1
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Kim K, Schwarz JM, Ben Amar M. A two-dimensional vertex model for curvy cell-cell interfaces at the subcellular scale. J R Soc Interface 2024; 21:20240193. [PMID: 39192725 DOI: 10.1098/rsif.2024.0193] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/20/2024] [Revised: 05/26/2024] [Accepted: 06/24/2024] [Indexed: 08/29/2024] Open
Abstract
Cross-sections of cell shapes in a tissue monolayer typically resemble a tiling of convex polygons. Yet, examples exist where the polygons are not convex with curved cell-cell interfaces, as seen in the adaxial epidermis. To date, two-dimensional vertex models predicting the structure and mechanics of cell monolayers have been mostly limited to convex polygons. To overcome this limitation, we introduce a framework to study curvy cell-cell interfaces at the subcellular scale within vertex models by using a parametrized curve between vertices that is expanded in a Fourier series and whose coefficients represent additional degrees of freedom. This extension to non-convex polygons allows for cells with the same shape index, or dimensionless perimeter, to be, for example, either elongated or globular with lobes. In the presence of applied, anisotropic stresses, we find that local, subcellular curvature or buckling can be energetically more favourable than larger scale deformations involving groups of cells. Inspired by recent experiments, we also find that local, subcellular curvature at cell-cell interfaces emerges in a group of cells in response to the swelling of additional cells surrounding the group. Our framework, therefore, can account for a wider array of multicellular responses to constraints in the tissue environment.
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Affiliation(s)
- Kyungeun Kim
- Department of Physics, Syracuse University , Syracuse, NY 13244, USA
| | - J M Schwarz
- Department of Physics, Syracuse University , Syracuse, NY 13244, USA
- Indian Creek Farm , Ithaca, NY 14850, USA
| | - Martine Ben Amar
- Laboratoire de Physique de l'Ecole Normale Supérieure, ENS, Université PSL, CNRS, Sorbonne Université, Université Paris Cité , 75005 Paris, France
- Institut Universitaire de Cancérologie, Faculté de Médecine, Sorbonne Université, 91 Boulevard de l'Hôpital , 75013 Paris, France
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2
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Divoux T, Agoritsas E, Aime S, Barentin C, Barrat JL, Benzi R, Berthier L, Bi D, Biroli G, Bonn D, Bourrianne P, Bouzid M, Del Gado E, Delanoë-Ayari H, Farain K, Fielding S, Fuchs M, van der Gucht J, Henkes S, Jalaal M, Joshi YM, Lemaître A, Leheny RL, Manneville S, Martens K, Poon WCK, Popović M, Procaccia I, Ramos L, Richards JA, Rogers S, Rossi S, Sbragaglia M, Tarjus G, Toschi F, Trappe V, Vermant J, Wyart M, Zamponi F, Zare D. Ductile-to-brittle transition and yielding in soft amorphous materials: perspectives and open questions. SOFT MATTER 2024. [PMID: 39028363 DOI: 10.1039/d3sm01740k] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 07/20/2024]
Abstract
Soft amorphous materials are viscoelastic solids ubiquitously found around us, from clays and cementitious pastes to emulsions and physical gels encountered in food or biomedical engineering. Under an external deformation, these materials undergo a noteworthy transition from a solid to a liquid state that reshapes the material microstructure. This yielding transition was the main theme of a workshop held from January 9 to 13, 2023 at the Lorentz Center in Leiden. The manuscript presented here offers a critical perspective on the subject, synthesizing insights from the various brainstorming sessions and informal discussions that unfolded during this week of vibrant exchange of ideas. The result of these exchanges takes the form of a series of open questions that represent outstanding experimental, numerical, and theoretical challenges to be tackled in the near future.
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Affiliation(s)
- Thibaut Divoux
- ENSL, CNRS, Laboratoire de physique, F-69342 Lyon, France.
| | - Elisabeth Agoritsas
- Department of Quantum Matter Physics (DQMP), University of Geneva, Quai Ernest-Ansermet 24, CH-1211 Geneva, Switzerland
| | - Stefano Aime
- Molecular, Macromolecular Chemistry, and Materials, ESPCI Paris, Paris, France
| | - Catherine Barentin
- Univ. de Lyon, Université Claude Bernard Lyon 1, CNRS, Institut Lumière Matière, F-69622 Villeurbanne, France
| | - Jean-Louis Barrat
- Laboratoire de Physique de l'Ecole Normale Supérieure, ENS, Université PSL, CNRS, Sorbonne Université, Université de Paris, F-75005 Paris, France
| | - Roberto Benzi
- Department of Physics & INFN, Tor Vergata University of Rome, Via della Ricerca Scientifica 1, 00133, Rome, Italy
| | - Ludovic Berthier
- Laboratoire Charles Coulomb (L2C), Université Montpellier, CNRS, Montpellier, France
| | - Dapeng Bi
- Department of Physics, Northeastern University, Boston, MA 02115, USA
| | - Giulio Biroli
- Laboratoire de Physique de l'Ecole Normale Supérieure, ENS, Université PSL, CNRS, Sorbonne Université, Université de Paris, F-75005 Paris, France
| | - Daniel Bonn
- Soft Matter Group, van der Waals-Zeeman Institute, University of Amsterdam, Science Park 904, 1098XH Amsterdam, The Netherlands
| | - Philippe Bourrianne
- PMMH, CNRS, ESPCI Paris, Université PSL, Sorbonne Université, Université Paris Cité, Paris, France
| | - Mehdi Bouzid
- Univ. Grenoble Alpes, CNRS, Grenoble INP, 3SR, F-38000 Grenoble, France
| | - Emanuela Del Gado
- Georgetown University, Department of Physics, Institute for Soft Matter Synthesis and Metrology, Washington, DC, USA
| | - Hélène Delanoë-Ayari
- Univ. de Lyon, Université Claude Bernard Lyon 1, CNRS, Institut Lumière Matière, F-69622 Villeurbanne, France
| | - Kasra Farain
- Soft Matter Group, van der Waals-Zeeman Institute, University of Amsterdam, Science Park 904, 1098XH Amsterdam, The Netherlands
| | - Suzanne Fielding
- Department of Physics, Durham University, South Road, Durham DH1 3LE, UK
| | - Matthias Fuchs
- Fachbereich Physik, Universität Konstanz, 78457 Konstanz, Germany
| | - Jasper van der Gucht
- Physical Chemistry and Soft Matter, Wageningen University & Research, Stippeneng 4, 6708WE Wageningen, The Netherlands
| | - Silke Henkes
- Lorentz Institute, Leiden University, 2300 RA Leiden, The Netherlands
| | - Maziyar Jalaal
- Institute of Physics, University of Amsterdam, Science Park 904, Amsterdam, The Netherlands
| | - Yogesh M Joshi
- Department of Chemical Engineering, Indian Institute of Technology, Kanpur 208016, Uttar Pradesh, India
| | - Anaël Lemaître
- Navier, École des Ponts, Univ Gustave Eiffel, CNRS, Marne-la-Vallée, France
| | - Robert L Leheny
- Department of Physics and Astronomy, Johns Hopkins University, Baltimore, Maryland 21218, USA
| | | | | | - Wilson C K Poon
- SUPA and the School of Physics and Astronomy, The University of Edinburgh, Peter Guthrie Tait Road, Edinburgh EH9 3FD, UK
| | - Marko Popović
- Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Str.38, 01187 Dresden, Germany
| | - Itamar Procaccia
- Dept. of Chemical Physics, The Weizmann Institute of Science, Rehovot 76100, Israel
- Sino-Europe Complex Science Center, School of Mathematics, North University of China, Shanxi, Taiyuan 030051, China
| | - Laurence Ramos
- Laboratoire Charles Coulomb (L2C), Université Montpellier, CNRS, Montpellier, France
| | - James A Richards
- SUPA and the School of Physics and Astronomy, The University of Edinburgh, Peter Guthrie Tait Road, Edinburgh EH9 3FD, UK
| | - Simon Rogers
- Department of Chemical and Biomolecular Engineering, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA
| | - Saverio Rossi
- LPTMC, CNRS-UMR 7600, Sorbonne Université, 4 Pl. Jussieu, F-75005 Paris, France
| | - Mauro Sbragaglia
- Department of Physics & INFN, Tor Vergata University of Rome, Via della Ricerca Scientifica 1, 00133, Rome, Italy
| | - Gilles Tarjus
- LPTMC, CNRS-UMR 7600, Sorbonne Université, 4 Pl. Jussieu, F-75005 Paris, France
| | - Federico Toschi
- Department of Applied Physics and Science Education, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands
- CNR-IAC, Via dei Taurini 19, 00185 Rome, Italy
| | - Véronique Trappe
- Department of Physics, University of Fribourg, Chemin du Musée 3, Fribourg 1700, Switzerland
| | - Jan Vermant
- Department of Materials, ETH Zürich, Vladimir Prelog Weg 5, 8032 Zürich, Switzerland
| | - Matthieu Wyart
- Department of Quantum Matter Physics (DQMP), University of Geneva, Quai Ernest-Ansermet 24, CH-1211 Geneva, Switzerland
| | - Francesco Zamponi
- Laboratoire de Physique de l'Ecole Normale Supérieure, ENS, Université PSL, CNRS, Sorbonne Université, Université de Paris, F-75005 Paris, France
- Dipartimento di Fisica, Sapienza Università di Roma, Piazzale Aldo Moro 5, 00185 Rome, Italy
| | - Davoud Zare
- Fonterra Research and Development Centre, Dairy Farm Road, Fitzherbert, Palmerston North 4442, New Zealand
- Nestlé Institute of Food Sciences, Nestlé Research, Vers Chez les Blancs, Lausanne, Switzerland
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3
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Hertaeg MJ, Fielding SM, Bi D. Discontinuous Shear Thickening in Biological Tissue Rheology. PHYSICAL REVIEW. X 2024; 14:011027. [PMID: 38994232 PMCID: PMC11238743 DOI: 10.1103/physrevx.14.011027] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Indexed: 07/13/2024]
Abstract
During embryonic morphogenesis, tissues undergo dramatic deformations in order to form functional organs. Similarly, in adult animals, living cells and tissues are continually subjected to forces and deformations. Therefore, the success of embryonic development and the proper maintenance of physiological functions rely on the ability of cells to withstand mechanical stresses as well as their ability to flow in a collective manner. During these events, mechanical perturbations can originate from active processes at the single-cell level, competing with external stresses exerted by surrounding tissues and organs. However, the study of tissue mechanics has been somewhat limited to either the response to external forces or to intrinsic ones. In this work, we use an active vertex model of a 2D confluent tissue to study the interplay of external deformations that are applied globally to a tissue with internal active stresses that arise locally at the cellular level due to cell motility. We elucidate, in particular, the way in which this interplay between globally external and locally internal active driving determines the emergent mechanical properties of the tissue as a whole. For a tissue in the vicinity of a solid-fluid jamming or unjamming transition, we uncover a host of fascinating rheological phenomena, including yielding, shear thinning, continuous shear thickening, and discontinuous shear thickening. These model predictions provide a framework for understanding the recently observed nonlinear rheological behaviors in vivo.
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Affiliation(s)
- Michael J Hertaeg
- Department of Physics, Durham University, Science Laboratories, South Road, Durham DH1 3LE, United Kingdom
| | - Suzanne M Fielding
- Department of Physics, Durham University, Science Laboratories, South Road, Durham DH1 3LE, United Kingdom
| | - Dapeng Bi
- Department of Physics, Northeastern University, Massachusetts 02115, USA
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4
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Huang J, Levine H, Bi D. Bridging the gap between collective motility and epithelial-mesenchymal transitions through the active finite voronoi model. SOFT MATTER 2023; 19:9389-9398. [PMID: 37795526 PMCID: PMC10843280 DOI: 10.1039/d3sm00327b] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 10/06/2023]
Abstract
We introduce an active version of the recently proposed finite Voronoi model of epithelial tissue. The resultant Active Finite Voronoi (AFV) model enables the study of both confluent and non-confluent geometries and transitions between them, in the presence of active cells. Our study identifies six distinct phases, characterized by aggregation-segregation, dynamical jamming-unjamming, and epithelial-mesenchymal transitions (EMT), thereby extending the behavior beyond that observed in previously studied vertex-based models. The AFV model with rich phase diagram provides a cohesive framework that unifies the well-observed progression to collective motility via unjamming with the intricate dynamics enabled by EMT. This approach should prove useful for challenges in developmental biology systems as well as the complex context of cancer metastasis. The simulation code is also provided.
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Affiliation(s)
- Junxiang Huang
- Department of Physics, Northeastern University, Boston, Massachusetts 02215, USA.
- Center for Theoretical Biological Physics, Northeastern University, Boston, Massachusetts 02215, USA
| | - Herbert Levine
- Department of Physics, Northeastern University, Boston, Massachusetts 02215, USA.
- Center for Theoretical Biological Physics, Northeastern University, Boston, Massachusetts 02215, USA
- Department of Bioengineering, Northeastern University, Boston, Massachusetts 02215, USA
| | - Dapeng Bi
- Department of Physics, Northeastern University, Boston, Massachusetts 02215, USA.
- Center for Theoretical Biological Physics, Northeastern University, Boston, Massachusetts 02215, USA
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5
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Khan AK, Muñoz-Castro G, Muñoz JJ. Single and two-cells shape analysis from energy functionals for three-dimensional vertex models. INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING 2023; 39:e3766. [PMID: 37551449 DOI: 10.1002/cnm.3766] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/30/2023] [Revised: 07/10/2023] [Accepted: 07/21/2023] [Indexed: 08/09/2023]
Abstract
Vertex models have been extensively used for simulating the evolution of multicellular systems, and have given rise to important global properties concerning their macroscopic rheology or jamming transitions. These models are based on the definition of an energy functional, which fully determines the cellular response and conclusions. While two-dimensional vertex models have been widely employed, three-dimensional models are far more scarce, mainly due to the large amount of configurations that they may adopt and the complex geometrical transitions they undergo. We here investigate the shape of single and two-cells configurations as a function of the energy terms, and we study the dependence of the final shape on the model parameters: namely the exponent of the term penalising cell-cell adhesion and surface contractility. In single cell analysis, we deduce analytically the radius and limit values of the contractility for linear and quadratic surface energy terms, in 2D and 3D. In two-cells systems, symmetrical and asymmetrical, we deduce the evolution of the aspect ratio and the relative radius. While in functionals with linear surface terms yield the same aspect ratio in 2D and 3D, the configurations when using quadratic surface terms are distinct. We relate our results with well-known solutions from capillarity theory, and verify our analytical findings with a three-dimensional vertex model.
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Affiliation(s)
- Ahmad K Khan
- Department of Mathematics, Universitat Politècnica de Catalunya, Barcelona, Spain
| | - Guillem Muñoz-Castro
- Department of Mathematics, Universitat Politècnica de Catalunya, Barcelona, Spain
| | - Jose J Muñoz
- Department of Mathematics, Universitat Politècnica de Catalunya, Barcelona, Spain
- Laboratori de Càlcul Numèric (LaCàN), Universitat Politècnica de Catalunya, Barcelona, Spain
- Centre Internacional de Mètodes Numèrics en Enginyeria (CIMNE), Barcelona, Spain
- Institut de Matemàtiques de la UPC-BarcelonaTech (IMTech), Barcelona, Spain
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6
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Mazarei M, Åström J, Westerholm J, Karttunen M. Effect of substrate heterogeneity and topology on epithelial tissue growth dynamics. Phys Rev E 2023; 108:054405. [PMID: 38115499 DOI: 10.1103/physreve.108.054405] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/20/2023] [Accepted: 09/20/2023] [Indexed: 12/21/2023]
Abstract
Tissue growth kinetics and interface dynamics depend on the properties of the tissue environment and cell-cell interactions. In cellular environments, substrate heterogeneity and geometry arise from a variety factors, such as the structure of the extracellular matrix and nutrient concentration. We used the CellSim3D model, a kinetic cell division simulator, to investigate the growth kinetics and interface roughness dynamics of epithelial tissue growth on heterogeneous substrates with varying topologies. The results show that the presence of quenched disorder has a clear effect on the colony morphology and the roughness scaling of the interface in the moving interface regime. In a medium with quenched disorder, the tissue interface has a smaller interface roughness exponent, α, and a larger growth exponent, β. The scaling exponents also depend on the topology of the substrate and cannot be categorized by well-known universality classes.
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Affiliation(s)
- Mahmood Mazarei
- Department of Physics and Astronomy, Western University, 1151 Richmond Street, London, Ontario, Canada N6A 3K7
| | - Jan Åström
- CSC Scientific Computing Ltd, Kägelstranden 14, 02150 Esbo, Finland
| | - Jan Westerholm
- Faculty of Science and Engineering, Åbo Akademi University, Vattenborgsvägen 3, FI-20500 Åbo, Finland
| | - Mikko Karttunen
- Department of Physics and Astronomy, Western University, 1151 Richmond Street, London, Ontario, Canada N6A 3K7
- Department of Chemistry, Western University, 1151 Richmond Street, London, Ontario, Canada N6A 5B7
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7
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Fielding SM, Cochran JO, Huang J, Bi D, Marchetti MC. Constitutive model for the rheology of biological tissue. Phys Rev E 2023; 108:L042602. [PMID: 37978678 DOI: 10.1103/physreve.108.l042602] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/07/2022] [Accepted: 09/13/2023] [Indexed: 11/19/2023]
Abstract
The rheology of biological tissue is key to processes such as embryo development, wound healing, and cancer metastasis. Vertex models of confluent tissue monolayers have uncovered a spontaneous liquid-solid transition tuned by cell shape; and a shear-induced solidification transition of an initially liquidlike tissue. Alongside this jamming/unjamming behavior, biological tissue also displays an inherent viscoelasticity, with a slow time and rate-dependent mechanics. With this motivation, we combine simulations and continuum theory to examine the rheology of the vertex model in nonlinear shear across a full range of shear rates from quastistatic to fast, elucidating its nonlinear stress-strain curves after the inception of shear of finite rate, and its steady state flow curves of stress as a function of strain rate. We formulate a rheological constitutive model that couples cell shape to flow and captures both the tissue solid-liquid transition and its rich linear and nonlinear rheology.
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Affiliation(s)
- Suzanne M Fielding
- Department of Physics, Durham University, Science Laboratories, South Road, Durham DH1 3LE, UK
| | - James O Cochran
- Department of Physics, Durham University, Science Laboratories, South Road, Durham DH1 3LE, UK
| | - Junxiang Huang
- Department of Physics, Northeastern University, Massachusetts 02115, USA
| | - Dapeng Bi
- Department of Physics, Northeastern University, Massachusetts 02115, USA
| | - M Cristina Marchetti
- Department of Physics, University of California, Santa Barbara, California 93106, USA
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8
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Sandoval M. Stiffening and dynamics of a two-dimensional active elastic solid. SOFT MATTER 2023; 19:6885-6895. [PMID: 37671426 DOI: 10.1039/d3sm00529a] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 09/02/2023]
Abstract
This work deals with the mechanical properties and dynamics of an active elastic solid defined as a two-dimensional network of active stochastic particles interacting by nonlinear hard springs. By proposing a discrete model, it is numerically found that when activity in the system is turned on, the active solid stiffens as a function of propulsion forces, thus deviating from equilibrium mechanics. To understand this effect a minimal stochastic model is offered, and a physical explanation based on spatial symmetry-breaking is put forward. In addition, the dynamics of the active solid in the absence of an external stress is also studied. From this, three main features are observed to emerge, namely, a collective behavior within the active solid, a time-density fluctuation, and oscillating dynamics of the internal stresses towards a steady state.
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Affiliation(s)
- Mario Sandoval
- Department of Physics, Complex Systems, Universidad Autonoma Metropolitana-Iztapalapa, Mexico City 09340, Mexico.
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9
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Zhao T, Yuan H. The analytical solution to the migration of an epithelial monolayer with a circular spreading front and its implications in the gap closure process. Biomech Model Mechanobiol 2023:10.1007/s10237-023-01723-4. [PMID: 37149822 DOI: 10.1007/s10237-023-01723-4] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/03/2022] [Accepted: 04/11/2023] [Indexed: 05/08/2023]
Abstract
The coordinated behaviors of epithelial cells are widely observed in tissue development, such as re-epithelialization, tumor growth, and morphogenesis. In these processes, cells either migrate collectively or organize themselves into specific structures to serve certain purposes. In this work, we study a spreading epithelial monolayer whose migrating front encloses a circular gap in the monolayer center. Such tissue is usually used to mimic the wound healing process in vitro. We model the epithelial sheet as a layer of active viscous polar fluid. With an axisymmetric assumption, the model can be analytically solved under two special conditions, suggesting two possible spreading modes for the epithelial monolayer. Based on these two sets of analytical solutions, we assess the velocity of the spreading front affected by the gap size, the active intercellular contractility, and the purse-string contraction acting on the spreading edge. Several critical values exist in the model parameters for the initiation of the gap closure process, and the purse-string contraction plays a vital role in governing the gap closure kinetics. Finally, the instability of the morphology of the spreading front was studied. Numerical calculations show how the perturbated velocities and the growth rates vary with respect to different model parameters.
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Affiliation(s)
- Tiankai Zhao
- Shenzhen Key Laboratory of Soft Mechanics & Smart Manufacturing, Department of Mechanics and Aerospace Engineering, Southern University of Science and Technology, Shenzhen, 518055, China.
| | - Hongyan Yuan
- Shenzhen Key Laboratory of Soft Mechanics & Smart Manufacturing, Department of Mechanics and Aerospace Engineering, Southern University of Science and Technology, Shenzhen, 518055, China
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10
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Li R, Moazzeni S, Liu L, Lin H. Micro and Macroscopic Stress-Strain Relations in Disordered Tessellated Networks. PHYSICAL REVIEW LETTERS 2023; 130:188201. [PMID: 37204891 PMCID: PMC10586522 DOI: 10.1103/physrevlett.130.188201] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/24/2022] [Accepted: 03/03/2023] [Indexed: 05/21/2023]
Abstract
We demonstrate that for a rigid and incompressible network in mechanical equilibrium, the microscopic stress and strain follows a simple relation, σ=pE, where σ is the deviatoric stress, E is a mean-field strain tensor, and p is the hydrostatic pressure. This relationship arises as the natural consequence of energy minimization or equivalently, mechanical equilibration. The result suggests not only that the microscopic stress and strain are aligned in the principal directions, but also microscopic deformations are predominantly affine. The relationship holds true regardless of the different (foam or tissue) energy model considered, and directly leads to a simple prediction for the shear modulus, μ=⟨p⟩/2, where ⟨p⟩ is the mean pressure of the tessellation, for general randomized lattices.
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Affiliation(s)
- Ran Li
- Department of Mechanical and Aerospace Engineering, Rutgers, The State University of New Jersey, 98 Brett Road, Piscataway, New Jersey 08854, USA
| | - Seyedsajad Moazzeni
- Department of Mechanical and Aerospace Engineering, Rutgers, The State University of New Jersey, 98 Brett Road, Piscataway, New Jersey 08854, USA
| | - Liping Liu
- Department of Mechanical and Aerospace Engineering, Rutgers, The State University of New Jersey, 98 Brett Road, Piscataway, New Jersey 08854, USA
- Department of Mathematics, Rutgers, The State University of New Jersey, 110 Frelinghuysen Road, Piscataway, New Jersey 08854, USA
| | - Hao Lin
- Department of Mechanical and Aerospace Engineering, Rutgers, The State University of New Jersey, 98 Brett Road, Piscataway, New Jersey 08854, USA
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11
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Staddon MF, Hernandez A, Bowick MJ, Moshe M, Marchetti MC. The role of non-affine deformations in the elastic behavior of the cellular vertex model. SOFT MATTER 2023; 19:3080-3091. [PMID: 37039037 DOI: 10.1039/d2sm01580c] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/04/2023]
Abstract
The vertex model of epithelia describes the apical surface of a tissue as a tiling of polygonal cells, with a mechanical energy governed by deviations in cell shape from preferred, or target, area, A0, and perimeter, P0. The model exhibits a rigidity transition driven by geometric incompatibility as tuned by the target shape index, . For with p*(6) the perimeter of a regular hexagon of unit area, a cell can simultaneously attain both the preferred area and preferred perimeter. As a result, the tissue is in a mechanically soft compatible state, with zero shear and Young's moduli. For p0 < p*(6), it is geometrically impossible for any cell to realize the preferred area and perimeter simultaneously, and the tissue is in an incompatible rigid solid state. Using a mean-field approach, we present a complete analytical calculation of the linear elastic moduli of an ordered vertex model. We analyze a relaxation step that includes non-affine deformations, leading to a softer response than previously reported. The origin of the vanishing shear and Young's moduli in the compatible state is the presence of zero-energy deformations of cell shape. The bulk modulus exhibits a jump discontinuity at the transition and can be lower in the rigid state than in the fluid-like state. The Poisson's ratio can become negative which lowers the bulk and Young's moduli. Our work provides a unified treatment of linear elasticity for the vertex model and demonstrates that this linear response is protocol-dependent.
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Affiliation(s)
- Michael F Staddon
- Center for Systems Biology Dresden, Dresden, Germany.
- Max Planck Institute for the Physics of Complex Systems, Dresden, Germany
- Max Planck Institute of Molecular Cell Biology and Genetics, Dresden, Germany
| | - Arthur Hernandez
- Department of Physics, University of California Santa Barbara, Santa Barbara, CA 93106, USA.
| | - Mark J Bowick
- Kavli Institute for Theoretical Physics, University of California Santa Barbara, Santa Barbara, CA 93106, USA.
| | - Michael Moshe
- Racah Institute of Physics, The Hebrew University of Jerusalem, Jerusalem, 91904, Israel
| | - M Cristina Marchetti
- Department of Physics, University of California Santa Barbara, Santa Barbara, CA 93106, USA.
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12
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Chen S, Markovich T, MacKintosh FC. Nonaffine Deformation of Semiflexible Polymer and Fiber Networks. PHYSICAL REVIEW LETTERS 2023; 130:088101. [PMID: 36898114 DOI: 10.1103/physrevlett.130.088101] [Citation(s) in RCA: 3] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/11/2022] [Accepted: 01/18/2023] [Indexed: 06/18/2023]
Abstract
Networks of semiflexible or stiff polymers such as most biopolymers are known to deform inhomogeneously when sheared. The effects of such nonaffine deformation have been shown to be much stronger than for flexible polymers. To date, our understanding of nonaffinity in such systems is limited to simulations or specific 2D models of athermal fibers. Here, we present an effective medium theory for nonaffine deformation of semiflexible polymer and fiber networks, which is general to both 2D and 3D and in both thermal and athermal limits. The predictions of this model are in good agreement with both prior computational and experimental results for linear elasticity. Moreover, the framework we introduce can be extended to address nonlinear elasticity and network dynamics.
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Affiliation(s)
- Sihan Chen
- Department of Physics and Astronomy, Rice University, Houston, Texas 77005, USA
- Center for Theoretical Biological Physics, Rice University, Houston, Texas 77005, USA
| | - Tomer Markovich
- Center for Theoretical Biological Physics, Rice University, Houston, Texas 77005, USA
- School of Mechanical Engineering, Tel Aviv University, Tel Aviv 69978, Israel
- Center for Physics and Chemistry of Living Systems, Tel Aviv University, Tel Aviv 69978, Israel
| | - Fred C MacKintosh
- Department of Physics and Astronomy, Rice University, Houston, Texas 77005, USA
- Center for Theoretical Biological Physics, Rice University, Houston, Texas 77005, USA
- Department of Chemical and Biomolecular Engineering, Rice University, Houston, Texas 77005, USA
- Department of Chemistry, Rice University, Houston, Texas 77005, USA
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13
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Picchi Scardaoni M. Energetic convenience of cell division in biological tissues. Phys Rev E 2022; 106:054405. [PMID: 36559362 DOI: 10.1103/physreve.106.054405] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/06/2022] [Accepted: 10/25/2022] [Indexed: 11/16/2022]
Abstract
A typical feature of living tissues is proliferation by division: it is a fundamental aspect of many biological processes, including embryonic development, morphogenesis, and cancer growth. Here, we study the energetics of cell division occurring in epithelia, highlighting the interplay of the key parameters ruling mitosis. We find the existence of a region, in the parameter space, which is independent of the cell elasticity and weakly dependent on the ratio between mother and daughter cells areas. In this region, cell division is energetically favorable. Our results may lead to an exact characterization of cells having anomalous proliferation.
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Affiliation(s)
- Marco Picchi Scardaoni
- Department of Civil and Industrial Engineering, University of Pisa, Largo Lucio Lazzarino 2, 56122 Pisa, Italy
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14
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Hopkins A, Chiang M, Loewe B, Marenduzzo D, Marchetti MC. Local Yield and Compliance in Active Cell Monolayers. PHYSICAL REVIEW LETTERS 2022; 129:148101. [PMID: 36240394 DOI: 10.1103/physrevlett.129.148101] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/08/2022] [Accepted: 08/23/2022] [Indexed: 06/16/2023]
Abstract
The rheology of biological tissue plays an important role in many processes, from organ formation to cancer invasion. Here, we use a multiphase field model of motile cells to simulate active microrheology within a tissue monolayer. When unperturbed, the tissue exhibits a transition between a solidlike state and a fluidlike state tuned by cell motility and deformability-the ratio of the energetic costs of steric cell-cell repulsion and cell-edge tension. When perturbed, solid tissues exhibit local yield-stress behavior, with a threshold force for the onset of motion of a probe particle that vanishes upon approaching the solid-to-liquid transition. This onset of motion is qualitatively different in the low and high deformability regimes. At high deformability, the tissue is amorphous when solid, it responds compliantly to deformations, and the probe transition to motion is smooth. At low deformability, the monolayer is more ordered translationally and stiffer, and the onset of motion appears discontinuous. Our results suggest that cellular or nanoparticle transport in different types of tissues can be fundamentally different and point to ways in which it can be controlled.
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Affiliation(s)
- Austin Hopkins
- Department of Physics, University of California Santa Barbara, Santa Barbara, California 93106, USA
| | - Michael Chiang
- SUPA, School of Physics and Astronomy, University of Edinburgh, Peter Guthrie Tait Road, Edinburgh EH9 3FD, United Kingdom
| | - Benjamin Loewe
- SUPA, School of Physics and Astronomy, University of Edinburgh, Peter Guthrie Tait Road, Edinburgh EH9 3FD, United Kingdom
| | - Davide Marenduzzo
- SUPA, School of Physics and Astronomy, University of Edinburgh, Peter Guthrie Tait Road, Edinburgh EH9 3FD, United Kingdom
| | - M Cristina Marchetti
- Department of Physics, University of California Santa Barbara, Santa Barbara, California 93106, USA
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15
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Sandoval M. Minimal model of an active solid deviates from equilibrium mechanics. THE EUROPEAN PHYSICAL JOURNAL B 2022; 95:154. [DOI: 10.1140/epjb/s10051-022-00421-x] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/22/2022] [Accepted: 09/06/2022] [Indexed: 09/02/2023]
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16
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Hernandez A, Staddon MF, Bowick MJ, Marchetti MC, Moshe M. Anomalous elasticity of a cellular tissue vertex model. Phys Rev E 2022; 105:064611. [PMID: 35854605 DOI: 10.1103/physreve.105.064611] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/22/2021] [Accepted: 06/07/2022] [Indexed: 06/15/2023]
Abstract
Vertex models, such as those used to describe cellular tissue, have an energy controlled by deviations of each cell area and perimeter from target values. The constrained nonlinear relation between area and perimeter leads to new mechanical response. Here we provide a mean-field treatment of a highly simplified model: a uniform network of regular polygons with no topological rearrangements. Since all polygons deform in the same way, we only need to analyze the ground states and the response to deformations of a single polygon (cell). The model exhibits the known transition between a fluid/compatible state, where the cell can accommodate both target area and perimeter, and a rigid/incompatible state. We calculate and measure the mechanical resistance to various deformation protocols and discover that at the onset of rigidity, where a single zero-energy ground state exists, linear elasticity fails to describe the mechanical response to even infinitesimal deformations. In particular, we identify a breakdown of reciprocity expressed via different moduli for compressive and tensile loads, implying nonanalyticity of the energy functional. We give a pictorial representation in configuration space that reveals that the complex elastic response of the vertex model arises from the presence of two distinct sets of reference states (associated with target area and target perimeter). Our results on the critically compatible tissue provide a new route for the design of mechanical metamaterials that violate or extend classical elasticity.
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Affiliation(s)
- Arthur Hernandez
- Department of Physics, University of California Santa Barbara, Santa Barbara, California 93106, USA
| | - Michael F Staddon
- Center for Systems Biology, Dresden 01307, Germany
- Max Planck Institute of Molecular Cell Biology and Genetics, Dresden 01307, Germany
- Max Planck Institute for the Physics of Complex Systems, Dresden 01187, Germany
| | - Mark J Bowick
- Department of Physics, University of California Santa Barbara, Santa Barbara, California 93106, USA
- Kavli Institute for Theoretical Physics, University of California Santa Barbara, Santa Barbara, California 93106, USA
| | - M Cristina Marchetti
- Department of Physics, University of California Santa Barbara, Santa Barbara, California 93106, USA
| | - Michael Moshe
- Racah Institute of Physics, The Hebrew University of Jerusalem, Jerusalem 91904, Israel
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17
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Tong S, Singh NK, Sknepnek R, Košmrlj A. Linear viscoelastic properties of the vertex model for epithelial tissues. PLoS Comput Biol 2022; 18:e1010135. [PMID: 35587514 PMCID: PMC9159552 DOI: 10.1371/journal.pcbi.1010135] [Citation(s) in RCA: 8] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/27/2022] [Revised: 06/01/2022] [Accepted: 04/25/2022] [Indexed: 12/13/2022] Open
Abstract
Epithelial tissues act as barriers and, therefore, must repair themselves, respond to environmental changes and grow without compromising their integrity. Consequently, they exhibit complex viscoelastic rheological behavior where constituent cells actively tune their mechanical properties to change the overall response of the tissue, e.g., from solid-like to fluid-like. Mesoscopic mechanical properties of epithelia are commonly modeled with the vertex model. While previous studies have predominantly focused on the rheological properties of the vertex model at long time scales, we systematically studied the full dynamic range by applying small oscillatory shear and bulk deformations in both solid-like and fluid-like phases for regular hexagonal and disordered cell configurations. We found that the shear and bulk responses in the fluid and solid phases can be described by standard spring-dashpot viscoelastic models. Furthermore, the solid-fluid transition can be tuned by applying pre-deformation to the system. Our study provides insights into the mechanisms by which epithelia can regulate their rich rheological behavior.
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Affiliation(s)
- Sijie Tong
- Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, New Jersey, United States of America
| | - Navreeta K. Singh
- Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, New Jersey, United States of America
| | - Rastko Sknepnek
- School of Science and Engineering, University of Dundee, Dundee, United Kingdom
- School of Life Sciences, University of Dundee, Dundee, United Kingdom
| | - Andrej Košmrlj
- Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, New Jersey, United States of America
- Princeton Institute of Materials, Princeton University, Princeton, New Jersey, United States of America
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