Traveling and standing waves mediate pattern formation in cellular protrusions


 Branden Carson
 4 days ago
 Views:
Transcription
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 previouslyunidentified 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 llornone type, ensuring similr wve mplitudes cross the medium. The wve bck is formed by downjump 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 BelousovZhbotinsky 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 zerovelocity 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 BYNC). wvepinning 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 llencompssing 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 FitzHughNgumo 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 phseplne digrms, the ctivtor nullcline displys n inverted Nshpe (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 lrgescle excursion in phse spce, which trnsltes to shrp upjump in ctivity, thus creting the wvefront (fig. S1A). This ctivtorinhibitor 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 ctivtorinhibitor 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 wvepinning 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 lowequilibrium excitble system to spred s trveling wve before switching to higherequilibrium 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 wvestopping threshold, this trnsformtion does not rise, s the diffusion grdient is shortlived inside trveling wve nd its contribution is counterblnced 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 wvespeed 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 trvelingtostnding 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 wvepinning 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 lowthreshold 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 newlyformed 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 (PTENnull cells), re known to crete elongted fingerlike 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 trvelingtostnding 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) Zoomedin 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 Actininhibited 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 PTENnull 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 fingerlike 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 fingerlike 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 PTENnull cells. This ws confirmed by incresing the ctivity of PTENnull 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 trvelingtostnding wve trnsformtion. For prticulr prmeter regime, wves expnded nd were extinguished to crete typicl protrusions (Fig. 4A nd movie S4), s seen in wildtype 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 fingerlike extensions (Fig. 4B nd movie S5). The cellulr protrusions were modeled using viscoelstic cell model in the levelset 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 twosmple KolmogorovSmirnov test reveled tht these two protrusive phenotypes belonged to different distributions (Mterils nd Methods). In onedimensionl 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 twodimensionl 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 MCF10A 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 ltrunculintreted wildtype (A) nd PTENnull (B) Dictyostelium cells. Imges of the cells re shown on the right, with the white circle mrked to follow ctivity t smll region. (C) Wildtype (wt) nd PTENnull cell morphology, with LimERFP. Scle br, 5 m. (D) PTENnull exmple showing ctin (left) nd signling (right) mrkers. Scle br, 25 m. (E) Actin dynmics in PTENnull cells. Arrows indicte smll ctin rings. This pnel is tken from (28) with permission. (F) Fctin wve pttern (GFPLimE) phenotype induced by RsCQ62L expression in PTENnull cells (scle br, 5 m) forming pncketype 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 twodimensionl 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) Levelset simultions from the ctivity in (A). (B) (Left) Kymogrphs of PTENnull 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) Levelset 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) Twodimensionl 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 MCF10A 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 MCF10A 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 PHAKT ctivity in trnsformed MCF10A cell. Activity persists t loction (dshed circle) without spreding for over 100 min. Scle br, 21 m. (H) Similr stnding ctivity from stochstic twodimensionl simultions. F A phse digrm of cellulr phenotypes revels n llencompssing 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 wvestopping 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% (PTENnull like, if stnding), or <10% (smller puncttype wves). Inside the stnding wve zone, with lower threshold thn the simulted PTENnull 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 wildtype cells nd PTENnull cells ( to d ), in terms of the wve phenotype. Similrly, recruitment of PKBA (protein kinse B, Akt homolog) rpidly converted wildtype 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 4,5bisphosphte [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 PTENnulls 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 zerowve 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 spcescle seprtion, i.e., the rtio of diffusion coefficients, directly. f Fig. 5. An llencompssing 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, PTENnull; 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 trvelingtostnding 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 ctivtorinhibitor 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 pncketype 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 PTENnull 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 trvelingtostnding 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 onedimensionl 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 onedimensionl 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 KolmogorovSmirnov test for protrusion durtions. The twodimensionl deterministic simultions of Fig. 4C were done using COMSOL Multiphysics 4.2 (Burlington, MA), using the sme prmeters s the MATLAB onedimensionl simultions, except tht c 2 = 4 nd epsilon = 0.4 were used. Wves were triggered using step input t the centrl point. The twodimensionl stochstic simultions of Fig. 4E were done using twodimensionl version of URDME. The prmeters were the sme s in the onedimensionl 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 levelset 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 levelset simultions cn be found in (13). Experimentl methods Dictyostelium Cells nd plsmids. The wildtype 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 wildtype nd gene knockout cell lines were cultured xeniclly in HL5 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 Fctin dynmics. For exogenous gene expressions, Dictyostelium cells were trnsformed with PH crc mcherry, RBDGFP (Rsbinding domin of mmmlin Rf1, green fluorescent protein), LimE coil  RFP (red fluorescent protein), or GFPLimE 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 eightwell Nunc LbTek coverslip chmber nd llowed to dhere for 10 min. Then, the HL5 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 (SigmAldrich; 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 timelpse confocl imges were cquired using Zeiss LSM780 singlepoint lser scnning confocl microscope (Zeiss Axio Observer with 780Qusr; 34chnnel spectrl, highsensitivity gllium rsenide phosphide detectors), illuminted by 488 nm (rgon lser) for GFP or by 561 nm (solidstte lser) for mcherry nd RFP. All experiments were performed in 40 /1.30 PlnNeoflur oil objective. The imges were processed using Fiji/ImgeJ [Ntionl Institutes of Helth (NIH)]. Kymogrphs were generted by customwritten MATLAB script. The LimEmRFP nd PH crc YFP (yellow fluorescent protein) expressing pten cells in Fig. 3D were imged in every 4s 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. MCF10A Cells. MCF10A cell (cquired from Iijim lbortory of Johns Hopkins University) nd Krs (G12V) MCF10A cell (generted by virl trnsfection) were grown t 37 C in 5% CO 2 using Dulbecco s modified Egle s medium/f12 medium (Gibco, # ) supplemented with 5% horse serum (Gibco, # ), epiderml growth fctor (EGF) (20 ng/ml) (SigmAldrich, #E9644), choler toxin (100 ng/ml) (SigmAldrich, #C8052), hydrocortisone (0.5 mg/ml) (SigmAldrich, #H0888), nd insulin (10 g/ml) (SigmAldrich, #I1882). Stble Krs (G12V) MCF10A cell line ws selected nd mintined in culture medium contining puromycin (2 g/ml) (Thermo Fisher Scientific, #A ) fter virus trnsfection. LYNFRB, FKBP INP54P, PHAKT, nd LIFEACT stble cell lines were sorted by fluorescence tgs fter virus trnsfection. Cells were trnsferred to 35mm glssbottom dishes (MtTek, #P35G C) or chmbered coverglss (LbTek, #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 CFPLynFRB nd mcherryfkbpinp54p were obtined from Inoue lbortory (Johns Hopkins University). GFP/RFPPHAKT, RFPLifeAct, pfuw2, pmdl, prsv, nd pcmv were obtined from Desiderio lbortory (Johns Hopkins University). pbabekrsg12v (#9052), pumvc (#8449), nd pcmvvsvg (#8454) constructs were obtined from Addgene. LynFRB, FKBP INP54P, PHAKT, nd LifeAct were subcloned into lentivirl expression plsmid pfuw2. Drugs. The EGF stock solution ws prepred by dissolving EGF (SigmAldrich, #E9644) in 10 mm cetic cid to finl concentrtion of 1 mg/ml. Insulin (SigmAldrich, #I1882) ws resuspended t 10 mg/ml in sterile ddh 2 O contining 1% glcil cetic cid. Hydrocortisone (SigmAldrich, #H0888) ws resuspended t 1 mg/ml in 200 proof ethnol. Choler toxin (SigmAldrich, #C8052) 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. Twentyfive milliliters of 293T cells ws seeded t /ml to 15cm 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 pcmvvsvg), 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 phosphtebuffered 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 LSM780Qusr (34chnnel spectrl, highsensitivity 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 Bioprotocol. 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. AbubkerShrif, 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. EdelsteinKeshet, Wvepinning nd cell polrity from bistble rectiondiffusion system. Biophys. J. 94, (2008). 12. M. Edwrds, H. Ci, B. AbubkerShrif, 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 RsPI(3,4)P 2 feedbck loops medite cell migrtion. Proc. Ntl. Acd. Sci. U.S.A. 115, E9125 E9134 (2018). 24. P. C. Fife, NonEquilibrium Dynmics in Chemicl Systems (Springer, 1984), pp G. Ermentrout, S. Hstings, W. Troy, Lrge mplitude sttionry wves in n excitble lterlinhibitory 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 forcebsed 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, Boundryinduced 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, Rectiondiffusion 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. EdelsteinKeshet, 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 rectiontrnsport 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, Selforgniztion 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 C0139 (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 BYNC).
Experiment 6: Friction
Experiment 6: Friction In previous lbs we studied Newton s lws in n idel setting, tht is, one where friction nd ir resistnce were ignored. However, from our everydy experience with motion, we know tht
More informationGraphs on Logarithmic and Semilogarithmic Paper
0CH_PHClter_TMSETE_ 3//00 :3 PM Pge Grphs on Logrithmic nd Semilogrithmic Pper OBJECTIVES When ou hve completed this chpter, ou should be ble to: Mke grphs on logrithmic nd semilogrithmic pper. Grph empiricl
More informationModule 2. Analysis of Statically Indeterminate Structures by the Matrix Force Method. Version 2 CE IIT, Kharagpur
Module Anlysis of Stticlly Indeterminte Structures by the Mtrix Force Method Version CE IIT, Khrgpur esson 9 The Force Method of Anlysis: Bems (Continued) Version CE IIT, Khrgpur Instructionl Objectives
More informationLectures 8 and 9 1 Rectangular waveguides
1 Lectures 8 nd 9 1 Rectngulr wveguides y b x z Consider rectngulr wveguide with 0 < x b. There re two types of wves in hollow wveguide with only one conductor; Trnsverse electric wves
More informationPolynomial Functions. Polynomial functions in one variable can be written in expanded form as ( )
Polynomil Functions Polynomil functions in one vrible cn be written in expnded form s n n 1 n 2 2 f x = x + x + x + + x + x+ n n 1 n 2 2 1 0 Exmples of polynomils in expnded form re nd 3 8 7 4 = 5 4 +
More informationPhysics 43 Homework Set 9 Chapter 40 Key
Physics 43 Homework Set 9 Chpter 4 Key. The wve function for n electron tht is confined to x nm is. Find the normliztion constnt. b. Wht is the probbility of finding the electron in. nmwide region t x
More informationBasic Analysis of Autarky and Free Trade Models
Bsic Anlysis of Autrky nd Free Trde Models AUTARKY Autrky condition in prticulr commodity mrket refers to sitution in which country does not engge in ny trde in tht commodity with other countries. Consequently
More information2 DIODE CLIPPING and CLAMPING CIRCUITS
2 DIODE CLIPPING nd CLAMPING CIRCUITS 2.1 Ojectives Understnding the operting principle of diode clipping circuit Understnding the operting principle of clmping circuit Understnding the wveform chnge of
More informationOperations with Polynomials
38 Chpter P Prerequisites P.4 Opertions with Polynomils Wht you should lern: Write polynomils in stndrd form nd identify the leding coefficients nd degrees of polynomils Add nd subtrct polynomils Multiply
More informationMathematics. Vectors. hsn.uk.net. Higher. Contents. Vectors 128 HSN23100
hsn.uk.net Higher Mthemtics UNIT 3 OUTCOME 1 Vectors Contents Vectors 18 1 Vectors nd Sclrs 18 Components 18 3 Mgnitude 130 4 Equl Vectors 131 5 Addition nd Subtrction of Vectors 13 6 Multipliction by
More informationMath 135 Circles and Completing the Square Examples
Mth 135 Circles nd Completing the Squre Exmples A perfect squre is number such tht = b 2 for some rel number b. Some exmples of perfect squres re 4 = 2 2, 16 = 4 2, 169 = 13 2. We wish to hve method for
More informationUse Geometry Expressions to create a more complex locus of points. Find evidence for equivalence using Geometry Expressions.
Lerning Objectives Loci nd Conics Lesson 3: The Ellipse Level: Preclculus Time required: 120 minutes In this lesson, students will generlize their knowledge of the circle to the ellipse. The prmetric nd
More informationProject 6 Aircraft static stability and control
Project 6 Aircrft sttic stbility nd control The min objective of the project No. 6 is to compute the chrcteristics of the ircrft sttic stbility nd control chrcteristics in the pitch nd roll chnnel. The
More informationRotating DC Motors Part II
Rotting Motors rt II II.1 Motor Equivlent Circuit The next step in our consiertion of motors is to evelop n equivlent circuit which cn be use to better unerstn motor opertion. The rmtures in rel motors
More informationHelicopter Theme and Variations
Helicopter Theme nd Vritions Or, Some Experimentl Designs Employing Pper Helicopters Some possible explntory vribles re: Who drops the helicopter The length of the rotor bldes The height from which the
More informationCOMPONENTS: COMBINED LOADING
LECTURE COMPONENTS: COMBINED LOADING Third Edition A. J. Clrk School of Engineering Deprtment of Civil nd Environmentl Engineering 24 Chpter 8.4 by Dr. Ibrhim A. Asskkf SPRING 2003 ENES 220 Mechnics of
More informationPROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY
MAT 0630 INTERNET RESOURCES, REVIEW OF CONCEPTS AND COMMON MISTAKES PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY Contents 1. ACT Compss Prctice Tests 1 2. Common Mistkes 2 3. Distributive
More informationWeek 11  Inductance
Week  Inductnce November 6, 202 Exercise.: Discussion Questions ) A trnsformer consists bsiclly of two coils in close proximity but not in electricl contct. A current in one coil mgneticlly induces n
More informationTreatment Spring Late Summer Fall 0.10 5.56 3.85 0.61 6.97 3.01 1.91 3.01 2.13 2.99 5.33 2.50 1.06 3.53 6.10 Mean = 1.33 Mean = 4.88 Mean = 3.
The nlysis of vrince (ANOVA) Although the ttest is one of the most commonly used sttisticl hypothesis tests, it hs limittions. The mjor limittion is tht the ttest cn be used to compre the mens of only
More informationVectors 2. 1. Recap of vectors
Vectors 2. Recp of vectors Vectors re directed line segments  they cn be represented in component form or by direction nd mgnitude. We cn use trigonometry nd Pythgors theorem to switch between the forms
More informationAll pay auctions with certain and uncertain prizes a comment
CENTER FOR RESEARC IN ECONOMICS AND MANAGEMENT CREAM Publiction No. 12015 All py uctions with certin nd uncertin prizes comment Christin Riis All py uctions with certin nd uncertin prizes comment Christin
More informationLecture 3 Gaussian Probability Distribution
Lecture 3 Gussin Probbility Distribution Introduction l Gussin probbility distribution is perhps the most used distribution in ll of science. u lso clled bell shped curve or norml distribution l Unlike
More information6.2 Volumes of Revolution: The Disk Method
mth ppliction: volumes of revolution, prt ii Volumes of Revolution: The Disk Method One of the simplest pplictions of integrtion (Theorem ) nd the ccumultion process is to determine soclled volumes of
More informationThe Velocity Factor of an Insulated TwoWire Transmission Line
The Velocity Fctor of n Insulted TwoWire Trnsmission Line Problem Kirk T. McDonld Joseph Henry Lbortories, Princeton University, Princeton, NJ 08544 Mrch 7, 008 Estimte the velocity fctor F = v/c nd the
More informationEcon 4721 Money and Banking Problem Set 2 Answer Key
Econ 472 Money nd Bnking Problem Set 2 Answer Key Problem (35 points) Consider n overlpping genertions model in which consumers live for two periods. The number of people born in ech genertion grows in
More informationTITLE THE PRINCIPLES OF COINTAP METHOD OF NONDESTRUCTIVE TESTING
TITLE THE PRINCIPLES OF COINTAP METHOD OF NONDESTRUCTIVE TESTING Sung Joon Kim*, DongChul Che Kore Aerospce Reserch Institute, 45 EoeunDong, YouseongGu, Dejeon, 35333, Kore Phone : 824286231 FAX
More informationSmall Business Networking
Why network is n essentil productivity tool for ny smll business Effective technology is essentil for smll businesses looking to increse the productivity of their people nd business. Introducing technology
More informationSmall Business Networking
Why network is n essentil productivity tool for ny smll business Effective technology is essentil for smll businesses looking to increse the productivity of their people nd business. Introducing technology
More informationg(y(a), y(b)) = o, B a y(a)+b b y(b)=c, Boundary Value Problems Lecture Notes to Accompany
Lecture Notes to Accompny Scientific Computing An Introductory Survey Second Edition by Michel T Heth Boundry Vlue Problems Side conditions prescribing solution or derivtive vlues t specified points required
More informationSmall Business Networking
Why network is n essentil productivity tool for ny smll business Effective technology is essentil for smll businesses looking to increse the productivity of their people nd processes. Introducing technology
More informationSpace Vector Pulse Width Modulation Based Induction Motor with V/F Control
Interntionl Journl of Science nd Reserch (IJSR) Spce Vector Pulse Width Modultion Bsed Induction Motor with V/F Control Vikrmrjn Jmbulingm Electricl nd Electronics Engineering, VIT University, Indi Abstrct:
More informationEE247 Lecture 4. For simplicity, will start with all pole ladder type filters. Convert to integrator based form example shown
EE247 Lecture 4 Ldder type filters For simplicity, will strt with ll pole ldder type filters Convert to integrtor bsed form exmple shown Then will ttend to high order ldder type filters incorporting zeros
More informationLINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES
LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES DAVID WEBB CONTENTS Liner trnsformtions 2 The representing mtrix of liner trnsformtion 3 3 An ppliction: reflections in the plne 6 4 The lgebr of
More informationSmall Business Networking
Why network is n essentil productivity tool for ny smll business Effective technology is essentil for smll businesses looking to increse the productivity of their people nd processes. Introducing technology
More informationDlNBVRGH + Sickness Absence Monitoring Report. Executive of the Council. Purpose of report
DlNBVRGH + + THE CITY OF EDINBURGH COUNCIL Sickness Absence Monitoring Report Executive of the Council 8fh My 4 I.I...3 Purpose of report This report quntifies the mount of working time lost s result of
More information** Dpt. Chemical Engineering, Kasetsart University, Bangkok 10900, Thailand
Modelling nd Simultion of hemicl Processes in Multi Pulse TP Experiment P. Phnwdee* S.O. Shekhtmn +. Jrungmnorom** J.T. Gleves ++ * Dpt. hemicl Engineering, Ksetsrt University, Bngkok 10900, Thilnd + Dpt.hemicl
More informationReview guide for the final exam in Math 233
Review guide for the finl exm in Mth 33 1 Bsic mteril. This review includes the reminder of the mteril for mth 33. The finl exm will be cumultive exm with mny of the problems coming from the mteril covered
More informationEQUATIONS OF LINES AND PLANES
EQUATIONS OF LINES AND PLANES MATH 195, SECTION 59 (VIPUL NAIK) Corresponding mteril in the ook: Section 12.5. Wht students should definitely get: Prmetric eqution of line given in pointdirection nd twopoint
More informationaddition, there are double entries for the symbols used to signify different parameters. These parameters are explained in this appendix.
APPENDIX A: The ellipse August 15, 1997 Becuse of its importnce in both pproximting the erth s shpe nd describing stellite orbits, n informl discussion of the ellipse is presented in this ppendix. The
More information9 CONTINUOUS DISTRIBUTIONS
9 CONTINUOUS DISTIBUTIONS A rndom vrible whose vlue my fll nywhere in rnge of vlues is continuous rndom vrible nd will be ssocited with some continuous distribution. Continuous distributions re to discrete
More informationUplift Capacity of KSeries Open Web Steel Joist Seats. Florida, Gainesville, FL 32611; email: psgreen@ce.ufl.edu
Uplift Cpcity of KSeries Open Web Steel Joist Sets Perry S. Green, Ph.D, M.ASCE 1 nd Thoms Sputo, Ph.D., P.E., M.ASCE 2 1 Assistnt Professor, Deprtment of Civil nd Costl Engineering, University of Florid,
More informationHow To Network A Smll Business
Why network is n essentil productivity tool for ny smll business Effective technology is essentil for smll businesses looking to increse the productivity of their people nd processes. Introducing technology
More informationwww.mathsbox.org.uk e.g. f(x) = x domain x 0 (cannot find the square root of negative values)
www.mthsbo.org.uk CORE SUMMARY NOTES Functions A function is rule which genertes ectl ONE OUTPUT for EVERY INPUT. To be defined full the function hs RULE tells ou how to clculte the output from the input
More informationEconomics Letters 65 (1999) 9 15. macroeconomists. a b, Ruth A. Judson, Ann L. Owen. Received 11 December 1998; accepted 12 May 1999
Economics Letters 65 (1999) 9 15 Estimting dynmic pnel dt models: guide for q mcroeconomists b, * Ruth A. Judson, Ann L. Owen Federl Reserve Bord of Governors, 0th & C Sts., N.W. Wshington, D.C. 0551,
More informationIntegration by Substitution
Integrtion by Substitution Dr. Philippe B. Lvl Kennesw Stte University August, 8 Abstrct This hndout contins mteril on very importnt integrtion method clled integrtion by substitution. Substitution is
More informationPROBLEMS 13  APPLICATIONS OF DERIVATIVES Page 1
PROBLEMS  APPLICATIONS OF DERIVATIVES Pge ( ) Wter seeps out of conicl filter t the constnt rte of 5 cc / sec. When the height of wter level in the cone is 5 cm, find the rte t which the height decreses.
More informationPHY 140A: Solid State Physics. Solution to Homework #2
PHY 140A: Solid Stte Physics Solution to Homework # TA: Xun Ji 1 October 14, 006 1 Emil: jixun@physics.ucl.edu Problem #1 Prove tht the reciprocl lttice for the reciprocl lttice is the originl lttice.
More informationReasoning to Solve Equations and Inequalities
Lesson4 Resoning to Solve Equtions nd Inequlities In erlier work in this unit, you modeled situtions with severl vriles nd equtions. For exmple, suppose you were given usiness plns for concert showing
More informationAn Integrateandfire Model of Prefrontal Cortex Neuronal Activity during Performance of Goaldirected Decision Making
Cerebrl Cortex doi:10.1093/cercor/bhi072 An Integrtendfire Model of Prefrontl Cortex Neuronl Activity during Performnce of Goldirected Decision Mking NOT FOR PUBLIC RELEASE Rndl A. Koene nd Michel E.
More informationExample A rectangular box without lid is to be made from a square cardboard of sides 18 cm by cutting equal squares from each corner and then folding
1 Exmple A rectngulr box without lid is to be mde from squre crdbord of sides 18 cm by cutting equl squres from ech corner nd then folding up the sides. 1 Exmple A rectngulr box without lid is to be mde
More informationWeek 7  Perfect Competition and Monopoly
Week 7  Perfect Competition nd Monopoly Our im here is to compre the industrywide response to chnges in demnd nd costs by monopolized industry nd by perfectly competitive one. We distinguish between
More informationVersion 001 Summer Review #03 tubman (IBII20142015) 1
Version 001 Summer Reiew #03 tubmn (IBII20142015) 1 This printout should he 35 questions. Multiplechoice questions my continue on the next column or pge find ll choices before nswering. Concept 20 P03
More informationFactoring Polynomials
Fctoring Polynomils Some definitions (not necessrily ll for secondry school mthemtics): A polynomil is the sum of one or more terms, in which ech term consists of product of constnt nd one or more vribles
More informationAREA OF A SURFACE OF REVOLUTION
AREA OF A SURFACE OF REVOLUTION h cut r πr h A surfce of revolution is formed when curve is rotted bout line. Such surfce is the lterl boundr of solid of revolution of the tpe discussed in Sections 7.
More informationWhy is the NSW prison population falling?
NSW Bureu of Crime Sttistics nd Reserch Bureu Brief Issue pper no. 80 September 2012 Why is the NSW prison popultion flling? Jcqueline Fitzgerld & Simon Corben 1 Aim: After stedily incresing for more thn
More informationWarmup for Differential Calculus
Summer Assignment Wrmup for Differentil Clculus Who should complete this pcket? Students who hve completed Functions or Honors Functions nd will be tking Differentil Clculus in the fll of 015. Due Dte:
More information5.2. LINE INTEGRALS 265. Let us quickly review the kind of integrals we have studied so far before we introduce a new one.
5.2. LINE INTEGRALS 265 5.2 Line Integrls 5.2.1 Introduction Let us quickly review the kind of integrls we hve studied so fr before we introduce new one. 1. Definite integrl. Given continuous relvlued
More informationUNIVERSITY OF NOTTINGHAM. Discussion Papers in Economics STRATEGIC SECOND SOURCING IN A VERTICAL STRUCTURE
UNVERSTY OF NOTTNGHAM Discussion Ppers in Economics Discussion Pper No. 04/15 STRATEGC SECOND SOURCNG N A VERTCAL STRUCTURE By Arijit Mukherjee September 004 DP 04/15 SSN 10438 UNVERSTY OF NOTTNGHAM Discussion
More informationHow To Set Up A Network For Your Business
Why Network is n Essentil Productivity Tool for Any Smll Business TechAdvisory.org SME Reports sponsored by Effective technology is essentil for smll businesses looking to increse their productivity. Computer
More informationDistributions. (corresponding to the cumulative distribution function for the discrete case).
Distributions Recll tht n integrble function f : R [,] such tht R f()d = is clled probbility density function (pdf). The distribution function for the pdf is given by F() = (corresponding to the cumultive
More informationThe International Association for the Properties of Water and Steam. Release on the Ionization Constant of H 2 O
The Interntionl Assocition for the Properties of Wter nd Stem Lucerne, Sitzerlnd August 7 Relese on the Ioniztion Constnt of H O 7 The Interntionl Assocition for the Properties of Wter nd Stem Publiction
More informationANALYSIS OF THERMAL STRATIFICATION IN THE PRIMARY CIRCUIT WITH THE CFX CODE
ANALYSIS OF THERMAL STRATIFICATION IN THE PRIMARY CIRCUIT WITH THE CFX CODE Ildikó Boros, Dr. Attil Aszódi Budpest University of Technology nd Economics, Institute of Nucler Techniques Abstrct The therml
More informationOr more simply put, when adding or subtracting quantities, their uncertainties add.
Propgtion of Uncertint through Mthemticl Opertions Since the untit of interest in n eperiment is rrel otined mesuring tht untit directl, we must understnd how error propgtes when mthemticl opertions re
More informationIntroducing Kashef for Application Monitoring
WextWise 2010 Introducing Kshef for Appliction The Cse for Reltime monitoring of dtcenter helth is criticl IT process serving vriety of needs. Avilbility requirements of 6 nd 7 nines of tody SOA oriented
More informationReview Problems for the Final of Math 121, Fall 2014
Review Problems for the Finl of Mth, Fll The following is collection of vrious types of smple problems covering sections.,.5, nd.7 6.6 of the text which constitute only prt of the common Mth Finl. Since
More information1. Find the zeros Find roots. Set function = 0, factor or use quadratic equation if quadratic, graph to find zeros on calculator
AP Clculus Finl Review Sheet When you see the words. This is wht you think of doing. Find the zeros Find roots. Set function =, fctor or use qudrtic eqution if qudrtic, grph to find zeros on clcultor.
More informationRate and Activation Energy of the Iodination of Acetone
nd Activtion Energ of the Iodintion of Acetone rl N. eer Dte of Eperiment: //00 Florence F. Ls (prtner) Abstrct: The rte, rte lw nd ctivtion energ of the iodintion of cetone re detered b observing the
More informationPHY 222 Lab 8 MOTION OF ELECTRONS IN ELECTRIC AND MAGNETIC FIELDS
PHY 222 Lb 8 MOTION OF ELECTRONS IN ELECTRIC AND MAGNETIC FIELDS Nme: Prtners: INTRODUCTION Before coming to lb, plese red this pcket nd do the prelb on pge 13 of this hndout. From previous experiments,
More informationtrademark and symbol guidelines FOR CORPORATE STATIONARY APPLICATIONS reviewed 01.02.2007
trdemrk nd symbol guidelines trdemrk guidelines The trdemrk Cn be plced in either of the two usul configurtions but horizontl usge is preferble. Wherever possible the trdemrk should be plced on blck bckground.
More informationt 3 t 4 Part A: Multiple Choice Canadian Association of Physicists 1999 Prize Exam
Cndin Assocition of Physicists 1999 Prize Exm This is three hour exm. Ntionl rnking nd prizes will be bsed on student s performnce on both sections A nd B of the exm. However, performnce on the multiple
More informationIntegration. 148 Chapter 7 Integration
48 Chpter 7 Integrtion 7 Integrtion t ech, by supposing tht during ech tenth of second the object is going t constnt speed Since the object initilly hs speed, we gin suppose it mintins this speed, but
More informationSection 74 Translation of Axes
62 7 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY Section 74 Trnsltion of Aes Trnsltion of Aes Stndrd Equtions of Trnslted Conics Grphing Equtions of the Form A 2 C 2 D E F 0 Finding Equtions of Conics In the
More informationA.7.1 Trigonometric interpretation of dot product... 324. A.7.2 Geometric interpretation of dot product... 324
A P P E N D I X A Vectors CONTENTS A.1 Scling vector................................................ 321 A.2 Unit or Direction vectors...................................... 321 A.3 Vector ddition.................................................
More informationHealth insurance exchanges What to expect in 2014
Helth insurnce exchnges Wht to expect in 2014 33096CAEENABC 02/13 The bsics of exchnges As prt of the Affordble Cre Act (ACA or helth cre reform lw), strting in 2014 ALL Americns must hve minimum mount
More informationEngineertoEngineer Note
EngineertoEngineer Note EE280 Technicl notes on using Anlog Devices DSPs, processors nd development tools Visit our Web resources http://www.nlog.com/eenotes nd http://www.nlog.com/processors or emil
More information1. In the Bohr model, compare the magnitudes of the electron s kinetic and potential energies in orbit. What does this imply?
Assignment 3: Bohr s model nd lser fundmentls 1. In the Bohr model, compre the mgnitudes of the electron s kinetic nd potentil energies in orit. Wht does this imply? When n electron moves in n orit, the
More informationUnderstanding Basic Analog Ideal Op Amps
Appliction Report SLAA068A  April 2000 Understnding Bsic Anlog Idel Op Amps Ron Mncini Mixed Signl Products ABSTRACT This ppliction report develops the equtions for the idel opertionl mplifier (op mp).
More informationEuler Euler Everywhere Using the EulerLagrange Equation to Solve Calculus of Variation Problems
Euler Euler Everywhere Using the EulerLgrnge Eqution to Solve Clculus of Vrition Problems Jenine Smllwood Principles of Anlysis Professor Flschk My 12, 1998 1 1. Introduction Clculus of vritions is brnch
More informationORBITAL MANEUVERS USING LOWTHRUST
Proceedings of the 8th WSEAS Interntionl Conference on SIGNAL PROCESSING, ROBOICS nd AUOMAION ORBIAL MANEUVERS USING LOWHRUS VIVIAN MARINS GOMES, ANONIO F. B. A. PRADO, HÉLIO KOII KUGA Ntionl Institute
More informationReversing Medications That Cause Bleeding
Reversing Medictions Tht Cuse Bleeding Dine M. Birnbumer, M.D., FACEP Professor of Medicine University of Cliforni, Los Angeles Senior Fculty Deprtment of Emergency Medicine HrborUCLA Medicl Center The
More information2. Transaction Cost Economics
3 2. Trnsction Cost Economics Trnsctions Trnsctions Cn Cn Be Be Internl Internl or or Externl Externl n n Orgniztion Orgniztion Trnsctions Trnsctions occur occur whenever whenever good good or or service
More informationEngineertoEngineer Note
EngineertoEngineer Note EE265 Technicl notes on using Anlog Devices DSPs, processors nd development tools Contct our technicl support t dsp.support@nlog.com nd t dsptools.support@nlog.com Or visit our
More informationNovel Methods of Generating SelfInvertible Matrix for Hill Cipher Algorithm
Bibhudendr chry, Girij Snkr Rth, Srt Kumr Ptr, nd Sroj Kumr Pnigrhy Novel Methods of Generting SelfInvertible Mtrix for Hill Cipher lgorithm Bibhudendr chry Deprtment of Electronics & Communiction Engineering
More informationINVESTIGATION OF THE EXTINGUISHING FEATURES FOR LIQUID FUELS AND ORGANIC FLAMMABLE LIQUIDS ATOMIZED BY A WATER FLOW
EPJ Web of Conferences 110, 01083 (2016) DOI: 10.1051/ epjconf/ 201611001083 C Owned by the uthors, published by EDP Sciences, 2016 INVESTIGATION OF THE EXTINGUISHING FEATURES FOR LIQUID FUELS AND ORGANIC
More informationAAPT UNITED STATES PHYSICS TEAM AIP 2010
2010 F = m Exm 1 AAPT UNITED STATES PHYSICS TEAM AIP 2010 Enti non multiplicnd sunt preter necessittem 2010 F = m Contest 25 QUESTIONS  75 MINUTES INSTRUCTIONS DO NOT OPEN THIS TEST UNTIL YOU ARE TOLD
More informationCypress Creek High School IB Physics SL/AP Physics B 2012 2013 MP2 Test 1 Newton s Laws. Name: SOLUTIONS Date: Period:
Nme: SOLUTIONS Dte: Period: Directions: Solve ny 5 problems. You my ttempt dditionl problems for extr credit. 1. Two blocks re sliding to the right cross horizontl surfce, s the drwing shows. In Cse A
More informationClearPeaks Customer Care Guide. Business as Usual (BaU) Services Peace of mind for your BI Investment
ClerPeks Customer Cre Guide Business s Usul (BU) Services Pece of mind for your BI Investment ClerPeks Customer Cre Business s Usul Services Tble of Contents 1. Overview...3 Benefits of Choosing ClerPeks
More informationTechniques for Requirements Gathering and Definition. Kristian Persson Principal Product Specialist
Techniques for Requirements Gthering nd Definition Kristin Persson Principl Product Specilist Requirements Lifecycle Mngement Elicit nd define business/user requirements Vlidte requirements Anlyze requirements
More informationHillsborough Township Public Schools Mathematics Department Computer Programming 1
Essentil Unit 1 Introduction to Progrmming Pcing: 15 dys Common Unit Test Wht re the ethicl implictions for ming in tody s world? There re ethicl responsibilities to consider when writing computer s. Citizenship,
More informationResearch Article Competition with Online and Offline Demands considering Logistics Costs Based on the Hotelling Model
Mthemticl Problems in Engineering Volume 4, Article ID 678, pges http://dx.doi.org/.55/4/678 Reserch Article Competition with Online nd Offline Demnds considering Logistics Costs Bsed on the Hotelling
More information1.00/1.001 Introduction to Computers and Engineering Problem Solving Fall 2011  Final Exam
1./1.1 Introduction to Computers nd Engineering Problem Solving Fll 211  Finl Exm Nme: MIT Emil: TA: Section: You hve 3 hours to complete this exm. In ll questions, you should ssume tht ll necessry pckges
More informationCHAPTER 11 Numerical Differentiation and Integration
CHAPTER 11 Numericl Differentition nd Integrtion Differentition nd integrtion re bsic mthemticl opertions with wide rnge of pplictions in mny res of science. It is therefore importnt to hve good methods
More informationHealth insurance marketplace What to expect in 2014
Helth insurnce mrketplce Wht to expect in 2014 33096VAEENBVA 06/13 The bsics of the mrketplce As prt of the Affordble Cre Act (ACA or helth cre reform lw), strting in 2014 ALL Americns must hve minimum
More informationSection 54 Trigonometric Functions
5 Trigonometric Functions Section 5 Trigonometric Functions Definition of the Trigonometric Functions Clcultor Evlution of Trigonometric Functions Definition of the Trigonometric Functions Alternte Form
More informationEasyMP Network Projection Operation Guide
EsyMP Network Projection Opertion Guide Contents 2 About EsyMP Network Projection Functions of EsyMP Network Projection... 5 Vrious Screen Trnsfer Functions... 5 Instlling the Softwre... 6 Softwre Requirements...6
More informationNetwork Configuration Independence Mechanism
3GPP TSG SA WG3 Security S3#19 S3010323 36 July, 2001 Newbury, UK Source: Title: Document for: AT&T Wireless Network Configurtion Independence Mechnism Approvl 1 Introduction During the lst S3 meeting
More informationData replication in mobile computing
Technicl Report, My 2010 Dt repliction in mobile computing Bchelor s Thesis in Electricl Engineering Rodrigo Christovm Pmplon HALMSTAD UNIVERSITY, IDE SCHOOL OF INFORMATION SCIENCE, COMPUTER AND ELECTRICAL
More informationExample 27.1 Draw a Venn diagram to show the relationship between counting numbers, whole numbers, integers, and rational numbers.
2 Rtionl Numbers Integers such s 5 were importnt when solving the eqution x+5 = 0. In similr wy, frctions re importnt for solving equtions like 2x = 1. Wht bout equtions like 2x + 1 = 0? Equtions of this
More informationCOMPARISON OF SOME METHODS TO FIT A MULTIPLICATIVE TARIFF STRUCTURE TO OBSERVED RISK DATA BY B. AJNE. Skandza, Stockholm ABSTRACT
COMPARISON OF SOME METHODS TO FIT A MULTIPLICATIVE TARIFF STRUCTURE TO OBSERVED RISK DATA BY B. AJNE Skndz, Stockholm ABSTRACT Three methods for fitting multiplictive models to observed, crossclssified
More informationEnterprise Risk Management Software Buyer s Guide
Enterprise Risk Mngement Softwre Buyer s Guide 1. Wht is Enterprise Risk Mngement? 2. Gols of n ERM Progrm 3. Why Implement ERM 4. Steps to Implementing Successful ERM Progrm 5. Key Performnce Indictors
More information