Motility-limited aggregation of mammary epithelialcells into fractal-like clustersSusan E. Leggetta,b,1, Zachary J. Neronhaa, Dhananjay Bhaskara, Jea Yun Sima, Theodora Myrto Perdikaria,and Ian Y. Wonga,b,2

aSchool of Engineering, Center for Biomedical Engineering, Brown University, Providence, RI 02912; and bPathobiology Graduate Program, BrownUniversity, Providence, RI 02912

Edited by David A. Weitz, Harvard University, Cambridge, MA, and approved July 22, 2019 (received for review April 11, 2019)

Migratory cells transition between dispersed individuals andmulticellular collectives during development, wound healing, andcancer. These transitions are associated with coordinated behav-iors as well as arrested motility at high cell densities, but remainpoorly understood at lower cell densities. Here, we show thatdispersed mammary epithelial cells organize into arrested, fractal-like clusters at low density in reduced epidermal growth factor(EGF). These clusters exhibit a branched architecture with a fractaldimension of Df = 1.7, reminiscent of diffusion-limited aggrega-tion of nonliving colloidal particles. First, cells display diminishedmotility in reduced EGF, which permits irreversible adhesion uponcell–cell contact. Subsequently, leader cells emerge that guide col-lectively migrating strands and connect clusters into space-fillingnetworks. Thus, this living system exhibits gelation-like arrest atlow cell densities, analogous to the glass-like arrest of epithelialmonolayers at high cell densities. We quantitatively capture thesebehaviors with a jamming-like phase diagram based on local celldensity and EGF. These individual to collective transitions repre-sent an intriguing link between living and nonliving systems, withpotential relevance for epithelial morphogenesis into branchedarchitectures.

jamming | gelation | collective migration

Transitions between individual and collective cell migra-tion are associated with tissue morphogenesis, wound heal-

ing, and tumor progression (1). In particular, the epithelial–mesenchymal transition (EMT) occurs when tightly adherentepithelial cells detach and disseminate as individual mesenchy-mal cells (2). Such behaviors are observed with clusters ofepithelial cells in vitro, which “scatter” individually upon addi-tion of epidermal growth factor (EGF) (3, 4) or hepatocytegrowth factor (HGF) (5–7). Similarly, a “partial” EMT is asso-ciated with leader cells that exhibit enhanced motility but retainsome cell–cell contacts for mechanical guidance of their fol-lowers (8–17). Instead, a reverse mesenchymal–epithelial tran-sition (MET) can occur when mesenchymal cells condense anddifferentiate into a compact epithelial tissue, associated withskeletal development in vivo (18). Analogous “swarming” behav-iors have been observed during neural crest development (19),neutrophil recruitment (20), and Dictyostelium aggregation (21),although these cells do not acquire strong cell–cell junctions.Overall, this emergence of complex spatial organization fromcollective cellular motion and cell–cell adhesion remains poorlyunderstood.

Nonliving soft matter systems such as colloidal particles ina fluid medium also exhibit collective phase behaviors due tointerparticle interactions (22). For instance, dispersed colloidsthat diffuse randomly but adhere irreversibly can aggregateinto highly branched, connected clusters with fractal-like archi-tectures (i.e., diffusion-limited aggregation) (23–25). This localarrest of particle dynamics at low densities corresponds to amacroscopic transition from a fluid-like solution to solid-likegel, which can be mapped to a “jamming” phase diagram con-trolled by particle density, interparticle adhesion, and shearstress (26). In the limit of higher packing densities, colloidal

systems also exhibit a jamming transition as they approachrandom close packing, analogous to a glass transition (27).Interestingly, closely packed epithelial monolayers also exhibitarrested dynamics as they approach confluent cell densities (28–39). Based on these results, a jamming-like phase diagram for cellmonolayers has been proposed based on cell density, cell–celladhesion, and cell speed (40), but the importance of cell densityremains unresolved since jamming can occur at constant density(35, 36, 39). Since both colloidal particles and living cells exhibita glass-like jamming transition at high densities, and colloidalparticles exhibit a gelation-driven jamming transition at low den-sities, an intriguing possibility is that living cells could also exhibitaggregation and arrest at low densities.

Here, we show that epithelial cells aggregate into multicellu-lar clusters with branched, fractal-like architectures when cul-tured at low densities in reduced EGF. Cluster formation wasobserved only with nontransformed mammary epithelial cells,which exhibited diminished proliferation and motility due togrowth factor dependence. Single-cell tracking revealed thatmigratory individuals adhered irreversibly to multicellular clus-ters and became immobilized. EMT induction resulted in theemergence of leader cells at the cluster periphery, guiding col-lective migration outward to connect clusters into spanning net-works. We constructed a phase diagram for cluster formation and

Significance

Individually migrating cells cluster into multicellular tissuesduring tissue formation, inflammation, and cancer. The cor-responding increase in cell density can result in arrestedmotion, analogous to the “jamming” of soft materials suchas glasses and gels. Here, we show that cells with reducedmotility and proliferation organize into branching clusters,reminiscent of aggregation in nonliving colloidal particles.Subsequently, “leader cells” guide collective migration to linkclusters together into spanning networks. These arresteddynamics occur at unusually low density and are reminiscentof gelation. Furthermore, increased motility and prolifera-tion recover a glass-like transition at higher density. Overall,arrested motion in living cells has striking similarities withnonliving colloidal particles, suggesting physical signatures ofclustering and branching morphogenesis in development anddisease.

Author contributions: S.E.L. and I.Y.W. designed research; S.E.L. performed research;S.E.L., Z.J.N., D.B., and I.Y.W. contributed new reagents/analytic tools; S.E.L., Z.J.N., D.B.,J.Y.S., T.M.P., and I.Y.W. analyzed data; and S.E.L. and I.Y.W. wrote the paper.y

The authors declare no conflict of interest.y

This article is a PNAS Direct Submission.y

Published under the PNAS license.y1 Present address: Department of Chemical and Biological Engineering, PrincetonUniversity, Princeton, NJ 08540.y

2 To whom correspondence may be addressed. Email: ian wong@brown.edu.y

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1905958116/-/DCSupplemental.y

Published online August 14, 2019.

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arrest based on local cell coordination number rather than over-all density. We further developed a computational particle-basedmodel that recapitulated cluster formation and arrest. Thesebiophysical phenomena are unanticipated by existing models ofcellular jamming and may yield insights into branching networkarchitectures formed in living tissues.

ResultsCell Proliferation and Motility Are Arrested in Reduced EGF Condi-tions. Human mammary epithelial cells (MCF-10A) were firsttracked with single-cell resolution in “growth” media supple-mented with 20 ng/mL EGF relative to “assay” media withreduced EGF (0.075 ng/mL). These cells were stably transfectedto undergo a controlled EMT through the Snail transcription fac-tor, using an inducible construct responsive to tamoxifen (OHT)(2, 41). Such EMT induction is associated with increased motil-ity and elongated morphologies over 48 to 72 h, due to increasedactin polymerization and membrane protrusion at the leadingedge (42). In growth media supplemented with OHT, dispersedcells exhibited relatively fast proliferation and formed continu-ous monolayers that occupied almost all of the available area by60 h (Fig. 1A and Movie S1). These cells were also highly motile

and constantly rearranged themselves, even at near-confluentcell densities, which remained qualitatively consistent over arange of initial cell densities (SI Appendix, Fig. S1 A and B).In contrast, in assay media with EGF reduced to 0.075 ng/mLand OHT, cells displayed significantly slower proliferation withlarge unoccupied regions after 60 h (Fig. 1B and Movie S2),even from higher initial densities (SI Appendix, Fig. S1 C andD). Despite these relatively sparse cell densities, initially dis-persed cells became associated with a multicellular cluster by24 h, which was statistically significant based on Ripley’s H func-tion (SI Appendix, Fig. S1E). The overall morphology of theseclusters remained consistent through 60 h, suggesting that cellmotility was arrested and rearrangements were limited. Interest-ingly, these clusters exhibited highly branched morphologies thatmerged into spanning networks over time (Fig. 1B).

For fluorescence imaging, these cells were also stably trans-fected with red fluorescent protein in the nucleus (H2BmCherry) as well as green fluorescent protein (GFP) in thecytoplasm. Proliferation rates were compared across conditionsby detecting the number of fluorescent nuclei present over theduration of the experiment. In growth media, MCF-10A cellsexhibited roughly exponential growth with an 11-fold increase in

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Fig. 1. Subconfluent mammary epithelial (MCF-10A) cells arrest into multicellular clusters in “assay media” with reduced EGF. (A and B) Cells in “growthmedia” (20 ng/mL EGF) organize into confluent monolayers (A), while cells in assay media (0.075 ng/mL EGF) organize into branched clusters at subconfluentdensity after 60 h (B). Merged images of live cells with fluorescent proteins in the nucleus (red, mcherry-H2B) and cytoplasm (gray, GFP cytoplasm). (C) Cellproliferation is exponential in growth media (blue), but occurs more slowly in assay media (orange). (D) Similarly, population-averaged cell speed decreasesat long times for both growth media (blue) and assay media (orange), which occurs consistently at varying initial cell seeding density (Inset, cells/mm2).Data are summarized from 3 independent experiments, with mean and SD indicated by solid lines and shaded regions, respectively. Statistical significance isindicated by black circles for all P values < 0.05 computed at hourly intervals (C and D).

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cell density over 60 h (Fig. 1C). In assay media, cell proliferationslowed and approached a plateau at 5-fold higher cell density by50 h, analogous to resource-limited logistic growth. Cell viabilityremained above 99% after 60 h (SI Appendix, Fig. S1 B and D),and no changes in the (phenol red) media color were observed,indicating minimal changes in pH. Individual cell speeds werethen determined by tracking nuclear motion of all cells over time,with an initial increase in speed over ∼10 h as cells adhered tothe substrate. In growth media, MCF-10A cells initially exhib-ited a population averaged speed of ∼25 µm/h, which graduallydecreased to ∼8 µm/h by 60 h, while cells in assay media startedout with average speeds of ∼10 µm/h and gradually slowed to∼2 µm/h by 60 h (Fig. 1D and SI Appendix, Fig. S2 A and B).This decrease in average speed occurred with comparable kinet-ics across all initial cell densities, suggesting that arrested motionwas not dependent on overall cell density (Fig. 1D, Inset). Over-all, these combined observations indicate that assay media withreduced EGF resulted in the formation of multicellular clustersand spanning networks despite diminished cell proliferation andmotility.

Dispersed Individuals Aggregate into Multicellular Clusters withFractal-Like Morphology. Clustering dynamics were then analyzedfrom single-cell tracks over the duration of the experiment. Ini-tially dispersed cells migrated randomly, but their motion wasarrested upon encountering other cells (Fig. 2A). Individual cellswere defined as cells whose nuclei were located at least 75 µmfrom any other nuclei. Based on this definition, roughly 10% to30% of individuals were typically observed at the start of theexperiment, which decreased to 5% or less over ∼20 h (Fig. 2B).These individuals exhibited random-walk–like migration, whichwas quantitatively analyzed from the mean-squared displacement(MSD) per cell 〈∆r2(τ)〉. These MSDs could be fitted to apower law of the form Kατ

α, where α= 1 corresponds to ran-dom diffusion-like motion and α= 2 corresponds to persistentballistic-like motion (43). Note that Kα is a generalized diffu-sion coefficient with units of µm2·h−α. The statistical distribu-tion of these fitted parameters was α= 1.23± 0.40 for n = 44individuals (from 3 independent experiments) (Fig. 2B, Inset).Nevertheless, it should also be noted that Kα also varied con-siderably with α, resulting in large variations in cell motility (SIAppendix, Fig. S2C). Although there was substantial cell-to-cellheterogeneity, individual cell migration was more random thandirected.

Multicellular clusters were then defined by computationallylinking nuclei located less than 75 µm apart, which was verifiedby segmenting cluster morphologies in the cytoplasmic channel(SI Appendix, Fig. S2 D and E). This relatively high size cutoffwas chosen to correctly associate elongated cells with a cluster,but more compact cells within a cluster were typically locatedless than 50 µm apart (37). For each field of view, ∼10 clus-ters were typically observed, a number which remained relativelystable over ∼24 h, indicating that cells aggregated irreversiblyand that clusters did not dissociate (Fig. 2C). Nevertheless, thenumber-averaged cluster size increased steadily over time, reach-ing∼100 cells by 60 h (Fig. 2D). This increase could be attributedto migratory individuals being “captured” at the cluster periph-ery, some proliferation events, and finally the merging of localclusters, resulting in a sharp decrease in the number of clustersafter 20 h (Fig. 2C).

At later times, cells organized into large clusters (>10 cells)that exhibited a dendritic, noncompact morphology with geomet-rically “rough” features at the periphery (Fig. 1B). In particular,cells at the interior were tightly connected with many neighbors,while cells at the periphery were less connected with fewer neigh-bors, indicating a decrease in local cell density with increasingradial distance. Both the cluster morphology and this radiallydecreasing density were strikingly reminiscent of fractal-like

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Fig. 2. Dispersed individuals exhibit random walks and adhere irreversiblyto form multicellular, fractal-like clusters. (A) Representative snapshots ofrandomly migrating individuals that arrested into clusters when cultured inassay media. Shown are merged images of cell cytoplasm (gray) and celltracks over 5 h (red line). (Scale bar, 50 µm.) (B) Fraction of individual cells(not associated with a cluster) decreases over time. (B, Inset) Mean-squareddisplacement computed for representative individual cells (n = 9) at earlytimes (dashed lines). Cells are from 3 independent experiments, indicatedby distinct colors. Power law with indicated α values is plotted for compari-son (solid black lines). (C and D) Number of clusters decreases over time (C),while average cluster size increases over time (D). (E) Mean radius of gyra-tion (squares) and SD (error bars) observed at each cluster size for n = 4,516total clusters. Radius of gyration scales with cluster size as a power law, withfractal dimension 1/Df derived from the slope of the log–log fit. A 95%confidence band (gray region) for the power-law parameters is estimatedusing a bootstrapping procedure with 1,000 iterations. E, Inset shows thesame data with linear scaling. Mean (solid lines) and SD (shaded regions)are computed across 3 independent experiments (B–D).

structures associated with the diffusion-limited aggregation ofcolloids (23–25). A quantitative signature of fractals is the frac-tal dimension Df , which scales the radius of gyration Rg with thecluster size M as a power law: Rg =M 1/Df . For these multicel-lular clusters, the number of nuclei per cluster scaled with theradius of gyration as Df = 1.74± 0.03 (95% confidence intervalfor power-law fit, ordinary nonparametric bootstrap analysis) forall clusters of at least 4 nuclei over varying initial densities andearly times (Fig. 2E). This fractal dimension was also evaluatedindependently using a box-counting analysis of the cell morphol-ogy over varying length scales, which yielded a similar Df =1.65± 0.04 (mean ± standard deviation [SD]) (SI Appendix,Fig. S2D). It should be noted that the first analysis of fractaldimension is based on the discrete spatial distribution of cellnuclei, whereas the second analysis is based on the continuousspatial distribution of cell cytoplasm. The slight differences maybe explained by the elongated cell morphologies at the periph-eral branches relative to nuclei positions. Remarkably, Df = 1.7is consistent with previous experimental and computational workon diffusion-limited aggregation of nonliving colloids, whichexhibit random diffusive motion but adhere irreversibly on

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contact (23–25). Thus, this motility-driven aggregation of liv-ing cells into stationary fractal-like clusters exhibits strikingsimilarities to the diffusion-limited aggregation of colloids.

Transient Collective Migration Connects Clusters into Spanning Net-works. Within stationary clusters, small groups of cells oftenexhibited transient but highly correlated motions (Fig. 3A). This“dynamic heterogeneity” was elucidated using the 4-point sus-ceptibility χ4(τ), which resolves correlated density fluctuationsin space and time (44). Typically, χ4(τ) exhibits a peak that cor-responds approximately to the characteristic size and timescaleof collective motion. For this analysis, cells were classified as“motile” with 0% overlap if they traveled more than 1 nucleardiameter (10 µm) over some time interval τ , analogous to“mobile” nonliving particles (45). This cutoff distance is compa-rable to the 15% of a cell diameter used elsewhere for jammingin epithelial monolayers (36). The self-overlap function 〈Q(τ)〉was calculated by averaging these overlap values for all cellswithin a time frame and then ensemble averaging over a rangeof start times (45). Essentially, 〈Q(τ)〉 represents the fractionof cells that have displaced less than 1 nuclear diameter aftersome time duration τ has elapsed. For instance, 〈Q(τ)〉= 1 atτ = 0 h but decreased to 〈Q(τ)〉= 0.5 at τ = 2 h at early times(<12 h), indicating that roughly half of the cells were motile(nonoverlapping) after 2 h (Fig. 3B). At later times (>12 h),〈Q(τ)〉 increased, corresponding to a decreasing fraction ofmotile cells and an overall arrest of cell migration over time.Next, to account for temporal correlations between motile cells,the 4-point susceptibility χ4(τ) =N [〈Q(τ)2〉− 〈Q(τ)〉2] was cal-culated from the moments of 〈Q(τ)〉 and the total number of

cells N . At early times (<12 h), the peak χ∗4(τ) occurred atτ ∼ 1.5 h, which corresponds to a characteristic lifetime of cor-related motion (Fig. 3B). At later times, χ∗4(τ) shifted slightlyto τ ∼ 2 h. The peak height χ∗4(τ) decreased from 6 cells to2 cells, which can be attributed to the increase in cell numbersover time.

Some highly motile cells (>7 µm/h) often exhibited an elon-gated morphology with directionally persistent motion, reminis-cent of a leader cell phenotype (Fig. 3C) (8, 9, 14). Indeed, theseleader-like cells exhibited highly concerted motion with followers(Fig. 3A) and retained some cell–cell contacts. Immunofluores-cence staining at the completion of the experiment revealedE-cadherin localization at cell–cell junctions, which was absentin growth media (Fig. 3C). Cells at the cluster peripheryalso exhibited increased vimentin expression, a classical mes-enchymal biomarker (46). These highly motile cells thereforerepresent a leader cell phenotype that retains both epithe-lial and mesenchymal biomarker expression, consistent with apartial EMT.

Over time, leader cells from adjacent clusters often guided col-lectively migrating strands to merge together into a continuousmulticellular structure. Once cell–cell contact occurred, leadercells lost their motile phenotype with front–back polarizationand reverted to an adherent epithelial morphology with cell–celljunctions, consistent with previous reports (9). As a consequence,isolated clusters became connected together as spanning, space-filling networks, analogous to the gelation of nonliving particles(26). The motion of these multicellular strands away from thecluster was highly persistent, which could be directly visualizedusing a kymograph analysis (Fig. 3D and SI Appendix, Fig. S3

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Fig. 3. Leader cells guide transient collective migration events that result in cluster merging. (A) Correlated cell motion within otherwise stationary clusters.Spots represent nuclei positions and lines represent tracks over 5 h for leader cells (blue) and nonleaders (green), overlaid onto live images of cells (GFPcytoplasm, gray). General direction of migration is denoted by white arrows, with dashed line indicating region between clusters quantified by kymographanalysis in D. (B) The self-overlap function 〈Q(τ )〉 and dynamic 4-point susceptibility function χ∗

4 (τ ) identify motile fractions of the population with somecharacteristic timescale of collective migration, which arrests over time; representative data are shown. (C) Leader cells exhibit a partial EMT phenotype(arrow) with coexpression of vimentin (red) and E-cadherin (green) at cell–cell junctions (asterisks). (Scale bar, 25 µm.) (D) Representative kymograph depictsthe intensity of the cytoplasm fluorescence at the leader cell front for 2 merging cells over time with respect to the dashed line in A; leader and followercells exhibit directionally persistent motion to merge together. (E) Leader cells (blue) exhibit persistent motion for longer durations relative to nonleadercells (green), significant at P< 0.01; black line indicates median value. (F) Leader cells migrate with faster velocities than nonleaders, significant at P< 0.01,and the population average is shown (mean and SD in orange solid line and shaded region, respectively). (G) Number of leaders increases with cluster size,highlighted with gray line representing the median number of leaders. Motion of leaders and nonleaders was manually quantified across 2 independentexperiments, n = 82 leaders and n = 102 nonleaders (D–G).

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A and B). The typical persistence time of these strands basedon lamellipodial extension was ∼4 h, which greatly exceededthe persistence time of 2.5 h for the largely random motionof other cells within the cluster (Fig. 3E). Indeed, these tran-sient bouts of leader-driven collective migration often reachedlamellipodial speeds of 10 µm/h or more, which greatly exceededthe population-averaged speed of <5 µm/h at later times, whenthe migration of most cells was largely arrested (Fig. 3F). Inter-estingly, the number of leader cells at the cluster peripheryincreased with increasing cluster size. For instance, smallerclusters of less than 10 cells were typically associated withonly 1 leader cell (Fig. 3G). Moreover, larger clusters with 10to 30 cells were associated with 2 or more leader cells, etc.The number of leader cells scales approximately linearly withthe radius of gyration, with an average separation of ∼245µm between leader cells along the periphery (SI Appendix,Fig. S3 C and D), in agreement with previous theoretical pre-dictions (10, 15). Overall, the emergence of leader cells atthe cluster periphery resulted in transient collective motionthat connected isolated clusters into a continuous, space-fillingarchitecture.

Clustering and Jamming Are Governed by Local Cell Density and EGF.Cells were observed to accumulate greater numbers of neighborsover time during the transitions from dispersed individuals toaggregated clusters. As a measure of local cell density, the num-ber of neighbors (“bonds”) 〈B(t)〉 was determined by countingall nuclei located within 75 µm of a given cell and then ensem-ble averaging over all cells within a field of view. For the lowestinitial density condition (∼30 cells/mm2), the transition fromdispersed individuals to aggregated clusters occurred at 〈B〉∼ 4when t ∼ 30 h (Fig. 4A). Similarly, for a higher-density condition(∼ 60 cells/mm2), this transition to aggregated clusters occurredat 〈B〉∼ 4 when t ∼ 20 h, while the subsequent formation ofspanning networks occurred at 〈B〉∼ 7 when t ∼ 40 h. This tran-sition has analogies to a percolation threshold associated withgelation (47), although it exceeds the critical bond number asso-ciated with an idealized square lattice (≈4), due to increasedconnectivity since clusters are linked by “strands” that are severalcells wide. In general, higher-density conditions formed clusterseven more rapidly (<20 h), which connected together soon after-ward (by 30 h). It should be noted that cells typically packedtogether more closely than 75 µm within epithelial monolayers,

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Fig. 4. Jamming-like phase diagram based on neighbor density and EGF. (A) Neighbor density is a readout of individual, clustered, and spanning phases.Each line indicates data from a representative well treated with 0.075 ng/mL EGF, for various initial starting densities with several replicates shown. (Right)Reference snapshots of live cells (GFP cytoplasm, gray) for each of the density regimes. (B) A jamming-like phase diagram defined with neighbor densityand EGF concentration. Gray circles indicate motile cells (>7 µm/h) and Xs indicate nonmotile cells (<7 µm/h) at experimentally tested EGF concentrations,while cell speed is interpolated across untested values. Replicate conditions for each EGF concentration and initial starting density were used to construct thephase diagram. (C) Clustering and spanning are captured by a computational model of self-propelled particles with varying polarization force (cell speed)at fixed adhesion (A = 0.03). (Right) Red particles represent individuals, while gray particles are associated with a cluster. (D) A jamming-like phase diagramdefined with neighbor density and cell polarization parameter recapitulates the behaviors observed in the experimental low-EGF regime, highlighted inpurple in B. Each pixel represents the average values (speed, neighbor number) for a distinct simulation.

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so that this analysis likely overestimates the number of cells indirect proximity at higher densities. Overall, 〈B(t)〉 captures thetransitions to clusters and spanning networks across conditions.

The experimental regime where clustering and arrested motil-ity occurred was characterized by systematically varying theinitial EGF concentration from 0 ng/mL up to 20 ng/mL ingrowth media. For low [EGF] < 0.5 ng/mL, cells were observedto form clusters (〈B〉< 4) and spanning networks (〈B〉< 7)(Fig. 4B), with proliferation arrested at subconfluent cell den-sities. Moreover, for [EGF] ≤ 0.075 ng/mL, cell motilitywas largely arrested for both clusters and spanning networks(<7 µm/h), while at [EGF] = 0.1 ng/mL, cell motility wasarrested only in spanning networks. At high [EGF] ≥ 0.5 ng/mLand above, cells typically migrated much more rapidly and didnot form stable clusters (Fig. 4C), but proliferated exponen-tially (SI Appendix, Fig. S4 A and B). Motility was consistentlyarrested at 〈B〉∼ 13, which correlated with the formation ofa confluent, space-filling monolayer. It should be noted thatthese EMT-induced cells showed weak E-cadherin expressionat cell–cell junctions for high concentrations of EGF, suggestingthat this slowdown in motility may occur through transient cell–cell contacts, consistent with contact inhibition of locomotionthrough active repulsion or random deflection (48). These resultsindicate that increasing EGF enhances both motility and prolif-eration and that cluster formation and arrested motility occur for[EGF] ≤ 0.1 ng/mL and 〈B〉< 13.

These transitions between motile individuals and arrestedclusters were recapitulated by a minimal model of self-propelledparticles that incorporated 3 mechanisms. First, each particlemoved randomly with a polarity force of constant magnitude,which changed to a randomly chosen direction at constant inter-vals (offset to different starting times) (SI Appendix, Fig. S4E).Second, pairs of particles adhered via a short-ranged attraction,but could not approach each other closer than a fixed parti-cle size (SI Appendix, Fig. S4F). This simplified interaction wasinspired by theoretical models of diffusion-limited aggregationin nonliving colloidal particles with strong adhesion (47) andneglects more complex friction-like biomolecular interactionsthat could occur at cell–cell or cell–matrix adhesions (35). Third,cells proliferated with a cell cycle of constant duration (offset todifferent starting times). However, cells also obey contact inhibi-tion of proliferation and could not divide if they had 4 or moreneighbors. For a representative simulation with fixed adhesion(A = 0.03) and varying polarization (P = 0.0025 to 0.0043), theinitially dispersed individuals transitioned over time to multi-cellular clusters and then to a spanning network (Fig. 4C andMovie S3). These values further corresponded to slower prolif-eration, decreasing fraction of individuals, decreasing numberof clusters, increasing cluster size, and a fractal dimension ofDf = 1.74± 0.003 (SI Appendix, Fig. S4 G–K), in good agree-ment with experiments (Figs. 1C and 2 B–E). The correspondingincrease in the ensemble-averaged number of neighbors 〈B〉associated with clusters (>2) and spanning networks (>4) wasqualitatively consistent with experimental observations (Fig. 4A).This minimal model was also modified to address leader cell for-mation based on a local curvature-dependent mechanism thatsustained cell polarization (SI Appendix, Fig. S3 E–H), anal-ogous to previous work (15). For simplicity, proliferation wascontrolled only by contact inhibition, so that particles contin-ued to proliferate until a spanning network was formed. Incomparison, some experiments with lower initial cell densitiesshowed minimal proliferation of clusters, indicating that pro-liferation may exhibit a more complicated time dependence(Fig. 4A). Future work will systematically explore how time-varying proliferation rates affect clustering in this minimalmodel.

More generally, particles in this regime exhibited a gradualarrest of cell motility with increasing ensemble-averaged number

of neighbors 〈B〉 (Fig. 4D and SI Appendix, Fig. S4 L and M), inqualitative agreement with the low-EGF regime (Fig. 4B). A sec-ond regime with slightly decreased adhesion (A = 0.02) resultedin a transition from subconfluent individuals to a confluentmonolayer with corresponding arrest of motility (SI Appendix,Fig. S4 N and O and Movie S4), consistent with the observedbehaviors in the higher-EGF regime (Fig. 4B). It should be notedthat the crossover between these 2 regimes could not be capturedby a single adhesion value in our model (SI Appendix, Fig. S4F)and likely occurred because the short-ranged adhesion was verysensitive to changing polarization value (A = 0.03). This theoret-ical result suggests that experimentally increasing EGF may bothincrease cell motility and decrease cell–cell adhesion, maintain-ing cells as individuals rather than clusters. Finally, it should benoted that this model is based on monodisperse particles, whichcluster together with uniform separations. However, living cellshave deformable shape and varying sizes, which results in var-ied nuclear separations. We have shown that a 75-µm thresholdis optimal for linking cells into clusters, but this likely over-counts the number of nearest neighbors relative to the minimalmodel.

Clustering Occurs with Decreased EGF Signaling for Growth-Factor–Dependent Mammary Epithelial Cells. These clustering behaviorswere most pronounced with OHT treatment, but qualitativelysimilar behaviors were observed under a number of experimen-tal conditions where EGF signaling was reduced. For instance,clustering also occurred with MCF-10A cells without EMTinduction, when cultured in assay media with a matched con-centration of dimethyl sulfoxide (DMSO) (corresponding to theconcentration needed to dissolve OHT) (SI Appendix, Fig. S5and Movies S5 and S6). Indeed, considerably fewer leader cellswere observed without EMT induction (DMSO), resulting inclusters that were more compact with reduced connectivity atlonger times (SI Appendix, Figs. S6 and S7) relative to con-ditions with EMT induction (OHT). Moreover, clustering alsooccurred after drug treatment to inhibit the EGF receptor (i.e.,gefitinib, in growth media) (SI Appendix, Fig. S8 and MoviesS7 and S8). Cell viability was minimally affected by culture inassay media and gefitinib relative to growth media (SI Appendix,Figs. S5 C and D and S8 B and G). It should be noted that theproliferation and migration of MCF-10A cells are highly depen-dent on growth factors (e.g., EGF) (49), unlike transformedbreast cancer cell lines, which are less sensitive to growth factorconcentration. For a meaningful comparison, a second growthfactor-dependent mammary epithelial cell line (hTERT-HME1)also exhibited clustering in reduced growth factor conditionsacross a range of starting densities and after gefitinib treat-ment (SI Appendix, Fig. S9). In contrast, clustering was notobserved with the highly metastatic breast adenocarcinoma cellline, MDA-MB-231, which is transformed and exhibits reducedsensitivity to EGF (SI Appendix, Fig. S10). These experimentssuggest that clustering occurs only for growth factor-dependentmammary epithelial cells and is driven by decreased EGFRsignaling, through either reduced EGF or chemical inhibition.

Discussion and ConclusionOur experiments reveal a 2-step process of multicellular self-organization and arrest in reduced-EGF conditions. First, mam-mary epithelial cells exhibit random individual migration thatis arrested upon cell–cell contact. As a consequence, multicel-lular clusters formed with dendritic morphologies and a fractaldimension Df ∼ 1.7. This scaling is reminiscent of diffusion-limited aggregation of nonliving colloidal particles with strongshort-ranged attraction (23–25). This clustering behavior consis-tently occurred for these EGF-dependent epithelial cells whenEGF concentrations were reduced or EGF receptor activity wasinhibited. In contrast, previous studies with more motile cells in

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higher-serum media have observed initially unstable clusteringthat progresses through a nucleation and growth mechanism (32,37). Second, small groups of cells exhibited transient collectivemigration within otherwise stationary clusters, which differedfrom diffusion-limited aggregation in nonliving colloidal gels.EMT induction enhanced the formation of motile groups atthe cluster periphery, driven outward by “leader cells” to con-nect isolated clusters into spanning networks. These leader cellswere mechanically coupled with a group of follower cells andexhibited highly correlated directional motion, which could beexplained by sustained polarization due to cell–cell junctionswith neighbors (14), but also a universal coupling of cell speedand persistence driven by actin retrograde flow (50) or EMT(2). Moreover, the number of leader cells increased with clustersize, with an average separation of ∼250 µm between leaders,comparable to previously reported values for straight migrationfronts (11, 17). The highly curved periphery of these fractal-like clusters likely enhances leader cell formation relative tostraight fronts, in good agreement with previous theoretical mod-els that postulate a curvature-dependent motility (10, 15), aswell as experiments with micropatterned fronts (9, 12, 13, 15).The critical cluster size for leader cell formation is likely to begoverned by a complex interplay of cytoskeletal tension and cell–cell and cell–matrix adhesions (14, 16, 17). Such cooperativerearrangements are also a signature of dynamic heterogeneityin supercooled liquids, colloidal glasses, and granular materials(27). Interestingly, mobile particles are often observed as quasi-1D “chains” (45), which are reminiscent of the single-file strandsof cells observed here. These collective behaviors are suggestiveof branched networks in epithelial, endothelial, and neuronaltissues (51).

Jamming in nonliving systems can occur at both high- and low-density regimes (glass transition and gelation, respectively), withadditional control parameters of temperature and shear (26).By analogy, jamming in fully confluent cell monolayers (at highserum) has been explained as a glass-like transition (29, 30, 33,35–37, 39) with control parameters of motility, cell–cell adhe-sion, and density (40). Our findings of arrested motility at lowcell density and low EGF are reminiscent of gelation, since cellsorganize into tenuous, space-filling networks (22). These resultsare fully consistent with the proposed jamming framework formonolayers, since reduced EGF slows cell motility, permittingarrest at a lower cell density (assuming adhesion remains con-stant). It is remarkable that this minimal physical analogy cancapture these emergent behaviors, given the phenotypic hetero-geneity of living cells. The similarities and differences between“active” living cells and “passive” nonliving particles representan exciting direction for future work, particularly the role of“aging”-like behaviors based on the gradual depletion of EGFand the maturation of cell–cell junctions, which could modu-late cellular activity and dissipation, respectively (35). Indeed,an important consideration for active living cells is that den-sity changes over time due to proliferation (29, 32, 37), whichmay be further modulated by cell–cell contacts through contactinhibition of proliferation (31).

In summary, we demonstrate that dispersed mammary epithe-lial cells cultured under low-EGF conditions migrate into aggre-gated clusters with fractal-like morphologies. Subsequently,groups of cells within the cluster exhibit transient and collectivemigration to link clusters together into spanning networks. Theselinkage events are often led by elongated leader cells, whichoccur more frequently after OHT treatment to activate Snailand EMT. These behaviors exhibit unexpected physical analo-gies with diffusion-limited aggregation in nonliving colloidalparticles. Indeed, EGF is an experimental control parameterto regulate cell speed and density, which tunes a jamming-liketransition at subconfluent densities. Overall, this comprehen-sive physical picture unifies collective cell migration and arrest

from sparse to confluent densities, with striking analogies togelation and the glass transition in nonliving soft matter. Thesephysical signatures of subconfluent cell aggregation and arrestmay have broader relevance to elucidate phenotypic transitions(e.g., EMT/MET) in development, wound-healing, and cancerprogression.

Materials and MethodsCell Culture. MCF-10A mammary epithelial cells stably transfected with aninducible Snail expression construct fused to an estrogen receptor responseelement were a gift from D. A. Haber (Massachusetts General Hospital,Boston, MA) (41). One variant overexpressed fluorescent proteins in thenucleus (mCherry-H2B) and cytoplasm (GFP) for live cell tracking. Cells wereroutinely cultured as described in SI Appendix. For these experiments, cellswere cultured on collagen-coated 96-well plates at varying EGF concentra-tions (0 to 20 ng/mL) with DMSO and OHT, with a total media volume of125 µL. No changes in media color were observed after 60 h. Cells testednegative for Mycoplasma contamination (PlasmoTest; Invivogen) and anabsence of additional cell culture contaminants was routinely verified byculturing cells in antibiotic-free media. At least 3 independent experimentswere conducted for each condition.

Time-Lapse Microscopy. Cell proliferation, clustering, and migration weremeasured with an inverted epifluorescence microscope (Nikon TiE) with alight-guide coupled white light illumination system (Lumencore Spectra-X3)under environmentally controlled conditions (37 ◦C, 5% CO2, humidified).Using Nikon Elements software, images were acquired every 15 min with12-bit resolution using an sCMOS camera (Andor Neo), a 10× PlanApo objective (NA 0.45, 4 mm working distance), a GFP/FITC Filter Set(Chroma 49002), and a TRITC/DSRed Filter Set (Chroma 49004). Imageswere recorded under consistent acquisition parameters (e.g., exposure time,camera gain/gamma control, and microscope aperture).

Image Analysis and Cell Tracking. Automated cell tracking based on fluo-rescent nuclei was performed using Bitplane Imaris 8.2. First, nuclei weredetected based on an estimated x− y diameter of 11.5 µm and thenlinked together across time points using an autoregressive motion algo-rithm, with a maximum displacement of 30.0 µm, a max gap size of 3 frames,and fill gaps enabled. Consistency in cell phenotype (e.g., confluency, clus-tered/nonclustered, morphology, network architecture) was first confirmedqualitatively across all experiments. Subsequently, 3 representative exper-iments were chosen for the bulk of semiautomated analyses (single-celltracking and manual verification) and 2 representative experiments werechosen for labor-intensive manual analyses (cluster threshold determinationand leader cell identification).

Proliferation and Velocity Analyses. Cell counts were determined fromdetected nuclei in each time-lapse image. The normalized cell count wascalculated by dividing cell counts at all times by the starting cell count.

Single-cell velocities were calculated based on the displacement of nucleiover some time interval, which was averaged over 1 h (4 time frames)to reduce noise from nuclear shape changes. Thus, the velocity vi(tj) =[ri(tj)− ri(tj−4)]/(tj − tj−4), where ri was the position of cell i at time framej (for j> 4); cells tracked for fewer than 5 frames were necessarily dis-carded. The population-averaged velocity was calculated by averaging overall single-cell velocities at a given time frame. Significance was computedat the P< 0.05 level using a paired-sample t test between conditions withn = 3 and was reported hourly for simplicity. The time interval used to cal-culate velocity was also varied (15 min, 30 min, and 2 h compared to 1 h) toshow that trends between conditions were robust.

Mean-Squared Displacement, Cluster Mass, and Fractal Dimension. The mean-squared displacements were determined as a function of time interval τas MSD(τ ) = 〈|ri(t + τ )− ri(t)|2〉, averaged over all early starting times tand computed separately for each individual cell i, using the open-sourceMATLAB code MSDANALYZER (52). To be considered in the analysis, a cellmust remain separated from all neighboring cells by greater than 75 µmfor at least 5 h over early times (0 h < t < 20 h) and subsequently MSD(τ )was computed for τ > 1 h to eliminate noise from nuclear motion and τ <10 h since individual cells quickly associated into clusters. The resulting curvewas fitted according to a power law of the form Kατ

α using the MATLABfunction FIT with option POWER1. The data were plotted alongside a line ofα = 1 and α = 1.5 to guide the eye. The mean and SD of fitted parameter αwere quantified for the resulting curves of individuals with high goodness

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of fit (R2 > 0.90) across 3 independent experiments (n = 44). The generalizeddiffusion coefficient Kα was plotted against exponent α for all individualsmeeting these criteria to show the spread of the data.

Multicellular clusters were defined at each time frame as groups of 2 ormore cells where neighboring pairs of nuclei were separated by 75 µm orless; anything not meeting these criteria was defined as a single cell. Thevalue 75 µm was chosen as the optimal nuclear linking distance based onvisual inspection of linked clusters across a range of threshold distances rel-ative to the actual cytoplasmic connectivity (detailed in SI Appendix). Thefraction of individuals was computed across the time lapse by taking thequotient of the number of single cells and the total number of cells. Clustersize was calculated by dividing the total number of cells in clusters by thetotal number of clusters.

For the purpose of computing the fractal dimension, multicellular clus-ters were defined at each time frame as groups of 4 or more cells wheresuccessive pairs of nuclei were separated by 75 µm or less, to reducethe noise associated with especially small clusters. For each cluster with a“mass” of M cells, the centroid was calculated as rcm = (1/M)

∑Mi ri . The

radius of gyration Rg was then calculated from R2g = (1/M)

∑Mi (ri − rcm)2.

The scaling of the radius of gyration Rg vs. M as a power-law fit withexponent 1/Df was first calculated from this dataset using the MATLABfunction FIT with option POWER1. Next, an ordinary, nonparametric boot-strap analysis was conducted to determine confidence intervals for thispower-law scaling using the R function BOOT. Briefly, the dataset was ran-domly resampled 1,000 times (with replacement runs) and each samplewas reanalyzed to empirically determine a 95% confidence interval for thepower-law fit.

Clustering was independently verified by direct segmentation of cellcytoplasm and nuclei using custom MATLAB code. First, cell nuclei weresegmented (by adaptive thresholding) and connected using mathematicalmorphology to obtain a rough foreground binary mask. Next, MATLAB’simplementation of the Chan–Vese algorithm was used to obtain segmentedcell boundaries. The number of nuclei inside each cluster was estimated bydividing the total area of the nuclear segmentation by typical nucleus size.Finally morphological features, including fractal dimension, were calculatedfor each segmented cluster. The segmentation quality was manually evalu-ated and only data obtained from correctly segmented images were used insubsequent analysis.

Dynamic 4-Point Susceptibility Function. Spatially heterogeneous dynamicswere quantified based on the self-overlap of cells, which defined motileor nonmotile subpopulations. A cutoff distance of L = 10 µm was selected,which corresponds to 1 nuclear diameter and roughly 20% of a cell diam-eter, comparable to previous analyses on epithelial monolayers (36). Thecutoff function wi is 1 if the displacement of cell i is less than L over sometime interval τ (nonmotile) and 0 otherwise (motile). An instantaneousself-overlap parameter is given by Q(L, τ ) = 1

N

∑Ni=1 wi . A 4-point suscep-

tibility function can be calculated from the moments of Q(L, t) averagedover all starting times and all cells: χ4(τ ) = 〈N〉[〈Q(L, τ )2〉− 〈Q(L, τ )〉2] (45),where 〈N〉 is the average number of cells present over the time rangeindicated.

Leader Cell Analysis. Leader cells were manually identified by 3 individu-als (S.E.L., Z.J.N., J.Y.S.) based on the following criteria: 1) Leader cells werelocated at the periphery of clusters, 2) leader cells drove collective migra-tion of follower cells (while maintaining physical contact) over at leastseveral hours, 3) leader cells exhibited elongation in the direction of cellmigration, and 4) leader cells tended to present ruffled lamellipodia and anenlarged cell size relative to follower cells. The combined number of lead-

ers identified across all individuals, or the total number of unique leadercells, was then used to determine the threshold at which multiple leadersemerge and the dependency on cluster size. Kymographs were constructedusing Icy (Institut Pasteur, 2011) and used to determine net leader speed(start to end position of leading edge over time range of interest) andpersistence time (the duration over which cells traveled along the samedirection).

Immunofluorescence Staining. Immunofluorescence staining protocols aredescribed in SI Appendix. Briefly, cells were fixed and labeled withE-cadherin primary antibody (mouse mAB; BD 610181) and vimentin pri-mary antibody (rabbit mAB; CST 5741) and then incubated with Hoechst(ThermoFisher H3569), goat anti-mouse, and goat anti-rabbit secondaryantibodies (Alexa Fluor 488 and 555; ThermoFisher). Cells were then washedseveral times with 1× PBS prior to imaging.

Local Neighbor Density and Jamming Phase Diagram. The ensemble-averagedlocal neighbor density was calculated for each cell at each time point byfinding the number of neighboring cells within a 75-µm radius and thenaveraging over all cells detected for that time point. The jamming phase dia-gram was then mapped by comparing this local neighbor density with theensemble averaged velocity 〈vrms〉 (as previously calculated), over a range ofexperimental EGF concentrations.

Computational Self-Propelled Particle Model. A minimal physical modeltreated cells as self-propelled particles which randomly polarize in newdirections and can interact with other cells through a tunable attractiveinteraction and a short-range repulsion corresponding to the cell radius. Par-ticle dynamics were simulated in a square domain with periodic boundaryconditions. Further details are available in SI Appendix.

Statistical Analysis. Experiments were repeated at least 3 times, with thenumber of independent experiments used for analysis indicated in the fig-ure legends. For major findings of population-averaged measurements overtime (e.g., normalized cell count [Fig. 1C and SI Appendix, Figs. S5E, S8C,S8H], cell speed [Fig. 1D and SI Appendix, Figs. S5F, S8D, S8I], fraction ofindividuals [Fig. 2B and SI Appendix, Fig. S6A], number of clusters [Fig.2C and SI Appendix, Fig. S6B], and cluster size [Fig. 2D and SI Appendix,Fig. S6C]), the mean and SD were computed across 3 biological replicatesfrom independent experiments. Statistical significance between groups wasdetermined and plotted at hourly intervals, where applicable, at the 5%significance level (P values<0.05) using the paired-sample t test. For signifi-cance testing between distributions of individual cells from manual analyses(e.g., leader/follower cells [Fig. 3 E and F and SI Appendix, Fig. S7 D–F ]),data from 2 representative experiments were pooled and results from the2-sample Kolmogorov–Smirnov test were reported (5% significance level).For a subset of findings, representative data from a single experiment wereshown for ease of visualization (4-point susceptibility function [Fig. 3B andSI Appendix, Fig. S7A], Ripley’s H function [SI Appendix, Figs. S1E, S5G, S8E,S8J]). In these cases, data were rigorously analyzed over several replicatesand across at least 2 experiments, to ensure consistency in the observedtrends.

ACKNOWLEDGMENTS. We thank C. Franck, T. M. Powers, and J. X. Tangfor careful readings; D. A. Haber for the inducible MCF-10A cell lines;and R. J. Giedt and R. Weissleder for the MDA-MB-231 GFP-H2B cellline. This work was supported by NIH Grants T32ES007272, P30GM110759,and R21CA212932 and by Brown University (DiMase Summer Fellowship,Karen T. Romer Undergraduate Research and Teaching Award, and Start-UpFunds).

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