated higher forces than those of already rounded cells in collagen gels36. Importantly, we find that these forces are required for successful completion of mitosis. When forces are unable to deform the surrounding matrix sufficiently to allow mitotic elongation, as in stiff and elastic hydrogels, mitosis fails. In these cases, our results suggest that mitotic progression is halted at metaphase due to activity of the SAC. Mitotic elongation may be required for proper alignment of chromosomes and stretch of the kinetochores, and thus inactivation of the SAC.

Extracellular protrusive force generation was found to arise from direct coupling of interpolar spindle to hydrogel and cytokinetic ring contraction. Extracellular forces generated from interpolar spindle elongation were in part driven by kinesin activity and are probably transmitted through astral microtubules5,31. While the role of the interpolar spindle in mitosis had previously been unclear5, these findings suggest that a key function is to generate protrusive extracellular forces. In addition to interpolar spindle elongation, lateral contraction of the cytokinetic ring also contributes to extracellular force generation. This mechanism was suggested by the finding that cellular volume was conserved during mitosis. Some previous studies found volume to decrease during mitosis for cells in suspension or cultured on 2D substrates20,23, while other studies found volume to be conserved through mitosis8,10. The difference may be due to the different culture conditions, and suggests that volume may be actively regulated during mitosis. We speculate that volume might be conserved when force generation is required for mitotic elongation. Interestingly, previous studies have found the mechanical properties of the actin cortex during mitosis to be anisotropic, with the cortex stiffer closer to the cleavage furrow, and softer at the poles37. This anisotropy may help direct viscoelastic flow of intracellular materials that are displaced by the ingression of the cleavage furrow towards pole deformation, rather than uniform expansion of the forming daughter cells. We note that there could be other possible mechanisms that contribute to extracellular force generation during mitosis, such as more complex mechanochemical signal transduction pathways. Broadly, this work provides a new perspective on the mechanics of cell division, revealing the extracellular forces that comprise an essential aspect of mitosis, allowing cell division to occur normally, or pathologically, in confining tissue microenvironments.

## Methods

### Alginate hydrogel preparation

Sodium alginate with a high molecular weight (MW; 280 kDa, LF20/40) was purchased from FMC Biopolymer, and was prepared as described previously16. High-MW alginate was irradiated by 8 Mrad cobalt source to produce low-MW alginate. Both high-MW and low-MW alginate were dialysed against deionized water for 2–3 days, filtered with activated charcoal, lyophilized and then reconstituted at 3 wt% in serum-free Dulbecco’s modified Eagle's medium (DMEM; Life Technologies). High-MW alginate was used to form stiff and elastic hydrogels, while low-MW alginate was used to form soft and viscoelastic hydrogels.

### Cell experiments

MDA-MB-231 human cells transfected with green fluorescent protein (GFP)-labelled α-tubulin and red fluorescent protein (RFP)-labelled histone were kind gifts from B. Weaver (University of Wisconsin-Madison). The cells were cultured in standard DMEM (Invitrogen) containing 10% fetal bovine serum (Hyclone) and 1% penicillin/streptomycin (Gibco/Thermo Fisher Scientific).

For cell encapsulation in hydrogels, cells were trypsinized, washed in serum-free DMEM and re-suspended as a single-cell suspension in serum-free DMEM at a concentration of 20 million per millilitre. The concentration of the cells was measured using a Coulter counter (Beckman Coulter). Fluorescent microbeads (Thermo Fisher, 0.2 µm bead size, 365 nm and 415 nm excitation and emission wavelength, or 660 nm and 680 nm excitation and emission wavelength) were used as a reporter for hydrogel displacements and prepared at ×10 the final concentration. Cells were then mixed with the prepared alginate and fluorescent microbeads, reconstituted in serum-free DMEM in one Luer lock syringe (Cole-Parmer). The cell–alginate–microbead solution in the syringe was then rapidly mixed with DMEM containing calcium sulfate in another syringe, connected by a female–female Luer lock coupler, and then deposited on cell culture plates. The final concentration of calcium sulfate was 24.4 mM for high-MW alginate and 28 mM for low-MW alginate. The final concentration of high-MW and low-MW alginate was 20 mg ml−1. The mixture was allowed to gel for 45 min. The diameter of hydrogels was 20 mm with 1 mm thickness. The hydrogels containing cells were then immersed in growth media until live-cell imaging.

For laser ablation and inhibition experiments, and comparison of mitosis success in soft viscoelastic versus stiff elastic hydrogels, cells were synchronized at prometaphase before being encapsulated in the hydrogels. Cells were first treated with thymidine at 2 mM for 1 day to arrest them at G1/S phase, released and then treated with S-trityl-l-cysteine (STLC) at 10 µM for 1 day to arrest them at prometaphase. The cells arrested at prometaphase were released from STLC when they were encapsulated in hydrogels.

For pharmacological inhibitor experiments, STLC, cytochalasin D, blebbistatin and M2I-1 were added to the medium at a final concentration of 0.4 µM (refs 32,33), 5 µg ml−1 (ref. 38), 100 µM (ref. 34) and 100 µM (ref. 22), respectively, approximately 4 h before imaging.

### Time-lapse fluorescence imaging of dividing cells

Live-cell imaging of dividing cells was conducted using confocal microscopy (Leica, SP8) under standard cell culture conditions (37 °C, 5% CO2). Cells undergoing division with their mitotic spindle parallel to the microscope slide were first located. Once dividing cells were found, they were imaged using a ×10/0.4NA objective or a ×25/0.95NA water-immersion objective at excitation wavelengths of 488 nm for GFP-labelled α-tubulin, 555 nm for RFP-labelled histone and 639 nm for the microbeads. It should be noted that dividing cells, especially those undergoing prometaphase or metaphase, are sensitive to laser intensity and duration. Therefore, to minimize the exposure to the laser, 2D imaging at the mid-plane of dividing cells was performed for the analysis of matrix deformation along with spindle length, cell length and width, at low laser intensities. For 3D volume imaging, a stack of the 2D images of GFP–tubulin alone along the z axis of dividing cells was performed.

### Laser ablation experiments

Live-cell imaging combined with laser ablation was performed using a multiphoton laser scanning confocal microscope (Zeiss, LSM 780) under cell culture conditions (37 °C, 5% CO2) at Stanford Cell Sciences Imaging Facility. Cells were imaged using a ×40/1.2NA water-immersion objective at excitation wavelengths of 405 nm for microbeads, 488 nm for GFP-labelled α-tubulin and 555 nm for RFP-labelled histone. Ablation of spindles was performed during live-cell imaging using Mai Tai DeepSee (Spectra-Physics) at a wavelength of 800 nm with 95–100% of maximum power of 2.4 W. Once mitotic cells entered anaphase, the region of ablation was selected, and laser ablation was used to sever interpolar spindles.

### Determination of mitotic stages

Specific cell division stages were determined manually using the following criteria: metaphase—chromosomes were lined up along the metaphase plate, and interpolar spindles were aligned along the polar axis; anaphase A—chromosome segregation was initiated; anaphase B—interpolar spindles were elongated and the cleavage furrow had formed; telophase/cytokinesis—cleavage furrow ingression was fully progressed and mid-bodies were formed. The cell length along the mitotic axis, the width perpendicular to the mitotic axis and the spindle length at the different stages of cell division were measured using Fiji.

### Normalization of cell length, width and interpolar spindle

The measured cell length, lateral contraction distance and spindle elongation were normalized as the ratio of changes relative to their maximum value during cell division, given by

$y norm = y t - y initial y final - y initial$
(1)

where y can be cell length, lateral contraction distance or spindle elongation. Normalized cell width was calculated as

$w norm = w final - w t w final - w initial$
(2)

### Calculation of matrix deformation

To obtain matrix deformation, or hydrogel displacements, first, drift of images was corrected by image registration using ImageJ. Matrix deformation was then calculated by tracking microbeads embedded in hydrogels, using a standard particle image velocimetry algorithm (PIVlab; open source code for MATLAB) with three passes (overall, 128 × 128, 64 × 64 and 32 × 32 pixel-size interrogation window with 50% overlap)39. The interrogation window sizes were scaled accordingly for different image sizes. Displacements inside cell boundaries were replaced by the averaged values of surrounding displacements. Maximum matrix deformation was selected within ~10 µm around the cells.

### Mitotic swelling experiments

For mitotic swelling experiments, STLC, a pharmacological inhibitor that inhibits the activity of the kinesin Eg5 and blocks mitotic progression, was added to the media at a concentration of 10 µM after encapsulation to arrest cells at mitosis40. Before adding the drug, cells and the surrounding microbeads were imaged to obtain a reference image. The cells were then imaged again 2–3 days after adding the drug. The two images were compared to calculate the hydrogel deformation associated with mitotic swelling.

### Spindle curvature measurements

The curvatures of interpolar spindles were analysed using ImageJ. Three points, two end and one middle points, were selected on each interpolar spindle and a circle to fit the three points was determined41. The curvature of an interpolar spindle was defined as an inverse of the radius of the circle.

### Statistical analysis

For statistical analysis, we used Student’s t-tests or Fisher’s exact test to compare two groups. One-way analysis of variance with Tukey’s multiple comparison or the Kruskal–Wallis test was used to compare more than two groups using GraphPad. Multiple linear regression analysis was performed to obtain the relation of matrix deformation with respect to relative change in spindle elongation or lateral contraction distance over mitotic progression (see Supplementary Note 2) using a customized code with MATLAB.

### 3D reconstruction of dividing cells and volume measurements

To measure the volumes of dividing cells, 3D imaging was performed using z stack confocal microscopy (Leica, SP8) with 1.5–2.2 µm intervals in the z axis between images in a stack, and the images were three-dimensionally reconstructed using IMARIS (Bitplane). The total cell area inside cell boundaries at each confocal section was calculated using ImageJ. Cell volumes were assessed by multiplying the total cell area and the z-axis intervals, and normalized by the volumes at T/C.

Kymographs were created using ImageJ with a stack of cell division images at different stages. Lines were drawn across fluorescent microbeads along the mitotic or perpendicular axis of dividing cells. Image pixels that the lines passed through were then reorganized to create a kymograph.

### Mechanical measurements of hydrogels

The initial modulus and stress relaxation of alginate gels were characterized with unconfined compression tests using a material testing machine (Instron 5848 MicroTester). Alginate gels without cells were formed between two glass plates and punched as a disc using a 6 mm biopsy punch (Miltex). The alginate gel discs were placed on the machine and compressed to 10% strain at a rate of 1 mm min−1, and then the strain was held for the stress relaxation test. The slope of the stress–strain curve up to a strain of 5–10% was measured as the initial modulus, and stress was recorded while strain was held constant. The stress relaxation timescale was measured as the time at which the stress is relaxed to half its original value16.

For storage modulus measurements and creep tests, rheology was performed using an AR-G2 stress-controlled rheometer (TA Instruments) equipped with 25-mm-diameter top and bottom plate, as has been described previously42,43. Alginate gels were directly deposited onto the bottom surface of the rheometer plate immediately after mixing with the crosslinker. The top plate was immediately brought down rapidly to form a 25-mm-diameter disc of sample with a thickness of ~400 μm between the rheometer plates. The exposed gel surface between the plates was enclosed by mineral oil (Sigma Aldrich). During gelation of alginate, the storage modulus was monitored at a strain of 0.01 and frequency of 1 rad s−1. Creep tests were performed after the storage modulus reached an equilibrium value. In creep tests, a constant stress of 100 Pa was applied for 1,200 s while strain in response to the stress was measured over time.

### Simulation

Computational simulations were performed using Abaqus (6.14) to test whether cell-generated forces would deform these matrices sufficiently for normal mitotic elongation to proceed. For a simulation model, a rectangular hexahedron with depth, width and height of 200, 200 and 100 (a.u.) was constructed including a hemispherical hole at the centre with a diameter of 30 (a.u.). The rectangular hexahedron and the hemispherical hole indicate a matrix and a cell, respectively (Supplementary Fig. 3a). The mechanical properties of the simulation model were determined by the experimentally obtained values. The initial elastic modulus of the soft, viscoelastic model and the stiff, elastic model was ~3 kPa and ~16 kPa, respectively (Supplementary Fig. 1a). The viscoelastic properties of the models were determined by fitting the creep measurements of soft, viscoelastic gels and stiff elastic gels with Prony’s series (Supplementary Fig. 1d). The Poisson ratio used for the model was 0.49. Outward stress of ~2.0 kPa, the upper bounds of the estimated maximum stress generated during mitosis, was applied to the polar regions of the models for 30 min, and displacement in response to the stress was obtained (Supplementary Fig. 3). This provides an estimation of the upper bound of matrix deformation during mitosis in these hydrogels. Displacement was scaled by comparing the diameter of the hemispherical hole to the actual cell sizes.

### Life Sciences Reporting Summary

Further information on experimental design is available in the Life Sciences Reporting Summary.

### Data availability

Data are provided in the Supplementary information or are available from the corresponding author upon request.

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## Acknowledgements

The authors thank the members of the Chaudhuri laboratory, J. Nelson (Stanford University) and D. Fletcher (University of California, Berkeley) for helpful discussions, and M. Levenston (Stanford University) for use of the rheometer. This work was supported by a Samsung Scholarship for S.N., and a grant from the National Science Foundation (CMMI-1536736) to O.C.

## Author information

### Affiliations

1. #### Department of Mechanical Engineering, Stanford University, Stanford, CA, USA

• Sungmin Nam
•  & Ovijit Chaudhuri

### Contributions

S.N. and O.C. designed the experiments and analysed the data. S.N. conducted the experiments and ran the simulations. S.N. and O.C. wrote the manuscript.

### Competing interests

The authors declare no competing interests.

### Corresponding author

Correspondence to Ovijit Chaudhuri.

## Supplementary information

1. ### Supplementary Information

Supplementary Figures 1–14, Supplementary Table 1, Supplementary Notes 1–3, Supplementary References 44–49

## Videos

1. ### Supplementary Video 1: Dividing cells deform the surrounding matrices as they progress mitosis

Single dividing MDA-MB-2.1 cell within a hydrogel exert outward forces and pushes away the surrounding hydrogels along the mitotic axis. Matrix deformation was visualized with microbeads embedded in the hydrogels. White arrows point to specific microbeads, which are notably displaced. Images were shown at the indicated mitotic stages. Scale bar is 10µm.

2. ### Supplementary Video 2: Another example of matrix deformation during mitosis

Another movie of cell division within a hydrogel. Scale bar is 10µm.

3. ### Supplementary Video 3: Matrix deformation associated with mitotic swelling.

Cells undergoing mitotic swelling expand their size and generate outward forces, consequently deforming the surrounding matrices. Cells are arrested at mitosis by introducing S-trityl-L-cysteine. Matrix deformation was visualized by microbeads embedded in the hydrogels. White arrows point microbeads, which are notably displaced. Scale bar is 10µm.

4. ### Supplementary Video 4: Cell at metaphase does not progress through division in stiff and elastic 3D hydrogels.

Single dividing MDA-MB-2.1 cell within stiff and elastic hydrogels does not progress through mitosis, failing to divide. Matrix deformation was visualized with microbeads embedded in the hydrogels. White arrows point to specific microbeads, which are notably displaced inwardly. Scale bar is 10µm.

5. ### Supplementary Video 5: Another example of failure of division of cell at metaphase in stiff and elastic 3D hydrogels.

Single dividing MDA-MB-2.1 cell within stiff and elastic hydrogels does not progress through mitosis, failing to divide. Matrix deformation was visualized with microbeads embedded in the hydrogels. White arrows point to specific microbeads, which are notably displaced inwardly. Scale bar is 10µm.

6. ### Supplementary Video 6: Cells dividing in stiff and elastic hydrogels often undergo cell death.

Example of cell at metaphase in a stiff and elastic gel, which fails to undergo mitosis and later undergoes apoptosis. Scale bar is 10µm.

7. ### Supplementary Video 7: 3D reconstruction of a dividing cell.

Time-lapse images of a dividing cell were three-dimensionally reconstructed. Scale bar is 10µm.

8. ### Supplementary Video 8: Demonstration of outward force generation by lateral contraction using water balloon.

A macroscopic analogy to the outward forces generated by lateral contraction of the cytokinetic ring would be squeezing of a spherical water balloon at its equator, which leads to longitudinal expansion at the poles. In the movie, a green balloon contains water inside and is wrapped with a cable. The green balloon and the cable represent a cell and cytokinetic ring. The cable was pulled by a hand. Red arrows indicate ingression of the balloon equator due to pulling of the cable. Black arrows indicate longitudinal expansion of the balloon due to the water flow induced by pulling of the cable.

9. ### Supplementary Video 9: Buckled interpolar spindles observed for cells dividing in 3D hydrogels.

Cells dividing in 3D hydrogels were confined by the surrounding hydrogels and often exhibited very curved spindles at the end of cell division, indicative of buckling under compression. Yellow arrows indicate compressive reaction forces from hydrogels, in response to spindle-driven forces. White arrow points high curvature of interpolar spindle. Scale bar is 10µm.

10. ### Supplementary Video 10: Cells dividing in 2D culture exhibit straight interpolar spindles.

Cells dividing on 2D cell culture plates were free from confinement and did not show curved spindles, in contrast to cells dividing in 3D hydrogels. Scale bar is 10µm.

11. ### Supplementary Video 11: Relaxation of hydrogel deformation after interpolar spindle ablation.

Laser ablation was used to sever the interpolar spindles for cells entering anaphase. Relaxation of hydrogel deformation was visualized by microbeads embedded in the hydrogels. The position of beads retracts immediately in the direction of the cell after the spindles were ablated. White line represents the region of laser ablation, and white arrowheads indicate spindles ablated. White arrows point to specific microbeads, which are notably displaced inwardly. Scale bar is 10µm.

12. ### Supplementary Video 12: Another example of laser ablation at anaphase, imaging from metaphase.

Single dividing cell within a hydrogel generate outward forces and pushes away beads embedded in the hydrogel along the mitotic axis, when the cell progresses mitosis from metaphase to anaphase B. However, the position of the beads immediately retracts in the direction of the cell after the spindle was severed by laser ablation. White line represents the region of laser ablation, and white arrowheads indicate spindles ablated. White arrows point to specific microbeads, which are notably displaced. Images were shown at the indicated mitotic stages. Scale bar is 10µm.