Traveling and standing waves mediate pattern formation in cellular protrusions

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1 SYSTEMS BIOLOGY Trveling nd stnding wves medite pttern formtion in cellulr protrusions Syk Bhttchry 1, Ttst Bnerjee 2,3, Yuchun Mio 3,4, Huiwng Zhn 3,5, Peter N. Devreotes 3, Pblo A. Iglesis 1,3 * The mechnisms regulting protrusions during moeboid migrtion exhibit excitbility. Theoreticl studies hve suggested the possible coexistence of trveling nd stnding wves in excitble systems. Here, we demonstrte the direct trnsformtion of trveling into stnding wve nd estblish conditions for the stbility of this conversion. This theory combines excitble wve stopping nd the emergence of fmily of stnding wves t zero velocity, without ltering diffusion prmeters. Experimentlly, we show the existence of this phenomenon on the cell cortex of some Dictyostelium nd mmmlin mutnt strins. We further predict templte tht encompsses spectrum of protrusive phenotypes, including pseudopodi nd filopodi, through trnsitions between trveling nd stnding wves, llowing the cell to switch between excitbility nd bistbility. Overll, this suggests tht previously-unidentified method of pttern formtion, in which trveling wves spred, stop, nd turn into stnding wves tht rerrnge to form stble ptterns, governs cell motility. INTRODUCTION Excitble wves hve been observed in vrious physiologicl settings, from rotting clcium wves in the crdic myocyte (1) to ctin polymeriztion wves during moeboid cell migrtion (2). The wvefront in n excitble medium is creted by nonliner ctivtor response to suprthreshold stimulus. This ultrsensitive response is ll-or-none type, ensuring similr wve mplitudes cross the medium. The wve bck is formed by down-jump in the ctivtor owing to delyed inhibitor response. The slow nture of the inhibitor cretes n ensuing refrctory period before the inhibitor returns to equilibrium, isolting n ctivity spike from subsequent triggers. Through diffusion, this spike propgtes cross djcent excitble elements, creting trveling wve. In systems with only ctivtor diffusion, the delyed inhibition llows the wve to spred without restriction in spce, s is chrcteristic of neurl wves (3). In contrst, interesting sptil phenomenon emerges with diffusive inhibitor (4). For exmple, if the rtio of inhibitor to ctivtor diffusion, 1, then one obtins diverse wve ptterns, s in the Belousov-Zhbotinsky rection (5). For» 1, lterl inhibition llows the formtion of stble stnding wves (6), creting ptterns similr to mny seen in nture, like the intricte involutions of seshells (7) or the tentcle ptterns of Hydr (8). Theoreticl studies demonstrte tht it is possible for trveling nd stnding wves to coexist by ltering the inhibitor diffusion, stlling trveling wves t the zero-velocity mrk, leding to the emergence of stnding wves (9). In this study, we show how direct trnsformtion from trveling to stnding wve cn occur without chnging diffusion prmeters. In this cse, zero wve speed or wve stopping is chieved through the nturl ccumultion of the inhibitor in spce (10), similr to the 1 Deprtment of Electricl nd Computer Engineering, Johns Hopkins University, 3400 N. Chrles St., Bltimore, MD 21218, USA. 2 Deprtment of Chemicl nd Biomoleculr Engineering, Johns Hopkins University, 3400 N. Chrles St., Bltimore, MD 21218, USA. 3 Deprtment of Cell Biology nd Center for Cell Dynmics, Johns Hopkins School of Medicine, 725 N. Wolfe St., Bltimore, MD 21205, USA. 4 Deprtment of Biologicl Chemistry, Johns Hopkins School of Medicine, 725 N. Wolfe St., Bltimore, MD 21205, USA. 5 Deprtment of Biologicl Chemistry, Johns Hopkins University School of Medicine, 725 N. Wolfe Street, Bltimore, MD 21205, USA. *Corresponding uthor. Emil: Copyright 2020 The Authors, some rights reserved; exclusive licensee Americn Assocition for the Advncement of Science. No clim to originl U.S. Government Works. Distributed under Cretive Commons Attribution NonCommercil License 4.0 (CC BY-NC). wve-pinning mechnism proposed for bistble systems (11), thus llowing the system to move from n initilly low equilibrium to permnent higher stble stte. This is chieved t intermedite levels of where trveling wves cn lso be sustined. Becuse ltertion of diffusion coefficients is chllenging, to the best of our knowledge, this direct trnsformtion hs not been demonstrted experimentlly. Our interest in this mechnism rose from recent observtions of wve propgtion in perturbed moeboid cells (10, 12). Both excitble (13) nd bistble systems (14) hve been proposed to ccount for cellulr protrusions during migrtion. The conflicting rguments regrding the roles of excitbility nd bistbility in regulting protrusive morphology stems mostly from the fct tht while some protrusions, such s the filopodium or the stble front in directed migrtion, cnnot be explined by trnsient trveling wves (15), others, such s the signling wves tht re continully observed on the cell cortex, cnnot rise from persistent ctivity tht is typicl of bistble systems. While in our erlier work we described how n excitble system model cn reproduce different trveling wve phenotypes (10), we did not consider persistent protrusive ctivity. Here, we illustrte tht trnsformtion from trveling to stnding wve llows excitbility nd bistbility to switch between one nother without drsticlly ltering system prmeters. This llows us to explin the vrious types of cell protrusions seen in migrting cells nd crete n ll-encompssing protrusive templte. Moreover, in the process, we describe potentilly previously unknown method of pttern formtion. RESULTS Trveling wves cn trnsform into stnding wves t the instnt of wve stopping The model we use to generte wve propgtion (4, 16) is inspired by the FitzHugh-Ngumo model of excitbility (17, 18), modified to ensure tht the species levels remin positive. It consists of n utoctlytic ctivtor (u) nd delyed inhibitor (v). du dt = D u 2 3 u 2 u 1 u 2 u(v r ) u dv dt = D v 2 v + ϵ( c 1 v + c 2 u) 1 of 9

2 The ultrsensitivity of the ctivtor mnifests through the coopertivity term, while the dely in the inhibitor is incorported through the vrible, resulting in time scle seprtion between the two components, creting distinguishble wve front nd bck. In phse-plne digrms, the ctivtor nullcline displys n inverted N-shpe (fig. S1A). As the slope of the inhibitor nullcline is vried, the system undergoes two Hopf bifurctions, pproximtely t the minimum nd mximum of the ctivtor nullcline (19). Between these two bifurction points, the equilibrium is unstble. The initil equilibrium is to the left of the minimum (fig. S1A) such tht the threshold of the system corresponds pproximtely to the verticl distnce between the equilibrium set point (v 0 ) nd the minimum of the ctivtor nullcline (v min ) (19). Chnges in r reflect externl stimuli tht lower the threshold of the system nd trigger lrge-scle excursion in phse spce, which trnsltes to shrp up-jump in ctivity, thus creting the wvefront (fig. S1A). This ctivtor-inhibitor model hs been used to recrete trveling wves observed on the cell cortex in different cell types (20 22). Our recent work hs lso suggested models for the underlying biochemicl signling network tht displys this type of ctivtor-inhibitor dynmics (10, 23). Using this model, we hve shown how different wve chrcteristics re ltered when perturbtions re introduced to the governing signling network, nd tht these model prmeters llow us to cpture spectrum of trveling wve phenotypes. The velocity of the excitble wvefront hs been the subject of extensive reserch using singulr perturbtion pproches (3). In onedimensionl spce, the wve velocity cn be completely determined by function of the initil level of the inhibitor, lso clled the controller species (24). This function is inversely proportionl to v 0, i.e., higher thresholds led to slower wve velocities nd vice vers. The wve speed nd wve stopping ply crucil role in determining the protrusive phenotype of cells (13). Specificlly, how fr the wve trvels before extinguishing determines the wve rnge nd, in turn, the size of the protrusion. If the dispersion of the inhibitor exceeds tht of the ctivtor, then, s wve trvels, the threshold levels continully increse in the surrounding, cusing the wve to slow down s it spreds (10), ultimtely stopping when criticl threshold is reched (Fig. 1A). Note tht this stopping is independent of the size of the simultion domin (fig. S1E). This is similr to the wve-pinning mechnism (11), with the key difference being tht the wve is extinguished upon stopping, insted of being pinned t higher stedy stte tht exists only in bistble system. However, s we show below, it is possible for n initilly low-equilibrium excitble system to spred s trveling wve before switching to higher-equilibrium stedy stte t the instnt of wve stopping. When wve is triggered, the sptil grdient in inhibition mkes the contribution of the diffusion term negtive. Rewriting the inhibitor eqution dv dt = ϵ ( c 1 v + c 2 u + 1 ϵ D v 2 v ) we see tht when the Lplcin is negtive, the inhibitor nullcline shifts to the right nd thereby lowers the threshold (Fig. 1, B nd C). As the stte moves round its trjectory, this grdient grdully subsides. If this shift is sufficiently lrge, then new, stble, higher equilibrium is trnsiently creted. This new equilibrium my ttrct the trjectory of the system, in which cse the stte remins t its A C D F Inhibitor (v) Inhibitor (v) d d i Activtor (u) i Activtor (u) B 2 v 2 v ii i Activtor (u) Activtor (u) new high stte (Fig. 1, D to F, nd movie S1). From the eqution bove, it is cler tht sufficiently lrge shift requires smll vlue of /D v, similr to conditions for stble stnding wve (25). The negtive Lplcin is necessry but not sufficient condition, s this trnsformtion lso requires tht the stte trjectory be ttrcted by the trnsient high equilibrium. This is controlled by vrious prmeters including the excursion time nd shpe of the ctivtor nullcline. For exmple, with the sme vlue of /D v, the cse of Fig. 1A ws unble to crete the stnding wve, owing to higher initil threshold. Altering the shpe of the ctivtor nullcline (fig. S1G) by incresing positive feedbck cn lso led to the formtion of stnding wves, s it increses the region of ttrction for the new equilibrium. During wve propgtion, when the wve velocity is greter thn the criticl wve-stopping threshold, this trnsformtion does not rise, s the diffusion grdient is short-lived inside trveling wve nd its contribution is counter-blnced by the ctivtion from surrounding spce. As the wve slows down, this grdient lsts longer becuse the new triggers re further prt in time (Fig. 1B). The wve stops when no new trigger occurs, t which point the grdient i ii iii b c d 3 0 Time (A.U.) 5 10 E ii Activtor (u) b c d ii iii 3 0 Time (A.U.) 5 10 iii iii Activtor (u) Fig. 1. Trnsformtion of trveling to stnding wve. (A) Kymogrph of wve tht trveled, stopped, nd extinguished (c 2 = 4.2 in inhibitor eqution). Dshed rrows indicte where the stopping occurred. (B) Time profiles of to d from the kymogrph in (A), plotting the Lplcin evolution t ech of these sptil points. (C) Illustrtion of how the nullclines re ltered by the Lplcin term. The three situtions correspond to the time instnts mrked in (B). The white circle denotes bifurction point; the equilibrium is stble if the inhibitor nullcline (red) is to the right of this. The blck circle denotes the stte, with the immedite trjectory shown by the dshed rrow. (D to F) Exmple of stnding wve. The pnels re s in (A) to (C) but for wve tht trnsformed into stnding wve on stopping (c 2 = 3.9 in inhibitor eqution). 2 of 9

3 is mximized (blck curve in Fig. 1, B nd E), enbling the wve to trnsform into stnding wve, shrply chnging wve speed ner the zero wve-speed mrk (fig. S1B). Although this trnsformtion cn lso be chieved by vrying diffusion coefficient (9), the criticl difference in our cse is tht wve stopping occurs without ltering diffusion prmeters. The trnsformed stnding wves form stble ptterns tht depend on system threshold The two stnding wves tht emerge from the initil trveling wve ultimtely spred out to form sptilly symmetric pttern (Fig. 2A), s predicted by theory tht specifies tht periodic solution is stble on circulr domin (25). In (25), conditions ensured tht trigger instntneously produced the stnding wve, s the high diffusion rtio ( = 100) could not sustin trveling wves (fig. S1C). Note tht the periodic pttern formtion occurs on n order of mgnitude slower time scle when compred to the time tken for the trveling wve to stop nd stnd (Fig. 2B). This time scle seprtion distinguishes trveling-to-stnding wve trnsitions from the formtion of stnding wves. Although two stnding wves re trveling during formtion, their velocity is gretly lower (fig. S1B) nd they ttrct or repel ech other to crete periodic sptil rrngement. We illustrte this in fig. S1F, where the spreding of one stnding wve brnch is noticebly repelled by nother. Two oppositely directed trveling wves however, merge nd nnihilte owing to the ccumulted inhibitor tht trils ech (4, 26). A stnding wve hs high level of inhibition surrounding it tht cuses trveling wve pproching it from either side to be extinguished (fig. S1F). This lso distinguishes the wve-pinning brnches proposed in (11) A C m m + n σ n D Inhibitor (v; A.U.) from these stnding wves, s the former do not spred out periodiclly in spce to crete stble pttern. The previous results were in deterministic setting with mnul trigger to crete the wve. In stochstic setting, however, stnding wve cn end (Fig. 2, C nd D), becuse sufficiently lrge rndom perturbtion my move the stte wy from the new equilibrium (Mterils nd Methods). Figure 2E shows tht low-threshold stnding wve is more stble to stochstic perturbtions. This occurs s the inhibitor grdient formed is stronger for lower threshold surroundings, which results in lrger threshold for the newly-formed equilibrium (fig. S1D). Stble, confined protrusions observed in mutnt cell types cn be recreted using stnding wves During cell migrtion, moeboid cells extend pseudopods, i.e., periodic protrusions of their cortex, to propel the cell forwrd. The extensions re controlled by wves of signling molecules tht orgnize ctin polymeriztion ner the membrne, creting protrusions tht lst round 60 s nd cover 5 to 25% of the cell cortex (movie S2) (21). Mutnt forms of Dictyostelium, such s those in which the tumor suppressor gene PTEN hs been deleted (PTEN-null cells), re known to crete elongted finger-like protrusions tipped by smll regions of elevted signl trnsduction nd cytoskeletl events (12, 27). Excitble wves cnnot crete these elongtions, which require wves tht neither spred nor die but persist t one prticulr region. Note tht the coexistence of trveling wves nd stble ptterns does not depend on the cytoskeleton becuse ltrunculin- treted cells, in which ctin polymeriztion is inhibited, lso disply both phenomen (15). These ptches re nomlous becuse, typiclly, responses of excitble systems oscillte, propgte, or extinguish. Activtor (u; A.U.) Probbility of stnding B E t=30 t=356 0 Time (A.U.) 450 c 2= 3.7 c 2= 3.8 c = Sigm (noise level) Fig. 2. Pttern formtion nd stbility. (A) (Left) Exmple of trveling-to-stnding trnsformtion on longer time scle. The pttern formtion is indicted using the vribles m nd n tht show equl spcing of stnding brnches on periodic domin. (Right) Zoomed-in version of the ctivity in the white dshed box. (B) Time evolution of the ctivity in the red dshed spce in (A), showing through the verticl lines the time tken to trvel nd stop versus the time tken to form the finl pttern. (C) Exmple of deterministic (top) nd stochstic (bottom) simultions, where noise (sigm) in the ltter cuses the stnding brnches to fll off. (D) Nullclines illustrting the flling off of the stble stte to return to the originl equilibrium (light red nullcline). (E) Averge of 40 simultions with different levels of noise (sigm) nd system threshold, which is controlled by the slope of the inhibitor nullcline (c 2 ). A lower slope corresponds to lower threshold nd vice vers. 2 3 of 9

4 Actin-inhibited moeboid signling wves (Mterils nd Methods) spin round the cell cortex (Fig. 3A), with the response t ny given point lsting bout 1 min. However, in the PTEN-null cells (Mterils nd Methods), wve cn linger t portion of the cortex for over 4 min (Fig. 3B). These persistent ptches, when coupled with the cytoskeleton, crete the elongted finger-like protrusions (Fig. 3C nd movie S3). These fingertips re lso ccompnied by ccumultion of signling mrkers (PH crc shown in Fig. 3D). Often, the signl trnsduction nd cytoskeletl events t the tips of the protrusions pper in the form of smll rings of ctin (Fig. 3E) tht continully push on the cell boundry (28). The rings suggest tht these finger-like protrusions were formed by wves tht stopped quickly fter being triggered but were not extinguished upon stopping. Tht is, the hole in the center of the rings suggests tht the wves trveled some distnce before the trnsformtion occurred. As lowering threshold increses wve rnge (13), it ws predicted tht lrger rings my pper fter lowering the threshold of PTEN-null cells. This ws confirmed by incresing the ctivity of PTEN-null cells using ctivted Rs. These cells displyed lrge, fluctuting rings t the edge of the cell [termed pncke cell (12)] often lsting indefinitely, until the cell finlly tore itself prt. Figure 3F shows the evolution of one of these lrge rings. These experimentl observtions were recreted in simultions using the trveling-to-stnding wve trnsformtion. For prticulr prmeter regime, wves expnded nd were extinguished to crete typicl protrusions (Fig. 4A nd movie S4), s seen in wild-type moeboid cells. In the stnding prmeter regime, however, ctivity persisted t one point in spce lsting longer in time, creting elongted, A B s t = 1 s t = 150 s finger-like extensions (Fig. 4B nd movie S5). The cellulr protrusions were modeled using viscoelstic cell model in the level-set frmework (Mterils nd Methods). The durtions of protrusions tht were similr in size (5 to 25% of the cortex) were quntified. The cells in Fig. 4B showed significntly longer durtion (Fig. 4C), lthough system turnround time ( ) ws not ltered. A two-smple Kolmogorov-Smirnov test reveled tht these two protrusive phenotypes belonged to different distributions (Mterils nd Methods). In one-dimensionl simultions, it ws difficult to pprecite the existence of smll rings t the tips of protrusions. With lowered threshold, however, the wve ws expected to expnd nd crete lrger stnding wve tht is, s previously demonstrted, more stble to stochstic perturbtions. We simulted the formtion of this pncke ring using two-dimensionl sptil simultion (Fig. 4D nd movie S6). The wve initilly trveled, stopped, nd then evolved into stnding wve. Therefter, the wve broke prt nd rerrnged to form stble periodic pttern in spce (t = 130 to 900 rbitrry time units). We conjecture tht, in experiments, we do not see the periodic rerrngement within rings of the pncke cells for two resons. First, the time scle for this to occur is over n order of mgnitude lrger. Second, the cell boundry hs n orgnizing effect on the wve, which does not llow it to brek up. In fig. S1E, we show through simultion, in which ctivtor diffusion ws sptilly limited, tht the stnding wve orgnized s stble ring t the boundry (movie S7). Aprt from Dictyostelium, we lso looked t trnsformed cells where KRsG12V oncogenic muttion ws introduced in MCF-10A epithelil cells (Mterils nd Methods). These cells similrly disply spontneous excitble wves on the cortex (Fig. 4E nd movie S8). t = 100 t = 200 t = 250 t = 250 t = 350 t = s C wt pten D pten (LimE) pten (PH crc ) E F pten pten Fig. 3. Trveling nd stnding phenotypes in cell migrtion. (A nd B) Kymogrph of PH crc signling mrker for ltrunculin-treted wild-type (A) nd PTEN-null (B) Dictyostelium cells. Imges of the cells re shown on the right, with the white circle mrked to follow ctivity t smll region. (C) Wild-type (wt) nd PTEN-null cell morphology, with LimE-RFP. Scle br, 5 m. (D) PTEN-null exmple showing ctin (left) nd signling (right) mrkers. Scle br, 25 m. (E) Actin dynmics in PTEN-null cells. Arrows indicte smll ctin rings. This pnel is tken from (28) with permission. (F) F-ctin wve pttern (GFP-LimE) phenotype induced by RsCQ62L expression in PTEN-null cells (scle br, 5 m) forming pncke-type cell. This pnel is tken from (12) with permission. 4 of 9

5 A B persisted for significntly longer durtions (Fig. 4F, dshed circle), displying the stnding phenotype (Fig. 4G nd movie S9). In ll these cses, membrne nd cytosolic mrker ws used to rule out membrne undultions (fig. S2B). These cells demonstrte tht both trnsient nd persistent ctivity levels re observed experimentlly. Twodimensionl sptilly stochstic simultions showed remrkbly similr wve phenotypes in which some wves trveled, while others lingered t one point for significntly longer durtions (Fig. 4H nd movie S10). C E G Protrusion durtion (A.U.) P < 1e t = 0 min b D 4 min 6 min 8 min t = 1 (A.U.) t = 12 t = 36 t = 48 t = 60 t = 110 t = 300 t = 500 t = 700 t = 900 Spce 0 Spce 0 Time (min) 400 We quntified the two-dimensionl wve ctivity through kymogrphs to study the spred nd durtion of these wves. On verge, these wves lsted round 10 to 20 min t prticulr point on the cortex (Fig. 4F). However, we observed numerous cses where the wve Spred (frction of cortex) Time (min) 120 H Activity lifetime (min) Spce 0 Time (A.U.) 300 Fig. 4. Simultions of the excitble system recreting experimentlly observed wve nd morphologicl phenotypes. (A) (Left) Kymogrphs of norml moeboidtype protrusions. The yellow dshed line indictes the trveling wve. (Right) Level-set simultions from the ctivity in (A). (B) (Left) Kymogrphs of PTEN-null type protrusion, showing significntly longer thin fingers of ctivity. The yellow dshed line hs much lower slope thn tht of (A), indictive of the slow velocity of stnding wve. (Right) Level-set simultions from the ctivity in (B), showing elongted protrusions. (C) Quntifiction of the durtion of ctivity obtined through simultions from prmeter sets of (A) nd (B). Ptches tht covered between 5 nd 25% of the domin size were quntified. P vlue obtined from t test for 180 protrusions. (D) Two-dimensionl deterministic simultions mnully triggering wve t the center of the domin to study the time evolution of sptil ctivity. (E) Imges nd kymogrph showing ctin ctivity in trnsformed MCF-10A cells. Trveling wves re seen in the imges (white rrow) nd in the kymogrph (dshed circle). Scle br, 50 m. (F) Quntifiction of wve durtions seen in trnsformed MCF-10A cells (three cells). Ech point corresponds to protrusion. The points in the dshed circle indicte those tht persisted longer thn trveling wves typiclly do. (G) Imges nd corresponding kymogrphs showing PH-AKT ctivity in trnsformed MCF-10A cell. Activity persists t loction (dshed circle) without spreding for over 100 min. Scle br, 21 m. (H) Similr stnding ctivity from stochstic two-dimensionl simultions. F A phse digrm of cellulr phenotypes revels n ll-encompssing protrusion templte Using phse digrm of excitble system prmeters, we chrcterized the regions where different protrusive phenotypes re observed (Fig. 5A). Two prmeters were chosen, one controlling the negtive feedbck from the inhibitor nd nother controlling the time scle seprtion such tht the lower left corner represented the lowest threshold nd the upper left corner represented the highest. The red region denotes the set of prmeters for which our initil stimulus ws unble to elicit ny response (subthreshold). The green region demrctes the region where wve triggered, spred, nd ws extinguished t the criticl wve-stopping threshold. The yellow region denotes the set of prmeters for which the wve, t the criticl threshold, trnsformed into stnding wve. To the left of the stnding wve region is the prmeter spce for which the wve did not stop in the finite rnge of the cortex, nd the two brnches of the trveling wve spred until they met nd nnihilted. As mentioned previously, lower vlue of /D v is necessry for stnding wve formtion. In this digrm, the diffusion coefficients were constnt, but ws vried. The stnding wve zone (yellow) seems to thin out s is lowered. However, this occurs s lowering lso cuses wves to spred further (lower threshold), nd owing to finite domin size, the wve ends meet to nnihilte, creting the oscilltory zone, before the stopping threshold is reched. The threshold of wve types ws lso ctegorized bsed on wve rnge (Fig. 5B). These wve types were mpped onto different regions of the phse digrm depending on whether the wves covered 20 to 30% of the cortex (moeboid, if not stnding), 10 to 20% (PTEN-null like, if stnding), or <10% (smller punct-type wves). Inside the stnding wve zone, with lower threshold thn the simulted PTEN-null cells, ws the pncke phenotype where the stnding wve covered lrger portion of the cortex. To the left of the stnding wve region were oscilltor cells (13) tht sustin wves tht do not stop or stnd but repper in periodic cycles. The wve rnges re overlid on the phse digrm using different color shdes (Fig. 5A). Informtion regrding the trnsitions between these phenotypes re lso embedded within this phse digrm. Rising the threshold of moeboid cells ( in Fig. 5A) resulted in smller wves. However, these my be t different plces of the phse digrm depending on which prmeter ws ltered. For exmple, incresing the time scle seprtion moved the cell closer to the unexcitble zone ( b in Fig. 5A). However, if negtive feedbck ws concomitntly decresed, the cell moved to the cusp of the stnding wve region ( c nd d in Fig. 5A). This chnge is consistent with the trnsition between wild-type cells nd PTEN-null cells ( to d ), in terms of the wve phenotype. Similrly, recruitment of PKBA (protein kinse B, Akt homolog) rpidly converted wild-type wve ptterns to punctte pttern ( to c ) tht genertes numerous elongted protrusions (10). 5 of 9

6 Time scle seprtion (ε) A B f d e c Negtive feedbck (c 2) Wve rnge - frction of cortex (threshold level) Unexcitble Trvel, die Trvel, stnd Oscillte b c d e f 1.0 e d Experimentlly, lowering phosphtidylinositol 4,5-bisphosphte [PI(4,5)P 2 ] levels leds to trnsformtion from moeboid to oscilltor cells (13) by incresing positive feedbck. In our digrm, similr trnsformtion tht bypssed the stnding wve region nd enters the oscilltor zone ( f in Fig. 5A) ws obtined by lowering negtive feedbck or by incresing positive feedbck (fig. S2A). Inside the stnding wve region, however, lowering threshold from PTEN-nulls led to lrger, more stble, stnding wves ( e in Fig. 5A), s is seen experimentlly in pncke cells (Fig. 3D) (12). The choice of negtive feedbck strength nd time scle seprtion s prmeters to explore the wve phenotypes ws rbitrry. The sme phenotypes nd trnsitions were lso obtined by vrying positive feedbck strength nd negtive feedbck strength (fig. S2A). It ws only necessry to choose prmeters tht hve direct effect on the threshold of the excitble system. DISCUSSION The existence of trveling nd stnding wves in excitble systems or in systems with limit cycle ttrctors hs been well documented (25, 29). It hs lso been suggested tht both ptterns cn coexist when diffusion coefficients re vried to relize zero-wve speed scenrio (9). However, it is unlikely to expect diffusion prmeters to be ltered in rel time; hence, this trnsformtion mechnism is difficult. Using the concept of wve stopping, we demonstrted how it is possible for trveling wve to convert into stnding wve without ltering the spce-scle seprtion, i.e., the rtio of diffusion coefficients, directly. f Fig. 5. An ll-encompssing protrusion templte. (A) Phse digrm showing different wve phenotypes through colors nd wve rnges through shdes. The letters correspond to the prticulr wve phenotypes. (B) Ctegorizing wve phenotype thresholds bsed on wve rnge, i.e., the frction of simultion domin occupied by the wve., moeboid; b nd c, punct/little wves; d, PTEN-null; e, pncke; f, oscilltor. b b c The grdul conversion of trveling wve to stnding wve without mnully ltering diffusion coefficients suggests possible method of pttern genertion. Most pttern formtion theories suggest tht ptterns rise spontneously becuse of n unstble sptilly homogeneous stte (6, 8), nd tht the resultnt spots my then rerrnge to form finl stble configurtion (30, 31). We hve shown tht it is possible for pttern to begin s continuous trveling wve tht ultimtely slows down, stops, nd trnsforms into discrete stnding wves tht then rerrnge to form the resulting pttern (Fig. 4D). Note tht Turing s instbility conditions (32) re not stisfied by our model (Mterils nd Methods); hence, in our system, pttern formtion occurs owing to combintion of lterl inhibition nd excitbility (25). In the context of cellulr signling dynmics, using this trvelingto-stnding trnsformtion, we were ble to recrete situtions in which ctivity on the cell cortex persisted t point in spce without spreding, in both Dictyostelium nd mmmlin mutnt strins. While we do not clim to reproduce every phenotype completely, this study suggests mechnism for both trnsient (trveling) nd persistent (stnding) ctivity on the cell cortex, phenomenon tht occurs often in cells, using n excitble system. The experiments provided here do not serve to rule out other possible mechnisms nd only motivte the need for model tht cn cpture ll wve phenotypes. An ctivtor-inhibitor system pproximtes the underlying biologicl signling network, nd more detiled model is necessry to completely recrete mutnt phenotypes such s migrtory or growth chrcteristics. The phse digrm of Fig. 5A provides n interesting insight into how cell phenotypes re normlly perceived. We hve previously rgued tht these cellulr protrusions lie on continuum nd re interchngeble by the overll stte of the signling nd cytoskeletl system (10). Here, we hve shown tht this continuum hs multiple dimensions nd tht n moeboid cell my trnsition to different phenotypes depending on which wy you go in trnsition digrm. One prticulr phenotype presents itself t multiple loctions on the phse digrm, nd so, the sme phenotype my suggest trnsitions into different phenotypes bsed on where it strted from. Simply put, one my not be ble to predict trnsition phenotype by merely studying prticulr cell stte. For exmple, cells in b nd c in Fig. 5A hve indistinguishble wve type. However, being t different loctions on the trnsition digrm, incresing the ctivity of such wve will crete different phenotypes. The phse digrm lso provides numerous trnsition predictions. For exmple, it suggests tht one cn move from pncke-type cell (12), which eventully frgments, to n ctive oscilltor cell by lowering negtive feedbck (13). Depending on the strengths of the feedbck loops ltered in the overll excitble network rchitecture, it is theoreticlly possible to trverse through ll these different phenotypes. In this study, we chieved this by mnipulting time scle seprtion (or positive feedbck) nd negtive feedbck. In cells, this would trnslte to ltering the threshold of the system by perturbing different nodes of the signl trnsduction system. For exmple, the moeboid to PTEN-null trnsition ( to d in Fig. 5A) could be chieved by lowering negtive feedbck through one node while simultneously incresing threshold through nother. To know the exct correspondences of the feedbck loops to biochemicl species, more detiled biochemicl excitble model is needed. It is lso worth noting tht these stnding wves only occur t the boundry of the cell nd not in the interior. It is likely tht surfce 6 of 9

7 contct lters cellulr threshold nd tht the edge of the cell hs different stte tht llows the stnding phenomenon to mnifest. Experimentlly, it would be interesting to lter the contct of the cells with the substrte to generte stnding ptterns inside the cell. Mny reserchers hve suggested the concept of bistbility s mens to explin protrusive ctivity tht do not propgte or die (15). Tht, in itself, cnnot explin the wve propgtion observed regulrly on the cortex however. Although, one study illustrtes how different ptterns cn rise s refrctory vrible is introduced to bistble model (33) even tht requires ltertions to the model for different phenotypes. The trveling-to-stnding wve bifurction theory provides semless wy to move within these phenotypes without hving to lter the system drsticlly. A trveling wve my thus nturlly persist for longer durtion t prticulr point, llowing cell to modulte its pseudopods. MATERIALS AND METHODS Simultion methods The excitble system equtions used to model the system were du dt = f(u, v ) = D u 2 3 u 2 u 1 u 2 u(v r ) u dv dt = g(u, v ) = D v 2 v + ϵ( c 1 v + c 2 u) The one-dimensionl simultions of Figs. 1 nd 2 ssumed periodic line of 600 points with dx = 0.05 in MATLAB (Ntick, MA). For Fig. 5, line of 1200 points ws used. Diffusion ws implemented using the centrl difference pproximtion. To dd Gussin white noise to the simultions, the SDE toolbox of MATLAB ws used (34). Stbility of stnding wves ws clculted by dding prticulr noise vrince nd checked fter fixed time intervl if the stnding wve still persisted. Deterministic wves were triggered by incresing the initil ctivtor concentrtion t point in spce. The exct level of noise ws smll enough to ensure tht new wve trigger did not initite. The following excitble system prmeters for the bove equtions were used: 1 = 0.167, 2 = 16.67, 3 = 167, 4 = 1.44, 5 = 1.47, c 1 = 0.1, c 2 = 4.2 (nonstnding), c 2 = 3.9 (stnding), epsilon = 0.52 (for Fig. 5, epsilon = 0.4), D u = 0.1, D v = 1. To determine whether Turing s instbility conditions hold (32), we note tht the three required conditions re () f u + g v < 0, (b) f u g v f v g u > 0, (c) D v f u + D u g v > 0, where the subscripts denote the prtil derivtives. When r = 0 nd using the prmeters listed bove, conditions (= 22.65) nd b (=2.84) re stisfied for stble equilibrium, but condition c (= 22.61) is not. The one-dimensionl simultions of Fig. 4 (A nd B) were done using the pckge URDME (35), which implements the next subvolume method nd llows better pproximtion of system intrinsic noise. For this purpose, the prmeters were scled from concentrtions to number of molecules using multipliction fctor of 18. Simultions were done on 314 points, with dx = 0.1. Nominl prmeters for both simultions were s follows: 1 = 0.167, 2 = 16.67, 4 = 1.44, 5 = 1.47, c 1 = 0.1, epsilon = 0.4, D u = 0.1 nd D v = 1. Prmeters for Fig. 4A were s follows: 3 = 167, c 2 = 2.1. Prmeters for Fig. 4B were s follows: 3 = 300.6, c 2 = 3.0. A smple size of 180 protrusions ws used to conduct the Student s t test, nd the Kolmogorov-Smirnov test for protrusion durtions. The two-dimensionl deterministic simultions of Fig. 4C were done using COMSOL Multiphysics 4.2 (Burlington, MA), using the sme prmeters s the MATLAB one-dimensionl simultions, except tht c 2 = 4 nd epsilon = 0.4 were used. Wves were triggered using step input t the centrl point. The two-dimensionl stochstic simultions of Fig. 4E were done using two-dimensionl version of URDME. The prmeters were the sme s in the one-dimensionl URDME simultion, except tht c 2 = 2.8. A circulr mesh of rdius 8 units ws creted, where the mximum llowed distnce between two nodes ws The cell movement simultions were crried out using viscoelstic cell membrne model (36), using the level-set toolbox of MATLAB (37), where the cell is modeled s circle tht is then subjected to stresses obtined from the ctivity from the wve simultions. This ctivity ws pplied to viscoelstic cell, norml to the cell membrne. The totl stresses included ctive stress from the wves, surfce tension, nd volume conservtion. Detils nd prmeter vlues for the level-set simultions cn be found in (13). Experimentl methods Dictyostelium Cells nd plsmids. The wild-type Dictyostelium discoideum cells of xenic AX2 strin were obtined from R. Ky lbortory (MRC Lbortory of Moleculr Biology, UK). The pten strin ws generted in our lbortory from prent AX2 strin nd ws described previously (27). Both wild-type nd gene knockout cell lines were cultured xeniclly in HL-5 medium t 22 C. Within 2 months of thwing the cells from the frozen stocks, the experiments were done. To visulize PIP3 dynmics, PH crc ws used s the biosensor. To visulize Rs ctivtion, RBD (the Rs binding domin of Rf1) ws used. LimE coil ws used to obtin newly polymerized F-ctin dynmics. For exogenous gene expressions, Dictyostelium cells were trnsformed with PH crc -mcherry, RBD-GFP (Rs-binding domin of mmmlin Rf1, green fluorescent protein), LimE coil - RFP (red fluorescent protein), or GFP-LimE coil plsmids by electroportion nd selected using either hygromycin B (50 g/ml) or G418 (20 g/ml), s per the ntibiotic resistnces of the vectors. Cell preprtion for microscopy. Growth phse cells were trnsferred to n eight-well Nunc Lb-Tek coverslip chmber nd llowed to dhere for 10 min. Then, the HL-5 medium ws replced with 450 l of development buffer (5 mm N 2 HPO 4, 5 mm KH 2 PO 4, supplemented with 2 mm MgSO 4 nd 0.2 mm CCl 2 ). The cells were treted with 4 mm (finl concentrtion) cffeine (Sigm-Aldrich; C0750) for 20 min to visulize more wves, s reported previously (38). To inhibit cytoskeletl input in signling dynmics, the ctin polymeriztion inhibitor ltrunculin A (Enzo Life Sciences; BML- T119) ws dded to cells t finl concentrtion of 5 M nd then cells were incubted for round 25 min. Confocl microscopy nd imge processing. The time-lpse confocl imges were cquired using Zeiss LSM780 single-point lser scnning confocl microscope (Zeiss Axio Observer with 780-Qusr; 34-chnnel spectrl, high-sensitivity gllium rsenide phosphide detectors), illuminted by 488 nm (rgon lser) for GFP or by 561 nm (solid-stte lser) for mcherry nd RFP. All experiments were performed in 40 /1.30 Pln-Neoflur oil objective. The imges were processed using Fiji/ImgeJ [Ntionl Institutes of Helth (NIH)]. Kymogrphs were generted by custom-written MATLAB script. The LimE-mRFP nd PH crc -YFP (yellow fluorescent protein) expressing pten cells in Fig. 3D were imged in every 4-s intervl. 7 of 9

8 The LimE is shown in Grys nd the PH crc is shown in Fire Invert LUT of Fiji/ImgeJ (NIH). The mjority of bckground cytosolic signl ws subtrcted in PH crc chnnel for clrity. MCF-10A Cells. MCF-10A cell (cquired from Iijim lbortory of Johns Hopkins University) nd Krs (G12V) MCF-10A cell (generted by virl trnsfection) were grown t 37 C in 5% CO 2 using Dulbecco s modified Egle s medium/f-12 medium (Gibco, # ) supplemented with 5% horse serum (Gibco, # ), epiderml growth fctor (EGF) (20 ng/ml) (Sigm-Aldrich, #E9644), choler toxin (100 ng/ml) (Sigm-Aldrich, #C-8052), hydrocortisone (0.5 mg/ml) (Sigm-Aldrich, #H-0888), nd insulin (10 g/ml) (Sigm-Aldrich, #I-1882). Stble Krs (G12V) MCF-10A cell line ws selected nd mintined in culture medium contining puromycin (2 g/ml) (Thermo Fisher Scientific, #A ) fter virus trnsfection. LYN-FRB, FKBP- INP54P, PH-AKT, nd LIFEACT stble cell lines were sorted by fluorescence tgs fter virus trnsfection. Cells were trnsferred to 35-mm glss-bottom dishes (MtTek, #P35G C) or chmbered coverglss (Lb-Tek, #155409PK) nd llowed to ttch overnight t 37 C in 5% CO 2 before imging. Cells were kept in phenol red free culture medium t 37 C in 5% CO 2 during microscope imging. Plsmids. Constructs of CFP-Lyn-FRB nd mcherry-fkbp-inp54p were obtined from Inoue lbortory (Johns Hopkins University). GFP/RFP-PH-AKT, RFP-LifeAct, pfuw2, pmdl, prsv, nd pcmv were obtined from Desiderio lbortory (Johns Hopkins University). pbabe-krsg12v (#9052), pumvc (#8449), nd pcmv-vsv-g (#8454) constructs were obtined from Addgene. Lyn-FRB, FKBP- INP54P, PH-AKT, nd LifeAct were subcloned into lentivirl expression plsmid pfuw2. Drugs. The EGF stock solution ws prepred by dissolving EGF (Sigm-Aldrich, #E9644) in 10 mm cetic cid to finl concentrtion of 1 mg/ml. Insulin (Sigm-Aldrich, #I-1882) ws resuspended t 10 mg/ml in sterile ddh 2 O contining 1% glcil cetic cid. Hydrocortisone (Sigm-Aldrich, #H-0888) ws resuspended t 1 mg/ml in 200 proof ethnol. Choler toxin (Sigm-Aldrich, #C-8052) ws resuspended t 1 mg/ml in sterile ddh 2 O nd stored t 4 C. All drug stocks except choler toxin were stored t 20 C. Virus genertion. Twenty-five milliliters of 293T cells ws seeded t /ml to 15-cm cell culture dishes on dy 1. Conventionl clcium phosphte trnsfection ws performed on dy 2 to deliver expressing nd pckging plsmids into 293T cells. pfuw2 (20 g), pmdl (9.375 g), prsv (9.375 g), pcmv plsmids (9.375 g) (or 10 g of pbabe, 9 g of pumvc, 1 g of pcmv-vsv-g), CCl 2 (250 l), nd ddh 2 O in totl volume of 2.5 ml were mixed with 2.5 ml of 2 Hepes (ph 7.05) nd incubted for 5 min. The trnsfection mix ws dded to the plted cells nd shken gently. Medium ws chnged fter 4 to 6 hours. For virus collection, the medium from infected cells ws collected on dy 5 nd spun t 1000 rpm for 3 min to remove the debris nd filtered through m filter followed by ultrcentrifugtion t 25,000 rpm for 90 min t 4 C in Beckmn ultrcentrifuge. The superntnt ws discrded nd the pellet ws dissolved in 70 l of phosphte-buffered sline overnight t 4 C to obtin concentrted virus, which ws stored s 25- l liquots t 80 C. Microscopy. Confocl microscopy ws crried out on Zeiss Axio Observer inverted microscope with either LSM780-Qusr (34-chnnel spectrl, high-sensitivity gllium rsenide phosphide detectors, GAsP) or LSM800 confocl module controlled by the Zen softwre. All live cell imging ws crried out in temperture/humidity/co 2 -regulted chmber. The signling/cytoskeletl wves on the cell ventrl surfce were obtined by cpturing the confocl slice of the very bottom of the cell. SUPPLEMENTARY MATERIALS Supplementry mteril for this rticle is vilble t content/full/6/32/ey7682/dc1 View/request protocol for this pper from Bio-protocol. REFERENCES AND NOTES 1. A. M. Pertsov, J. M. Dvidenko, R. Slomonsz, W. T. Bxter, J. Jlife, Spirl wves of excittion underlie reentrnt ctivity in isolted crdic muscle. Circ. Res. 72, (1993). 2. P. N. Devreotes, S. Bhttchry, M. Edwrds, P. A. Iglesis, T. Lmpert, Y. Mio, Excitble signl trnsduction networks in directed cell migrtion. Annu. Rev. Cell Dev. Biol. 33, (2017). 3. J. J. Tyson, J. P. Keener, Singulr perturbtion theory of trveling wves in excitble medi ( review). Physic D 32, (1988). 4. S. Bhttchry, P. A. Iglesis, Controlling excitble wve behviors through the tuning of three prmeters. Biol. Cybern. 113, (2019). 5. A. M. Zhbotinsky, M. D. Eger, I. R. Epstein, Refrction nd reflection of chemicl wves. Phys. Rev. Lett. 71, (1993). 6. A. Gierer, H. Meinhrdt, A theory of biologicl pttern formtion. Kybernetik 12, (1972). 7. H. Meinhrdt, The Algorithmic Beuty of Se Shells (Springer Science & Business Medi, 2009). 8. A. M. Turing, The chemicl bsis of morphogenesis. Philos. Trns. R. Soc. Lond. B Biol. Sci. 237, (1952). 9. J. Dockery, J. Keener, Diffusive effects on dispersion in excitble medi. SIAM J. Appl. Mth. 49, (1989). 10. Y. Mio, S. Bhttchry, T. Bnerjee, B. Abubker-Shrif, Y. Long, T. Inoue, P. A. Iglesis, P. N. Devreotes, Wve ptterns orgnize cellulr protrusions nd control corticl dynmics. Mol. Syst. Biol. 15, e8585 (2019). 11. Y. Mori, A. Jilkine, L. Edelstein-Keshet, Wve-pinning nd cell polrity from bistble rection-diffusion system. Biophys. J. 94, (2008). 12. M. Edwrds, H. Ci, B. Abubker-Shrif, Y. Long, T. J. Lmpert, P. N. Devreotes, Insight from the mximl ctivtion of the signl trnsduction excitble network in Dictyostelium discoideum. Proc. Ntl. Acd. Sci. U.S.A. 115, E3722 E3730 (2018). 13. Y. Mio, S. Bhttchry, M. Edwrds, H. Ci, T. Inoue, P. A. Iglesis, P. N. Devreotes, Altering the threshold of n excitble signl trnsduction network chnges cell migrtory modes. Nt. Cell Biol. 19, (2017). 14. H. Meinhrdt, Orienttion of chemotctic cells nd growth cones: Models nd mechnisms. J. Cell Sci. 112, (1999). 15. S. Mtsuok, M. Ued, Mutul inhibition between PTEN nd PIP3 genertes bistbility for polrity in motile cells. Nt. Commun. 9, 4481 (2018). 16. S. Bhttchry, D. Bisws, G. A. Enciso, P. A. Iglesis, Control of chemotxis through bsolute concentrtion robustness, in 2018 IEEE Conference on Decision nd Control (CDC) (IEEE, 2018), pp R. FitzHugh, Impulses nd physiologicl sttes in theoreticl models of nerve membrne. Biophys. J. 1, (1961). 18. J. Ngumo, S. Arimoto, S. Yoshizw, An ctive pulse trnsmission line simulting nerve xon. Proc. IRE 50, (1962). 19. S. Bhttchry, P. A. Iglesis, The threshold of n excitble system serves s control mechnism for noise filtering during chemotxis. PLOS ONE 17, e (2018). 20. C. Shi, C.-H. Hung, P. N. Devreotes, P. A. Iglesis, Interction of motility, directionl sensing, nd polrity modules recretes the behviors of chemotxing cells. PLOS Comput. Biol. 9, e (2013). 21. C.-H. Hung, M. Tng, C. Shi, P. A. Iglesis, P. N. Devreotes, An excitble signl integrtor couples to n idling cytoskeletl oscilltor to drive cell migrtion. Nt. Cell Biol. 15, (2013). 22. M. Tng, M. Wng, C. Shi, P. A. Iglesis, P. N. Devreotes, C.-H. Hung, Evolutionrily conserved coupling of dptive nd excitble networks medites eukryotic chemotxis. Nt. Commun. 5, 5175 (2014). 23. X. Li, M. Edwrds, K. F. Swney, N. Singh, S. Bhttchry, J. Borleis, Y. Long, P. A. Iglesis, J. Chen, P. N. Devreotes, Mutully inhibitory Rs-PI(3,4)P 2 feedbck loops medite cell migrtion. Proc. Ntl. Acd. Sci. U.S.A. 115, E9125 E9134 (2018). 24. P. C. Fife, Non-Equilibrium Dynmics in Chemicl Systems (Springer, 1984), pp G. Ermentrout, S. Hstings, W. Troy, Lrge mplitude sttionry wves in n excitble lterl-inhibitory medium. SIAM J. Appl. Mth. 44, (1984). 8 of 9

9 26. P. A. Iglesis, P. N. Devreotes, Nvigting through models of chemotxis. Curr. Opin. Cell Biol. 20, (2008). 27. M. Iijim, P. Devreotes, Tumor suppressor PTEN medites sensing of chemottrctnt grdients. Cell 109, (2002). 28. T. J. Lmpert, N. Kmprd, M. Edwrds, J. Borleis, A. J. Wtson, M. Trntol, P. N. Devreotes, Sher force-bsed genetic screen revels negtive regultors of cell dhesion nd protrusive ctivity. Proc. Ntl. Acd. Sci. U.S.A. 114, E7727 E7736 (2017). 29. T. Kohsokbe, K. Kneko, Boundry-induced pttern formtion from uniform temporl oscilltion. Chos 28, (2018). 30. H. Meinhrdt, Pttern formtion in biology: A comprison of models nd experiments. Rep. Prog. Phys. 55, 797 (1992). 31. S. Kondo, T. Miur, Rection-diffusion model s frmework for understnding biologicl pttern formtion. Science 329, (2010). 32. T. Leppänen, Current Topics in Physics: In Honor of Sir Roger J Elliott (World Scientific, 2005), pp W. R. Holmes, A. E. Crlsson, L. Edelstein-Keshet, Regimes of wve type ptterning driven by refrctory ctin feedbck: Trnsition from sttic polriztion to dynmic wve behviour. Phys. Biol. 9, (2012). 34. U. Picchini, SDE toolbox: Simultion nd estimtion of stochstic differentil equtions with Mtlb (2007); B. Drwert, S. Engblom, A. Hellnder, URDME: A modulr frmework for stochstic simultion of rection-trnsport processes in complex geometries. BMC Syst. Biol. 6, 76 (2012). 36. L. Yng, J. Effler, B. L. Kutscher, S. E. Sullivn, D. N. Robinson, P. A. Iglesis, Modeling cellulr deformtions using the level set formlism. BMC Syst. Biol. 2, 68 (2008). 37. I. M. Mitchell, The flexible, extensible nd efficient toolbox of level set methods. J. Sci. Comput. 35, (2008). 38. Y. Ari, T. Shibt, S. Mtsuok, M. J. Sto, T. Yngid, M. Ued, Self-orgniztion of the phosphtidylinositol lipids signling system for rndom cell migrtion. Proc. Ntl. Acd. Sci. U.S.A. 107, (2010). Acknowledgments Funding: This work ws supported, in prt, by DARPA under contrct number HR C-0139 (P.A.I.), NIH/NIGMS R35 GM (P.N.D.), nd AFOSR MURI FA (P.N.D.). Author contributions: S.B. performed ll theoreticl nlysis nd simultions; T.B., Y.M., nd H.Z. crried out the experimentl imging under the supervision of P.N.D. P.A.I. directed the study. S.B. nd P.A.I. wrote the pper, which ws pproved by ll uthors. Competing interests: The uthors declre tht they hve no competing interests. Dt nd mterils vilbility: All dt needed to evlute the conclusions in the pper re present in the pper nd/or the Supplementry Mterils. Simultions files re vilble by request from P.A.I. Experimentl mterils re vilble by request from P.N.D. Additionl dt relted to this pper my be requested from the uthors. Submitted 16 July 2019 Accepted 26 June 2020 Published 7 August /scidv.y7682 Cittion: S. Bhttchry, T. Bnerjee, Y. Mio, H. Zhn, P. N. Devreotes, P. A. Iglesis, Trveling nd stnding wves medite pttern formtion in cellulr protrusions. Sci. Adv. 6, ey7682 (2020). 9 of 9

10 Trveling nd stnding wves medite pttern formtion in cellulr protrusions Syk Bhttchry, Ttst Bnerjee, Yuchun Mio, Huiwng Zhn, Peter N. Devreotes nd Pblo A. Iglesis Sci Adv 6 (32), ey7682. DOI: /scidv.y7682 ARTICLE TOOLS SUPPLEMENTARY MATERIALS REFERENCES PERMISSIONS This rticle cites 33 rticles, 8 of which you cn ccess for free Use of this rticle is subject to the Terms of Service Science Advnces (ISSN ) is published by the Americn Assocition for the Advncement of Science, 1200 New York Avenue NW, Wshington, DC The title Science Advnces is registered trdemrk of AAAS. Copyright 2020 The Authors, some rights reserved; exclusive licensee Americn Assocition for the Advncement of Science. No clim to originl U.S. Government Works. Distributed under Cretive Commons Attribution NonCommercil License 4.0 (CC BY-NC).

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