Rigid tumours contain soft cancer cells

Palpation utilizes the fact that solid breast tumours are stiffer than the surrounding tissue. However, cancer cells tend to soften, which may enhance their ability to squeeze through dense tissue. This apparent paradox proposes two contradicting hypotheses: either softness emerges from adaptation to the tumour’s microenvironment or soft cancer cells are already present inside a rigid primary tumour mass giving rise to cancer cell motility. We investigate primary tumour explants from patients with breast and cervix carcinomas on multiple length scales. We find that primary tumours are highly heterogeneous in their mechanical properties on all scales from the tissue level down to individual cells. This results in a broad rigidity distribution—from very stiff cells to cells softer than those found in healthy tissue—that is shifted towards a higher fraction of softer cells. Atomic-force-microscopy-based tissue rheology reveals that islands of rigid cells are surrounded by soft cells. The tracking of vital cells confirms the coexistence of jammed and unjammed areas in tumour explants. Despite the absence of a percolated backbone of stiff cells and a large fraction of unjammed, motile cells, cancer cell clusters show a heterogeneous solid behaviour with a finite elastic modulus providing mechanical stability. Cervix and breast carcinomas are highly heterogeneous in their mechanical properties across scales. This heterogeneity provides the tumour with stability and room for cell motility.

Palpation utilizes the fact that solid breast tumours are stiffer than the surrounding tissue. However, cancer cells tend to soften, which may enhance their ability to squeeze through dense tissue. This apparent paradox proposes two contradicting hypotheses: either softness emerges from adaptation to the t um ou r's m ic roenvironment or soft cancer cells are already present inside a rigid primary tumour mass giving rise to cancer cell motility. We investigate primary tumour explants from patients with breast and cervix carcinomas on multiple length scales. We find that primary tumours are highly heterogeneous in their mechanical properties on all scales from the tissue level down to individual cells. This results in a broad rigidity distribution-from very stiff cells to cells softer than those found in healthy tissue-that is shifted towards a higher fraction of softer cells. Atomic-force-microscopy-based tissue rheology reveals that islands of rigid cells are surrounded by soft cells. The tracking of vital cells confirms the coexistence of jammed and unjammed areas in tumour explants. Despite the absence of a percolated backbone of stiff cells and a large fraction of unjammed, motile cells, cancer cell clusters show a heterogeneous solid behaviour with a finite elastic modulus providing mechanical stability.
Early on, tumour biology recognized that cancer cells undergo dedifferentiation towards a more disordered and thus softer cytoskeleton 1 . Evidence for soft cells already inside primary tumours is, however, only circumstantial and cell lines differ from clinical samples [2][3][4][5] . Cell softening is associated with the downregulation of keratin after partial or complete epithelial-mesenchymal transition 6,7 ; these cells migrate more efficiently through dense environments 8 , until nuclear jamming hinders further movement 9,10 . For metastatic cells extracted from extracellular fluids such as pleural effusions, softness correlates with clinical pathology 5 . Similarly, cytobrushes indicate cancer cell softening in oral Article https://doi.org/10.1038/s41567-022-01755-0 ((1.4 ± 1.3) × 10 5 Pa) and have similar fluidity (for both, n = 4). The cervical epithelium is primarily surrounded by rigid connective tissue and smooth muscle cells. For effective cancer cell proliferation, it is sufficient if the tumour just matches or slightly exceeds the resistance of the surrounding microenvironment 16 . Moreover, cervical tissue is highly active showing functional and structural changes, for example, during the menstrual cycle, which is reflected in variably altered viscoelastic properties 37 . This demonstrates that carcinomas do not have to be drastically stiffer than the healthy surrounding tissue 38 .
With the atomic force microscopy (AFM) technique, we measured the elasticity maps of the same live tumour explants with cellular resolution to capture the local, heterogeneous distribution of stiffness. We see a log-normal distribution in stiffness (Fig. 1e), also seen in single-cell AFM measurements 39,40 . For breast cancer, we observe a stiffening in the median Young's Modulus from E BN = 132 Pa (n = 16) to E BC = 288 Pa (n = 13) moving from healthy to cancerous tissue. In the cervix, we see a drop in median stiffness from E CN = 570 Pa (n = 5) for healthy cervix to E CC = 385 Pa (n = 7) in cervical cancer. These values integrate over the differences between stroma and cancer cell clusters. Within the cancer cell clusters, we observe smaller stiff regions surrounded by softer cells, both spanning several hundreds of micrometres (Fig. 1c). The regions of soft cells are percolated within the observed sections. Moreover, the stiff regions remain in isolated areas. Since we have only sections of the tumour, it remains unclear whether the rigid regions are percolated or remain islands in three dimensions. The soft and stiff regions within the cell clusters show a solid behaviour with finite elastic moduli. This mechanically stable behaviour permits the understanding that fibrotic stroma does not solely contribute to the rigidity of a solid tumour.

Carcinomas contain cells that are softer than healthy cells
Suspended cells lose all stimuli from their microenvironment and enter an unperturbed ground state, so that changes must be due to expression changes. This ground state was characterized in step-stress experiments with an optical stretcher (OS) 41 (Fig. 3). For cervical tumours, neighbouring normal epithelial tissue was used as a reference. For breast cancer, it was benign lesions (fibroadenoma, FA) and primary human mammary epithelial cells (HMEpC) from breast reductions. All the samples were in culture for a short time, as primary cells soften with time in culture (Extended Data Fig. 1). As the HMEpC controls were in culture longer than the tumour samples, the observed differences underestimate the relative softening of cancer cells.
The relative deformation of measured cells also follows a log-normal distribution. From the cumulative distribution of 13 breast cancer samples (n = 6,526), compared with two FA samples (n = 186) and one HMEpC sample (n = 358), we find that the breast cancer cells are the softest with a median relative deformation of MD BC = 0.024 (FA cells, MD FA = 0.015; healthy cells, HMEpCMD HMEpC = 0.018) ( Fig. 3 and Extended Data Fig. 2). All the differences are significant.
An increase in soft cancer cells with respect to healthy cells characterizes both breast and cervical tumours (Extended Data Table 1). The cells from tumours are more heterogeneous, that is, they display a broader log-normal distribution, with a large fraction of cells that are just as stiff as those in normal tissues. Primary carcinoma contains soft as well as rigid cells, which could stem from cancer deregulation or the broad spectrum of epithelial and mesenchymal states may cause the variability in cytoskeletal expression that cause this heterogeneous behaviour. In cancer cell clusters, these cells separate in regions of soft and rigid cells, as shown by our AFM measurements. The stiffening of breast tissue felt by palpation can be attributed to the different compositions of healthy and cancerous breast tissue, as soft fat cells get replaced by large volumes of cancer cells. These cancer cells are a lot stiffer than fat cells, even if they are softer than the healthy epithelial cells that they originate from in the first place. cancer 11 . In contrast, circulating breast cancer cells are slightly stiffer than the surrounding white blood cells 12 . Fine-needle aspirations of breast tumours show that solid tumours contain well-defined soft regions 13,14 , although it remains unclear if these soft areas are composed of cancer cells or extracellular matrix (ECM).
Our recent research indicates that the fluid or solid behaviour of cancer cell clusters in breast and cervical tumours is modulated by cell unjamming 15 . Cell proliferation requires a cell cluster resistant enough to divide against a typically firm surrounding stroma 16,17 , yet proliferation fluidizes tissues 18,19 . Cancer cell spheroids can spread like a fluid droplet 20 , and yet their shapes and sorting behaviours are not solely governed by surface tension 21 . Many cell aggregates exhibit features of glassiness or jamming, suggesting the mechanical impact of solid-fluid transitions on tissue bulk behaviour 22,23 . Because fluid-like and solid-like tissues have different mechanisms for proliferation 24 , migration, self-organization and cohesion (that is, cancer cell escape) 25 , these uncertainties prevent us from fully understanding the initial metastatic cascade.
Nonetheless, a breast tumour is undoubtedly a rigid mass, as already stated in the ancient Egyptian medical text Ebers Papyrus. Neoplastic tissue, composed of cancer cell clusters surrounded by enhanced, stiff and often fibrotic stroma, appears as a rigid mass 26 with respect to the healthy surrounding tissue. Pathologists use excessive ECM deposition as a marker for poor prognosis, since it is a strong tumour promoter 27,28 . Tumour progression seems to simultaneously require rigid and soft properties of cancerous tissue and individual cancer cells. This apparent paradox, as discussed recently 17 , is solved by us by unique multiscale mechanical measurements on patient-derived tumour explants.
Moreover, cancer cells are highly mechanosensitive and mechanically adapt to their microenvironment. Mechanical changes may be directly caused through mechanosensitive responses of the cytoskeleton or through expression changes by cellular mechanotransduction [29][30][31] . This may cause cell stiffening after the cancer cell has left the tumour cell mass into the stiff ECM. Moreover, fluid unjammed cancer cell clusters may induce cancer cell softening or softening may be the cause of unjamming. The question remains if mechanical changes already start in the tumour mass or rather occur only when the cells leave into the stroma.

Multiscale tumour mechanics
Starting on the bulk-tissue level, we quantified macroscopic viscoelasticity with tabletop magnetic resonance elastography (MRE) 32-34 on centimetre-sized vital tumour explants, from cervical and mammary carcinomas (Figs. 1 and 2). Soft tissues and cells often exhibit a distinctive power-law viscoelastic response 35 , described by the complex shear modulus derived from the fractional Maxwell model. We extract the stiffness/elastic resistance (µ) and its power-law exponent (α) as a measure for fluidity/dissipation. The cells and tissues are highly complex compound materials, and no analytic constitutive model exists to describe their multiscale mechanical behaviour. Thus, mechanical constants measured with diverse techniques on different scales cannot be quantitatively compared 36 . Nevertheless, the measured mechanical behaviours can be compared and correlated.
The MRE data confirm the medical practice that breast cancer can be identified by palpation; breast tumour explants (n = 5) with a median mechanical resistance of µ BC = (2.9 ± 1.9) × 10 5 Pa are clearly (p = 0.021, Kolmogorov-Smirnov (KS) test) stiffer than healthy breast tissue with µ BN = (163 ± 77) Pa (n = 3), whereas the fluidity is similar (Fig. 1b). In the tumour explants, the fibrotic stroma may contribute to an increase in stiffness, whereas healthy epithelial breast tissue is surrounded by connective tissue and very soft fat tissue, which may dominate the averaged bulk stiffness measured by MRE.

Cancer cell unjamming modulates stiffness of cell clusters
In tumour explants, we have found unjammed as well as jammed regions by vital cell tracking. To understand how such regions affect the global tumour behaviour, we use cell spheroids to illustrate the mechanical properties of jammed and unjammed tissues. We have recently shown that MDA-MB-436 spheroids, a model for breast cancer, consist of unjammed cells that can move, whereas MCF-10A spheroids, a model for epithelial cells, consist of jammed, non-moving cells 15 . Spheroid fusion experiments demonstrate that spheroids with motile cells behave like a fluid and jammed cell clusters have properties of an amorphous solid 15 . This exemplifies the fact that tissue fluidity, as an emergent collective cell behaviour, is a key modulator of tissue stiffness. We performed force-indentation experiments on spheroids with an AFM instrument. MCF-10A spheroids have an elastic modulus of 88(±63) Pa compared with 135(±38) Pa for single cells. MDB-MB-436 spheroids dropped from a single-cell elastic modulus of 570(±300) Pa to 111(±72) Pa for spheroids (Fig. 4). The motile cells in MDB-MB-436 spheroids oppose much less external loads compared with individual cells. The jammed MCF-10A spheroids also lose some of their individual strength but only 36% compared with the 80% value of MDA-MB-436. We tracked the fusion of nine pairs of MCF-10A spheroids and ten pairs of MDA-MB-436 spheroids. The fusion progress rate-measured as Δ(cosθ)/Δt-between 24 and 36 h after fusion start, was significantly different between the two fusion experiments. Qualitatively, MCF-10A fusions virtually arrest, in contrast to the ongoing MDA-MB-436 fusions (Fig. 4). Together with the AFM measurements, this establishes the fact that tissue fluidity rather than direct individual cell stiffness impacts the mechanical stability of cell clusters. Single-cell stiffness may be more of a determinant of tissue fluidity. Already, the two cell lines show broad log-normal distributions for their cell stiffness. Furthermore, the primary tissue samples are even more heterogeneous. Our AFM-based cell elasticity maps of cervix and breast carcinoma display rigid and soft regions. The soft areas are unjammed and the rigid ones are jammed. If the rigid cells do not form a percolated backbone, how can the tissue maintain a mechanically stable behaviour with a finite elastic modulus?  (1), intraductal tumour growth (2) and fatty tissue (3) can be identified. b, MRE of breast tumours. Centimetre-sized vital pieces of primary breast tumours (n = 5) and healthy breast tissue (n = 3) were measured. The bulk stiffness µ of tumours is higher compared with healthy tissue, as expected from palpation (p = 0.021, KS test), and the difference in fluidity α is not significant. c,f, AFM maps of local tissue elasticity for breast tumour and healthy breast tissue. Tissue elasticity maps measured for 0.5 × 0.5 mm 2 areas in vital tissue explants with 10 µm resolution reveal domains of several hundreds of micrometres in size with distinctively higher or lower local elasticity leading to a heterogeneous structure. d, Median breast tissue stiffness; the median Young's modulus rises from E BN = 132 Pa (n = 16) to E BC = 288 Pa (n = 13) moving from healthy to cancerous tissue (p < 0.01, MWU test). e, Distribution of local Young's moduli of breast cancer tissue (dark blue) and control tissue (light blue) from the maps shown in c and f. Both tissues show a log-normal distribution in stiffness (red fit lines), with the tumour showing a much wider, heterogeneous distribution in stiffness. The box plots show quartiles 1, 2 and 3 (box) and 5%/95% (whiskers); n.s., not significant; *p < 0.05, **p < 0.01. Article https://doi.org/10.1038/s41567-022-01755-0

Regions of rigid cells surrounded by soft, motile cells
We used the measured stiffness distribution from patient samples and a vertex-based model 42,43 used for the unjamming transition 44-48 to disentangle the influence of a stiff and soft fraction of cancer cells on tissue mechanics (Fig. 5) 49 . Our qualitative simulations show-in a reductionist situation only-how the interplay of soft and rigid cells can assume different states of tissue fluidity depending on the distribution of soft and rigid cancer cells. For a tissue of mechanically homogeneous cells, the collective fluidity is controlled by the cell shape index P 0 , which describes the interaction between cellular cortical tension (that is, effective cell stiffness) and cell adhesion. Cells with large tension tend towards a stiff, round shape, whereas when it is low, cells tend towards a softer, elongated shape 15,50,51 . Thus, round, stiff cells with a shape parameter smaller than the critical P 0 collectively assume the state of an amorphous, jammed solid, whereas elongated, soft cells with a larger shape parameter are motile in a cooperative fluid, unjammed state.
We have stratified the fraction of soft and stiff cancer cells in our model 49 based on the measured stiffness distributions, since the elastic modulus of a single cell is linearly proportional to P 0 for small deformations 52 . Since carcinoma shows a mixed epithelial and mesenchymal phenotype 7 and different cadherins can bind to each other 53 , we assume that differences in P 0 between the cancer cells are predominantly caused by changes in the cancer cell's cortical tension and differences in cell-cell adhesion are less important. Cancer cell unjamming is further modulated by mechanical effects of the nucleus 15 and other effects. We have recently identified the mechanism of 'second-order rigidity' as a key driver 54,55 , which works in the same fashion in two and three dimensions. Since all our results are verified by experimental data, we visualized the mechanism in a qualitative two-dimensional model for clarifying how mechanical heterogeneity influences the tissue mechanics. In our model, the mechanical bulk property of the cell collective is determined by computing the shear modulus G (ref. 56 ). In terms of mechanical stability, G is finite in a fully jammed, solid tissue, but vanishes in a completely unjammed, fluid one.
We computed G as a function of the mean and standard deviation (s.d.) of P 0 and found three distinct mechanical phases (Fig. 5a): a fully unjammed (fluid) phase where the shear modulus of the tissue Centimetre-sized pieces of vital primary tumour explants of cervical tumours (n = 4) and healthy cervix tissue (n = 4) were measured. On the bulk-tissue level, tumours are similar compared with healthy tissue. The tumour is not significantly stiffer or more fluid than the surrounding tissue (n.s.; KS test). c,f, AFM-based map of local tissue elasticity of cervix carcinoma and healthy control tissue. Tissue elasticity measured for patches of 1 × 1 mm 2 with 10 µm resolution from vital tissue explants reveals that the cancer cell clusters are heterogeneously divided into domains of several hundreds of micrometres with high or low local elastic strength. d, Median cervix tissue stiffness; in the cervix, we see a drop in median stiffness from E CN = 570 Pa (n = 5) for healthy cervix to E CC = 385 Pa (n = 7) in cervical cancer (p < 0.01, MWU test). e, Histogram of the distribution of local Young's moduli from AFM measurements of cervix cancer tissue and control tissue, from the maps shown in c and f. Both tissues show a log-normal distribution in stiffness (red fit lines), with the tumour only displaying softer cells than the cells from the healthy tissue. The box plots show quartiles 1, 2 and 3 (box) and 5%/95% (whiskers). n.s., not significant; **p < 0.01. Article https://doi.org/10.1038/s41567-022-01755-0 remained zero; a partially jammed, heterogeneous phase where tension percolation gives a finite bulk stiffness; and a solid phase. This tissue classification is also well founded in our AFM and OS data, independent of the simulations. The edge tensions have been calculated based on P 0 values of two neighbouring cells (Methods). Stiff cells (P 0 < 3.812) form the jammed regions (Fig. 4b), whereas soft cells are responsible for fluid regions. In the heterogeneous solid phase, the tension network self-organizes into a percolated structure, yet the rigid, jammed cells do not percolate. This suggests that a small fraction of jammed islands in a fluid sea, as observed in our patient-derived tumour explants, are sufficient to give rise to a finite shear modulus. In the fully solid phase, both tension and stiff cells form percolating networks. In our AFM data ( Fig. 1), we find-for tumours-the same stiff islands surrounded by soft cells with a finite bulk modulus analogous to the heterogeneous solid state in our simulations. In the healthy tissue samples, we find a more homogeneous stiffness distribution, suggesting a more solid-like behaviour.
The solid heterogeneous phase is clearly determined by the fraction of rigid cells in the tissue, f r (Methods). The pure fluid phase only exists for f r < 0.24, also visible when plotting the shear modulus over f r (Fig. 5c). The heterogeneous solid spans 0.24 < f r < 0.48, and the fully solid phase corresponds to f r > 0.48. The dependence of tissue mechanics 49 on f r reveals that tumour heterogeneity, that is, variance σ(P 0 ), fosters the rigidification of a tumour, which can be seen by the positive slope of the phase boundaries (Fig. 5a). This suggests that a tissue can rigidify with increasing cellular heterogeneity, as evident in the widening of the stiffness distribution of single cancer cells.
With further simplified dynamical vertex model simulations where every cell experiences an active propulsive force 57 (Methods), we elucidate the effect of heterogeneity and the fraction of rigid cells on tissue fluidity. The long-time migration behaviour is described by the self-diffusivity D eff (ref. 58 ). The fluid phase is characterized by a finite value of D eff , and D eff becomes vanishingly small (Methods) as the solid state is approached at f r = 0.48. This suggests that the heterogeneous solid (0.24 < f r < 0.48) is jammed with respect to small perturbations but can be fluidized when subject to a large propulsive force. In the solid phase (f r > 0.48), the cells are jammed and diffusive motion is completely hindered due to the contact percolation of rigid cells.
Our patient-derived stiffness data positions within the phase diagram with respect to the fraction of rigid cells f r confirm the classification by looking at the spatial stiffness maps measured by AFM (Methods). This permits the categorization of breast and cervical tumour samples with respect to our phase diagram (Fig. 5e). We find that breast and cervical cancer samples are located within the heterogeneous solid phase permitting a large fraction of unjammed, motile The spatial organization of cancer cells in clusters with soft, motile and rigid, jammed regions within a tumour causes the counterintuitive result that many soft cells can exist within the solid mass without destroying its mechanical stability as a solid that resists the microenvironment. Even where no backbone of stiff cells permeates the bulk, the tissue can spontaneously self-organize a spanning tension network that maintains rigidity. This heterogeneous solid phase explains how a tumour is able to simultaneously provide mechanical stability and cancer cell motility through the presence of soft, unjammed cells.

Soft cancer cells induce multicellular streaming
We use vital cancer cell tracking to confirm that there are both jammed islands and motile, unjammed areas in patient-derived tumour explants (12 cervix and 4 mamma carcinomas). In half the samples, we find unjammed as well as jammed regions in cancer clusters (examples in Fig. 6 and Supplementary Videos 1 and 2). The rigid jammed cancer cell clusters act as dynamic obstacles that lead to percolated tension networks and transiently channel soft, unjammed cells into parallel streams that wind through the cancer cell clusters (Fig. 6).
Our reductionist dynamic vertex model 45,57 makes obvious the effect of motile cells in a mechanically heterogeneous microenvironment. By calculating the velocity correlation between a motile cell and surrounding cells, we characterize the collective streaming behaviour (Fig. 6a-c). For a fluid state at f r = 0.12, the correlations indicate that up to 3-4 other unjammed cancer cells tend to 'follow' the motile one. Lateral to the invading cell, the correlations are weaker and vanish at around a single-cell diameter. This directional anisotropy results in the formation of a cellular stream of cancer cells (Fig. 6d-f) 57 . The stream anisotropy decreases with f r and disappears for the solid states at f r > 0.48 (Extended Data Fig. 4). These results reveal that mechanical heterogeneity due to the presence of soft cancer cells has a strong tendency to enhance the collective stream-like behaviour, as found in our vital cancer cell tracking observations.

Heterogeneity allows solid tumours and motile cancer cells
Despite the fact that cancer is a systemic disease, particularly the fact that metastasis quintessentially depends on biomechanical changes at the cell and tissue level 59-61 , the black-and-white characterization of tumour masses as stiff and cancer cells as soft demonstrates a lack of a comprehensive, detailed picture of mechanics in tumour biology. For the development of a malignant tumour, cancer cells have to move, proliferate and displace dense healthy tissue. Previously, the importance of cellular mechanical changes has been recognized when cancer cells leave the tumour cell mass and enter the surrounding stroma 25 . Cancer cell unjamming triggered by cell softening already boosts metastasis through a collective motility transition in cancer cell clusters within the tumour. Article https://doi.org/10.1038/s41567-022-01755-0 cell clusters causes a three-dimensional volume flow of cancer cells out of the depth of cancer cell clusters to the boundary, which is much more efficient than just the two-dimensional dissociation of cancer cells from the cluster surface. Moreover, multicellular streams exit the tumour with the ability to form collective clusters, enhancing their ability to survive outside the tumour mass and enhance metastatic cascade 65,66 .
The emergent cooperative properties of a heterogeneous tissue induced by cancer cell softening cannot be understood by studying the molecular properties of single metastatic cells and may play a critical role in cancer invasiveness. The interplay of mechanical heterogeneity and cancer cell unjamming regulates the stiffness of cancer cell aggregates and simultaneously permits cell motility. To our knowledge, this property of the heterogeneous solid state goes beyond previously reported states of active matter. Deregulation and dedifferentiation as well as the spectrum of epithelial-mesenchymal transition, which are part of any malignant transformation, most probably cause a broad mechanical heterogeneity together with a shift to softer cells. Thus, we expect that the observed mechanical changes occur inherently with early neoplasm.
The ability for cancer cell unjamming may be part of the initial difference between benign tumours that grow locally and malignant, invasive tumours. To overcome the complexity and heterogeneity, the universal physics underlying the mechanical processes in the progression of solid tumours, which is agnostic to the molecular details of different tumour entities, may provide a more general perspective on cancer development as a systemic disease than the molecular cell perspective alone. Since the described processes relate to the initial steps of cancer cell spreading, they may become important predictors of patient outcome complementary to genetic signatures. As described here, pathological mechanical changes driven by emergent effects, which cannot be directly related to a simple molecular cause, are a missing link in understanding cancer and will ultimately lead to new diagnostics as well as therapy.

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Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. Primary tissue samples. The remaining tissue with vital cells were used for cancer cell extraction. To dissolve a tissue into individual cells, the tissue samples were sliced into pieces of about 1 mm thickness and put into a gentleMACS Unique C tube (Miltenyi Biotec) containing 5 ml Dulbecco's modified Eagle medium (DMEM)/Ham's F12 medium supplemented with either 1.60 mg ml -1 collagenase P (Roche) and 20 µg ml -1 DNAse for breast cancer samples or 0.25 mg ml -1 collagenase 1A (Sigma), 0.25 mg ml -1 pronase (Roche) and 20 µg ml -1 DNAse for cervical cancer samples. The C tubes were mounted onto the gentleMACS dissociator and stirred using a customized dissociation routine (that is, 30 s mixing at varying speeds). The suspension was then incubated at 37 °C for 30-60 min. This step was repeated twice until no tissue clusters were visible. The single-cell suspension was then centrifuged first at 40×g to collect the remaining cell clusters and cell debris. This was followed by another centrifuge step at 300×g for 5-10 min. The pellet containing the cells from the tumour sample was then resuspended and the cells were cultured first in DMEM/Ham's F12 supplemented with 10% foetal calf serum and ×1 penicillin/streptomycin/amphotericin B for 24 h; afterwards, a serum-free medium was used for culture (HuMEC medium (Gibco) for breast cancer cells and defined keratinocyte serum-free medium (Gibco) for cervical cancer cells, each supplemented with ×1 penicillin/streptomycin/amphotericin B). These media are optimized for epithelia cell culture and promote the growth of epithelia cells as well as suppress other cell types by the supplementation of growth factors and other components 67 . Fine-needle aspiration biopsy was used to obtain the samples from malignant breast tumours and FAs, a benign lesion of breast tissue. In contrast to core biopsy, where a small cylinder of tissue is obtained, a smaller needle (21 gauge) was used. By exerting a negative pressure, mainly loosely attached cells and sheets of epithelial cells were collected due to the capillary effect when inserted into FAs 68 . The obtained cells were shortly cultured as described above and all the vital cells were used for measurement. Primary human mamma epithelial cells (Invitrogen) and HMEpC (PromoCell) were cultured according to the protocols provided.

Biomechanical measurements of cancer cells using automated microfluidic OS.
The OS is a two-beam laser trap, enabling biomechanical studies without physical contact 4,69 . Two opposing infrared laser beams form a stable trap. The suspended cells were injected and delivered through a microfluidic device to the centre of the trap. They were probed in a creep experiment where they were subjected to a step-stress profile, stretching with a high laser power for 2 s and holding for another 2 s with low power afterwards to observe their relaxation behaviour. The measurements were fully automated. Phase-contrast image sequences taken during the measurements were analysed using custom-made image analysis software to extract the time course of the observed relative cell deformation. The measured cells remain viable after stretching 70 . The applied laser power exerted a peak tensile stress of about 20 Pa on the cells causing an elongation of 0.5-7.0% of the cell diameter along the laser axis. Actively contracting cells can lead to an underestimation of deformation, but is more common in cancer cells ( Supplementary Fig. 1); therefore, it might result in a slight underestimation of the difference. Log-normal distributions were fitted to the data and the significance was tested with the Mann-Whitney U-test (MWU test). The width of the distributions was characterized by the interquartile range with the difference between the first and third quartile.
Biomechanical measurements of cancer tissues using AFM. The tissue samples from breast and cervix carcinomas and adjacent healthy tissues were obtained during routine tumour resections. The tissue samples were measured within hours after resection. The tissue samples were chopped into 400-µm-thin slices with a McIlwain tissue chopper. Subsequently, the slices were glued (Histoacryl, B. Braun) onto microscope slides followed by the measurement of elastic strength (Young's modulus). The AFM used is a NanoWizard 4 instrument with 300 µm HybridStage ( JPK) combined with an Axio Zoom.V16 instrument (ZEISS). A CONT (NanoWorld) contact-mode cantilever was modified with a 6-µm-diameter polystyrene bead to increase the contact area. The force ramps were recorded with the following parameters: maximum force, 7.5 nN; z speed, 20 µm s -1 ; z length, 30 µm; capture rate, 2,048 Hz; imaging area, up to 1 × 1 mm 2 was split into smaller squares of 200-250 µm side length to fit into the piezo range of the hybrid stage. The maps were recorded with 10 µm data-point spacing. The AFM data were first analysed with the JPK data processing software (version 7.1.18) to calculate the Young's modulus using a Hertz fit to the smoothed and baseline-corrected force-indentation curves. The data were post-processed with a custom-written MATLAB program (MathWorks, version 2018b) to fit a log-normal distribution (2,000-10,000 data points per sample). The significance was tested with the MWU test. By using simultaneous fluorescence microscopy of DNA-stained cell nuclei, we assured that our maps, which we obtained, are from tumour areas with cancer cell clusters and not from the surrounding ECM.

Biomechanical measurements of single cells and MTS. MCF-10A
and MDA-MB436 cells were cultured in cell culture flasks (TPP) for single-cell AFM measurements. Multicellular tumour spheroids (MTS) were formed on UltraPure agarose gels in a 96-well plate. Here 20,000 cells are added to each well, as they cannot adhere to the agarose; they adhere only to the other cells present and form the MTS. MDA-MB-436 cells were cultured in 90% DMEM (without sodium pyruvate), 10% foetal calf serum and 1% 10,000 U ml -1 penicillin/streptomycin. MCF-7 cells were cultured in 88% Eagle's minimal essential medium supplemented with 10 µg ml -1 insulin and 1 mM sodium pyruvate, 10% foetal calf serum/foetal bovine serum, 1% non-essential amino acids and 1% penicillin/streptomycin. The cells and MTS were measured with a CellHesion 200 instrument ( JPK) and a tipless cantilever (Arrow TL1, NanoWorld). Single cells and MTS were directly measured after passaging into a Petri dish (TPP), and still being only weakly adherent to reduce the influence of the substrate. The CellHesion instrument is equipped with a custom climate chamber to provide 37 °C and 5% CO 2 during the measurements.
Biomechanical measurements of cancer tissues using MRE. Table-top MRE measurements were carried out on 8 mm punch biopsies from the same tissues described earlier. The setup consists of a tabletop magnetic resonance imaging scanner (Pure Devices) with a 10 mm bore and 0.5 T permanent magnet that was customized by an additional gradient amplifier (DC-600, Pure Devices) and a piezoelectric driver controlled by a magnetic resonance imaging system (Piezosystem Jena) covering the frequency range between 200 and 6,000 Hz. The tissue samples were placed at the bottom of 7 mm glass tubes protected from evaporation by the addition of a cotton wool ball soaked in phosphate-buffered saline at the top of the tube and sealed by a plastic plug with a silicon shock absorber at the bottom and a single slice of polyvinyl chloride at the top. The glass tubes with the samples Article https://doi.org/10.1038/s41567-022-01755-0 were coupled from the top to the piezo driver and the section with the sample was positioned within the bore of the magnetic resonance imaging scanner, which was heated to 37 °C. The vibrations from the piezo actuator are constrained in axial motion and coupled via the glass walls into the sample. A detailed overview of the imaging sequences and motion-encoding gradients is described elsewhere 6 . In brief, the data acquisition time for each frequency was approximately 8 min; a frequency range of 1-6 kHz was covered in 500 Hz intervals, resulting in 11 measurement points and a total runtime of approximately 1.5 h. The following acquisition parameters were used: repetition time, 500 ms; echo time, 42 ms; slice thickness, 3.00 mm; matrix size, 56 × 56; field of view, 8.40 × 8.40 mm 2 resulting in a voxel size of 0.15 × 0.15 × 3.00 mm 3 . The acquired data were unwrapped and Fourier transformed in time to extract complex-valued wave images for each driving frequency. The wave profiles for deflection parallel to the cylinder axis were created and fitted by the analytical solution of shear waves in a z-infinite cylinder 6 , resulting in the complex wavenumber k* = k′ + ik″. Based on the fact that the shear-wave speed c and shear-wave penetration rate a can be derived for each frequency, we get These parameters were directly fitted by a viscoelastic fractional element model to derive shear-modulus-related parameters.
Here µ and α are two independent variables; µ represents a measure of tissue stiffness and the power-law variable α is directly translated to the phase angle of the complex shear modulus G* by multiplication with π/2. More details can be found elsewhere 32,34 . The KS test was used to check the significance.

Spheroid fusion experiments.
Spheroids were formed with the same protocol as the AFM measurements. Two spheroids were transferred into a single well and observed with phase contrast microscopy over several hours. The used Leica DM IRB instrument was equipped with a custom climate chamber to provide 37 °C and 5% CO 2 during the measurement. Spheroids were fitted with two circles, the angle θ is the angle between the line connecting the two centre points and the radius from one centre to the intersection of the two circles. The progress of spheroid fusion was tracked over time and calculated as Δ(cosθ)/Δt in the time period of 24-36 h of fusion. The significance was checked with the KS test.

Vertex model of a mechanically heterogeneous tissue.
We use the vertex model to understand the collective mechanical behaviour of dense tumour aggregates. In the vertex model, a two-dimensional confluent epithelial tissue is governed by the energy function 42,71-75 , where cell areas {A i } and perimeters { Pi } are functionals of the positions of vertices {r i }. Also, K A and K P are the area and perimeter elasticities, respectively. The quadratic term in A i results from resistance to cell volume changes 71,73 . Changes to cell perimeters are related to the deformation of actomyosin cortex 71,73 . The term K P P 2 i corresponds to the energy cost of deforming the cortex. The linear term, −2K P P i 0 P i , is the effective line tension by cell i, which gives rise to a 'preferred perimeter' P i 0 . The value of P i 0 emerges from an interplay of cell-cell adhesion and cortical tension 71 . Here we assume the preferred cell area A 0 does not vary from cell to cell and is set to be the average area per cell ( A i 0 = A). The energy can be nondimensionalized by choosing K P A as the energy unit and √A as the length unit: where a i = A i /A and p i = P i /√A are the rescaled area and perimeter of the ith cell, respectively. Also, κ A = K A A/K P is the rescaled cell area elasticity and p i 0 = P i 0 /√A is the preferred cell shape index 76 . In this model, cell stiffness is determined by tension τ m on cell-cell junctions (edges). For an edge with length l m , the tension is given by 49,56,77 where p i and p j are the rescaled perimeters of cells i and j, respectively, adjacent to edge m. As a result, cell stiffness is directly tuned by the preferred cell shape indices. To capture the experimental heterogeneities in single-cell stiffness and cell-cell interactions 14 To initialize the simulation, Voronoi cells 58 are used to provide a set of initial vertex positions. Then, each cell is assigned a value of p 0 drawn from a log-normal distribution. The set of p 0 values remains as quenched variables. We use a combination of FIRE (fast inertial relaxation engine) and conjugate-gradient algorithms 81,82 to minimize the tissue energy under periodic boundary conditions with a fixed equilibrium cell area A 0 =Ā = 1. This algorithm produces stable states where the net residual force on vertices is less than 10 -8 . For this work, we simulate tissues with N = 400 cells. Each tissue is characterized by a mean (µ p0 ) and s.d. (σ p0 ) of single-cell p 0 values. We have systematically studied a large range of these parameters: µ p0 = 3.75-3.90; σ p0 = 0.05 to -0.20. Following a previous theoretical study 49 , we define the fraction of rigid cell f r as the fraction of cells with p 0 < 3.812, which can be written as Here ℱ μ,σ (p 0 ) is the distribution function of p 0 . For a log-normal-distributed p 0 , the fraction of rigid cells is analytically given by Here erfc is the complementary error function and µ and σ are the control parameters of the log-normal distribution for which the mean and s.d. can be calculated as Calculating mechanical response at the tissue level. At the tissue level, its mechanical response is characterized by shear modulus G. A non-zero G corresponds to a solid-like tissue, whereas G vanishes for a fluid state. We obtain G by calculating the linear response to an infinitesimal affine strain γ via the Born-Huang formulation 83 In the above equation, Ξ iμ is the derivative of the force on vertex i with respect to strain given by where r iµ is the position of vertex i and µ = x, y is the Cartesian index. Also, A total = ∑ N i A i is the total area of the tissue. Also, M is the Hessian Article https://doi.org/10.1038/s41567-022-01755-0 matrix given by the second derivative of tissue energy E with respect to position vectors of vertices i and j (refs. 49,56 ): Using effective diffusivity to characterize dynamics of cell motion. We use self-diffusivity D s = lim t→∞ ⟨Δr (t) 2 ⟩/ (4t) to distinguish between the solid and fluid states 57 , where lim t→∞ ⟨Δr (t) 2 ⟩ is the mean square displacement. For practicality, we calculate D s using simulation runs of 2 × 10 5 time steps at step size Δt = 4 × 10 -2 using Euler's method with propulsive force v 0 = 0.05 and rotational noise of cells D r = 1 under periodic boundaries. We present the self-diffusivity in units of D 0 = v 2 0 /(2D r ), which is the free-diffusion constant of an isolated cell. Then, D eff = D s /D 0 serves as a dynamical order parameter that distinguishes a fluid state from a solid state. The simulations are performed in the Surface Evolver program.
Accounting for measured distributions of cell stiffness. The experimental data for breast and cervix cancer cells are well described by log-normal distributions. The rigidity of the tissue with p 0 values given by these distributions can be obtained. We stress that only the relative cell stiffness can be inferred from the experimental data, but not the actual value of =p 0 for each cell. This is because p 0 is controlled by the interplay between cortical tension and cell-cell adhesion, whereas the OS only measures the mechanical tension, that is, the stiffness of the single-cell cortex in the absence of any cell-cell interactions. These data do not infer the effective tension that the cells experience in a confluent tissue. Nonetheless, this analysis provides an understanding of how soft cancer cells impact the mechanical behaviour of tissues. In particular, it suggests the possibility that a tumour-containing cells that are on average softer than that of the healthy tissue-could actually still exhibit rigidity at the collective tissue level due to the broadness of distribution.
The fraction of rigid cells can be extracted from the relative deformations of various tissue types. For each cell type, we use the mean deformation value of non-cancer cells (Fig. 3) to define the single-cell rigidity threshold for both non-cancer and cancer phenotypes. The f r value for each cell type is then calculated by computing the fraction of cells with deformation values smaller than the mean deformation value of their non-cancer counterpart. This allows the mapping of each cell type in each tumour category to the solid-fluid nexus predicted within the theory.
Heterogeneity and cellular invasion. Here we use a dynamic vertex model 45 to simulate a tissue where only a single cell is invasive to study the effect of heterotypic cellular environment on cell migration. The invading cell has a propulsive force v 0 along a polarity vector n , which undergoes random rotational diffusion 58 at a slow rate. This mimics the directional motility of a metastatic cell under the influence of strong chemotactic signals 84  The polarity vector undergoes random rotational diffusion as where θ i is the polarity angle that defines n and η i (t) is a white-noise process with zero mean and variance 2D r . The value of angular noise D r determines the memory of stochastic noise in the system, giving rise to a persistence timescale τ = 1/D r for the polarization vector n . The timescale τ = 1/D r controls the persistence of cell motion.
We numerically simulate the model using molecular dynamics by performing 10 5 integration steps at a step size of Δt = 10 -2 using Euler's method with propulsive force v 0 = 0.4 and rotational noise D r = 0.01 for N = 400 cells under periodic boundaries. With heterogeneity, tissues in the range of 0 < f r < 1 become accessible and cells moving through them must interact with rigid as well as soft neighbouring cells along the path of invasion. This results in a highly intermittent migration dynamics for the invading cell.
Emergence of stream-like collective behaviour. To reveal the motility patterns in a heterogeneous tissue, we assume a log-normal distribution of p 0 values based on the broad distribution of cell deformations from the OS. Since cancer cells are also shown to be more contractile and the contractility mechanism is also responsible for cell traction, we assume that soft cells (p 0 > 3.812) have larger motility than rigid cells (p 0 < 3.812).
To analyse the motion of the invading cell, we always orient the invading cell such that its direction of motion is along the x axis, that is, we choose a frame of reference in which the invading cell is located at (x = 0, y = 0). To correlate the motion of the invading cell and other cells in its vicinity, we compute the correlation function as 23 Positions are in units of mean cell diameter. When f r = 0.11, directly behind the invading cell, the correlations are long ranged indicating that up to 3-4 cells tend to 'follow' the invading cell. However, lateral to the invading cell, the correlations are weaker and decay beyond the one-cell diameter. This directional anisotropy is indicative of a cellular stream forming behind the invading cell. With an increase in f r , the anisotropy disappears. To further quantify the behaviour of stream anisotropy with changing f r , we calculate the correlation lengths ξ x , ξ y in the x and y directions, respectively, where the correlation C vv (x, y) decays to a threshold of 10 -3 . Then, the stream anisotropy parameter is given by the ratio ξ x /ξ y and we plot it as a function of f r (Supplementary Fig. 5).

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Simulation data and code are available from D.B. (d.bi@northeastern. edu) on reasonable request.

Corresponding author(s): Josef A. Käs
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