SUPPLEMENTARY INFORMATION

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1 SUPPLEMENTARY INFORMATION DOI: 1.138/NPHYS2355 Mehanial waves during tissue expansion Supplementary Figures 2.2 Relative mrna levels hours 3 hours 7 hours.8.6 vimentin ZO1 β-atenin E-adherin paxillin vinulin β-atin Fig. S1. Relative hanges in mrna levels during monolayer expansion. Results are plotted as mean SD of two independent experiments, eah performed in tripliate. For eah independent experiment, the ells from 5 independent ell sheets were olleted. NATURE PHYSICS Mamillan Pulishers Limited. All rights reserved.

2 a Veloity.5 Strain Rate 1/s x Trations 25 d e Bleistatin fcontrol Fig. S2. Bleistatin inreases veloity of monolayer expansion ut arogates tration generation, monolayer stress transmission and wave propagation. Kymographs of veloity (a), strain rate (), trations (), and monolayer stress omponent (d). Bleistatin prevented stress fier formation. (e) Control. (f) Bleistatin treatment. Sale ar = 2 m. 212 Mamillan Pulishers Limited. All rights reserved.

3 a Veloity.5 Strain Rate 1/s x Trations 25 d Fig. S3. Effet of a sudden disruption of ell ell juntions with EGTA during monolayer expansion. ymo raphs o veloity (a), strain rate ( ), trations (), and monolayer stress omponent (d). EGTA was added at 28 min and removed at 325 min (dashed lines). 212 Mamillan Pulishers Limited. All rights reserved.

4 a Veloity.5 Strain Rate 1/s x Trations 25 d Fig. S4. Dynami ehavior of an asymmetrially expanding ell sheet. Kymographs of veloity (a), strain rate (), trations (), and monolayer stress omponent (d). a Strain Rate 1/s x 1-4 Strain Rate 1/s x Fig. S5. Boundary effets of PIV. When PIV is applied to the monolayer oundaries, ells only oupy a fration of the interrogation window. The veloity vetor orresponding to that interrogation window has one ontriution from the motile monolayer and another ontriution from the immoile sustrate. As a onsequene, the resulting veloity is smaller than the real monolayer veloity. This effet leads to a systemati gradient of veloity at the very edge of the monolayer and, as suh, to an artifatual negative strain rate. We have suppressed this artifatual data from Fig. 3 and Supplementary Fig. S4. (a) and () show the unedited data orresponding to Fig. 3 and Supplementary Fig. S4, respetively. 212 Mamillan Pulishers Limited. All rights reserved.

5 Supplement 1: Monolayer Stress Mirosopy and material properties of the ell monolayer. To otain the distriution of stresses throughout the monolayer, MSM uses the map of tration fores together with the two-dimensional alane of fores that is demanded y Newton s laws 1. Beause this fore alane must e oeyed at every instant, the omputation of monolayer stresses from tration fores does not depend on whether the monolayer is elasti, visous, or visoelasti. Without loss of generality, we thus hose to treat the monolayer as a 2D elasti homogeneous material with Young s modulus E and Poisson s ratio. In our previous report 1 we used preisely the same approah with assigned to e.5, ut in that report we asserted - inorretly - that if the tration fores are speified then reovered stress distriution within the monolayer is independent of onstitutive properties of the monolayer. While that statement is rigorously true in one-dimensional systems, it is not true in twodimensional systems, wherein there are two degrees of freedom in the external trations ( and ) ut three omponents to the stress tensor (,, ). In that ase, the oupling etween trations and stresses does not depend on the Young s modulus ut does depend on the Poisson s ratio. That eing the ase, we assessed the sensitivity of stress maps to the assumed value of y omputing,, and from a set of experimental tration fields and with ranging from to.5 (Fig. S6). These results demonstrate that variations in have negligile effets on and ut sustantial effets on as approahed zero. This oupling etween and has its origin in the oundary onditions of zero normal displaement at the top and ottom edges of the field of view, ut has little influene when >.4. Moreover, our findings of wave propagation and pattern formation during monolayer growth are restrited to the diretion of expansion and are thus largely independent of the hoie of. Therefore, throughout this report we assume inompressiility ( =.5). 212 Mamillan Pulishers Limited. All rights reserved.

6 a Phase Contrast Trations Trations y 1 y x -1 yy xy d e f ν = ν =.5 ν =.4 g h i ν =.2 4 j k l 2 m n o Fig. S6. Dependene of omputed monolayer stresses on assumed Poisson s ratio. Phase ontrast image (a), tration maps (, ), and monolayer stress maps (d o) for different values of Poisson s ratio. 212 Mamillan Pulishers Limited. All rights reserved.

7 Supplement 2: Monolayer Stress Mirosopy and ell height Monolayer Stress Mirosopy performs a formal two-dimensional alane of line tension that is onverted to familiar units of stress y using the average height (h) of the monolayer. To aount for hanges in h during monolayer expansion we used onfoal mirosopy. We performed z-stak onfoal imaging on MDCK ells staly expressing LifeAt-GFP and omputed the average fluoresene intensity for eah time point and for eah z plane, where rakets denote average over x and y within the monolayer. We alulated h as the harateristi height otained from fitting an exponential deay to the tail of. The average ell height was roughly homogeneous aross the diretion of expansion (Fig. S7) and exhiited a sharp derease with time that stailized at 8 µm (Fig. S7a). a Monolayer height (μm) Time (minutes) 2 Monolayer height (μm) 15 1 minutes 15 minutes 3 minutes 525 minutes Distane from enter (μm) Fig. S7. Monolayer height. (a) Average ell height during monolayer expansion. Error ars represent the SD. () Average ell height as a funtion of the distane from the monolayer midline at different time points. Dashed lines are mean SD. 212 Mamillan Pulishers Limited. All rights reserved.

8 Supplement 3: one-dimensional model of monolayer expansion Using the model desried in Fig. 4, we onsider here four distint loading senarios and analyze the extent to whih eah of these senarios aptures the entral features of our oservations inluding 1) onstant veloity of the monolayer leading edge, 2) propagation of a strain rate front, 3) monolayer stress uildup, and 4) temporal reiteration of oth strain-rate propagation and stress uildup. In every ase, a numerial solution was otained y using Newton-Raphson reiterations. In eah node, residual fore ontriutions from loads along with fores of disrete elements that join at that node (springs and dashpot) were required to satisfy Eq. 1 (main text). We first onsidered the simple ase where only the two front ells (one on eah side of the monolayer) generate self-propelling fores (Fig. S8, Supplementary movie 9). We simulated this senario y applying two fores of onstant magnitude and opposite sign to the front nodes of the monolayer. In response to these fores, leading ells egin strething immediately, ut visous frition with the sustrate prevents instantaneous transmission of the leading fore to the follower ells (Fig. S8d). Consequently, the further a ell is from the leading edge, the longer that ell takes to egin strething. This delay translates into a pulse of strain rate that penetrates the monolayer and attenuates progressively (Fig. S8). The system tends asymptotially to a state in whih stress and strain of eah ell are equal aross the monolayer. Aordingly, the system never displays a stress uildup nor onstant veloity of the leading edges of the monolayer, thus showing that an ative leader/passive follower senario does not apture the experimental features we oserved. We next onsidered the ase in whih all ells have the apaity of generating selfpropelling fores simultaneously. (Fig. S9, Supplementary movie 1). We simulated this senario y applying outward pointing fores of onstant magnitude at eah node of the monolayer at the same instant of time. Under these irumstanes, eah half of the monolayer egins moving outwards uniformly and ohesively (Fig. S9). The elasti stress is highest at the enter of the monolayer and dereases towards its edges (Fig. S9d), thus ausing a progressive derease of ell veloities away from the monolayer midline (Fig. S9). When the model was run in suh a onfiguration we ould simulate onstant veloity at the leading edges (Fig. S9), stress uildup in its 212 Mamillan Pulishers Limited. All rights reserved.

9 entral regions (Fig. S9d), and outward propagation of a strain rate pulse (Fig. S9), all features we oserved experimentally. Nevertheless, the model is not ale in suh irumstanes to reprodue an initial phase where stress and strain rate propagate inward from the leading edges towards monolayer midline. a t t Veloity Strain Rate d Elasti Stress -4x1 4x /s Fig. S8: (a) Cartoon illustrating epithelial dynamis indued y migrating ells loated at the leading edges only. t instant the two front ells are sujet to self propelling fores of same modulus and opposite diretion. () Kymograph of epithelial veloity field vs. time; () kymograph of epithelial strain rate field vs. time; (d) kymograph of epithelial elasti stress field vs. time. We next onsidered a third ase in whih a ell is ale to generate a self-propelling fore only if its immediate front neighor has deformed more than a given strain (Fig. S1, Supplementary movie 11). This is equivalent to assume that eah ell an only aquire a motile phenotype if it senses a fore larger than on its front interellular juntion. This type of mehanism has een reently invoked to explain the olletive migration of ell lusters 2,3 and the underlying physis an e understood in analogy to unjamming phenomena 4-6. Under suh an assumption, the monolayer ahieves its initial expansion y suessive spreading of eah ell row, whih results in a strain rate pulse propagating away from the leading edge (Fig. S1). One the strain rate pulse reahes the monolayer midline, the system tends to its equilirium onfiguration in whih stress dereases away from the monolayer midline (Fig. S1d). However, the rate at whih every ell tends to its equilirium 212 Mamillan Pulishers Limited. All rights reserved.

10 length also dereases with the distane from the midline, thus giving rise to outward propagation of a strain rate pulse (Fig. S1). a t t t t t t t Veloity Strain Rate x1-4 4x /s 1/s d Elasti Stress Fig. S9: (a) Cartoon illustrating epithelial dynamis indued y ells that migrate y generating outward self propelling fores simultaneously at the initial instant t o. () Kymograph of epithelial veloity field vs. time; () kymograph of epithelial strain rate field vs. time; (d) kymograph of epithelial elasti stress field vs. time. Unjamming y itself aptures entral features of monolayer expansion inluding propagation of a strain rate pulse away from and ak to the leading edge, stress uildup, and linear leading edge veloity, ut it does not exhiit spatiotemporal reiteration of loal differentials of stress and strain rate. Instead, the system tends asymptotially to an equilirium state in whih the length of eah ell dereases with the distane from the leading edge. Spatiotemporal reiteration of loal differentials in mathematial models of tissue patterning is typially otained y onsidering two ompeting phenomena oupled through a time delay. In traditional reation-diffusion models, the ompeting phenomena are ativation and inhiition of morphogens and the time delay originates from their distint diffusion and reation rates 7,8. This idea was reently generalized to the ase of mehanohemial systems, where the ompeting phenomena are diffusion and ontratile advetion and the time delay originates in their distint rates 9, Mamillan Pulishers Limited. All rights reserved.

11 a t t 2 >t 1 t n >t n-1 t 2 >t 1 t 1 >t t t 1 >t Veloity Strain Rate d Elasti Stress x1-4 4x /s Fig. S1: (a) Cartoon illustrating epithelial dynamis indued y ells that migrate y generating outward self propelling fores only when their immediate front neighor has deformed more than a given threshold strain. () Kymograph of epithelial veloity field vs. time; () kymograph of epithelial strain rate field vs. time; (d) kymograph of epithelial elasti stress field vs. time. Here we propose an alternative mehanism in whih the two ompeting phenomena are two reently desried non-linear responses of the ytoskeleton to streth 11 : reinforement, whih auses inreases of stiffness 12, and fluidization, whih auses dereases of stiffness 13,14. While oth phenomena remain poorly understood, they have een shown to oexist ut play out over different time sales. For example, after a single streth-unstreth yle ells fluidize suddenly, ut hundreds of seonds later they reinfore slowly 13,14. In order to introdue reinforement and fluidization in our simple 1D model, we uild upon the unjamming senario previously analyzed (Fig. S1), now allowing to vary with time. Speifially, we assume that stiffness of eah spring is onstant only for strains elow an aritrary threshold. As soon as is reahed, a period of reinforement egins in whih inreases with time for a duration. This reinforement phase is followed y a fluidization phase during whih dereases with time for a duration (Fig. 4). Simulations show that for the first 4 min no ell reahes and thus the evolution of the monolayer (Fig. 4, Supplementary movie S8) is idential to the ase desried previously (Fig. S1). Progressively from the monolayer midline towards the leading 212 Mamillan Pulishers Limited. All rights reserved.

12 edge, ells egin to undergo strains greater than ε and thus eliit an outward wave of reinforement. As this wave propagates outwards, it auses a progressive stalling paralleled y uildup of stress, as in a tug-of-war, toward the enter. The reinforement wave is followed y a fluidization wave with a delay. This seond wave enales further ell strething and thus auses a seond pulse of positive strain rate to propagate outwards and a seond peak of stress at the monolayer midline. These simulations estalish that yles of reinforement and fluidization triggered y a strain threshold result in reiterated propagation of strain rate fronts through the monolayer and osillations of interellular stress at the monolayer midline. Simulation parameters The values of F, k, and were hosen to apture the experimental kymographs of veloity, stress, and strain-rate while remaining within the order of magnitude reported in the literature. Speifially, we hose F=4 nn, k= 4 nn, and =9.2 nn.min/µm. Variales with units of fore were transformed to more familiar units of stress y assuming a ross-setional area of 1 µm 2. Strain thresholds were hosen to e and 1. We note that as long as the model aptures the generation and propagation of waves. During a reinforement/fluidization yle, the maximum stiffness was set to 3 times its aseline value. Referenes 1 Tame, D. T. et al. Colletive ell guidane y ooperative interellular fores. Nat Mater 1, (211). 2 Carmona-Fontaine, C. et al. Contat inhiition of loomotion in vivo ontrols neural rest diretional migration. Nature 456, (28). 3 Weer, G. F., Bjerke, M. A. & Desimone, D. W. A Mehanoresponsive Cadherin-Keratin Complex Direts Polarized Protrusive Behavior and Colletive Cell Migration. Developmental ell (211). 4 Angelini, T. E. et al. From the Cover: Glass-like dynamis of olletive ell migration. Proeedings of the National Aademy of Sienes of the United States of Ameria 18, (211). 5 Keys, A. S., Aate, A. R., Glotzer, S. C. & Durian, D. J. Measurement of growing dynamial length sales and predition of the jamming transition in a granular material. Nat Phys 3, (27). 6 Trappe, V., Prasad, V., Cipelletti, L., Segre, P. N. & Weitz, D. A. Jamming phase diagram for attrative partiles. Nature 411, (21). 212 Mamillan Pulishers Limited. All rights reserved.

13 7 Kondo, S. & Miura, T. Reation-diffusion model as a framework for understanding iologial pattern formation. Siene (New York, N.Y 329, (21). 8 Turing, A. M. The Chemial Basis of Morphogenesis. Philosophial Transations of the Royal Soiety of London. Series B, Biologial Sienes 237, (1952). 9 Bois, J. S., Juliher, F. & Grill, S. W. ttern formation in ative fluids. Physial review letters 16, 2813 (211). 1 Howard, J., Grill, S. W. & Bois, J. S. Turing's next steps: the mehanohemial asis of morphogenesis. Nature reviews 12, (211). 11 Kroy, K. & Glaser, J. The glassy wormlike hain. New J. Phys., 416 (27). 12 Riveline, D. et al. Foal ontats as mehanosensors: externally applied loal mehanial fore indues growth of foal ontats y an mdia1-dependent and ROCK-independent mehanism. The Journal of ell iology 153, (21). 13 Trepat, X. et al. Universal physial responses to streth in the living ell. Nature 447, (27). 14 Bursa, P. et al. Cytoskeletal remodelling and slow dynamis in the living ell. Nat Mater 4, (25). 212 Mamillan Pulishers Limited. All rights reserved.