Active instability of cell-cell junctions at the onset of tissue fluidity
aa r X i v : . [ c ond - m a t . s o f t ] J a n Active instability of cell-cell junctions at the onset of tissue fluidity
Matej Krajnc, ∗ Tomer Stern,
2, 3 and Cl´ement Zankoc Joˇzef Stefan Institute, Jamova 39, SI-1000 Ljubljana, Slovenia Department of Molecular Biology, Princeton University, Princeton, NJ, USA Department of Chemical and Biological Engineering, Princeton University, Princeton, NJ, USA
Active in-plane remodeling is an important mechanism of tissue morphogenesis, yet its underlyingphysics is not understood. We study a nonlinear mechanical model of active cell-junction instability,which predicts universal critical collapse kinetics of its length during cell intercalation. At the onsetof fluidity, where the energy landscape flattens, this collapse can get stabilized by a limit cycle. Thistype of junction-length oscillations is different from quasioscillations, which appear around a stablefixed point even in the linear-elastic regime. Interestingly, in cuspy regions of the energy landscape,the instability often causes junctions to ”condense” around the cusps.
Introduction.—
The ability of cells to rearrange is cru-cial for tissue deformations and flows during develop-ment, wound healing, and cancer [1–3]. In confluentepithelia, cells rearrange through T1 transitions, wherepairs of initially neighboring cells get separated by in-tercalation of adjacent cell pairs (Fig. 1A and Ref. [4]).While in passive materials, e.g., the soap foam, T1 tran-sitions are induced by shear stresses or flows appliedthrough system’s boundaries [5, 6], in tissues, they canalso be driven locally.A well-known example are active cell intercalationsduring convergent extension, which help define the head-tail body axis and play a crucial role in shaping the bodyplan in early embryos [7–9]. Despite its geometric sim-plicity, an active cell-intercalation event is driven by a rel-atively complex biomechanical machinery that includesmultiple actomyosin structures, i.e., the medial and thejunctional actomyosin [10]. While cell intercalations dur-ing the convergent extension are typically planar polar-ized so as to define the directions of convergence andextension, in some systems cells can also intercalate inrandom directions due to active junctional noise [11, 12].To describe active junctional remodeling, we recentlygeneralized a generic model of active contractile elements,first proposed by Dierkes et al. [13], to solid tissues dom-inanted by the linear elasticity [14]. We showed that afeedback between junctional contractions and a buildupof active tension may dominate junctional stability andgive rise to various types of dynamics including junctionalcollapse and quasioscillations [14]. However, at the onsetof convergent extension, tissues transition from a solid-like to a fluid-like behavior [15]. Around this transition,the local energy landscape of confluent tissues flattensand the energy barriers for T1 transitions vanish [16–18].How may dynamics of individual active cell-cell junctionsbe influenced by these highly nonlinear collective effects?Furthermore, what biophysical parameters determine thestability of the junctions and what governs collapse ki-netics during cell intercalation?We address these intriguing questions by studying a ∗ [email protected] nonlinear mechanical model of an active cell-cell junction,which predicts universal collapse kinetics. Using a linearstability analysis, we identify the transition to junctionalinstability and then extend our analysis to the full nonlin-ear behavior. We find that the energy-landscape flatten-ing gives rise to supercriticality, establishing a periodictrajectory around an unstable fixed point. These oscilla-tions are different from quasioscillations, which evolve inthe presence of noise around a stable fixed point [14]. Ad-ditionally, junctions can stabilize their lengths at cuspsof the energy landscape. Universal collapse dynamics.—
We start by a simplifiedcoarse-grained description of a cell-cell junction. In theoverdamped limit, the junction contraction rate obeys η ˙ l ( t ) = − γ ( t ) − f ( l ), where ∆ γ ( t ) is the active junc- FIG. 1. (A) Snapshots of a T1 transition during GBE in
Drosophila . Red and purple arrows point to a collapsing andto an extending junction, respectively. (B) The collapse ki-netics in a model that assumes constant active tension (gray)and our model (red). (C) Schematic of the vertex model ofan active junction. Junction length and active tension aredenoted by l ( t ) and ∆ γ ( t ), respectively. Subscripts 1 and 2denote the two vertices of the active junction. (D) The elasticrestoring force f vs. junction length l in the quasistatic limit.Solid lines are polynomial fits f ( l ) = P i =1 a i ( l − l ) i . (E) Ra-tio between the third-order and the linear elastic coefficient | a /a | as a function of the preferred cell perimeter p . tional tension, and f ( l ) is an elastic restoring force thateffectively describes the elastic response of the surround-ing tissue to the active tension; η is the friction coeffi-cient. We follow the generic model of active contractileelements [13] and assume a linear relation between ∆ γ ( t )and the local concentration of myosin, defined as thenumber of motors per junction length, c ( t ) = N ( t ) /l ( t ).In particular, ∆ γ ( t ) = α [ c ( t ) − c ], where c and α arethe ambient myosin concentration and a constant pro-portionality factor describing the efficiency of motors,respectively. The total rate of change of the myosin con-centration then reads ˙ c ( t ) = ˙ N ( t ) /l ( t ) − N ( t ) ˙ l ( t ) /l ( t ) .Assuming a simple motor-actin binding and undbind-ing dynamics at a rate 1 /τ m , described by ˙ N ( t ) =[ − N ( t ) + c l ( t )] /τ m , we arrive at a first-order dynamicsequation for the active junctional tension, which reads∆ ˙ γ ( t ) = − τ m ∆ γ ( t ) − [ γ + ∆ γ ( t )] ˙ l ( t ) l ( t ) . (1)Here, γ is the tension at the ambient concentration ofmyosin motors. The first term in Eq. (1) describes ten-sion relaxation due to myosin turnover, whereas the sec-ond term describes a feedback between junction contrac-tion and the change of junctional tension.At the time point of junction collapse, t = t c , l ( t )vanishes, causing ∆ γ ( t ) to diverge. As a consequence,Eq. (1) simplifies to (d / d t )(∆ γ ( t ) l ( t )) = 0, implying∆ γ ( t ) ∝ /l ( t ). In turn, since the elastic restoring force f ( l ) is negligible compared to ∆ γ ( t ), η ˙ l ( t ) = − γ ( t ),which allows decoupling the equations for ˙ l ( t ) and ∆ ˙ γ ( t )and deriving the governing equation for the dynamics ofjunctional length: ˙ l ( t ) ∝ l ( t ) − . The solution of thisequation describes universal kinetics close to the point ofjunction collapse: l ( t ) ∝ ( t c − t ) / . (2)For comparison, these dynamics are fundamentally dif-ferent from a relaxation of junction length if the ac-tive tension was assumed independent from the junctionlength. For example, under a high and constant activity∆ γ = const . , the contraction rate ˙ l = − γ/η is con-stant as well and depends on the interplay between activetension and friction, and l ( t ) = l − (2∆ γ/η ) t (Fig. 1B). Vertex model.—
Since the active junction interacts withthe surrounding cells, their collective mechanics could af-fect the dynamics of the junction. To describe this, wenext employ the Area- and perimeter-elasticity (APE)vertex model, which represents the tissue by a planartiling of polygonal cells, parametrized by the positions ofvertices r i = ( x i , y i ) [19–22]. In our case, the activity isapplied on a single junction as an active force dipole, de-scribed by the tension ∆ γ ( t ), while the surrounding cellsare considered passive for simplicity (Fig. 1C).The potential energy of the tissue reads W = k A P k [ A k ( t ) − A ] + k p P k [ p k ( t ) − p ] [23], where thesums go over all the cells and describe cell-area elasticity ( A k and A being the actual and the preferred cell ar-eas, respectively) and cell-perimeter elasticity ( p k and p being the actual and the preferred cell perimeters, respec-tively). The parameters k A and k p are the correspondingmoduli. Due to strong friction with the environment, thevertex dynamics are best described by the overdampedequation of motion, which reads η ˙ r i ( t ) = F i ( t ) = −∇ i W ( r ) − ∆ γ ( t ) ∇ i l ( t ) . (3)Here, the first term describes conservative forces thatdrive the system towards the minimum of W , whereasthe second term describes the active forces, which onlycontribute to the motion of the vertices correspondingto the active junction; r = { r , r ... } . In turn, ∆ γ ( t )evolves in time according to Eq. (1), where the rate ˙ l ( t ) =[ r ( t ) − r ( t )] · [ F ( t ) − F ( t )] /l ( t ); subscripts 1 and 2 re-fer to the two vertices of the active junction (Fig. 1C).We choose √ A , η/k p , and k p √ A as the units oflength, time, and tension, respectively, and we assumethat cells are nearly incompressible by setting k A A /k p =100. We study tissues consisting of N = 324 cells, start-ing with a regular honeycomb cell tiling, followed by ir-regular ordered, and finally, disordered tilings. Nonlinear elasticity.—
Describing the junction me-chanics within the whole-tissue vertex model allows usto capture the collective response of cells to the de-formation of the active junction. This response con-tributes a restoring force f ( l ), which acts on the activejunction, opposing its deformation. As p approaches p hex = 2 / · / ≈ . inordered tissues flattens around the rest junction length l = 3 − / √ ≈ .
62 and the elasticity gets dominatedby nonlinearities (Fig. 1D and Refs. [16–18]). In particu-lar, the contribution of the third-order elastic term rela-tive to the linear term grows rapidly when p approaches p hex (Fig. 1E). This highly nonlinear behavior could im-portantly contribute to the dynamics of the active junc-tion and could even affect the nature of its instability.Therefore, bellow, we particularly focus on the vicinityof p = p hex . Junction collapse.—
We start our analysis of the ver-tex model by studying the linear stability of the sys-tem’s fixed point ρ = (cid:0) r (0) , ∆ γ (0) (cid:1) , where r (0) = (cid:16) r (0)1 , r (0)2 ... r (0) N v (cid:17) and ∆ γ (0) are the vertex positions andthe active tension at the fixed point, respectively. For p ≤ p hex one of the fixed points corresponds to a regularhoneycomb lattice of cells, where A k = 1 and p k = p hex ,for all cells k , whereas l = l and ∆ γ (0) = 0, for the ac-tive junction. Considering a small perturbation from thisfixed point, δ ρ = ( δ r , δ r ..., δ r N v , ∆ γ ), the linearizedsystem [Eqs. (1) and (3)] reads δ ˙ ρ = J δ ρ , where J is theJacobian matrix of the energy W + ∆ γl . The system isunstable if the maximal real part of eigenvalues λ k of J ,Λ Re = max k [Re( λ k )], is positive and stable otherwise.We numerically diagonalize the Jacobian matrices inthe region of the (1 /τ m , p , γ ) parameter space closeto p = p hex and identify the critical surface γ ∗ = FIG. 2. (A-C) Maximal real and imaginary parts of the eigen-values of J , Λ Re and Λ Im , respectively, versus γ (panel A),1 /τ m (panel B), and p (panel C). In panel A: 1 /τ m = 0 . p = 3 .
65; In panel B: γ = 1 and p = 3 .
65; Inpanel C: 1 /τ m = γ = 0 .
11. (D) Stability diagram in the(1 /τ m , p , γ ) parameter space showing the critical bifurca-tion surface γ ∗ (1 /τ m , p ). (E, F) Junction length l versustime t c − t (panel E) and junction contraction rate ˙ l versusjunction length l (panel F) for 25 combinations of parame-ters, chosen randomly from the regime of junction collapse.Dashed lines show l ( t ) ∼ ( t c − t ) / and ˙ l ( l ) ∼ l − . γ ∗ (1 /τ m , p ), where the system undergoes a Hopf bifur-cation. The results show that the fixed point becomeslinearly unstable either upon increasing γ (Fig. 2A) ordecreasing 1 /τ m (Fig. 2B). In addition, close to p = p hex ,the stability also depends on p (Fig. 2C and D).The linear relation between the critical γ ∗ and 1 /τ m at p = p hex (Fig. 2D) can be estimated by the coarse-grained model. In particular, close to the point of thevanishing linear response, we can write ˙ l ( t ) = − γ ( t ),whereas ∆ ˙ γ ( t ) still obeys Eq. (1). The stability conditionTr( J ) <
0, where J = ((0 , − , (0 , − /τ m + 2 γ /l )),yields an estimate for the bifurcation line, γ ∗ = l / (2 τ m ),which agrees well with the critical γ ∗ obtained within thevertex model at p = p hex (gray line in Fig. 2D).To study the kinetics of junction collapse, we next sim-ulate the original system [Eqs. (1) and (3)] in time upona small initial perturbation. First, in the stable regime,where Λ Re ≤
0, the junction relaxes its length directlyback to l = l if additionally Λ Im = max k Im( λ k ) = 0. Incontrast, if Λ Im = 0, the junction undergoes dampedtransient oscillations to the fixed point. As we haverecently shown, junctional noise [11, 25] sustains andeven amplifies these transient oscillations, giving rise tononperiodic (quasi)oscillations or the so-called quasicy-cle (Ref. [14] and Supplemental Material, Fig. S1 [26]).The junction collapses in the unstable regime, whereΛ Re > l ( t ) ∼ ( t c − t ) / (Fig. 2E),whereas the contraction rate ˙ l ( t ) ∝ l ( t ) − (Fig. 2F).These relations agree with the predictions of the coarse-grained model [Eq. (2)]. Furthermore, since the elasticrestoring force, which depends on the degree of order aswell as on p [16–18], is negligible close to t = t c since itis dominated by the divergent active tension, the samecritical dynamics are found in disordered tissues–even for p > p hex (Supplemental Material, Fig. S2 [26]). Limit cycle.—
It is possible that the highly nonlinearresponse in the vicinity of p = p hex stabilizes an orbitaround an unstable fixed point so as to form a properlimit cycle of junctional oscillations. In dynamical sys-tems, such a behavior appears when the Hopf bifurca-tion becomes supercritical. To explore this possibility,we examine the vertex model at values of p close to p hex . Indeed, we find cases where a limit cycle of junc-tional oscillations is stabilized (Fig. 3A-C). In particular,such dynamics can be found in the region of the param-eter space adjacent to the stable regime just below thepoint of the local energy-landscape flattening, which oc-curs at p = p hex (Fig. 3D-F). The amplitude of lengthoscillations increases when approaching the transition tojunction collapse, whereas the frequency increases with FIG. 3. (A,B) Junction length l (panel A) and activetension ∆ γ (panel B) versus time t at (1 /τ m , p , γ ) =(0 . , p hex , . l, ∆ γ )-plane, cor-responding to trajectories shown in panels A and B. (D-F) Phase diagrams in the (1 /τ m , γ ) planes at p = 3 . , . , and p hex . Gray curves show the surface of bifurcation.(G) Tissue snapshots showing junctions that do/do not oscil-late at p = p hex when considered active (orange/gray edges).The number of non-hexagonal cells increases from left to right. both 1 /τ m and γ . Overall, both the accessible rangeof amplitudes as well as that of frequencies are quite re-markable (Supplemental Material, Fig. S3 [26]).Since the regular honeycomb tiling becomes linearlyunstable at p = p hex , the junctional dynamics in orderedtissues for p > p hex need to be studied in irregular tis-sue samples. Due to irregularity, the results from suchtissues may depend on the choice of the active junction,rendering the analysis quite challenging. To capture allpossible dynamic behaviors at a given value of p , weperform multiple simulations using the same initial cellconfiguration, each time applying activity on a differentrandomly chosen junction. We find that upon increasing p beyond p hex , junction collapse becomes more commonwhile less junctions develop a limit cycle (SupplementalMaterial, Fig. S4 [26]). This is because a nonlinear elas-tic response, which is needed to stabilize a limit cycle,weakens with an increasing p , up until even the energybarriers for T1 transitions vanish at p ≈ .
81 and thelandscape becomes completely flat [18].In contrast to ordered tissues, disordered tissues loseboth their linear elastic response as well as the energybarriers for T1 transitions at p ≈ .
81 [17]. This affectsthe existence of the limit cycle, which is seen through thefraction of edges that can develop a limit cycle within agiven tissue at p ≈ p hex . This fraction rapidly decreaseswith an increasing degree of disorder (Fig. 3G). In turn,by studying highly disordered tissues with hexagon frac-tion of only about 0.6, we find that a considerable frac-tion of all junctions within a given tissue can establish alimit cycle when considered active for values of p around3 .
81, consistent with disordered tissues losing their linearelastic response at p ≈ .
81 [17].
Condensation at cusps.—
Interestingly, the limit cycleis not the only mechanism by which the system can pre-vent junction collapse following divergence from an un-stable fixed point. We discover that in ordered tissues for p hex < p < .
81, junctions in some cases ”condense” at acouple of well-defined lengths (Fig. 4A). By closely exam-ining dynamics of a representative junction (Fig. 4B), wefind that these lengths correspond to cusps in the energylandscape–regions where the landscape transitions fromflat to nonflat and the elastic restoring force becomes fi-nite (Fig. 4C and Refs. [18, 27]). It appears that thesecusps can prevent collapse of unstable junctions, however,rather than catching the system into a limit cycle, theymerely stop junction contraction/expansion and keep theactive junction at a constant length, located close to thecusps (Fig. 4D). In the (1 /τ m , γ ) phase diagram, this be-havior is observed in the region between the collapse andstable regimes, somewhat replacing the limit cycle, whichappears in the same regime in the honeycomb case at p ≈ p hex (Figs. 4E and 3F). Finally, junctional noise [28]can cause the system to switch between the two cusps ina step-wise manner (Fig. 4D). Discussion.—
We showed that while junctions’ col-lapse, driven by the juncitonal actomyosin, may followuniversal critical kinetics, other types of dynamics de-
FIG. 4. (A) Length l at t = 2000, versus the initial lengthfor 100 randomly chosen active junctions in an ordered tissueat p = 3 .
75. (C, D) Energy landscape and restoring force(panel C) and length dynamics (panel D) for the examinedactive junction shown in panel B. Gray dashed lines denotelocations of the cusps for the examined junction. In panelsA-D, (1 /τ m , γ ) = (0 . , . /τ m , γ ) plane at p = 3 .
75 for the examined junction. pend strongly on collective cell properties. In nonlinearand cuspy regions of the energy landscape, junctional in-stability can turn into a limit cycle or cause condensationof junctions at the cusps. Overall, our results demon-strate possible types of junctional movements due to theactivity of the junctional actomyosin (Supplemental Ma-terial, Movies M1-M7 [26]). Furthermore, they illustratehow complex active dynamics can arise in highly nonlin-ear regions of the energy landscape close to tissue fluidity.Our work opens new avenues of possibilities for futurestudies. In particular, further experiments are neededto test our predictions in different biological contexts.For instance, cell rearrangements during germband ex-tension (GBE) in the early
Drosophila embryo are drivenby active contractions due to planar polarized localiza-tion of myosin motors at the junctions [7–9]. By ana-lyzing data on junctional dynamics in this system [29],we do find cases of junction collapse with similar kineticsto that predicted by our model (Supplemental Material,Fig. S5 [26]). However, we also observe a relaxation-like decrease of junction length, which may indicate thatsome of the analyzed cell intercalations, i.e., those thatare located away from the myosin-rich parasegmentalboundaries [30], may be driven by shear stresses exertedby the neigboring cells rather than by the actomyosin ac-tivity. Furthermore, l ( t ) in GBE are often staircase-likedue to pulsatile contractions of the medial actomyosin fol-lowed by a ratchet-like stalling of junction length [10, 31].Including some of these biomechanical mechanisms in ourmodel may lead to quite complex behaviors such as mul-tistability of junction length, where multiplicative noisecould also play an important role. Apart from general-izing the active-tensions dynamics, it will also be neces-sary to improve the description of the viscoelasticity ofcell-cell junctions–possibly along the lines of the recentlyproposed model by Staddon et al. [32]. ACKNOWLEDGMENTS
We thank Simon ˇCopar for the suggestion to look atthe critical dynamics of junction collapse, Ricard Alert, Eric Wieschaus, Stas Shvartsman, and other members ofthe Princeton’s Gastrulation club for fruitful discussions,and Jan Rozman, and Primoˇz Ziherl for critical readingof the manuscript. We acknowledge the financial supportfrom the Slovenian Research Agency (research projectNo. Z1-1851 and research core funding No. P1-0055). [1] R. Etournay, M. Popovi´c, M. Merkel, A. Nandi,C. Blasse, B. Aigouy, H. Brandl, G. Myers, G. Salbreux,F. J¨ulicher, S. Eaton, and H. McNeill, eLife , e07090(2015).[2] R. J. Tetley, M. F. Staddon, D. Heller, A. Hoppe,S. Banerjee, and Y. Mao, Nat. Phys. , 1195 (2019).[3] L. Oswald, S. Grosser, D. M. Smith, and J. A. K¨as, J.Phys. D: Appl. Phys. , 483001 (2017).[4] D. Weaire and N. Rivier, Contemp. Phys. , 59 (1984).[5] A. M. Kraynik, Annu. Rev. Fluid Mech. , 325 (1988).[6] R. H¨ohler and S.Cohen-Addad, J. Phys. Condens. Matter , R1041 (2005).[7] C. Bertet, L. Sulak, and T. Lecuit, Nature , 667(2004).[8] D. Kong, F. Wolf, and J. Grosshans, Mech. Dev. , 11(2017).[9] M. Rauzi, Philos. Trans. R. Soc. B , 20190552 (2020).[10] M. Rauzi, P.-F. Lenne, and T. Lecuit, Nature , 1110(2010).[11] S. Curran, C. Strandkvist, J. Bathmann, M. de Gennes,A. Kabla, G. Salbreux, and B. Baum, Dev. Cell , 480(2017).[12] A. Mongera, P. Rowghanian, H. J. Gustafson, E. Shel-ton, D. Kealhofer, E. K. Carn, F. Serwane, A. A. Lucio,J. Giammona, and O. Camp`as, Nature , 401 (2018).[13] K. Dierkes, A. Sumi, J. Solon, and G. Salbreux, Phys.Rev. Lett , 148102 (2014).[14] C. Zankoc and M. Krajnc, Biophys. J. , 13541 (2020).[16] D. B. Staple, R. Farhadifar, J.-C. R¨oper, B. Aigouy,S. Eaton, and F. J¨ulicher, Eur. Phys. J. E , 117 (2010).[17] D. Bi, J. Lopez, J. Schwarz, and M. L. Manning, Nat.Phys. , 1074 (2015).[18] P. Sahu, J. Kang, G. Erdemci-Tandogan, and M. L. Man- ning, Soft Matter , 1850 (2020).[19] R. Farhadifar, J.-C. R¨oper, B. Aigouy, S. Eaton, andF. J¨ulicher, Curr. Biol. , 2095 (2007).[20] A. Fletcher, M. Osterfield, R. Baker, and S. Y. Shvarts-man, Biophys. J. 106, 2291 (2014).[21] S. Alt, P. Ganguly, and G. Salbreux, Philos. Trans. RoyalSoc. B , 20150520 (2017).[22] D. L. Barton, S. Henkes, C. J. Weijer, and R. Sknepnek,PLoS Comp. Biol. , e1005569 (2017).[23] The potential energy of the tissue in its full form reads W = P k k A ( A k − A ) + P k k p p k − P ij αl ij + P ij γ l ij .Here k A and k p are the moduli of the cell- and perimeterelasticity, respectively, α is the adhesion strength, and γ is the tension in the junctional actomyosin. By definingthe preferred perimeter as p = ( α − γ ) / (4 k p ), we canwrite W = P k (cid:2) k A ( A k − A ) + k p ( p k − p ) (cid:3) .[24] The value p hex ≈ . , 3209 (2020).[26] See Supplemental Material at http://link.aps.org/...[27] D. M. Sussman, J. M. Schwarz, M. C. Marchetti, andM. L. Manning, Phys. Rev. Lett. 120, 058001 (2018).[28] Noise is described by an additional term in the equationfor active tension [Eq. (1)]: ∆ ˙ γ noise = p σ /τ m ξ ( t ),which describes the white noise with long-time variance σ ; h ξ ( t ) i = 0, h ξ ( t ) ξ ( t ′ ) i = δ ( t − t ′ ).[29] T. Stern, S. Y. Shvartsman, and E. F. Wieschaus, PLOSComput. Biol. , 1 (2020).[30] R. J. Tetley, G. B. Blanchard, A. G. Fletcher,R. J. Adams, and B. Sanson, eLife , e12094 (2016).[31] R. Fernandez-Gonzalez and J. A. Zallen, Phys. Biol. ,045005 (2011).[32] M. F. Staddon, K. E. Cavanaugh, E. M. Munro,M. L. Gardel, and S. Banerjee, Biophys. J. , 1739(2019). r X i v : . [ c ond - m a t . s o f t ] J a n Active instability of cell-cell junctions at the onset of tissue fluidity:Supplemental material
Matej Krajnc, Tomer Stern,
2, 3 and Cl´ement Zankoc Joˇzef Stefan Institute, Jamova 39, SI-1000 Ljubljana, Slovenia Department of Molecular Biology, Princeton University, Princeton, NJ, USA Department of Chemical and Biological Engineering, Princeton University, Princeton, NJ, USA
FIG. S1. (A, B) Upon a perturbation, junction length relaxes directly back to the fixed point if Λ Im = 0 (panel A), andundergoes damped transient oscillations to the fixed point if Λ Im = 0 (panel B). Panel A: p = 3 .
7, 1 /τ m = 0 .
01, and γ = 0 . p = 3 .
7, 1 /τ m = 15 .
85, and γ = 0 .
09. (C, D) Junction length l (panel C) and active tension ∆ γ (panel D) versus time t at p = 3 .
65 and 1 /τ m = γ = 0 . l, ∆ γ ) plane. The trajectory iscalled quasicycle and corresponds to transient oscillations to the stable fixed point, sustained and amplified by junctional noise.Noise is described by an additional term in the equation for active tension [Eq. (1) of the main text]: ∆ ˙ γ noise = p σ /τ m ξ ( t ),which describes the white noise with long-time variance σ and h ξ ( t ) i = 0, h ξ ( t ) ξ ( t ′ ) i = δ ( t − t ′ ). The result shown in panelsA-C corresponds to σ = 0 . l versus time t c − t (where t c is the time point of collapse) for active junctions in disordered tissuesat p = 3 . , . , and 3 .
85 (panels A, B, and C, respectively). Each panel shows trajectories for about 20 pairs of parameters1 /τ m and γ , chosen randomly from the regime of junction collapse. Dashed lines show l ( t ) ∼ ( t c − t ) / . FIG. S3. Amplitude (panel A) and frequency (panel B) of oscillations in the (1 /τ m , γ )-plane for p = p hex .FIG. S4. (A) Maximal real part of eigenvalues of the Jacobian matrix for a regular honeycomb cell tiling described by theAPE model. The regular honeycomb becomes linearly unstable at p ≈ p hex . (B) Irregular ordered tissue samples generatedto study active junctional dynamics in the regime 3 . ≤ p ≤ .