University of Dundee

Dense active matter model of motion patterns in confluent cell monolayers Henkes, Silke; Kostanjevec, Kaja; Collinson, J. Martin; Sknepnek, Rastko; Bertin, Eric

Published in: Nature Communications

DOI: 10.1038/s41467-020-15164-5

Publication date: 2020

Licence: CC BY

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Citation for published version (APA): Henkes, S., Kostanjevec, K., Collinson, J. M., Sknepnek, R., & Bertin, E. (2020). Dense active matter model of motion patterns in confluent cell monolayers. Nature Communications, 11, [1405]. https://doi.org/10.1038/s41467-020-15164-5

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https://doi.org/10.1038/s41467-020-15164-5 OPEN Dense active matter model of motion patterns in confluent cell monolayers ✉ ✉ ✉ Silke Henkes 1,2 , Kaja Kostanjevec3, J. Martin Collinson3, Rastko Sknepnek 4,5 & Eric Bertin6

Epithelial cell monolayers show remarkable displacement and velocity correlations over distances of ten or more cell sizes that are reminiscent of supercooled liquids and active nematics. We show that many observed features can be described within the framework of

1234567890():,; dense active matter, and argue that persistent uncoordinated cell motility coupled to the collective elastic modes of the cell sheet is sufficient to produce swirl-like correlations. We obtain this result using both continuum active linear elasticity and a normal modes formalism, and validate analytical predictions with numerical simulations of two agent-based cell models, soft elastic particles and the self-propelled Voronoi model together with in-vitro experiments of confluent corneal epithelial cell sheets. Simulations and normal mode analysis perfectly match when tissue-level reorganisation occurs on times longer than the persistence time of cell motility. Our analytical model quantitatively matches measured velocity correlation functions over more than a decade with a single fitting parameter.

1 School of Mathematics, University of Bristol, Bristol BS8 1TW, United Kingdom. 2 Institute of Complex Systems and Mathematical Biology, University of Aberdeen, Aberdeen AB24 3UE, United Kingdom. 3 School of Medicine, Medical Sciences and Nutrition, University of Aberdeen, Aberdeen AB25 2ZD, United Kingdom. 4 School of Science and Engineering, University of Dundee, Dundee DD1 4HN, United Kingdom. 5 School of Life Sciences, University of ✉ Dundee, Dundee DD1 5EH, United Kingdom. 6 Université Grenoble Alpes and CNRS, LIPHY, F-38000 Grenoble, France. email: [email protected]; [email protected]; [email protected]

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ollective cell migration is of fundamental importance in correlation patterns in the cell motion, with correlation lengths embryonic development1–4, organ regeneration and exceeding ten or more cell sizes. Inspired by the theory of sheared C 5 41,42 wound healing . During embryogenesis, robust regulation granular materials , we develop a normal modes formalism for of collective cell migration is key for formation of complex tissues the linear response of confluent cell sheets to active perturbations and organs. In adult tissues, a paradigmatic model of collective (see Fig. 1a), and derive a displacement correlation function with cell migration is the radial migration of corneal epithelial cells a characteristic length scale of flow patterns. Using numerical across the surface of the eye6,7. Major advances in our under- simulations of models for cell sheets, including a soft disk model standing of collective cell migration have been obtained from as well as a self-propelled Voronoi model (SPV)37,38, we show in vitro experiments on epithelial cell monolayers4,8–11. A key that our analytical model provides an excellent match for both observation is that collective cell migration is an emergent, types of simulations up to a point where substantial flow in the strongly correlated phenomenon that cannot be understood by sheet begins to subtly alter the correlation functions (Fig. 1b). At studying the migration of individual cells12. For example, forces the level of linear elasticity, we are able to make an analytical in a monolayer are transmitted over long distances via a global prediction for the velocity correlation function and the mean tug-of-war mechanism9. The landscape of mechanical stresses is velocity in a generic cell sheet. We test our theoretical predictions, rugged with local stresses that are correlated over distances which apply to any confluent epithelial cell sheet on a solid spanning multiple cell sizes10. These strong correlations lead to substrate dominated by uncoordinated migration, with time-lapse the tendency of individual cells to migrate along the local observations of corneal epithelial cells grown to confluence on a orientation of the maximal principal stress (plithotaxis10,13) and a tissue culture plastic substrate. We find very good agreement tendency of a collection of migrating epithelial cells to move between experimental velocity correlations and analytical pre- towards empty regions of space (kenotaxis14). Furthermore, such dictions and are, thus, able to construct fully parametrized soft coordination mechanisms lead to propagating waves in confined disk and SPV model simulations of the system (Fig. 1c) that clusters15,16, expanding colonies17 and in colliding monolayers18, quantitatively match the experiment. Garcia, et al.40 observed which all occur in the absence of inertia. similar correlations and proposed a scaling theory based on Active matter physics19,20 offers a natural framework for coherently moving cell clusters, and either cell–substrate or describing subcellular, cellular and tissue-level processes. It stu- cell–cell dissipation. Our approach generalises their result for dies the collective motion patterns of agents each internally able cell–substrate dissipation, and we recover both the scaling results to convert energy into directed motion. In the dense limit, and also find quantitative agreement with the experiments pre- motility leads to a number of unexpected motion patterns, sented in ref. 40. including flocking21, oscillations22, active liquid crystalline20, and – arrested, glassy phases23. In silico studies22,24 27, in the dense Results regime have been instrumental in describing and classifying Model overview. We model the monolayer as a dense packing of experimentally observed collective active motion. soft, self-propelled agents that move with overdamped dynamics, Continuum active gel theories28,29 are able to capture many and where the main source of dissipation is cell–substrate friction. aspects of cell mechanics30, including spontaneous flow of cor- The equations of motion for cell centers are tical actin31 and contractile cell traction profiles with the sub- ζr_ Fact Fint; strate32. In some cases, cell shapes form a nematic-like texture4,33 i ¼ i þ i ð1Þ and topological defects present in such texture have been argued ζ – fi Fact 4 where is the cell substrate friction coef cient, i is the net to assist in the extrusion of apoptotic cells . To date, however, the – fl motile force resulting from the cell substrate stress transfer, and cell-level origin of the heterogeneity in ow patterns and stress Fint profiles in cell sheets is still poorly understood. Many epithelial i is the interaction force between cell i and its neighbours. Commonly used interaction models are short-ranged pair forces tissues show little or no local nematic order or polarization, and 43 37,38 even where order is present, the local flow and stress patterns with attractive and repulsive components and SPV models . only follow the continuum prediction on average, while indivi- Here we only require that the intercell forces can be written as the fl gradients of a potential energy that depends on the positions of dual patterns are dominated by uctuations. This suggests that int cell centres, F ¼À∇r Vðfr gÞ. Furthermore, we neglect cell active nematic and active gel approaches capture only part of the i i j picture. division and extrusion for now, but we will reconsider the issue Confluent cell monolayers exhibit similar dynamical behaviour when we match simulations to experiment below. The precise Fact to supercooled liquids approaching a glass transition. One form and molecular origin of the active propulsion force i is a observes spatio-temporally correlated heterogeneous patterns in topic of ongoing debate, and interactions between cells through cell displacements34 known as dynamic heterogeneities35,a flocking, nematic alignment, plithotaxis and kenotaxis have all hallmark of the glass transition36 between a slow, albeit flowing been proposed. What is clear, however, is that all alignment liquid phase and an arrested amorphous glassy state. The notion mechanisms occur over a substantial background of uncoordi- that collectives of cells reside in the vicinity of a liquid to solid nated motility, and therefore, as a base model, we assume that the transition provides profound biological insight into the active cell forces undergo random, uncorrelated fluctuations in Fact actn^ n^ mechanisms of collective cell migration. By tuning the motility direction. With i ¼ F i, where i is the unit vector that 37–39 θ and internal properties of individual cells, e.g. cell shape or makes an angle i with the x -axis of the laboratory frame, the cell–cell adhesion40, a living system can drive itself across this angular dynamics is transition and rather accurately control cell motion within the 1 θ_ η ; η η 0 δ δ 0 ; ð2Þ sheet. This establishes a picture in which tissue-level patterning is i ¼ i h iðÞt jðt Þi ¼ τ ij ðÞt À t not solely determined by biochemistry (e.g. the distribution of morphogens) but is also driven by mechanical cues. where τ sets a persistence time scale, and different cells are not In this paper, we show that the cell-level heterogeneity, that is coupled (Fig. 1a). This dynamics is equivalent to active Brownian variations in size, shape, mechanical properties or motility particles44, and in isolation, model cell motion is a persistent between individual cells of the same type inherent to any cell random walk. At sufficiently low driving, such models form active monolayer, together with individual, persistent, cell motility and glasses23,45–47, where the system moves through a series of local soft elastic repulsion between neighbouring cells leads to energy minima (i.e. spatial configurations of cells) on the time

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a bc Mechanism Long-range, 'swirly' Experiment: velocity correlations corneal epithelial cells Long-time motion through energy landscape = 2000 Phase contrast Soft disk,  with PIV solid = 200 100 μm Soft disk, Vertex Matched vertex simulation  liquid solid Matched soft disk simulation

50 m/h μ 25

Active fluctuations No spatial active Velocity, in potential well force correlations 0

Fig. 1 Active elasticity leads to correlated velocity fields. a Mechanisms at the origin of the active elastic theory in the energy landscape (top), inside an energy minimum (bottom left), and between particles (bottom right). b Velocity fields in simulated cell sheets. Top—System-spanning correlations in a solid soft disk system at τ = 2000. Bottom left—liquid soft disk system at τ = 200, and bottom right—SPV model simulation at τ = 200, cell outlines in white. c Velocity fields in experimental cell sheets. Top—sample experimental velocity field, overlaid over phase-contrast image of the cell sheet. Bottom left—particle-based best fit model to the experiment, including divisions and extrusions (visible as dark red arrows). Bottom right—SPV model-based best fit model to the experiment.

τ ζ 2τ= 37,45 scale of the alpha-relaxation time α, which diverges at dynamical where Teff ¼ v0 2, consistent with previous work .In arrest. Fig. 2a, the leftmost column is for a simulated thermal system, We now develop a linear response formalism. As shown in with properties that are nearly indistinguishable from the τ = 0.2 Fig. 1a, on time scales below τα, the self-propulsion reduces to a results. In the opposite, high persistence limit when τ → ∞,we fl 2 2 stochastic, time-correlated force uctuating inside a local energy obtain instead Eν ¼ ζ v =4λν, i.e. a divergence of the contribu- τ ≪ τ 0 minimum. If the persistence time scale α, the full dynamics tion of the lowest modes. A predominance of the lowest modes in can be described by a statistical average over long periods active driven systems was also noted in ref. 22,37,49. fluctuating around different energy minima, by assuming that the τ It thus becomes clear that for large values of , Teff can no brief periods during which the system rearranges do not longer be interpreted as temperature since the fluctuation- 47 contribute appreciably (see also ref. ). We linearize the dissipation theorem is no longer valid. An analogous result to interaction forces in the vicinity of an energy minimum, i.e. a the τ → ∞ limit has been obtained in granular material with an fi r0 mechanically stable or jammed con guration f i g by introducing externally applied shear41,42, showing that the mechanisms at δr r r0 = act∕ζ i ¼ i À i . After introducing the active velocity v0 F , Eq. play are generic, and that tuning τ allows active systems to bridge (1) becomes between features of thermal systems and (self-)sheared systems X 37,47 ζδr_ ζ n^ K δr ; (see also ref. ). However, the glass transition lies on a curve of i ¼ v0 i À ij Á j τ 46 ð3Þ constant Teff with moderate contributions (Fig. 2 and ref. ), j making Teff a convenient parameter, in spite of its lack of genuine ∂2 r K VðÞf ig 48 thermodynamic interpretation. where ij ¼ ∂r ∂r jfr0g is the dynamical matrix , organised as i j j In order to make connections to experiments on cells sheets, 2 × 2 blocks corresponding to cells i and j. In this limit, we can we compute several directly measurable quantities. One measure solve the dynamics exactly, see Supplementary Note 1. that is easily extracted from microscopy images is the velocity field, using particle image velocimetry (PIV)50. We compute Normal mode formulation fi 〈∣v q ∣2〉 = . Assuming that there are a suf cient the Fourier space velocity correlationP function, ( ) * N iqÁr0 number of intercell forces to constrain the tissue to be elastic at 〈v(q) ⋅ v (q)〉, with vðqÞ¼1=N e j δr_ , where the fr0g short time scales, the dynamical matrix has 2N independent j¼1 j j ν are the positions of the cell centres at mechanical equilibrium. normal modes ξ with positive eigenvalues λν. If we project Expanding over the normal modes, and taking into account the Eq. (3) onto the normal modes, we obtain statistical independence of the time derivatives a_ ν of the modes fi ζa_ ν ¼Àλνaν þ ην; ð4Þ amplitudes for different modes, we rst derive (see Supplemen- 2 2 P tary Note 1) the mode correlations ha_ i¼v =½Š21ðÞþ λντ=ζ ,so δr ξν ν 0 where aν ¼ i i Á i and the self-propulsionP force has been that in Fourier space, we obtain η ζ n^ ξν projected onto the modes, ν ¼ v0 i i Á i . The self- X 2 propulsion then acts like a time-correlated Ornstein-Uhlenbeck 2 v 2 0 v q 0 ξ q ; noise (see Supplementary Note 1), with hηνðtÞην0 ðt Þi ¼ hj ð Þj i¼ j νð Þj ð6Þ 2ð1 þ λντ=ζÞ 1 ζ2 2 0 =τ δ ν 2 v0 expðÀjt À t j Þ ν;ν0 . We can integrate Eq. (4) and obtain ν the moments of aν. In particular, the mean energy per mode is where ξν(q) is the Fourier transform of the vector ξ . given by i ζ 2τ Continuum elastic formulation. In most practical situations, it is 1 λ 2 v0 ; Eν ¼ νhaνi¼ ð5Þ impossible to extract either the normal modes or their eigenva- 2 41ðÞþ λντ=ζ lues. While it is possible to do so in, e.g. colloidal particle explicitly showing that equipartition is broken due to the mode- experiments51,52, the current methods are strictly restricted to dependence induced by λν in Eq. (5). In the limit τ → 0, we thermal equilibrium, and also require an extreme amount of data. ζ 2τ= : = recover an effective thermal equilibrium, Eν ! v0 4 ¼ Teff 2, Fortunately, the results above are easily recast into the language of

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ab 5.0 –1.0 –1.0 4.0

–1.5 –1.5 ) ) 3.0 eff eff T T –2.0 –2.0 log ( log ( 2.0

–2.5 –2.5 1.0

0.0 –100112 33 –1 2 Thermal log ( ) log ( ) log (  )

τ τ T ζv2 τ= Fig. 2 Glassy dynamics. Alpha-relaxation time α as a function of the persistence time and effective temperature eff ¼ 0 2. The gray scale indicates τ ϕ = T = T p : log α. a Soft disk model at 1, the leftmost column is for a thermal system at eff. b SPV model at 0 ¼ 3 6 (see Methods). solid state physics53. We rewrite Eq. (3)as cutoff a is of the order of the cell size. Eq.ffiffiffi (10) shows that the X pτ τ → ∞ ζu_ R ζ n^ R D R R0 u R0 ; correlation length of the system scales as . In the limit , ð Þ¼ v0 ð ÞÀ ð À Þ ð Þ ð7Þ 〈∣v q ∣2〉 54 R0 ( ) diverges at low q, as was found in ref. . The dominant v 2 =ξ2 u R scaling hjji1 is the same as results from the scaling where ( ) denotes the elastic deformations from the equilibrium Ansatz for cell–substrate dominated coordinated motion obtained R D R R0 positions in the solid, and ð À Þ is the continuum dyna- by ref. 40. mical matrix. The normal modes of the system are now simply While Eq. (10) is elegant, correlations of cell velocities Fourier modes with expressed in the Fourier space are not easy to interpret. Àiζωuðq; ωÞ¼Factðq; ωÞÀDðqÞuðq; ωÞ; ð8Þ Therefore, we derive a more intuitive, real-space expression for R P the correlation of velocities of cells separated by r,defined as Fact q; ω ζ 1 n^ R; iωt iqÁR D q Z where ð Þ¼ v0 À1 dt R ð tÞe e and ( ) are Fourier transforms of the active force and the dynamical matrix, r 1 2r v r r v r : Cvvð Þ¼ 2 d 0h ð 0 þ ÞÁ ð 0Þi ð12Þ respectively. Note that we assume that the system has a finite L volume, so that Fourier modes are discrete. At scales above the In the infinite size limit L → ∞, the real-space correlation n^ðR; Þ fi r cell size a, the noise t is spatially uncorrelated, and we nd function Cvv( ) can be evaluated from the Fourier correlation the noise correlators (see Supplementary Note 2) 〈∣v(q)∣2〉 as τ Z 2 2 2 Fact q; ω Fact q; ω0 π ζ 2 δ ω ω0 : a q r h ð ÞÁ ðÀ Þi ¼ 2 N v0 2 ð þ Þ r 2q v q 2 Ài Á : þ ðÞτω Cvvð Þ¼ 2 d hj ð Þj i e ð13Þ 1 ð2πÞ N ð9Þ Using Eq. (11), one finds the explicit result (see Supplementary The dynamical matrix D(q) has two independent eigenmodes Note 2) in two dimensions, one longitudinal q^ with eigenvalue B + μ "# q^? μ μ a2v2 K ðr=ξ Þ K ðr=ξ Þ and one transverse one with eigenvalue ,whereB and are C ðrÞ¼ 0 0 L þ 0 T ; ð14Þ vv π ξ2 ξ2 the bulk and shear moduli, respectively. We can then decompose 4 L T our solution into longitudinal and transverse parts, uðq; ωÞ¼ fi q; ω q^ q; ω q^? where K0 is the modi ed Bessel function of the second kind. Note uLð Þ þ uTð Þ . We are interested in the equal time, fi that this expression describes velocity correlations for r > a, with Fourier transform of the velocity, which we nd to be (see Sup- ∕ξ ≫ plementary Note 2) a the cell size. For r L,T 1, i.e. for distance much larger than the "#correlation lengths, 2 rffiffiffiffiffi ! 2 Nv0 1 1 =ξ =ξ hjjvðqÞ i¼ þ ; ð10Þ a2v2 π eÀr L eÀr T 2 1 þðξ qÞ2 1 þðξ qÞ2 C ðrÞ 0 þ ; ð15Þ L T vv 4π 2r ξ3=2 ξ3=2 L T where we have introduced the longitudinal and transverse cor- 40 ξ2 μ τ=ζ ξ2 μτ=ζ i.e. as expected and consistent with the results of ref. , Cvv relation lengths L ¼ ðÞB þ and T ¼ . Note that there are subtle differences in prefactors between expressions for decays exponentially at large distances. velocity correlation function in Fourier space (cf., Eq. (6), Eq. (10) Comparison to simulations and Eq. (52) in Supplementary Note 2), and that in two dimen- . We proceed to compare predictions sions, [μ∕ζ] = [B∕ζ] = L2T−1. As discussed in detail in Supple- made in the previous section to the correlation function measured mentary Note 2, these differences are due the use of discrete vs. in numerical simulations of an active Brownian soft disk model, fi continuum Fourier transforms and are important for comparison as well as to an SPV model. The active Brownian model is de ned Fact n^ with simulations and experiments. Finally, the mean-square by Eq. (1) and Eq. (2), with self-propulsion force i ¼ v0 i and P F velocity of the particles hjjv 2i¼h1 jv j2i decreases with active pair interaction forces ij that are purely repulsive. We simulate a N i i confluent sheet in this model by setting the packing fraction to correlation time as "#ϕ = 1 in periodic boundary conditions. The SPV model is the v2a2 1 ÀÁ1 ÀÁ same as introduced in refs. 37,38, and assumes that every cell is jjv 2 ¼ 0 log 1 þ ξ2q2 þ log 1 þ ξ2 q2 ; ð11Þ fi 8π ξ2 L m ξ2 T m de ned by the Voronoi tile corresponding to its centre. For this L T  : model, we choose the dimensionless shape factor p0 ¼ 3 6, put- = π∕ where qm 2 a is the maximum wavenumber and the high-q ting the passive system into the solid part of the phase

4 NATURE COMMUNICATIONS | (2020) 11:1405 | https://doi.org/10.1038/s41467-020-15164-5 | www.nature.com/naturecommunications NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-020-15164-5 ARTICLE diagram37,55, and we employ open boundary conditions. Please on a tissue culture plastic substrate (see Methods section and see the method section for full details of the numerical models Supplementary Movie 5). We use PIV to extract the velocity fields and simulation protocols. corresponding to collective cell migration (Fig. 4a, Fig. 1cand ζ 2τ= fi The effective temperature Teff ¼ v0 2 has emerged as a good Supplementaryqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Movie 6). We rst extract a mean velocity of predictor of the active glass transition45,54, at least at low τ, and v ¼ hjvðr; tÞj2i ¼ 12 ± 2μm=h (n = 5 experiments, see Fig. 4e), we use it together with τ itself as the axes of our phase diagram. fl The liquid or glassy behaviour of the model can be characterised consistent with the typical mean velocities of con uent epithelial by the alpha-relaxation time τα. Fig. 2 provides a coarse-grained cell lines grown on hard substrates. To reduce the effects of varying phase diagram where τα is represented in grey scale as a function mean cell speed at differentqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi times and in different experiments, we τ fi 2 of persistence time and Teff. For a xed persistence time, the use vðr; tÞ¼vðr; tÞ= hjvðr; tÞj ir, i.e., the velocity normalized by system is liquid at high enough temperature and glassy at low its mean-square spatial average at that moment in time. Using temperature, as expected. Now fixing the effective temperature, direct counting, we find an area per cell of 〈A〉 ≈ 380 μm2 corre- the system becomes more glassy when τ increases. This non- sponding to a particle radius of 〈σ〉 ≈ 11 μm, setting the micro- trivial result is consistent with the recent RFOT theory of the scopic length scale a. To compare the experimental result to our active glass transition46 and related simulation results46,47. It can τ theoretical predictions, we perform a Fourier transform on the PIV be partly understood from the fact that v0 decreases when fi v q 2 increases at fixed T , meaning that the active force decreases and velocity eld and compute hjjð Þ i, shown in Fig. 4b. Using the eff results from Eq. (11) we can rewrite Eq. (10)as it becomes more difficult to cross energy barriers. In contrast, "# 54  existing mode coupling theories of the active glass transition N v 2 1 1 only apply in the small τ regime. We note that the features of the hjjvðqÞ 2i¼ 0 þ ; ð16Þ 2 v ξ2 2 ξ2 2 active glass transition of the soft disk model and the SPV model 1 þ Lq 1 þ Tq are very similar. ξ2 ξ2 where L and T are the longitudinal and transverse squared cor- As is apparent from Fig. 1b, the growing correlation length relation lengths (with units of μm2)defined below Eq. (10). As can with increasing τ is readily apparent as swirl-like motion (see also = be seen from Eq. (11), the ratio v0 v is a function only of the Supplementary Movies 1-4). Fig. 3a-c shows the Fourier velocity dimensionless ratios ξ ∕a and ξ ∕a. If we further make the plausible correlation 〈∣v(q)∣2〉 measured in the numerical simulations for L T 〈∣v q ∣2〉 2 assumption that the ratio of elastic moduli is the same as in the soft different values of v0, after normalizing ( ) by v0N. In panel μ + ∕μ = ξ ξ = disk simulations, ( B) 4.3, the correlation lengths L and T A, we show that for soft disks at low Teff 0.005, where the are not independent. Therefore, we are left with a single fitting system is solid, the correlation function develops a dramatic 1∕q2 ξ fi ξ = τ parameter, T. The best t to the theory is obtained with T 100 slope as increases (dots), exactly in line with our modes μm as indicated by the solid black line in Fig. 4a, with the interval predictions (lines). We can determine the bulk and shear moduli of confidence denoted by dashed lines. The q = 0 intercept of the of the soft disk system (B = 1.684 ± 0.008, μ = 0.510 ± 0.004, see = correlation function gives a ratio v0 v  10, corresponding to the Methods section) and then draw the predictions of Eq. (10)on high activity limit where most self-propulsion is absorbed by the the same plot (dashed lines). At low q, where the continuum elastic deformation of the cells. Consistent with this, on the elastic approximation is valid, we have excellent agreement, and dimensionless plot Fig. 3d we are located at the point at larger q, the peak associated with the static structure factor ξ ∕ ≈ 〈 〉∕ ≈ τ → T a 5, v v0 0.1, on the right, strongly active side. The becomes apparent (in the limit 0, the correlation function deviations between theory and experiment in the tail of the dis- reduces to S(q), see Supplementary Fig. 2). In panel C, we show tribution are not due to loss of high-q information in imaging, as the same simulation results for the SPV model (dots), far as we could determine, and the disappearance of the peak at accompanied by the continuum predictions (dashed lines) using = μ = 55  : high q is particularly striking in this context. B 7.0 and 0.5, as estimated from ref. for p0 ¼ 3 6. We did In Fig. 4c, we show the real-space velocity correlations for the not compute normal modes for the SPV model. Note that due to ξ = μ ≪ μ + experiments, and the analytical prediction Eq. (14) with T 100 B, the contribution of the transverse correlations μm. We obtain a very good fit for experiments 1, 2, 3 and 7, but dominate the analytical results in both cases. In panel B, we show fi τ = experiments 5 and 6 have signi cantly longer-ranged correlations. the soft disk simulation at 20 when the transition to a liquid is This indicates that the precise value of the correlation length is crossed as a function of Teff. Deviations from the normal mode very sensitive to the exact experimental conditions that are not predictions become apparent only at the two largest values of Teff, τ τ simple to accurately control. The qualitative features of the when > α (Fig. 2a), and even in these very liquid systems, a correlation function are, however, robust. Note that experiments significant activity-induced correlation length persists. In Supple- − τ 5 and 6 have the same mean density as experiments 1 3, 7. mentary Fig. 3, we show that for all and both soft disk and SPV To match experiments and simulations, we consider the temporal models, our predictions remain in excellent agreement with the 〈v ⋅ v 〉 = τ ≲ τ autocorrelation function (t) (0) in Fig. 4d. As Eq. (57) in simulations for Teff 0.02, where α. In Fig. 3d, we show the Supplementary Note 2 shows, it is a complex function with a mean-square velocity normalized by v0 as a functionffiffiffi of the characteristic inverse S-shape that also depends on the moduli and ξ = pτ dimensionless transverse correlation length T a  , for all q .Usingthevalueofξ2 ¼ 104μm2 extracted from fitting hjjvðqÞ 2i our simulations, using a = σ, the particle radius. The dramatic m T and the ratio (μ + B)∕μ = 4.3, we used different μ∕ζ and τ compatible drop corresponds to elastic energy being stored in distortions of ξ2 μσ2=ζτ with T ¼ to obtain the best (numerically integrated) the sheet, and it is in very good agreement with our analytical fi fi prediction in Eq. (11) (solid lines). In Supplementary Fig. 4, we analytical t to the experimental result. We settled on a best t autocorrelation time of τ = 2.5 h and μ∕(σ2ζ) = 60.5 h−1 (black compare our numerical results for the spatial velocity correlations fi to the analytical prediction Eq. (14). The data and the predictions line in Fig. 4d). Experiments 5 and 6 have signi cantly longer autocorrelation times, and we achieve a good fit to experiment 5 are in reasonable agreement. τ = μ∕ζ for 5 20 h and the same (light gray line). This is consistent with the longer spatial correlations observed inp Fig.ffiffiffiffiffiffiffiffiffi4d, where the Comparison to experiment ξ = μ τ =τ μ . We now compare our theoretical light grey line corresponds to T 100 m 5  283 m. predictions and numerical simulations with experimental data There is also potentially weak local cell alignment in the obtained from immortalised human corneal epithelial cells grown experiment, not considered in the present theory.

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ab100 100 cd100 0 = 20 10

–1 –1 0 0 0 2 10 2 2 10   

0 N N N  / / / / 〉 〉 〉 T 〉 –1 T 2 2 2 –2 eff –2 2 | | | eff v ) 10 ) ) 10 10 〈

q q –1 0.002 q 0.002 ( ( 10 ( 0.2 0.2 √ Soft disk v v 0.005 v 0.005 〈| –3 2.0 〈| 0.01 〈| –3 2.0 0.01 Vertex Analytic, soft disk 10 20 0.02 10 20 0.02 Analytic, vertex 200 0.05 200 0.05 –4 2000 T = 0.005 0.1 –4 2000 T = 0.005 0.1 10 eff 10 eff 10–2 10–1 100 10–1 100 10–1 100 10–1 100 101  ⎜q⎜ ⎜q⎜ ⎜q⎜ T/a

Fig. 3 Behaviour of velocity correlation functions. a Velocity correlations in Fourier space for the soft disk model for different τ at Teff = 0.005 in the glassy phase and b: for changing Teff at τ = 20. Dots correspond to the simulated velocity correlation function, solid lines are the results from the normal mode expansion Eq. (6), and the dashed lines are the continuum elasticity predictions Eq. (10). c Fourier velocity correlations for the SPV model, together with analytical prediction. d Mean-square velocity as a function of ξT∕a for both soft disks and the SPV model and analytical predictions (solid lines). All v2 T =τ fi correlation functions have been normalized by 0 ¼ 2 eff , and there are no t parameters in the predictions.

b d 100 f 102 Theory 102 Soft 1 2 Simulations Dividing T = 5000 a 〉 〉 Vertex Comparison to 2 2 〉  = 10,000 2

T  2

–1  1 2 〈 1  /  = 20,000 〈 experiment 10 10 1 T 〉 Theory 10 N

2 N

/ μ ( m ) 1 = 20 / (0) 〉 2 〉 2 2 v

| = 5.0 | ) 0 3 2 = 2.5 ) 0 q

q 10

10 ) ·

( Experiment –2 400 ( t 3

5 (

v Simulation

10 v v 〈| 5 Soft 〈| Theory-soft

6 〈| Vertex –1

–1 Experiment Theory-vertex 7 6 10 10 Exp. fit –3 7 m) 10 μ 300 –2 –1 –2 –1 y ( 10 10 0246 10 10 Time (h) –1 ⎜q⎜(μm–1) ⎜q⎜(μm ) 200 c eg0 0 20 10 10 Theory 100 200 300

〉 15 –1 2

μ  10 )

x ( m) / 〉 〉 m/h) 2 μ  ( Correlation length 1 〈 (0) 10

–1 〉 v  = 100 μm 10 2 2 –2 / T  10 〈 υ ) ·

r 3 (

5 P( Experiment v 5 Soft 〈| 6 –3 Dividing 0 10 7 Experiment Vertex 10–2 123576 0 200 400 600 Experiment Soft 10–1 100 Vertex

Distance (microns) Dividing υ/ 〈2〉

Fig. 4 Experimental results and comparison to theory and simulations. a Cell sheet image overlaid with PIV arrows, and best fit length scale ξT. 2 b Experimental Fourier velocity correlation function normalised by mean velocity (dots), and best fit to theory with a single stiffness parameter ξT ¼ μτ=ζ. c Experimental real-space velocity correlation function (dots) and theoretical prediction. d Velocity autocorrelation function for the experiments, theoretical predictions for different τ, and simulation results for the same soft disk and SPV model simulations as in f. e Mean velocity magnitude for the experiments, and the soft disk, dividing soft disk and SPV model simulations. f Numerical results for soft, soft dividing, and SPV model interactions (solid lines), and À2 À1 −1 theoretical predictions (dashed lines) for the same τ = 2.5h, k ¼ 55μm h and v0 = 90 μmh (see text). g Normalized velocity distribution from experiment (black with grey confidence interval), and the three simulated models.

We can now fully parametrise particle and SPV model Our results are in quantitative agreement with ref. 40 for simulations to the experiment as follows. Our results for v and confluent but still motile cells, with a reported maximum = = ξ = μ the ratio v0 v can be combined to give an initial estimate of v0 correlation length of 100 m, and a cell crawling speed that 120 μmh−1. Then, the normalised time autocorrelation function of drops by a factor of 10 from ~ 90 μmh−1 at low density to this the cell velocities is only a function of ξ and τ,andwecanuseitto point of maximal correlation. Note also that we have an elastic determinetheelasticmoduli.Then,finally, we can determine time scale (k∕ζ)−1 ≈ 0.02h, much shorter than our correlation time the appropriate model parameters: In Fig. 4b, the red and blue scale τ = 2.5h, confirming again that we are in the strongly active dashed lines show Eq. (10)withB and μ chosen with the same ratio regime. as in the previous particle (respectively, SPV) simulations. From As can be seen in Supplementary Movie 5, a significant number these values, we can approximate the parameter values k∕ζ = μ∕σ2 for of divisions take place in the epithelial sheet during the 48h of the the particle model and K∕ζ = μ∕〈A〉2, Γ∕ζ = μ∕〈A〉 for the SPV model. experiment. While it is difficult to adapt our theory to include The solid red and blue curves in Fig. 4bshowthebestfit divisions, we can simulate our particle model with a steady-state simulations that we obtain this way, for k∕ζ = Γ∕ζ = 55 h−1, K∕ζ = division and extrusion rate at confluence using the model μ −2 −1 = μ −1 56,57 0.454 m h and v0 90 mh , and snapshots are shown in developed in refs. . With a typical cell cycle time of τ Fig. 1c (see also Supplementary Movies 7, 9 and 10). The red and div ¼ 48h, we obtain results (green line) that are very similar to blue dashed lines in Fig. 4d show the autocorrelations of our the model without division (red line), suggesting that typical cell matched simulations for soft disk and vertex simulations, division rates do not change the velocity correlations noticeably respectively. (see also Supplementary Movie 8). This result is consistent with

6 NATURE COMMUNICATIONS | (2020) 11:1405 | https://doi.org/10.1038/s41467-020-15164-5 | www.nature.com/naturecommunications NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-020-15164-5 ARTICLE the observed separation of motility time scale τ and division time Simulations. The main simulations consist of N = 3183 particles simulated with τ either soft repulsion, or the SPV model (in literature also refered to as the Active scale div. We have also considered the effect of weak polar alignment between cells, using the model from ref. 22. For weak Vertex Model (AVM)) potential, using SAMoS (https://github.com/sknepneklab/ τ ≥ τ SAMoS). The interaction potential for soft harmonic disks is Vi ¼ alignment with time scales v that do not lead to global P k ðσ þ σ Àjr À r jÞ2 if ∣r − r ∣ ≤ σ + σ and 0 otherwise. To emulate a con- flocking of the sheet, which we do not observe, 〈∣v(q)∣2〉 does not j 2 i j j i j i i j fi fi fluent cell sheet,P we used periodic boundary conditions at packing fraction ϕ = 1, change signi cantly, though we nd somewhat longer autocorre- ϕ πσ2= 2 where ¼ i i L and thus double-counts overlaps. We also introduce 30% lation times. Finally, the simulations can also give us information polydispersity in radius, to emulate cell size heterogeneity. At this density, the about the velocity distribution function, a quantity that is not model at zero activity is deep within the jammed region (ϕ > 0.842) and has a accessible from our theory. In Fig. 4d, we show the experimental significant range of linear response. The disk simulations fitted to experiment also normalised velocity distribution (black line with grey confidence include a short-range attractive region as in57 with ε = 0.15. fi fi For the SPV, cells are defined as Voronoi polygonal tiles around cell centers, interval), together with the distribution we nd from the best t and the multiparticle interaction potential is given by simulations (coloured lines). As can be seen, there is an excellent K 2 Γ 2 Vi ¼ 2 ðAi À A0Þ þ 2 ðPi À P0Þ , where Ai is the area of the tile, and Pi is its Γ fi match in particular with the SPV model simulation. The particle perimeter, K and are the area and perimeter stiffness coef cients and A0 and P0 model with division has additional weight in the tail due to the are reference area and perimeter, respectively. SPV is confluent by construction,pffiffiffiffiffiffi  = particular division algorithm implemented in the model (over- and its effective rigidity is set by the dimensionless shape parameter p0 ¼ P0 A0, fl  : with a transition from a solid to a uid that occurs for p0  3 812. We simulate the lapping cells pushing away from each other).  : 37,55 model at p0 ¼ 3 6, well within the solid region at zero activity , and also introduce 30% variability in A0. AVM was implemented with open boundary conditions, and we use a boundary line tension λ = 0.3 to avoid a fingering Discussion instability at the border that appears especially at large τ. Both models are simulated with overdamped active Brownian dynamics In this study, we have developed a general theory of motion in ζr_ n^ ∇ n^ θ ; θ i ¼ v0 i À r Vi, where the orientation vector i ¼ðcos i sin iÞ follows dense epithelial cell sheets (or indeed other dense active assem- _ i θ ¼ η , hη ðtÞη ðt0Þi ¼ 1 δ δðt À t0Þ. Equations of motions are integrated using a blies58) that only relies on the interplay between persistent active i i i j τ ij fi first order scheme with time step δt = 0.01. Simulations are 5 × 104 time units long, driving and elastic response. We nd an emerging correlation with snapshots saved every 50 time units, and the first 1250 time units of data are length that depends only on elastic moduli, the substrate friction discarded in the data analysis. coefficient and scales with persistence time as τ1∕2. While we found an excellent match between theory and simulations, further Velocity correlations and glassy dynamics. We compute the velocity correlation experimental validations with different cell lines and on larger function for a given simulation directly from particle positions and velocities by first computing the Fourier transform. Then for a given q and configuration, the systems should be performed. Note that without a substrate, the 2 * 59 correlation function is ∣v(q)∣ = v(q) ⋅ v (q), of which we then take a radial q mechanisms of cell activity are very different . More generally, average, followed by a time average. The procedure is identical for the experimental including cell–cell dissipation in addition to cell–substrate dis- PIV fields using the grid positions and velocities, with N = 54 × 40 grid points as sipation could significantly modify scalings40, a known result in normalization. We compute the α-relaxation time from the self-intermediate P q r r ; 1 i Áð j ðt0þtÞÀ j ðt0Þ continuum models of dry (substrate dissipation) vs. wet (internal scattering function Sðq tÞ¼hN j e i ; q , where the angle brackets t0 j j¼q dissipation) active materials. Due to the suppression of tangential indicate time and radial averages. At q = 2π∕σ, we determine τα as the first time slipping between cells, it could also be responsible for the dis- point where S(q, t) < 0.5, bounded from above by the simulation time. appearance of the high-q peak in the velocity correlations. The assumption of uncoordinated activity between cells is a strong one, Normal mode analysis. The normal modes are the eigenvalues and eigenvectors of 2 the Hessian matrix Kij = ∂ V({ri})∕∂ri∂rj, evaluated at mechanical equilibrium. We and it will be interesting to extend the theory by including dif- fi = = 5 ferent local mechanisms of alignment22,49. From a fundamental rst equilibrate the t 2500 snapshot with v0 0 for 2 × 10 time steps, equivalent 22 to a steepest descent energy minimization. We made sure that results are not point of view, our theoretical results (and also the results of ref. ) sensitive to the choice of snapshot as equilibration starting point (with the are examples of a larger class of non-equilibrium steady-states that exception of the deviations apparent in Fig. 3 at τ = 2000). For the soft disk model, 60 n^ ^t can be treated using a linear response formalism . each individual ij contact with contact normal ij and tangential ij vectors con- n^ ´ n^ f ^t ´^t tributes a term Kij ¼Àk ij ij þj ijj ij ij to the ij 2 × 2 off-diagonal element of − 48 the matrix and Kij is added to the ii diagonal element . We use the NumPy eigh Methods function (numpy.linalg.eigh). As the system is deep in the jammed phase, with the exception of two translation modes, all eigenvalues of the HessianP are positive. Experiment. Spontaneously immortalised, human corneal epithelial cells (HCE-S) 1 iqÁr j We compute the Fourier spectrum of mode ν through ξν ðqÞ¼ e j ξ and (Notara & Daniels, 2010) were plated into a 12-well plate using growth medium N j ν ξ q 2 ξ q ξ q à consisting of DMEM/F12 (Gibco) Glutamax, 10% fetal bovine serum, and 1% then j ν ð Þj ¼ ν ð ÞÁð ν ð ÞÞ . The 2 × 2 continuum Fourier space dynamical ∘ D q q^ + μ 2 penicillin/streptomycin solution (Gibco). The medium was warmed to 37 C prior matrix ( ) has one longitudinal eigenmode along with eigenvalue (B )q q^? μ to plating and the cells were kept in a humidified incubator at 37 °C and an and one transverseP P eigenmode along with eigenvalue . We compute fl D q 1 iqα rj;α Àiqβ rj;β α β atmosphere of 5% CO2 overnight until the cells reached con uence. Before imaging ð Þαβ ¼ N j l e Hjl;αβe , where the greek indices , correspond to x the cells were washed with PBS and the medium was replaced with fresh medium or y and there is no sum implied. We diagonalise the resulting matrix, and choose buffered with HEPES. The cells were imaged using a phase-contrast Leica DM IRB the longitudinal eigenvector as the one with the larger projection onto q^, and from inverted microscope enclosed in a chamber which kept the temperature at 37 °C. there the longitudinal and transverse eigenvalues λL(q) and λT(q) after a radial q fi + μ λ 2 = μ The automated time-lapse imaging setup took an image at 10 minute intervals at a average. We t B as the slope of L(q)vsq up to q 1.5, and the same for fi fi μ λ magni cation of ×10, corresponding to a eld of view of 867 × 662 m that was and T(q) (Supplementary Fig. 3). saved at a resolution of 1300 × 1000 pixels. The total experimental run time for each culture averaged 48 h, or 288 separate images. The collected data consists of seven experimental imaging runs, of which number 3 and 4 were consecutive on Data availability the same well-plate (number 4 was not used in this article). Cell extrusions were Data supporting the findings of this manuscript are available from the corresponding counted at three time points during the experimental run by direct observation and authors upon reasonable request. counting from the still image (extruded cells detach from the surface and round up, appearing as white circles above the cell sheet in phase contrast). From this data, a Code availability typical cell number of N = 1400, and a typical cell radius of r = 10.95 μm were extracted. Cell movements were determined using Particle Image Velocimetry Simulation and analysis code used in this study are available under an open source (GNU (PIV), using an iterative plugin for ImageJ (https://sites.google.com/site/ GPL v3.0) licence at: https://github.com/sknepneklab/SAMoS (https://doi.org/10.5281/ qingzongtseng/piv). At the finest resolution (level 3), it provides displacement zenodo.3616475). vectors on a 54 × 40 grid, corresponding to a resolution of 16 μm in the x and the y- direction, i.e. slightly less than 1 cell diameter. The numerous extruded cells and Received: 31 March 2019; Accepted: 7 February 2020; the nucleoli inside the nuclei acted as convenient tracer particles for the PIV allowing for accurate measurements.

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