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) Ã 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. Ã 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.