ARTICLES PUBLISHED ONLINE: 21 SEPTEMBER 2015 | DOI: 10.1038/NPHYS3471 A density-independent rigidity transition in biological tissues Dapeng Bi1, J. H. Lopez1, J. M. Schwarz1,2 and M. Lisa Manning1,2* Cell migration is important in many biological processes, including embryonic development, cancer metastasis and wound healing. In these tissues, a cell’s motion is often strongly constrained by its neighbours, leading to glassy dynamics. Although self-propelled particle models exhibit a density-driven glass transition, this does not explain liquid-to-solid transitions in confluent tissues, where there are no gaps between cells and therefore the density is constant. Here we demonstrate the existence of a new type of rigidity transition that occurs in the well-studied vertex model for confluent tissue monolayers at constant density. We find that the onset of rigidity is governed by a model parameter that encodes single-cell properties such as cell–cell adhesion and cortical tension, providing an explanation for liquid-to-solid transitions in confluent tissues and making testable predictions about how these transitions dier from those in particulate matter. mportant biological processes such as embryogenesis, find a new type of rigidity transition that is not controlled by the tumorigenesis and wound healing require cells to move density, but instead by a dimensionless target shape index that is Icollectively within a tissue. Recent experiments suggest that specified by single-cell properties. This rigidity transition possesses when cells are packed ever more densely, they start to exhibit several hallmarks of a second-order phase transition. These findings collective motion1–3 traditionally seen in non-living disordered provide a novel explanation for liquid-to-solid transitions in tissues systems such as colloids, granular matter, or foams4–6. that remain at constant density. Collective behaviours in these non-living systems are governed The vertex model, which agrees remarkably well with by a `jamming' transition from a fluid-like state to a solid-like state experimental data for confluent monolayers13–21, approximates that occurs as the density approaches a critical packing density of the monolayer as a collection of adjacent columnar cells. The 14,16 particles φc (ref. 6). This is also called a rigidity transition, because mechanical energy of a single cell labelled `i' is given by : the material becomes linearly stable with respect to infinitesimal D − 2 C 2 C perturbations and begins to support shear stresses. Many of these Ei KAi .Ai Ai0/ ξiPi γiPi (1) effects are also seen in self-propelled particle (SPP) models7. In SPP models, overdamped particles experience an active force that The first term results from a combination of three-dimensional causes them to move at a constant speed, and particles change cell incompressibility and the monolayer's resistance to height 15,23 direction owing to interactions with their neighbours or an external fluctuations or cell bulk elasticity . Then KAi is a height elasticity, bath. To model cells with a cortical network of actomyosin and and Ai and Ai0 are the actual and preferred cell cross-sectional areas. adhesive molecules on their surfaces, particles interact as simple The second term in equation (1) is quadratic in the cell repulsive disks or spheres, sometimes with an additional short-range cross-sectional perimeter Pi and models the active contractility 8,9 attraction . These models also exhibit a rigidity/jamming transition of the actin–myosin subcellular cortex, with elastic constant ξi at φc <1 (refs 1,8,10,11), and near the transition point they exhibit (ref. 14), and the last term represents an interfacial tension γi collective motion8 that is very similar to that seen in experiments12. set by a competition between the cortical tension and the energy 18,24 An important open question is whether the density-driven of cell–cell adhesion between two contacting cells. γi can be rigidity transition in SPP models explains the collective behaviour positive if the cortical tension is greater than the adhesive energy, or observed in non-proliferating confluent biological tissues, where negative if the adhesion dominates. It is also possible to incorporate there are no gaps between cells and the packing fraction φ is strong feedback between adhesion and cortical tension in this fixed at precisely unity. For example, zebrafish embryonic explants term18,25. As only the effective forces—the derivatives of the energy are confluent three-dimensional tissues where the cells divide with respect to the degrees of freedom—are physically relevant, D − 2 C − 2 slowly—and therefore the number of cells per unit volume remains equation (1) can be rewritten as Ei KAi .Ai Ai0/ ξi.Pi Pi0/ , nearly constant. Nevertheless, these tissues exhibit hallmarks of where Pi0 D−γi=.2ξi/ is an effective target shape index. glassy dynamics, such as caging behaviour and viscoelasticity. As discussed in ref. 16, when all single-cell properties are equal D D D D Furthermore, ectoderm tissues have longer relaxation timescales (KAi KA, ξi ξ, Ai0 A0, Pi0 P0), the total mechanical energy of a than mesendoderm tissues, suggesting ectoderm tissues are closer tissue containing N cells can be non-dimensionalized: to a glass transition, despite the fact that both tissue types have N N 1 Q 2 the same density . This indicates that there should be an additional 1 X X .pi −p0/ " D E D .aQ −1/2 C (2) parameter controlling the mechanical response in confluent tissues. K A2 i i r A 0 iD1 iD1 In this work, we study confluent monolayers using the vertex p 13–22 model , to determine how tissue mechanical response varies with where aQ i DAi=A0 and pQi DPi= A0 are the rescaled shape functions single-cell properties such as adhesion and cortical tension. We for area and perimeter. r DKAA0/ξ is the inverse perimeter modulus 1Department of Physics, Syracuse University, Syracuse, New York 13244, USA. 2Syracuse Biomaterials Institute, Syracuse, New York 13244, USA. *e-mail: [email protected] NATURE PHYSICS | ADVANCE ONLINE PUBLICATION | www.nature.com/naturephysics 1 © 2015 Macmillan Publishers Limited. All rights reserved ARTICLES NATURE PHYSICS DOI: 10.1038/NPHYS3471 abvalue of p0 and r tested, we obtained the distribution of energy 1.5 r = 0.5 barrier heights ρ.1"/. The functional form of the distribution r =0.1 becomes universal (Fig. 1a) when scaled by the mean energy 100 1.0 barrier height 1".r,p0/. The rescaled distribution is fitted well by a ) r =1 ε k k−1 Δ Δ −1 k-gamma distribution (k x exp.−kx/=.k − 1/W) with x D 1"=1" / 10 Δ 0.5 r =2 and kD2.2±0.2. The k-gamma distribution has been observed ( ρ 10−2 30–32 εεε r =10 in many non-biological disordered systems , and generically Δ 31,32 10−3 0.0 results from maximizing the entropy subject to constraints . This 0123456 3.0 3.2 3.4 3.6 3.8 4.0 confirms that the distribution of energy barriers depends on the Δε Δε p x = / 0 single-cell properties p0 and r only through its average 1". Figure 1b shows the dependence of 1" on p0 for various values Figure 1 | Energy barriers for local cellular rearrangements. a, Illustration of r. At p0 .3.8, the energy barriers are always finite—that is, cells of a T1 transition in a confluent tissue and the normalized distribution ρ of must put in some amount of work to deform and rearrange. Here normalized energy barrier heights 1"=1" for a large range of parameters the tissue behaves like a solid; it is a rigid material with a finite D D (r 0.5,1,2 and p0 3.2–3.7). They have a universal shape that is fitted well yield stress. As p0 is increased, the energy barriers decrease and by a k-gamma distribution (solid line), indicating that 1" completely become vanishingly small in the vicinity of p0 ≈ 3.8, so that cell specifies the distribution and describes the mechanical response. b, 1" as rearrangements cost almost no energy. This suggests that the vertex function of the target shape index p0 for various values of the inverse model may undergo a critical rigidity transition near p0 ≈3.8. perimeter modulus r. To test this hypothesis, we searched for a scaling collapse p based on theories for continuous phase transitions near a critical and p0 DP0= A0 is the target shape index or a preferred perimeter- point, such as the Ising model for ferromagnetism. Figure 1b to-area ratio. For simplicity, we focus on p0 >0. For p0 <0 there are demonstrates that r sets the overall scale of 1" as well as the two regimes: a static regime with results identical to those presented `sharpness' of the transition, whereas p0 controls the distance to the − ∗ here for 0<p0 <ppent, and a coarsening regime that is not consistent transition. This suggests that the trio .r1", r, p0 p0 / is analogous with most observations of biological tissues.p p Geometrically, to .m, h, T − Tc/ in the Ising model. Therefore, our scaling ansatz hex D 4 ≈ a regular hexagon correspondsp topp0 2 2 3 3.72 and a is that the order parameter r1".r, p0/ vanishes at the critical pent D − 1=4 ≈ D ∗ regular pentagon to p0 2 5.5 2 5/ 3.81. point p0 p0 , with fluctuations controlled by r. In that case, near In non-biological materials, bulk quantities such as shear/bulk the critical point the order parameter should obey the universal modulus, shear viscosity and yield stress are often used to describe scaling form33: the mechanical response to external perturbations.