PHYSICAL REVIEW X 9, 011029 (2019)
Multicellular Rosettes Drive Fluid-solid Transition in Epithelial Tissues
Le Yan* Kavli Institute for Theoretical Physics, University of California, Santa Barbara, California 93106, USA † Dapeng Bi Department of Physics, Northeastern University, Massachusetts 02115, USA
(Received 7 June 2018; revised manuscript received 12 November 2018; published 12 February 2019)
Models for confluent biological tissues often describe the network formed by cells as a triple- junction network, similar to foams. However, higher-order vertices or multicellular rosettes are prevalent in developmental and in vitro processes and have been recognized as crucial in many important aspects of morphogenesis, disease, and physiology. In this work, we study the influence of rosettes on the mechanics of a confluent tissue. We find that the existence of rosettes in a tissue can greatly influence its rigidity. Using a generalized vertex model and the effective medium theory, we find a fluid-to-solid transition driven by rosette density and intracellular tensions. This transition exhibits several hallmarks of a second-order phase transition such as a growing correlation length and a universal critical scaling in the vicinity of a critical point. Furthermore, we elucidate the nature of rigidity transitions in dense biological tissues and other cellular structures using a generalized Maxwell constraint counting approach, which answers a long-standing puzzle of the origin of solidity in these systems.
DOI: 10.1103/PhysRevX.9.011029 Subject Areas: Biological Physics, Soft Matter
I. INTRODUCTION However, in many tissues [26], higher-order vertices Multicellular organization in tissues is important to where four or more cells meet can occur. A prominent understanding many aspects of biology and medicine, such example occurs in Drosophila embryogenesis, where a 1 – as embryonic development, disease generation, and pro- mixture of T junctions [27 31] (vertices where four cells – gression. Particularly in fully confluent epithelial tissues, meet) and multicellular rosettes [32 37] (vertices with five or where cells are densely packed and have tight adherent more cells) have been observed during the elongation of the junctions between them, the behavior and response of cells body axis. It has been further shown that, as the body length can be highly collective and differ from the single-cell-level of the embryo doubles due to a series of cellular rearrange- behavior. Many models have been proposed to understand ments, a majority of cells in the developing epithelium the emergence of this collective behavior and the biophysi- participate in rosette formation [33,36]. The morphogenesis cal properties of tissues. In the past two decades, a class of of the Drosophila eye is also facilitated by rosettes [38].In cell-based models known as the vertex model has been zebrafish lateral-line development, the lateral line is com- developed to study tissue mechanics [1–25]. In the vertex posed of mechanosensory organs called neuromasts, which are formed from rosettes composed of 20 or more cells model, each cell is represented as a deformable polygonal – inclusion, with edges and vertices shared by neighboring [39 41]. In other vertebrates, rosettes have been observed in cells. This class of models often assumes that exactly three the neural plate of chick [42] and mouse [43] embryos, in the cells meet at any vertex in an epithelial tissue and fourfold development of the mouse visceral endoderm [44],the vertices occur only as an intermediate state during a T1 kidney tubule [45], and the pancreas [46]. They are also rearrangement. observed in adults and in vitro such as adult mammalian brains and cultured neural stem cells [47,48]. The importance of cellular rosettes has been widely *[email protected] recognized. They have been proposed as an efficient † [email protected] mechanism for tissue remodeling and growth during the body elongation in the Drosophila embryo [33,35,36] and Published by the American Physical Society under the terms of are thought to be crucial for the orderliness of collective the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to cell migration in the mouse visceral endoderm [26,44]. the author(s) and the published article’s title, journal citation, Cancer pathologists visually inspecting histologic samples and DOI. from tumors use cellular rosettes as strong indications of
2160-3308=19=9(1)=011029(18) 011029-1 Published by the American Physical Society LE YAN and DAPENG BI PHYS. REV. X 9, 011029 (2019)
malignancy such as medulloblastoma and retinoblastoma area constraints ðKAA0Þ=ðKPÞ. When p0 ≤ p0, the tissue [49]. However, compared to the large body of live-imaging behaves like a rigid solid with a finite shear modulus. and molecular studies [26], there are surprisingly few Above p0, the tissue becomes soft and fluidlike with a modeling [44] and even less physical understanding of vanishing shear modulus. rosettes regarding how they influence tissue mechanics. Here, we develop a generalized vertex model and an III. MANIPULATION OF CELLULAR TOPOLOGY effective medium theory that takes into account the AND THE CREATION OF HIGHER-ORDER presence of rosettes. We also make experimentally testable VERTICES predictions regarding the strong correlations and the inter- play between cellular topology and mechanical tensions in In developmental and in vitro examples, the general a tissue. We find that the tissue rigidity is precisely mechanism for rosette formation is the contraction of controlled by the density of higher-order vertices including actomyosin networks in cells [26]. This mechanism can rosettes and T1 junctions as well as a single parameter that be manifested in several ways. For example, in Drosophila tunes intracellular tensions in the tissue. The results show body-axis elongation, rosettes are due to planar polarized that a tissue behaving as a fluid can be rigidified by the constriction [28,29,31,33,34,36], while in gastrulation and creation of just a few higher-order vertices. The transition neural tube closure, actomyosin structures in the apical cell surface constrict to form rosettes [39–41]. Also, when cells between fluid and rigid states exhibit many hallmarks of a delaminate or extrude from epithelia [51–54], a vertex with second-order phase transition, such as a growing correla- more than four edges can be left behind that becomes a tion length and a universal critical scaling in the vicinity of center of a rosette. Similarly, a multicellular rosette can also a critical point. Furthermore, we offer a unifying perspec- form as the result of a wound closure [55,56]. tive on rigidity transitions in dense confluent tissues and Here, instead of focusing on the origin of cellular rosettes cellular materials by elucidating the nature of this transition which can be varied for different processes, we assume that using a generalized Maxwell constraint counting. they have been created via one of the observed mechanisms and ask how their presence affects the mechanics at the II. GENERALIZED VERTEX MODEL tissue level. In practice, we create rosettes and T1 junctions We begin with the most generic form of the vertex model via a simple protocol of the random collapse of edges. for a homogeneous tissue [2], where the total energy is During this process, an edge is chosen at random, and its given by a sum over the mechanical cost of deforming length is reduced to zero. The two vertices on the end of the individual cells: edge are then merged into a single vertex. The fractions of T1 vertices and rosettes generated in this random protocol XF turn out to be consistent with the fractions in the fly embryo 2 2 U ¼ ½KAðAα − A0Þ þ KPðPα − P0Þ : ð1Þ epithelial tissue during the germ band extension [33].We α¼1 carry out this process while making sure the number of cells F remains constant. These moves model the convergence of The first term results from a combination of three- vertices in developmental processes and cell-extrusion dimensional cell incompressibility and the monolayer’s events. resistance to height fluctuations or cell bulk elasticity [3,8], Under periodic boundary conditions, the network will where KA is a height elasticity, Aα is the cross-sectional always obey Euler’s characteristic formula V − E þ F ¼ 0, area (apical) of cell α, and A0 is the preferred area for the where V and E are the number of vertices and edges, cell. The second term in Eq. (1) is quadratic in the cell respectively. The density of higher-order vertices is cap- cross-sectional perimeter Pα and models the active con- tured by the average vertex coordination number, given by tractility of the actin-myosin subcellular cortex, with elastic constant KP [2], and P0 is an effective target shape index, Z ¼ 2E=V: ð2Þ representing an interfacial tension set by a competition between the cortical tension and the energy of cell-cell Beginning with a tissue at Z ¼ 3 (i.e., containing only adhesion [5,8] between two contacting cells. In Eq. (1), the trijunctions), we choose an arbitrary value of Z between sum is over all F number of cells in the tissue. The cell 3 and 6 by applying a series of edge-collapse moves. areas (fAαg) and perimeter (fPαg) are fully determined by Representative simulation snapshots are shown in Fig. 1. the positions of vertices fRig. Because of a combination of After Z is changed, the energy is minimized using the cortical tension and cell-cell adhesion [2,50], each cell conjugate-gradient method. For simplicity, no additional contributes an effective line tension [10] to adjacent edges, topological changes are performed (e.g., T1 transitions, cell controlled by P0, which can also be interpreted as a pffiffiffiffiffi divisions, or cell apoptosis or extrusion) during the min- geometrical nondimensional shape index p0 ¼ P0= A0. imization, which corresponds to looking at when higher- It is demonstrated that [2,8,18] the tissue undergoes a phase order vertices are formed but not immediately resolved transition at p0 ¼ p0 ≈ 3.81, independent of the strength of [31,33–36,57] or in systems where they are persistent for
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FIG. 1. Introducing higher-order vertices into the vertex model. In conventional vertex models, all vertices have threefold coordination. Here, we apply a protocol that randomly collapses cell edges to create vertices with coordination higher than 3. Fourfold vertices are called “T1 junctions,” and fivefold or higher vertices are termed rosettes. We characterize the state of a tissue by the average vertex coordination number Z. Simulation snapshots show states with different values of Z. Squares, five-pointed stars, and six-pointed stars indicate vertices with a coordination of 4, 5, or 6, respectively. Disks correspond to vertices with a coordination of 7 or more. extended periods of time [43,44,58,59]. We also perform However, Fig. 2(a) further shows that a fluidized tissue an alternative set of simulations in which T1 and T2 at Z ¼ 3 can be rigidified when Z is increased. The value of (cell apoptosis) transitions are allowed as shown in Fig. 11 Z where the tissue rigidifies depends on p0. The boundary which do not affect our findings. We apply the protocol between rigid and nonrigid states follows a line of transition 3 using two types of initial states at Z ¼ : (i) random tissue points Zcritðp0Þ given by networks, where cell shapes are obtained from a random Voronoi tessellation [9,24], and (ii) an ordered hexagonal 3 p0 ≤ p0; Z ðp0Þ¼ ð4Þ tiling [2,4], where every cell is a regular hexagon. crit 3 þ B½p0 − p0 p0 >p0;
IV. TISSUE MECHANICAL RIGIDITY where the slope of the boundary separating solid and fluid TRANSITIONS states (B ≈ 3.85) is obtained through empirical fitting. In We begin by probing the mechanical response of the Appendix B, we develop a mean-field model which tissue as a function of the average coordination number Z provides an accurate prediction for the numerical value of B. and the nondimensional cell shape index p0. In order to determine whether the tissue is mechanically rigid, we To better understand how the creation of higher-order analyze the zero modes of the Hessian matrix vertices can rigidify a tissue, we analyze the behavior of the fraction of zero modes f0 ≡ N0=ð2VÞ as a function of Z for 2 ∂ U various p0 >p0 in Fig. 2(b). Using the relation of Zcritðp0Þ Miμjν ¼ ; ð3Þ [Eq. (4)], we then replot fðλ ¼ 0Þ vs Z − Z in the inset in ∂R μ∂R ν crit i j Fig. 2(b) and show that all curves can be collapsed onto a where R and R are positions of vertices i and j, universal master curve. This result suggests that the nature i j of the rosette-driven rigidity transition may be universal respectively, while μ and ν are Cartesian coordinates. across different p0 values. Strikingly, f0 decreases faster The Hessian has dimensions of 2V × 2V and a eigenspec- than the increasing number of topological constraints Z, trum of 2V eigenvalues fλ ¼ ω2g. The presence of zero k k which betrays the Maxwell counting theorem [60] in eigenvalues indicates the loss of rigidity, which would traditional rigidity transitions. correspond to floppy modes or zero modes that allow deformation of the system without changes in the total A. Nature of the transition energy. The number of floppy modes N0 can therefore be used as a measure to distinguish between the rigid and fluid To explain the nature of the transition between the states. Mathematically, the number of floppy modes is just fluid and rigid states in Fig. 2(a), we employ a genera- the nullity of M, i.e., number of linearly independent lized version of the Maxwell constraint counting theo- solutions to MjδRi¼0. rem [60]: The rigidity is lost when the number of In Fig. 2(a), the rigid states (N0 ¼ 0) and the fluid states independent constraints no longer matches the number (N0 > 0) are shown for simulation results at all value pairs of degrees of freedom (d.o.f.) (2V for the model), and the (p0, Z). At Z ¼ 3, the tissue undergoes a rigidity transition number of floppy modes is given by the difference. It at p0 ¼ p0 ≈ 3.818 [p0 ≈ 3.722 for the hexagonally is tempting to simply prescribe one constraint per cell for ordered initial state (Fig. 9)]. This result recapitulates each area term and each perimeter term in Eq. (1). previous results on trijunction-only tissues [2,4,8]. However, this approach results in an erroneous counting of
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(a) (b) (c)
(d)
FIG. 2. Tissue mechanical properties. (a) The density of higher-order vertices Z and the nondimensional cell shape index p0 together control the mechanical rigidity of the tissue. Here, green data points indicate states with no floppy modes, i.e., mechanically rigid, and purple points highlight states with a finite number of floppy modes and zero shear modulus. When only trijunctions are allowed (Z ¼ 3), rigidity occurs only for tissues with p0 ≤ p0. However, when a sufficient number of higher-order vertices are present (Z>3), tissues can be rigidified even at p0 >p0. The dashed line indicates the phase boundary between rigid and fluid states, given by Zcritðp0Þ [Eq. (4)]. (b) The fraction of floppy modes f0 as a function of Z at various values of p0 (from left to right, p0 ¼ 3.829, 3.83,3.832, 3.834, 3.836, 3.838, 3.84, and 3.842). Inset: When the same data from (b) are plotted as a function of ΔZ ¼ Z − Zcritðp0Þ, all data collapse onto a single universal curve, indicating that the rigidity transition occurs at a line of critical points given by Zcritðp0Þ. (c) For the data points in (b), the actual number of floppy modes is compared against the prediction from the Maxwell constraint counting [Eq. (8)]. (d) Response of the tissue after a single edge collapse at Z ¼ 3. The displacement vector fields of all vertices are shown. The red dot indicates the location of the collapsed edge near the center of the tissue.
N0 ¼ 2V − 2F ¼ð4 − ZÞV, which deviates from the mechanical line tension [10] for an edge m shared by cell above numerical evidence and suggests that the floppy α and β and lm is the edge length. The second term in modes should always be present as long as Z<4. Eq. (6) can be rewritten as To accurately perform constraint counting [61], we apply XE τ the rank-nullity relation to the Hessian matrix: m eˆ⊥eˆ⊥; ð7Þ l m m m¼1 m z}|{d:o:f: no: ofzfflfflfflffl}|fflfflfflffl{ indep: constraints where eˆ⊥ ¼ Rðπ=2Þeˆ is a 90° rotation of the edge unit N0 ≡ nullityðMÞ¼ 2V − rankðMÞ ð5Þ m m vector eˆm ¼ lm=lm. Therefore, terms in this sum [Eq. (7)] τ 0 and count the number of independent constraints for are positive definite only for edges with m > .Asa mechanical equilibrium by calculating rankðMÞ. consequence, only edges with a positive tension contribute To illustrate how this result applies to the vertex model, to rankðMÞ, which would result in E − E0 number of we consider the Hessian of the energy without the area independent constraints, where E0 is the number of edges with zero tension τ 0. contribution, i.e., set KA ¼ 0 in Eq. (1). A complete m ¼ Together, we obtain the rank of the Hessian matrix to be calculation for rankðMÞ for an arbitrary value of KA is performed in Appendix C. To calculate its rank, we first F þ E − E0 and the Maxwell rigidity criterion for the rewrite the Hessian [Eq. (3)]: vertex model
d:o:f: no: of indep: constraints XF ∂ ∂ XE ∂2 z}|{ zfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflffl{ pα pα lm 2 − − Miμ;jν ¼ KP þ τm : ð6Þ N0 ¼ V ½ðE E0ÞþF : ð8Þ ∂R μ ∂R ν ∂R μ∂R ν α¼1 i j m¼1 i j Equation (8) means that both changing the topology of the The first term in Eq. (6) is positive definite and contributes network (e.g., changing Z, E) or changing the mechanical a total count of F to rankðMÞ. The second term sums over state (E0) can influence the rigidity of the tissue. For all E edges where τm ¼ðpα − p0Þþðpβ − p0Þ is the example, at Z ¼ 3, the Euler formula dictates 2V ¼ E þ F
011029-4 MULTICELLULAR ROSETTES DRIVE FLUID-SOLID … PHYS. REV. X 9, 011029 (2019) and, hence, N0 ¼ E0, which suggests that zero modes B. Nonlocal response near the transition emerge due to edges with zero tension. The rigidity transition is associated with a diverging Next, we put this generalized Maxwell relation to the test mechanical length scale at the transition. As higher-order by directly comparing the predicted N0 using Eq. (8) to the vertices are created using the edge collapse move, each move number of zero modes calculated from the Hessian for results in a topology change in the network where different Z and p0 in the fluid phase. We obtain an excellent V → V − 1, E → E − 1, and Z → Z þðZ − 2Þ=ðV − 1Þ. agreement between the theoretical prediction and simula- These changes reduce the number of zero modes by tion data [Fig. 2(c)]. The agreement between the two modifying the d.o.f. and constraints according to Eq. (8). measures in Fig. 2(c) implies that the entire rigidity line Furthermore, rosettes and T1 junctions change the mean in Eq. (4) is isostatic; i.e., the number of d.o.f. matches the polygon geometry and induce tension, which has nonlocal number of constraints. The slope of this rigidity line can be effects on other cells at a distance. To capture this effect, we obtained through a mean-field model proposed in start from a state at Z ¼ 3 and measure the tissue response to Appendix B, which is plotted as the dashed line in an edge collapse that creates a single fourfold vertex. Fig. 2(a). There are several special cases of constraint Figure 2(d) shows the displacement map of all vertices in counting which are detailed in Appendix C and Table I. response to a single edge collapse. Closer to the transition First, there is a fixed upper limit to the number of point of p0 ¼ 3.81, the response is highly nonlocal and independent constraints, i.e., that they cannot exceed the involves a majority of vertices, with the displacement vector d.o.f. [i.e., rank M ≤ 2V]. There is also a lower threshold ð Þ forming spatially extended swirl-like patterns. At p0 values 0 on the number of constraints. When KA ¼ , the number of further away from the transition, the response is more constraints cannot drop below F, which means that, when localized. We quantify this growing length scale closer to 0 KA ¼ , it is not possible for a tissue to have zero modes p by a spatial correlation function of the displacement field 6 0 0 when Z ¼ . When KA > , the number of constraints C r d 0 · d r shown in Fig. 10(a). This growing 2 dð Þ¼h ð Þ ð Þi cannot drop below F, which means that a tissue will length scale is highly reminiscent of the diverging dynamical ≥ 4 always be rigid for a state with Z . length scale approaching a jamming or glass transition [62].
(a) (b) (c)
Effective medium theory
(d) (e) (f)
Effective medium theory
FIG. 3. Shear modulus scaling in the rigid regime. (a) The shear modulus G as a function of Z − 3 for a range of p0 values. Here, the initial states at Z ¼ 3 are disordered. (b) When data from (a) are replotted according to the critical scaling ansatz [Eq. (9)], they collapse onto two distinct branches. The solid lines are predictions from the effective medium theory (EMT) [Eq. (18)]. (c) Average tension of the tissue as a function of p0 for various values of Z as indicated by the red arrow. Here, Z ¼ 3, 3.2, 3.4, 3.6, 3.8, 4, 4.2. The inset shows that the shear modulus is always a linear function of the tension, regardless of the value of Z. The dashed line indicates a linear relationship. (d) The shear modulus as a function of Z − 3 for a range of p0 values. Here, the initial states at Z ¼ 3 are an ordered hexagonal packing. At Z ¼ 3, the rigidity transition occurs at p0 ¼ p0 ≈3.722 [2]. (e) Rescaled shear modulus data according to the scaling ansatz [Eq. (9)] show that all data collapse onto two distinct branches. The black solid lines are predictions from the effective medium theory [Eq. (18)]. (f) Average tension of the tissue as a function of p0 for various values of Z as indicated by the red arrow. Here, Z ¼ 3, 3.03, 3.05, 3.07, 3.1, 3.12, 3.14, 3.17. The inset shows that the shear modulus is always a linear function of the tension, regardless of the value of Z. Here, the initial states at Z ¼ 3 are an ordered hexagonal packing. The dashed line indicates a linear relationship.
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C. Critical scaling of the shear modulus tension, uniform on all edges. Now, replacing the tension τ We next probe the tissue mechanics within the rigid on edge m with km ¼ m=lm, a random variable obeying the phase by analyzing the shear modulus. The shear modulus probability distribution G (defined in Appendix A) as a function of Z for different p k Pδ k − k¯ 1 − P δ k ; 11 values of p0 is shown in Fig. 3(a). The functional ð Þ¼ ð Þþð Þ ð Þ ð Þ dependence of G on Z separates into two regimes based ≤ → 3 which characterizes both edges of zero internal tension and on the value p0.Forp0 p0, G is finite as Z and ¯ the ones of the order of k (a more general distribution is increases with increasing Z.Forp0 >p0, as expected from the behavior of zero modes, G vanishes at the rigidity considered in Appendix D), results in a scattering potential: transition line Z ðp0Þ. In the limit of large Z, G becomes 8 crit > − ⊥ ⊥ ∈ ∂ less dependent on Z. <> ðkm keffÞemμemν if i ¼ j m; Given the hallmarks of a critical point observed for ¯ ⊥ ⊥ V ¼ Mm − M ¼ −ðk − k Þe μe ν if i ≠ j ∈ ∂m; > m eff m m p0 p ;Z 3 [8] and the rapidly growing correlation > ð ¼ 0 ¼ Þ : 0 length scale near the transition, we propose a critical- otherwise; scaling ansatz for the shear modulus: ð12Þ ΔZ − f where ∂m is the set of vertices defining the edge m. G ¼jp0 p0j g ϕ : ð9Þ jp0 − p0j The Green’s function of the perturbed system G can be written in terms of the Green’s function of the effective ¯ ¯ 2 −1 gþðyÞ and g−ðyÞ are the branches of the crossover scaling medium G ¼ðM − ω IÞ : function for p0 ≤ p0 and p0 >p0, respectively. Here, y ¼ ϕ 2 −1 ¯ ¯ ¯ ΔZ=jp0 − p0j serves as the crossover scaling variable G ¼ðM − ω IÞ ¼ G þ GT G; ð13Þ ϕ with exponent . In Fig. 3(b), we replot all data using the P − f Δ − ϕ T −V −GV¯ n rescaled variables G=jp0 p0j and Z=jp0 p0j . The where ¼ n≥0ð Þ sums over all multiple- best collapse is obtained with exponents f ¼ 1.5 0.1 and scattering contributions of the edge m. The EMT assumes ϕ ¼ 1.05 0.05. Furthermore, the branches of the cross- the effective medium resembles the random medium if on over scaling function lead to two distinct mechanical average the replacementR does not effect the mechanical ≤ ≪ 1 ¯ regimes: (I) When p0 p0, in the limit of y , g−ðyÞ¼ propagation, i.e., pðkÞGðkÞdk ¼ G: f const or, equivalently, G ∝ jp0 − p0j . (II) When y ≫ 1, ∝ f=ϕ ¯ the two branches merge or g−ðyÞ¼gþðyÞ y , which PT ðkm ¼ kÞþð1 − PÞT ðkm ¼ 0Þ¼0: ð14Þ means the shear modulus becomes independent of cell shapes and depends only on the network topology This assumption results in the self-consistency equation or G ∝ ΔZf=ϕ. given by We also perform the scaling analysis on states initialized ¯ from the hexagonal tiling. We calculate G for all data k − k −k P eff 1 − P eff 0; 15 1 ¯ − þð Þ 1 − ¼ ð Þ shown in Fig. 9 and plot them as a function of Z and p0 þðk keffÞGm keffGm values in Figs. 3(d) and 3(e). Testing the same scaling G¯ 2ˆ⊥ G¯ ˆ⊥ − 2ˆ⊥ G¯ ˆ⊥ ansatz [Eq. (9)] gives a good scaling collapse and yields where Gm ¼h im ¼ em · ii · em em · ij · em. The 1 0 05 ϕ 1 0 05 ≈ 3 722 f ¼ . , ¼ . , and p0 . . perimeters contribute F=V ¼½ðZ − 2Þ=2 independent con- G¯ M¯ straints on each vertex, so, on vertex i,trh ii;ω¼0 ¼ V. EFFECTIVE MEDIUM THEORY − 2 2 2 ½ðZ Þ= þkeffðZ= ÞGm ¼ d.SoGm ¼ðh=keffÞ, where To better understand the scaling relations for a shear h ¼½ð6 − ZÞ=Z (a more strict derivation is done for the modulus, we develop an effective medium theory (EMT) honeycomb lattice in Appendix D). Inserting it into the self- [63–69] near the critical point ðZ ¼ 3;p0Þ. To capture the consistency equation (15), we have the effective tension nature of the tension-induced rigidity, we map the random ¯ P−h ≥ 6−Z tension network described by the Hessian in Eq. (6) to a k 1−h P Pc ¼ h ¼ Z ; keff ¼ ð16Þ uniformly stressed medium whose Hessian is given by 0 otherwise:
M¯ Mtopo Mss ∈ 0 1 3 ¼ þ keff : ð10Þ As P ½ ; , Z ¼ is the minimal coordination for a vertex network to be stabilized by tension. The number of topo Here, M maps to the Hessian term dependent only on zero modes is N0 ¼ EðPc − PÞ¼E½ð6 − ZÞ=Z − EP ¼ the topology of the tissue network which is given by the 2V − F − ðE − E0Þ. first term in Eq. (6), and Mss maps to the tension- The elastic modulus is related to the Fourier transform of ’ G 1 dependent term [second term in Eq. (6)]. keff is the effective the Green s function as limq→0Cijklqiqk jl;ω¼0 ¼ [70].
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As shown above, near the rigidity threshold, the Green’s G¯ ∼ −1 −2 function of the effective medium is singular as keff q . So the shear modulus scales as the effective tension, ∝ ∼ ¯ G keff k.
A. Effective medium theory predicts the critical scaling observed in simulations As shown in the insets in Figs. 3(c) and 3(f), the theory predicts that the shear modulus scales as the effective ∼ tension on edges, G keff, which is determined by a self- consistency equation and proportional to the mean tension FIG. 4. A phase diagram in Z − p0 space. The critical scaling of Δ ≪ in the network k¯ when the Maxwell criterion is saturated G suggests three distinct mechanical regimes. When Z ϕ ¯ jp0 − p0j , the tissue is rigid with a shear modulus that is N0 ¼ 0. In the limit k → 0, the average tension vanishes dominated by cellular mechanics which is encoded in the shape with the geometric frustration, ϕ index p0. ΔZ ≫ jp0 − p0j , the topology of the cell network ¯ f dominates tissue rigidity where the shear modulus has a power- G ∝ k ∼ jp0 − p0ðZÞj : ð17Þ law dependence on the average vertex coordination Z. The tissue Inserting Eq. (4), the scaling relation for the modulus becomes fluidlike as floppy modes emerge at the line given by 8 Zcritðp0Þ. > Z − 3 f > − f 1 < jp0 p0j þ p0
011029-7 LE YAN and DAPENG BI PHYS. REV. X 9, 011029 (2019) vanishes at the formation of a ventral furrow and the onset of suggests that the correlations between junctional tension and a fast germ-band extension event. cellular topologies may be more universal. For future work, We also predict a high degree of correlation between we will build on these general results to specifically model higher-order vertices and adjacent edge tensions. A repre- the feedback mechanism that leads to rosette formation in sentative example is shown in Figs. 5(a) and 5(b). Here, T1 Drosophila embryogenesis, where both an increased rosette junctions and rosettes are marked in blue, and cell edges are count and increased tensions are predicted to emerge from drawn with widths proportional to the edge tensions. We the myosin positive-feedback loop [34,78]. Finally, the consistently observe higher-order vertices coincide with criticality of this rigidity transition results in a nonlocal nearby high edge tensions. To quantify this correlation, response to perturbations near the critical point, as shown in edges are grouped according to the degree of their adjacent Fig. 2(d). We thus expect the laser ablation [78] of cell edges vertices (Z1 and Z2). This grouping results in three different to induce a similar long-range effect to collapsing edges near categories of edges: (i) edges associated with only trijunc- the onset of tissue rigidity. The spatial extension of the tions (Z1 ¼ Z2 ¼ 3), (ii) edges associated with T1 junctions response could even be used as an indicator of how far the (at least one adjacent vertex is a T1 junction), and (iii) edges tissue is from the rigidity transition. associated with rosettes (at least one adjacent vertex is a In this work, we have focused mainly on the mechanics of rosette; i.e., Z1 > 4 or Z2 > 4). The relative fractions for tissues with stable rosettes. However, rosettes in actual these edge types are plotted as functions of Z in the inset in tissues can vary greatly in their lifetime. This variation Fig. 5(c). Interestingly, the statistics show very good agree- ranges from rosettes that are transient and resolve quickly to ment (Fig. 12) with the cell topologies measured during form new structures to rosettes that persist for an extended Drosophila embryo elongation [33]. Next, for edge types, period of time or may not resolve at all. For example, the we compare their tensions to the mean tension of the entire lifetime of higher-order vertices is thought to be linked to the tissue, which is shown in Fig. 5(c). This comparison predicts Hippo pathway and the force-sensitive protein Ajuba that, as Z increases, the edges near higher-order vertices [59,79,80]. Our predictions are therefore applicable to the consistently carry more tension compared to trijunction mechanical response of the tissue at timescales shorter than vertices. In particular, at the onset of higher-order vertices the rosette resolution time. In the version of the vertex model appearing, tensions associated with T1 junctions are in the here, we have not included biological feedback of cells in range of 1.5 0.3 times the tension near trijunctions. Again, response to mechanical stress. However, in Drosophila these numbers match up well with the experimental results embryo elongation, for example, myosin motors can slide from Drosophila embryo elongation, where tensions asso- more on actin in response to higher tensions, and, as a result, ciated with edges pointing along the anterior-posterior (AP) the observed cell perimeters can change [12]. At the same direction are typically 1.7 times [34] compared to edges time, myosin recruitment is stimulated to sustain the pointing in the dorsal-ventral (DV) direction. While a more perimeter at high tensions. These feedback events occur systematic study is warranted to model the elongation at a characteristic timescale corresponding to the relaxation processes in Drosophila development, our minimal model and resolution of rosettes. We would expect a breakdown of
(a) (b) (c)
FIG. 5. Spatial correlation between higher-order vertices and tensions. (a) At p0 ¼ 3.8 and Z ≈ 3.01, a few higher-order vertices are present, and their creation causes the tension in the tissue to increase from Z ¼ 3. Here, higher-order vertices are indicated by blue markers, and the thickness of cell edges is proportional to the tension magnitude. (b) Holding p0 ¼ 3.8 constant and further introducing higher-order vertices brings the tissue to Z ≈ 3.06. There is a significant degree of correlation between higher-order vertices and heightened local tensions (c) Relative edge tension due trijunctions (red), T1 junctions (blue), and rosettes (black) plotted as a function of Z at fixed p0 ¼ 3.8. The solid lines show the median values of each group, and shaded regions indicate the standard deviation. The relative tension is defined as the edge tension scaled by the mean tension in the entire tissue hτðZÞi, which scales as ðZ − 3Þ1.5. Inset: The relative populations of tri-, T1, and rosette vertices as a function of Z.
011029-8 MULTICELLULAR ROSETTES DRIVE FLUID-SOLID … PHYS. REV. X 9, 011029 (2019) the static rigidity theory in these regimes. For the Drosophila (T1 rearrangement, rosette formation, etc.) take over the embryo, this corresponds to a timescale of several minutes response, so that the system behaves as a liquid whenever a [81]. Modeling dynamical movements and transient finite energy is injected to it. This prediction is supported formation þ resolution of rosettes will be an interesting by ongoing work [77], which appears to show that, in the avenue for future research. marginal range of Voronoi-based cell models, T1 rear- Another important aspect of this work is the use of a rangements can be easily triggered with a perturbation generalized Maxwell constraint counting argument to whose magnitude vanishes with the system size. This explain why rigidity transitions can occur in tissues in behavior could very well suggest an example where the general. Whereas conventional counting arguments suggest Maxwell constraint counting and linear-response theory [82] that the vertex model is always underconstrained, we fail to predict the loss of the rigidity. show such not to be the case through an accurate counting of mechanical constraints. For example, conventional con- ACKNOWLEDGMENTS straint counting would predict that a cellular material should The authors acknowledge Sebastian Streichan for pro- be stable only for Z>4. However, most epithelial tissues viding helpful comments, suggestions, and preliminary can be stable at precisely Z ¼ 3. This work explains why experimental data. The authors also thank Boris Z ¼ 3 states are stabilized and further offers a unifying Shraiman, Matthieu Wyart, Cristina Marchetti, Mark perspective on why a rigidity transition can be expected at all Bowick, Jen Schwarz, Jin-Ah Park, Lisa Manning, in a cellular material by using a Maxwell constraint counting Xiaoming Mao, Frederick MacKintosh, Eric DeGiuli, approach. Our understanding of this rigidity transition is Karen Kasza, and Bulbul Chakraborty for helpful discus- based on a generalized Maxwell counting that properly sions. This research was supported in part by National counts the critical role of tension. It is a natural extension of Science Foundation (NSF) Grant No. PHY-1748958, the stress-driven rigidity transition in packings and fiber 2 – National Institutes of Health (NIH) Grant networks close to the critical topology with Z ¼ d [66,83 No. R25GM067110, and the Gordon and Betty Moore 85]. The appearance of the tension can also be viewed as a Foundation Grant No. 2919.01. The authors acknowledge result of the geometric incompatibility of the material metric the support of the Northeastern University Discovery as studied in a continuum treatment of the vertex model by Cluster. Moshe and co-workers [18]. The similar insight of tension and geometric compatibility was also proposed in a recent study by Merkel and Manning [11] on the 3D Voronoi-based APPENDIX A: SIMULATION METHODS cell model, a variation of the vertex model [9,17,86]. We simulate tissues composed of N 625 cells under ¼ pffiffiffiffi The constraint counting developed here can be easily periodic boundary conditions with box size L ¼ N ¼ 25, generalized to the Voronoi-based cell models. In those which is commensurate with the value of the preferred cell 2F models, the d.o.f. are cell centers (whose number is ) area of unity. We use Kp ¼ 1 for the data shown in this rather than cell vertices, and the cell shapes are obtained paper. The value of p0 is varied between 3.6 and 4.3. We from a Voronoi tessellation of the cell centers. For the carry out the protocol using two different types of initial same energy in Eq. (1), the number of constraints is the states at Z ¼ 3: (i) random tissue networks, where cell same as that given by Eq. (8), i.e., maxð2F; E − E0 þ FÞ, shapes are obtained from the random Voronoi tessellations which leads to a constraint counting that is given by in the Voronoi-based cell model [9], and (ii) an ordered − 2 − 2 − N0 Nss ¼ F maxð F; E E0 þ FÞ. When edges hexagonal tiling [2,4]. After initialization, the vertex carry tensions and the number of constraints is greater positions in the tissue are evolved to the energy minimum than the d.o.f. E − E0 þ F>2F, we expect there to be using the Broyden-Fletcher-Goldfarb-Shanno method [88] redundant constraints which will result in Nss states of self- until the residual force magnitude on all vertices is less than stress. Interestingly, when 2F>E− E0 þ F, the counting 10−8. The simulations methods are based on Ref. [8]. leads to N0 ¼ 0, indicating that the system is always The shear modulus is calculated by considering the marginal and, thus, the lack of a transition to fluid in linear response of the tissue to an infinitesimal affine strain the linear response, as recently observed and studied in γ.Itisgivenby[89] Ref. [87]. However, the solid-fluid transition is still ∂2 evidently shown in the Voronoi-based model with self- U −1 G G − G − Ξ μM Ξ ν: ¼ affine nonaffine ¼ ∂γ2 i iμjν j ðA1Þ propelling cells [9,24]. The nature of this observed tran- γ¼0 sition is determined by two aspects. First, there is still a Ξ transition to a self-stressed state when the number of edges Here, iμ is the derivative of the force on vertex i with under tension, E − E0, becomes larger than the number of respect to the strain, given by cells F at the same threshold p0 ¼ 3.81. Second, the ∂2U marginal nature of the system when p0 > 3.81 makes Ξ ≡ iμ ∂γ∂ : ðA2Þ the linear response fragile and plastic nonlinear processes riμ γ¼0
011029-9 LE YAN and DAPENG BI PHYS. REV. X 9, 011029 (2019) X APPENDIX B: MEAN-FIELD MODEL 1 Aα ¼ jg jh αðR × L Þ · z FOR DETERMINING THE SLOPE 4 mi m i m m;i OF THE RIGIDITY LINE X 1 ⊥ ¼ − jg jh αR · L ; The rigidity transition line in the ðp0;ZÞ plane can be 4 mi m i m expressed as Eq. (4) or its inverse X m;i Pα ¼ jhmαjLm: ðC3Þ −1 p0ðZÞ¼p0ðZ ¼ 3ÞþB ðZ − 3Þ; ðB1Þ m
B−1 ∂ ∂ where ¼ p0ðZÞ= ZjZ¼3, which captures how the critical preferred perimeter changes when the number of First, it is instructive to calculate the total force on each edges per cell changes. We then take a mean-field approach vertex, which is given by the gradient of Eq. (C1): and replace p0ðZÞ with the perimeter of a regular n-sided polygon (with area of unity) of n ¼ 2Z=ðZ − 2Þ sides: ∂ XF ∂ ∂ ffiffiffi U Aα Pα p fiμ ≡ − ¼ KAAα þ KPðPα − P0Þ : 2 π ∂R μ ∂R μ ∂R μ Pn ¼ n tanð =nÞðB2Þ i α¼1 i i B−1 ∂ ∂ ≈ 0 260 ðC4Þ to obtain ¼ Pn= ZjZ¼3 . . This mean-field ap- proximation serves well to predict the slope of the transition line [Fig. 2(a)]. However, the properties of regular polygons Based on the definitions of the area and perimeter are not helpful for predicting the value of the transition [Eq. (C3)], we obtain the geometric derivatives point itself [p0ðZ ¼ 3Þ¼3.81ðdisorderedÞ; 3.72ðhexÞ], which is an emergent collective property of the system. ∂ 1 X Aα − ⊥ ∂ ¼ 2 jgmijhmαLmemμ and APPENDIX C: GENERALIZED MAXWELL Riμ m X CONSTRAINT COUNTING FOR THE ∂Pα VERTEX MODEL ∂ ¼ jhmαjgmiemμ: ðC5Þ Riμ m 1. Hessian (dynamical matrix) of the vertex model We begin by calculating the Hessian matrix ofP the total Therefore, using Eq. (C5), it is possible to rewrite Eq. (C4) energy, which is given by Eq. (1). Now, since α Aα ¼ const for a confluent tissue not undergoing cell number in terms of a sum over all edges: changes, it is equivalent to work with the simpler form of energy: XE 1 ⊥ f μ ¼ − − K σ L jg je μ þK τ g e μ ; ðC6Þ 1 XF i 2 A m m mi m P m mi m 2 − 2 m¼1 U ¼ 2 ½KAAα þ KPðPα P0Þ : ðC1Þ α¼1
In Eq. (C1), we also multiply the energy by a factor of 1=2 where we have separated the force on each vertex into two for convenience when taking derivatives. mutually orthogonal components: a component along the ˆ ≡ The network of cells is given by vertices (at positions edge vector em Lm=Lm and a component perpendicular to the edge vector given by eˆ ⊥ ¼ Rðπ=2Þeˆ , or the edge fRig) and edges (specified by edge vectors Lm). The m m relation between edges and vertices is given by the directed vector rotated by π=2. The magnitudes of force are given by adjacency matrix gmi. Edge vectors can be calculated using X X τ ¼ jh αjðPα − P0Þ¼½ðPα − P0ÞþðPβ − P0Þ ; Lm ¼ gmiRi; ðC2Þ m m α i X σm ¼ hmαAα ¼ Aα − Aβ: ðC7Þ where gmi is nonzero if vertex i is on edge m;itisþ1 if α vertex i forms the head of edge vector m and −1 if it is the tail. The relation between edges and facets (cells) is given τ by edge-facet adjacency matrices hmα, which is þ1 ð−1Þ if KP m is the line tension due to the mismatch between the edge m goes counterclockwise (clockwise) around facet α actual perimeters of cell α and β from the preferred and zero otherwise. In this notation, both the area and the perimeter. KAσm=2 is the pressure due to the difference perimeter can be written in vertices and edges: between the areas of cells α and β.
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The Hessian is a 2V × 2V matrix given by M ¼ SCST 1 KPH −2KAΣ ∂ 2 k ⊥ k ⊥ T U ¼ðS jS Þ 1 Σ 1 ˜ ðS jS Þ : ðC12Þ Mjν;iμ ≡ 2KA KPT þ 4KAH ∂R ν∂R μ j i X ∂τ ∂ k ⊥ m emμ Here, we define the augmented matrix S ≡ ðS jS Þ and a ¼ K g e μ þ τ g P ∂ mi m ∂ m mi 2E × 2E block matrix m Rjν Rjν ⊥ 1 ∂σ ∂ 1 m ⊥ Lmμ K H − K Σ − KA Lmμjgmijþ σmjgmij ; ðC8Þ P 2 A 2 ∂Rjν ∂Rjν C ¼ 1 1 ˜ : ðC13Þ 2 KAΣ KPT þ 4 KAH which can be written explicitly as C is block antisymmetric. XE XE 2. Constraint counting Mjν;iμ ¼ KPenνgnj Hnm emμgmi m¼1 n¼1 We apply the rank-nullity theorem to the Hessian matrix ⊥ ⊥ in order to accurately count the number of constraints [61]: þ KPenνgnj Tnm emμgmi KA ⊥ ˜ ⊥ rankðMÞþnullityðMÞ¼2V: ðC14Þ þ 4 enνgnj Hnm emμgmi M KA Σ ⊥ The rank of give the number of independent constraints þ 2 enνgnj nm emμgmi for force balance [61,67]. The nullity of M is the dimen- sionality of the null space of M, i.e., the number of KA ⊥ − e νg Σ e μg : C9 δ 0 2 n nj nm m mi ð Þ solutions to Mj Rii¼ . These correspond to the number of ways to infinitesimally perturb the vertices such that the energy of the system does not change. This is exactly the Equation (C9) has been written in a symmetric form number of zero modes in the Hessian, which we term N0. 2 2 where each term is a ( V × V) matrix decomposed into a Therefore, the constraint counting becomes product of three matrices ð2V × EÞðE × EÞðE × 2VÞ. Here, the E × E matrices are defined as no: of indep: constraints z}|{d:o:f: zfflfflfflffl}|fflfflfflffl{ nullityðMÞ¼N0 ¼ 2V − rankðMÞ : ðC15Þ XF H h αh α ; nm ¼ j n m j To calculate the rank of M, we first note that it has a α¼1 trivial upper bound of 2V (in which case N0 ¼ 0). XF ˜ Additionally, since M is a product of S and C and S is H ¼ L L h αh α; nm n m n m full rank, it follows that [90] rankðMÞ¼rankðCÞ, and by α¼1 τ application of Guttman rank additivity of a block matrix T ¼ m δ ; [91] we can write down a general relation nm L nm m Σ σ δ 1 nm ¼ m nm: ðC10Þ ˜ rankðMÞ¼rankðKPHÞþrank KPT þ 4 KAH The matrices H and H˜ give the relationship between 1 2 −1Σ †Σ edges and facets. T and Σ are diagonal matrices which þ 4 KAKP H : ðC16Þ give the tension and pressure, respectively, on each edge. The geometric property of edges and the relationship In Eq. (C16), H† is the Morse-Penrose pseudoinverse of H, 2 between edges and vertices are captured by the V × E since below we show that it is always less than full rank and matrices therefore not invertible. Since H can be expressed as a product of matrices with lesser rank F [Eq. (C10)], it k always holds that rank H F. The reason that the Sn;j;ν ¼ enνgnj; ð Þ¼ ⊥ ⊥ pseudoinverse can be done without changing the actual Sn;j;ν ¼ enνgnj; ðC11Þ dimension of the C matrix is that one can show that the rank of Σ is also F, and these F directions are not independent of which are mutually independent by definition. In matrix the F directions in H. form, Eq. (C9) can be written using Eqs. (C11) and (C10) in In the case of KA ¼ 0, rankðMÞ¼rankðHÞþrankðTÞ. matrix form as a product of block matrices: The rank of T is precisely given by the number of edges
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z}|{d:o:f: zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{no of indep constraints N0 ¼ 2V − min ½2V;maxð2F; E − E0 þ FÞ : ðC19Þ
We explicitly test this in a system of F ¼ 625 cells at fixed Z ¼ 3. In Fig. 6, the number of zero modes is first calculated directed from the Hessian matrix and plotted for states at various values of p0. The quantity 2V − ðE − E0 þ FÞ is also plotted as a function of p0 for the same states, and they closely track the behavior of N0 until 2V − ðE − E0 þ FÞ becomes larger than the upper limit of zero modes, 2V − 2F. Equation (C18) also allows us to make interesting predictions. Using the fixed topological relations E ¼ 2 − 2 2 FIG. 6. Maxwell counting in the vertex model is given by ZV= and F ¼ðZ ÞV= , Eq. (C18) becomes a con- N0 ¼ 2V − maxð2F; E − E0 þ FÞ. dition for the fraction of zero modes in the system f0, which is given by τ 0 − with m > [Eq. (C10)], which we define as E E0 in the N0 4 − Z 0 f0 ≡ ∈ 0; : ðC20Þ main text. Hence, the constraint counting for when KA ¼ 2V 2 is given by Equation (C18) suggests that, when Z>4, no zero modes should be present in the system. At Z ¼ 3, the fraction of N0 ¼ 2V − min ½2V;ðE − E0 þ FÞ : ðC17Þ zero modes is at most 1=2. In Table I, we list the Maxwell counting for both KA ¼ 0 The minimum function is used in Eq. (C17), because the and KA > 0 cases, with the rigidity criterion as well as the rank of a matrix cannot exceed its largest dimension. number of zero modes in the system for all possible cases. When KA > 0, the rank M can take on a continuum of values; however, it is still possible to give bounds for the rank of M. First, even when all tensions and pressures APPENDIX D: EFFECTIVE MEDIUM THEORY DETAILS vanish, i.e., τm ¼ 0, σm ¼ 0 on all edges, the ranks of both H and H˜ still remain at F. This result provides the lower We derive the effective medium theory for the case ≥ 2 bound of rankðMÞ F. On the other hand, even when all with only perimeter constraints, KA ¼ 0 and KP > 0.To tensions and pressures are finite, the second term in approximate the scattering behavior of the Hessian k kT ⊥ ⊥T Eq. (C16) cannot exceed E þ F. We therefore obtain the M ¼ KPS HS þ KPS TS , we propose the following limits for the constraint counting Eq. (C15): effective medium:
M¯ Mtopo Mss ¼ þ keff ; ðD1Þ 2V − ðE þ FÞ ≤ N0 ≤ 2V − 2F: ðC18Þ topo k kT with M mapping to the cell topology term KPS HS Mss When zero tension edges are taken into account, we obtain and keff corresponding to the stress contribu- ⊥ ⊥T the more general result tion KPS TS .
TABLE I. An overview of constraint counting in the vertex model. The cases for when KA ¼ 0 and KA > 0 are presented separately. The rigidity criterion is the minimum value of Z needed to ensure rigidity. Note that the counts of edges (E), vertices (V), and cells (F) are related through the Euler characteristic (V − E þ F ¼ 0) and the definition for vertex coordination Z ¼ 2E=V.
a Model N ¼rankðMÞ Rigidity criterion (N >N ) N0 ¼N −N constraint d:o:f: constraint ( d:o:f: constraint ð3 − ZÞV þ E0 if ð2 − 6=ZÞE
aExcluding the d¼2 number of trivial transitional zero modes.
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1. General stress distribution We consider a general distribution of random stresses on edges km ¼ðτm=LmÞ:
pðkÞ¼PρðkÞþð1 − PÞδðkÞ; ðD2Þ where function ρðkÞ is normalized in k>0 with a single stress scale k¯: 1 k ρðkÞ¼ ρˆ : ðD3Þ k¯ k¯ FIG. 7. Primitive cells of the honeycomb lattice.
Following the standard coherent potential approximation for each unit cell l at Rl ¼ Rðl1;l2Þ ¼ l1a1 þ l2a2, where a procedure [67], the self-consistent equation of the effective are primitive translation vectors. Setting the edge length of ffiffiffi pffiffi ffiffiffi pffiffi medium reads p 1 3 p 1 3 hexagons to unity, a1 ¼ 3ð2 ; 2 Þ and a2 ¼ 3ð− 2 ; 2 Þ. Z ∞ k − k −k In Fourier space, P dkρðkÞ eff þð1 − PÞ eff ¼ 0; X − 0 1 þðk − k ÞG 1 − k G iq·Rl eff m eff m uq ¼ ule ; ðD4Þ l 1 X P u ¼ u eiq·Rl ; ðD7Þ G¯ ⊥ G¯ ⊥ l q where Gm ¼h im ¼ iμ;jνgmieiμ gmjejν is singular as N q 6 − Gm ¼ðh=keffÞ with h ¼½ð ZÞ=Z . Expanding near van- the elastic energy change becomes ishing stress to the first order of keff, we get Z 1 X h P − h ∞ 1 Δ M¯ ¯ ρˆ U ¼ 2 uq · −q;q0 · uq0 ; ðD8Þ keff ¼ k ;a¼ dx ðxÞ: ðD5Þ 2N aP 1 − h 0 x q;q0 where N is the number of unit cells. For a homogeneous 2. Gm on honeycomb lattice honeycomb lattice,
We derive Gm on a honeycomb lattice, where Z ¼ 3. The ¯ M 0 ¼ Nδ 0 M : ðD9Þ honeycomb lattice contain two particles per unit cell, as q;q q;q q shown in Fig. 7, so the displacement is described by a four- We first need to compute the band structure for the first dimensional vector: term in the Hessian of the cell perimeter constraints when 0 keff ¼ . Each unit cell contains a hexagonal cell on a ul ¼ðul;A;x;ul;A;y;ul;B;x;ul;B;yÞðD6Þ lattice. The six edges of a cell, labeled in Fig. 7, are
pffiffiffi pffiffiffi pffiffi pffiffi 3 − 3 3 1 − 3 3 3 1 0 1 0 −1 − ið 2 qxþ2qyÞ − ið 2 qxþ2qyÞ B1;q ¼ð ; ; ; Þ; B2;q ¼ 2 e ; 2e ; 2 ;2 ; pffiffiffi pffiffiffi pffiffi pffiffi pffiffi pffiffi pffiffi pffiffi 3 − 3 3 1 − 3 3 3 − 3 1 − 3 − 3 − 3 ið 2 qxþ2qyÞ − ið 2 qxþ2qyÞ − i qx i qx 0 i qx 0 − i qx B3;q ¼ 2 e ; 2e ; 2 e ;2e ; B4;q ¼ð ;e ; ; e Þ; pffiffiffi pffiffiffi pffiffiffi pffiffiffi pffiffi pffiffi pffiffi pffiffi pffiffi pffiffi 3 − 3 1 − 3 3 − 3 −3 1 − 3 −3 3 1 3 − 3 −3 1 − 3 −3 − i qx − i qx ið 2 qx 2qyÞ ið 2 qx 2qyÞ − − ið 2 qx 2qyÞ ið 2 qx 2qyÞ B5;q ¼ 2 e ; 2e ; 2 e ;2e ; B6;q ¼ 2 ; 2; 2 e ;2e : ðD10Þ
The corresponding Hessian is X6 Mtopo Mtopo topo 1 12 M ¼ B ⊗ B − ¼ ; ðD11Þ q m;q l; q Mtopo Mtopo m;l¼1 21 2 where
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0 ffiffiffi ffiffiffi ffiffiffi pffiffi 1 3 p p 1 p 3 3 i2qy 2 ð1 − cos 3qxÞ 3i 2 sin 3qx − sin 2 qxe topo topo B C M1 ¼ M2 ¼ @ ffiffiffi ffiffiffi pffiffi ffiffiffi pffiffi A; ðD12Þ p 1 p 3 3 3 1 p 3 3 −i2qy − 3i 2 sin 3qx − sin 2 qxe 2 þ 2 cos 3qx − 2 cos 2 qx cos 2 qy
0 ffiffiffi ffiffiffi ffiffiffi pffiffi 1 3 p p 1 p 3 3 −i2qy 2 ð1 − cos 3qxÞ − 3i 2 sin 3qx − sin 2 qxe topo topo B C M12 ¼ M21 ¼ @ ffiffiffi ffiffiffi pffiffi pffiffi pffiffi A: ðD13Þ p 1 p 3 3 3 3 3 −i2qy 2 −i2qy −i3y − 3i 2 sin 3qx − sin 2 qxe −cos 2 qx þ 2 cos 2 qxe − e
There are three zero bands and one acoustic branch for each unit cell. Besides a normalization prefactor, they are g g f f ψ˜ ψ˜ ψ˜ ψ˜ 0;1 ¼ ˆ ; 0;2 ¼ ˆ ; 0;3 ¼ ˆ ; a ¼ ˆ ; ðD14Þ e−iq·e1 cðqÞg −e−iq·e1 cðqÞg e−iq·e1 f −e−iq·e1 f P P pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P 3 ˆ 3 ⊥ − ˆ 3 − ˆ where f eˆ eiq·ej , g eˆ e iq·ej , and c q − 1= 3 2η q e iq·ej with η q P ¼ j¼1 j ¼ j¼1 j ð Þ¼ f ½ þ ð Þ g j¼1 ð Þ¼ 1 ˆ ˆ iq·ðej−ekÞ 2 j≠k e . The acoustic branch corresponds to dilation modes of cells, obeying the dispersion relation topo λa ¼ 2½3 − ηðqÞ . 0 Include the internal tension keff > contribution to the Hessian matrix, X3 Mss Mss ss 1 12 k M ¼ k D ⊗ D − ¼ k ; ðD15Þ eff q eff j;q j; q eff Mss Mss j¼1 21 2 where vectors D correspond to the perpendicular directions
D1;q ¼ð−1; 0; 1; 0Þ; pffiffiffi pffiffiffi pffiffi pffiffi 1 − 3 3 3 − 3 3 1 3 ið 2 qxþ2qyÞ − ið 2 qxþ2qyÞ − ; D2;q ¼ 2 e ; 2 e ; 2 ; 2 pffiffiffi pffiffiffi pffiffi pffiffi 1 − − 3 3 3 − − 3 3 1 3 ið 2 qxþ2qyÞ ið 2 qxþ2qyÞ − − D3;q ¼ 2 e ; 2 e ; 2 ; 2 ; ðD16Þ
3 Mss Mss I 1 ¼ 2 ¼ 2 ; ðD17Þ
0 pffiffi pffiffi pffiffi 1 1 − 3 3 3 − 3 3 −1 − e i2qy cos q −i e i2qy sin q ss ss @ 2 2 x 2 2 x A M12 ¼ M21 ¼ pffiffi pffiffi pffiffi : ðD18Þ 3 − 3 3 3 − 3 3 − i2qy − i2qy i 2 e sin 2 qx 2 e cos 2 qx
The band structure of Mss includes one zero band, one acoustic branch, and two optical branches: f g g f ϕ˜ ϕ˜ ϕ˜ ϕ˜ 0 ¼ ˆ ; a ¼ ˆ ; o;1 ¼ ˆ ; o;2 ¼ ˆ ðD19Þ −e−iq·e1 f e−iq·e1 cðqÞg −e−iq·e1 cðqÞg e−iq·e1 f pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi λss 0 λss 3 − 3 2η 2 λss 3 3 2η 2 λss 3 corresponding to 0 ¼ , a ¼ 2 f½ þ ðqÞ = g, o;1 ¼ 2 þf½ þ ðqÞ = g, and o;2 ¼ . Mtopo Mss 0 Notice that the only acoustic branch of q is the only zero band of q . So when there is internal tension keff > , the full Hessian matrix
M¯ Mtopo Mss qðkeffÞ¼ q þ keff q ðD20Þ is invertible:
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Z 2 d q U U† M¯ −ω2I −1U U† Gm ¼ D1;−q · q q½ qðkeffÞ q q ·D1;q; ðD21Þ 1BZ v1 where
U ¼ðϕ0; ϕ ; ϕ 1; ϕ 2Þ q a o; o; 1 fg gf pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ − ˆ − ˆ − ˆ − ˆ : ðD22Þ 2 3 − ηðqÞ −e iq·e1 f e iq·e1 cðqÞg −e iq·e1 cðqÞg e iq·e1 f
1 So Gmjω¼0 ¼ð =keffÞ. necessary to stabilize the structure, as illustrated in the phase diagram with internal tension as a control parameter 3. Phase diagram far from Z =3 in the right panel in Fig. 8.AboveZc, the topology alone can rigidify the structure, and the solid-liquid transition pffiffiffiffiffiffi In the main text, we derive the phase diagram near the happens along the line Z − Z ∼ −k as predicted in 3 c critical point ðZ ¼ ;p0Þ, illustrated by the left panel in Ref. [66], shown in Fig. 8. In this new topology-dominated Fig. 8. Though we do see a crossover from the perimeter- region, the shear modulus crosses over to a linear depend- dominated region to the topology-dominated region, the ence on the junction connectivity Z − Z . nature of the rigidity does not change in the vicinity of c trijunction tissues Z ¼ 3. As long as the number of (a) (b) constraints 2F<2V and Z
FIG. 10. (a) Spatial autocorrelation of the vertex displacement field after a single edge is pinched at Z ¼ 3. The dot on each
curve indicates the location of the negative dip in Cd⃗ðrÞ which defines the swirl size ξd. (b) The swirl size ξd associated with a FIG. 8. Left: Phase diagram Z − p0. Right: Phase diagram 3 − 4 0 6 0 single edge pinch at Z ¼ . In the rigid tissue, the swirl size spans Z k, where Zc ¼ for KA > and Zc ¼ for KA ¼ . The on the order of the tissue size but decreases in the fluid tissue. solid-fluid boundary is shown by the blue line. The red lines This result suggests that, near the fluid-solid transition, a single separate the regions where mechanical moduli are dominated by pinched edge has long-range effects, whereas deeper in the fluid different control parameters. state the effect of pinching is much more local.
(a) (b)
FIG. 9. (a) Application of the edge pinch to an initially ordered hexagonal lattice. (b) The corresponding Z − p0 phase diagram. The color bar gives the fraction of floppy modes in the tissue.
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