1
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Golkov R, Shokef Y. Many-body interactions between contracting living cells. THE EUROPEAN PHYSICAL JOURNAL. E, SOFT MATTER 2024; 47:14. [PMID: 38372851 PMCID: PMC10876807 DOI: 10.1140/epje/s10189-024-00407-w] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/15/2023] [Accepted: 01/25/2024] [Indexed: 02/20/2024]
Abstract
The organization of live cells into tissues and their subsequent biological function involves inter-cell mechanical interactions, which are mediated by their elastic environment. To model this interaction, we consider cells as spherical active force dipoles surrounded by an unbounded elastic matrix. Even though we assume that this elastic medium responds linearly, each cell's regulation of its mechanical activity leads to nonlinearities in the emergent interactions between cells. We study the many-body nature of these interactions by considering several geometries that include three or more cells. We show that for different regulatory behaviors of the cells' activity, the total elastic energy stored in the medium differs from the superposition of all two-body interactions between pairs of cells within the system. Specifically, we find that the many-body interaction energy between cells that regulate their position is smaller than the sum of interactions between all pairs of cells in the system, while for cells that do not regulate their position, the many-body interaction is larger than the superposition prediction. Thus, such higher-order interactions should be considered when studying the mechanics of multiple cells in proximity.
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Affiliation(s)
- Roman Golkov
- Department of Mechanical Engineering, Shamoon College of Engineering, Ashdod, 77245, Israel
| | - Yair Shokef
- 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.
- Center for Computational Molecular and Materials Science, Tel Aviv University, Tel Aviv, 69978, Israel.
- International Institute for Sustainability with Knotted Chiral Meta Matter, Hiroshima University, Higashi-Hiroshima, Hiroshima, 739-8526, Japan.
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2
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Yang H, Berthier E, Li C, Ronceray P, Han YL, Broedersz CP, Cai S, Guo M. Local response and emerging nonlinear elastic length scale in biopolymer matrices. Proc Natl Acad Sci U S A 2023; 120:e2304666120. [PMID: 37252962 PMCID: PMC10265995 DOI: 10.1073/pnas.2304666120] [Citation(s) in RCA: 5] [Impact Index Per Article: 5.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/20/2023] [Accepted: 04/16/2023] [Indexed: 06/01/2023] Open
Abstract
Nonlinear stiffening is a ubiquitous property of major types of biopolymers that make up the extracellular matrices (ECM) including collagen, fibrin, and basement membrane. Within the ECM, many types of cells such as fibroblasts and cancer cells have a spindle-like shape that acts like two equal and opposite force monopoles, which anisotropically stretch their surroundings and locally stiffen the matrix. Here, we first use optical tweezers to study the nonlinear force-displacement response to localized monopole forces. We then propose an effective-probe scaling argument that a local point force application can induce a stiffened region in the matrix, which can be characterized by a nonlinear length scale R* that increases with the increasing force magnitude; the local nonlinear force-displacement response is a result of the nonlinear growth of this effective probe that linearly deforms an increasing portion of the surrounding matrix. Furthermore, we show that this emerging nonlinear length scale R* can be observed around living cells and can be perturbed by varying matrix concentration or inhibiting cell contractility.
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Affiliation(s)
- Haiqian Yang
- Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA02139
| | - Estelle Berthier
- Arnold-Sommerfeld-Center for Theoretical Physics and Center for NanoScience, Ludwig-Maximilians-Universität München, MünchenD-80333, Germany
| | - Chenghai Li
- Department of Mechanical and Aerospace Engineering, University of California, San Diego, La Jolla, CA92093
| | - Pierre Ronceray
- Aix Marseille University, CNRS, CINAM, Turing Center for Living Systems, 13288Marseille, France
| | - Yu Long Han
- Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA02139
| | - Chase P. Broedersz
- Arnold-Sommerfeld-Center for Theoretical Physics and Center for NanoScience, Ludwig-Maximilians-Universität München, MünchenD-80333, Germany
- Department of Physics and Astronomy, Vrije Universiteit Amsterdam, Amsterdam1081 HV, The Netherlands
| | - Shengqiang Cai
- Department of Mechanical and Aerospace Engineering, University of California, San Diego, La Jolla, CA92093
| | - Ming Guo
- Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA02139
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3
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Benoist F, Saggiorato G, Lenz M. Generic stress rectification in nonlinear elastic media. SOFT MATTER 2023; 19:2970-2976. [PMID: 37014008 PMCID: PMC10131159 DOI: 10.1039/d2sm01606k] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Indexed: 05/17/2023]
Abstract
Stress propagation in nonlinear media is crucial in cell biology, where molecular motors exert anisotropic force dipoles on the fibrous cytoskeleton. While the force dipoles can be either contractile or expansile, a medium made of fibers which buckle under compression rectifies these stresses towards a biologically crucial contraction. A general understanding of this rectification phenomenon as a function of the medium's elasticity is however lacking. Here we use theoretical continuum elasticity to show that rectification is actually a very general effect in nonlinear materials subjected to anisotropic internal stresses. We analytically show that both bucklable and constitutively linear materials subjected to geometrical nonlinearities rectify small forces towards contraction, while granular-like materials rectify towards expansion. Using simulations, we moreover show that these results extend to larger forces. Beyond fiber networks, these results could shed light on the propagation of stresses in brittle or granular materials following a local plastic rearrangement.
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Affiliation(s)
- Félix Benoist
- Université Paris-Saclay, CNRS, LPTMS, 91400, Orsay, France.
| | | | - Martin Lenz
- Université Paris-Saclay, CNRS, LPTMS, 91400, Orsay, France.
- PMMH, CNRS, ESPCI Paris, PSL University, Sorbonne Université, Université de Paris, F-, 75005, Paris, France
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4
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Lerner E, Bouchbinder E. Anomalous linear elasticity of disordered networks. SOFT MATTER 2023; 19:1076-1080. [PMID: 36661121 PMCID: PMC9906635 DOI: 10.1039/d2sm01253g] [Citation(s) in RCA: 4] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 09/14/2022] [Accepted: 01/06/2023] [Indexed: 06/17/2023]
Abstract
Continuum elasticity is a powerful tool applicable in a broad range of physical systems and phenomena. Yet, understanding how and on what scales material disorder may lead to the breakdown of continuum elasticity is not fully understood. We show, based on recent theoretical developments and extensive numerical computations, that disordered elastic networks near a critical rigidity transition, such as strain-stiffened fibrous biopolymer networks that are abundant in living systems, reveal an anomalous long-range linear elastic response below a correlation length. This emergent anomalous elasticity, which is non-affine in nature, is shown to feature a qualitatively different multipole expansion structure compared to ordinary continuum elasticity, and a slower spatial decay of perturbations. The potential degree of universality of these results, their implications (e.g. for cell-cell communication through biological extracellular matrices) and open questions are briefly discussed.
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Affiliation(s)
- Edan Lerner
- Institute for Theoretical Physics, University of Amsterdam, Science Park 904, Amsterdam 1098 XH, The Netherlands.
| | - Eran Bouchbinder
- Chemical and Biological Physics Department, Weizmann Institute of Science, Rehovot 7610001, Israel.
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5
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Little J, Levine AJ, Singh AR, Bruinsma R. Finite-strain elasticity theory and liquid-liquid phase separation in compressible gels. Phys Rev E 2023; 107:024418. [PMID: 36932516 DOI: 10.1103/physreve.107.024418] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/21/2022] [Accepted: 12/21/2022] [Indexed: 06/18/2023]
Abstract
The theory of finite-strain elasticity is applied to the phenomenon of cavitation observed in polymer gels following liquid-liquid phase separation of the solvent, which opens a fascinating window on the role of finite-strain elasticity theory in soft materials in general. We show that compressibility effects strongly enhance cavitation in simple materials that obey neo-Hookean elasticity. On the other hand, cavitation phenomena in gels of flexible polymers in a binary solvent that phase separates are surprisingly similar to those of incompressible materials. We find that, as a function of the interfacial energy between the two solvent components, there is a sharp transition between cavitation and classical nucleation and growth. Next, biopolymer gels are characterized by strain hardening and even very low levels of strain hardening turn out to suppress cavitation in polymer gels that obey Flory-Huggins theory in the absence of strain hardening. Our results indicate that cavitation is, in essence, not possible for polymer networks that show strain hardening.
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Affiliation(s)
- Justin Little
- Department of Physics and Astronomy, University of California, Los Angeles, California 90095, USA
| | - Alex J Levine
- Department of Physics and Astronomy, University of California, Los Angeles, California 90095, USA
- Department of Chemistry and Biochemistry, University of California, Los Angeles, California 90095, USA
- Department of Computational Medicine, David Geffen School of Medicine, University of California, Los Angeles, California 90095, USA
| | - Amit R Singh
- Department of Mechanical Engineering, Birla Institute of Technology and Science, Pilani, RJ 333031, India
| | - Robijn Bruinsma
- Department of Physics and Astronomy, University of California, Los Angeles, California 90095, USA
- Department of Chemistry and Biochemistry, University of California, Los Angeles, California 90095, USA
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6
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Goren S, Levin M, Brand G, Lesman A, Sorkin R. Probing Local Force Propagation in Tensed Fibrous Gels. SMALL (WEINHEIM AN DER BERGSTRASSE, GERMANY) 2023; 19:e2202573. [PMID: 36433830 DOI: 10.1002/smll.202202573] [Citation(s) in RCA: 6] [Impact Index Per Article: 6.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/25/2022] [Revised: 10/31/2022] [Indexed: 06/16/2023]
Abstract
Fibrous hydrogels are a key component of soft animal tissues. They support cellular functions and facilitate efficient mechanical communication between cells. Due to their nonlinear mechanical properties, fibrous materials display non-trivial force propagation at the microscale, that is enhanced compared to that of linear-elastic materials. In the body, tissues are constantly subjected to external loads that tense or compress them, modifying their micro-mechanical properties into an anisotropic state. However, it is unknown how force propagation is modified by this isotropic-to-anisotropic transition. Here, force propagation in tensed fibrin hydrogels is directly measured. Local perturbations are induced by oscillating microspheres using optical tweezers. 1-point and 2-point microrheology are combined to simultaneously measure the shear modulus and force propagation. A mathematical framework to quantify anisotropic force propagation trends is suggested. Results show that force propagation becomes anisotropic in tensed gels, with, surprisingly, stronger response to perturbations perpendicular to the axis of tension. Importantly, external tension can also increase the range of force transmission. Possible implications and future directions for research are discussed. These results suggest a mechanism for favored directions of mechanical communication between cells in a tissue under external loads.
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Affiliation(s)
- Shahar Goren
- School of Chemistry, Raymond & Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, P.O. Box 39040, Tel Aviv, 6997801, Israel
- School of Mechanical Engineering, The Iby and Aladar Fleischman Faculty of Engineering, Tel Aviv University, P.O. Box 39040, Tel Aviv, 6997801, Israel
- Center for Physics and Chemistry of Living Systems, Tel Aviv University, P.O. Box 39040, Tel Aviv, 6997801, Israel
- Center for Light-Matter Interactions, Tel Aviv University, P.O. Box 39040, Tel Aviv, 6997801, Israel
| | - Maayan Levin
- School of Chemistry, Raymond & Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, P.O. Box 39040, Tel Aviv, 6997801, Israel
- Center for Physics and Chemistry of Living Systems, Tel Aviv University, P.O. Box 39040, Tel Aviv, 6997801, Israel
| | - Guy Brand
- School of Chemistry, Raymond & Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, P.O. Box 39040, Tel Aviv, 6997801, Israel
| | - Ayelet Lesman
- School of Mechanical Engineering, The Iby and Aladar Fleischman Faculty of Engineering, Tel Aviv University, P.O. Box 39040, Tel Aviv, 6997801, Israel
- Center for Physics and Chemistry of Living Systems, Tel Aviv University, P.O. Box 39040, Tel Aviv, 6997801, Israel
| | - Raya Sorkin
- School of Chemistry, Raymond & Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, P.O. Box 39040, Tel Aviv, 6997801, Israel
- Center for Physics and Chemistry of Living Systems, Tel Aviv University, P.O. Box 39040, Tel Aviv, 6997801, Israel
- Center for Light-Matter Interactions, Tel Aviv University, P.O. Box 39040, Tel Aviv, 6997801, Israel
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7
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Mao X, Shokef Y. Introduction to force transmission by nonlinear biomaterials. SOFT MATTER 2021; 17:10172-10176. [PMID: 34755159 DOI: 10.1039/d1sm90194j] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/13/2023]
Abstract
Xiaoming Mao and Yair Shokef introduce the Soft Matter themed collection on force transmission by nonlinear biomaterials.
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Affiliation(s)
- Xiaoming Mao
- Department of Physics, University of Michigan, Ann Arbor, Michigan, 48109, USA.
| | - Yair Shokef
- School of Mechanical Engineering, Sackler Center for Computational Molecular and Materials Science, and Center for Physics and Chemistry of Living Systems, Tel Aviv University, Tel Aviv, 69978, Israel.
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8
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Sirote C, Shokef Y. Mean-field interactions between living cells in linear and nonlinear elastic matrices. Phys Rev E 2021; 104:024411. [PMID: 34525613 DOI: 10.1103/physreve.104.024411] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/26/2021] [Accepted: 07/28/2021] [Indexed: 01/23/2023]
Abstract
Living cells respond to mechanical changes in the matrix surrounding them by applying contractile forces that are in turn transmitted to distant cells. We consider simple effective geometries for the spatial arrangement of cells, we calculate the mechanical work that each cell performs in order to deform the matrix, and study how that energy changes when a contracting cell is surrounded by other cells with similar properties and behavior. Cells regulating the displacements that they generate are attracted to each other in a manner that does not depend on the cell's rigidity. Whereas cells regulating the active stress that they apply repel each other. This repulsion depends on the cell's bulk modulus in spherical geometry, while in cylindrical geometries the interaction depends also on their shear modulus. In nonlinear, strain-stiffening matrices, for displacement regulation, in the presence of other cells, cell contraction is limited due to the divergence of the shear stress. For stress regulation, the interaction energy drops at the nonlinear stiffening stress. Our theoretical work provides insight into matrix-mediated interactions between contractile cells and on the role of their mechanical regulatory behavior.
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Affiliation(s)
- Chaviva Sirote
- Department of Biomedical Engineering, Tel Aviv University, Tel Aviv 69978, Israel
| | - Yair Shokef
- School of Mechanical Engineering, Tel Aviv University, Tel Aviv 69978, Israel.,Sackler Center for Computational Molecular and Materials Science, Tel Aviv University, Tel Aviv 69978, Israel.,Center for Physics and Chemistry of Living Systems, Tel Aviv University, Tel Aviv 69978, Israel
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9
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Alisafaei F, Chen X, Leahy T, Janmey PA, Shenoy VB. Long-range mechanical signaling in biological systems. SOFT MATTER 2021; 17:241-253. [PMID: 33136113 PMCID: PMC8385661 DOI: 10.1039/d0sm01442g] [Citation(s) in RCA: 24] [Impact Index Per Article: 8.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/11/2023]
Abstract
Cells can respond to signals generated by other cells that are remarkably far away. Studies from at least the 1920's showed that cells move toward each other when the distance between them is on the order of a millimeter, which is many times the cell diameter. Chemical signals generated by molecules diffusing from the cell surface would move too slowly and dissipate too fast to account for these effects, suggesting that they might be physical rather than biochemical. The non-linear elastic responses of sparsely connected networks of stiff or semiflexible filament such as those that form the extracellular matrix (ECM) and the cytoskeleton have unusual properties that suggest multiple mechanisms for long-range signaling in biological tissues. These include not only direct force transmission, but also highly non-uniform local deformations, and force-generated changes in fiber alignment and density. Defining how fibrous networks respond to cell-generated forces can help design new methods to characterize abnormal tissues and can guide development of improved biomimetic materials.
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Affiliation(s)
- Farid Alisafaei
- Center for Engineering Mechanobiology, University of Pennsylvania, Philadelphia, PA 19104, USA. and Department of Materials Science and Engineering, School of Engineering and Applied Science, University of Pennsylvania, Philadelphia, PA 19104, USA
| | - Xingyu Chen
- Center for Engineering Mechanobiology, University of Pennsylvania, Philadelphia, PA 19104, USA. and Department of Materials Science and Engineering, School of Engineering and Applied Science, University of Pennsylvania, Philadelphia, PA 19104, USA
| | - Thomas Leahy
- Center for Engineering Mechanobiology, University of Pennsylvania, Philadelphia, PA 19104, USA. and Department of Bioengineering, School of Engineering and Applied Science, University of Pennsylvania, Philadelphia, PA 19104, USA and McKay Orthopaedic Research Laboratory, University of Pennsylvania, Philadelphia, PA 19104, USA
| | - Paul A Janmey
- Center for Engineering Mechanobiology, University of Pennsylvania, Philadelphia, PA 19104, USA. and Institute for Medicine and Engineering, University of Pennsylvania, 3340 Smith Walk, Philadelphia, PA 19104, USA and Departments of Physiology, and Physics & Astronomy, University of Pennsylvania, Philadelphia, PA 19104, USA
| | - Vivek B Shenoy
- Center for Engineering Mechanobiology, University of Pennsylvania, Philadelphia, PA 19104, USA. and Department of Materials Science and Engineering, School of Engineering and Applied Science, University of Pennsylvania, Philadelphia, PA 19104, USA
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10
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Wang H, Xu X. Continuum elastic models for force transmission in biopolymer gels. SOFT MATTER 2020; 16:10781-10808. [PMID: 33289764 DOI: 10.1039/d0sm01451f] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/12/2023]
Abstract
We review continuum elastic models for the transmission of both external forces and internal active cellular forces in biopolymer gels, and relate them to recent experiments. Rather than being exhaustive, we focus on continuum elastic models for small affine deformations and intend to provide a systematic continuum method and some analytical perspectives on the study of force transmission in biopolymer gels. We start from a very brief review of the nonlinear mechanics of individual biopolymers and a summary of constitutive models for the nonlinear elasticity of biopolymer gels. We next show that the simple 3-chain model can give predictions that fit well the shear experiments of some biopolymer gels, including the effects of strain-stiffening and negative normal stress. We then review continuum models for the transmission of internal active forces that are induced by a spherically contracting cell embedded in a three-dimensional biopolymer gel. Various scaling regimes for the decay of cell-induced displacements are identified for linear isotropic and anisotropic materials, and for biopolymer gels with nonlinear compressive-softening and strain-stiffening elasticity, respectively. After that, we present (using an energetic approach) the generic and unified continuum theory proposed in [D. Ben-Yaakov et al., Soft Matter, 2015, 11, 1412] about how the transmission of forces in the biogel matrix can mediate long-range interactions between cells with mechanical homeostasis. We show the predictions of the theory in a special hexagonal multicellular array, and relate them to recent experiments. Finally, we conclude this paper with comments on the limitations and outlook of continuum modeling, and highlight the need for complementary theoretical approaches, such as discrete network simulations, to force transmission in biopolymer gels and phenomenological active gel theories for multicellular systems.
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Affiliation(s)
- Haiqin Wang
- Technion - Israel Institute of Technology, Haifa, 32000, Israel.
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11
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Zheng Y, Fan Q, Eddy CZ, Wang X, Sun B, Ye F, Jiao Y. Modeling multicellular dynamics regulated by extracellular-matrix-mediated mechanical communication via active particles with polarized effective attraction. Phys Rev E 2020; 102:052409. [PMID: 33327171 DOI: 10.1103/physreve.102.052409] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/03/2020] [Accepted: 11/02/2020] [Indexed: 01/23/2023]
Abstract
Collective cell migration is crucial to many physiological and pathological processes such as embryo development, wound healing, and cancer invasion. Recent experimental studies have indicated that the active traction forces generated by migrating cells in a fibrous extracellular matrix (ECM) can mechanically remodel the ECM, giving rise to bundlelike mesostructures bridging individual cells. Such fiber bundles also enable long-range propagation of cellular forces, leading to correlated migration dynamics regulated by the mechanical communication among the cells. Motivated by these experimental discoveries, we develop an active-particle model with polarized effective attractions (APPA) to investigate emergent multicellular migration dynamics resulting from ECM-mediated mechanical communications. In particular, the APPA model generalizes the classic active-Brownian-particle (ABP) model by imposing a pairwise polarized attractive force between the particles, which depends on the instantaneous dynamic states of the particles and mimics the effective mutual pulling between the cells via the fiber bundle bridge. The APPA system exhibits enhanced aggregation behaviors compared to the classic ABP system, and the contrast is more apparent at lower particle densities and higher rotational diffusivities. Importantly, in contrast to the classic ABP system where the particle velocities are not correlated for all particle densities, the high-density phase of the APPA system exhibits strong dynamic correlations, which are characterized by the slowly decaying velocity correlation functions with a correlation length comparable to the linear size of the high-density phase domain (i.e., the cluster of particles). The strongly correlated multicellular dynamics predicted by the APPA model is subsequently verified in in vitro experiments using MCF-10A cells. Our studies indicate the importance of incorporating ECM-mediated mechanical coupling among the migrating cells for appropriately modeling emergent multicellular dynamics in complex microenvironments.
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Affiliation(s)
- Yu Zheng
- Department of Physics, Arizona State University, Tempe, Arizona 85287, USA
| | - Qihui Fan
- Beijing National Laboratory for Condensed Matte Physics and CAS Key Laboratory of Soft Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
- School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
| | - Christopher Z Eddy
- Department of Physics, Oregon State University, Corvallis, Oregon 97331, USA
| | - Xiaochen Wang
- Beijing National Laboratory for Condensed Matte Physics and CAS Key Laboratory of Soft Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
- School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
| | - Bo Sun
- Department of Physics, Oregon State University, Corvallis, Oregon 97331, USA
| | - Fangfu Ye
- Beijing National Laboratory for Condensed Matte Physics and CAS Key Laboratory of Soft Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
- School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
- Wenzhou Institute, University of Chinese Academy of Sciences, Wenzhou, Zhejiang 325001, China
| | - Yang Jiao
- Department of Physics, Arizona State University, Tempe, Arizona 85287, USA
- Materials Science and Engineering, Arizona State University, Tempe, Arizona 85287, USA
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12
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Zheng Y, Nan H, Liu Y, Fan Q, Wang X, Liu R, Liu L, Ye F, Sun B, Jiao Y. Modeling cell migration regulated by cell extracellular-matrix micromechanical coupling. Phys Rev E 2020; 100:043303. [PMID: 31770879 DOI: 10.1103/physreve.100.043303] [Citation(s) in RCA: 22] [Impact Index Per Article: 5.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/25/2019] [Indexed: 01/24/2023]
Abstract
Cell migration in fibrous extracellular matrix (ECM) is crucial to many physiological and pathological processes such as tissue regeneration, immune response, and cancer progression. During migration, individual cells can generate active pulling forces via actomyosin contraction, which are transmitted to the ECM fibers through focal adhesion complexes, remodel the ECM, and eventually propagate to and can be sensed by other cells in the system. The microstructure and physical properties of the ECM can also significantly influence cell migration, e.g., via durotaxis and contact guidance. Here, we develop a computational model for two-dimensional cell migration regulated by cell-ECM micromechanical coupling. Our model explicitly takes into account a variety of cellular-level processes, including focal adhesion formation and disassembly, active traction force generation and cell locomotion due to actin filament contraction, transmission and propagation of tensile forces in the ECM, as well as the resulting ECM remodeling. We validate our model by accurately reproducing single-cell dynamics of MCF-10A breast cancer cells migrating on collagen gels and show that the durotaxis and contact guidance effects naturally arise as a consequence of the cell-ECM micromechanical interactions considered in the model. Moreover, our model predicts strongly correlated multicellular migration dynamics, which are resulted from the ECM-mediated mechanical coupling among the migrating cell and are subsequently verified in in vitro experiments using MCF-10A cells. Our computational model provides a robust tool to investigate emergent collective dynamics of multicellular systems in complex in vivo microenvironment and can be utilized to design in vitro microenvironments to guide collective behaviors and self-organization of cells.
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Affiliation(s)
- Yu Zheng
- Department of Physics, Arizona State University, Tempe, Arizona 85287, USA
| | - Hanqing Nan
- Materials Science and Engineering, Arizona State University, Tempe, Arizona 85287, USA
| | - Yanping Liu
- College of Physics, Chongqing University, Chongqing 401331, China
| | - Qihui Fan
- Beijing National Laboratory for Condensed Matte Physics and CAS Key Laboratory of Soft Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China.,School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
| | - Xiaochen Wang
- Beijing National Laboratory for Condensed Matte Physics and CAS Key Laboratory of Soft Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China.,School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
| | - Ruchuan Liu
- College of Physics, Chongqing University, Chongqing 401331, China
| | - Liyu Liu
- College of Physics, Chongqing University, Chongqing 401331, China
| | - Fangfu Ye
- Beijing National Laboratory for Condensed Matte Physics and CAS Key Laboratory of Soft Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China.,School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
| | - Bo Sun
- Department of Physics, Oregon State University, Corvallis, Oregon 97331, USA
| | - Yang Jiao
- Department of Physics, Arizona State University, Tempe, Arizona 85287, USA.,Materials Science and Engineering, Arizona State University, Tempe, Arizona 85287, USA
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13
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Elastic Anisotropy Governs the Range of Cell-Induced Displacements. Biophys J 2020; 118:1152-1164. [PMID: 31995739 DOI: 10.1016/j.bpj.2019.12.033] [Citation(s) in RCA: 27] [Impact Index Per Article: 6.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/28/2019] [Revised: 12/22/2019] [Accepted: 12/27/2019] [Indexed: 12/20/2022] Open
Abstract
The unique nonlinear mechanics of the fibrous extracellular matrix (ECM) facilitates long-range cell-cell mechanical communications that would be impossible for linear elastic substrates. Past research has described the contribution of two separated effects on the range of force transmission, including ECM elastic nonlinearity and fiber alignment. However, the relation between these different effects is unclear, and how they combine to dictate force transmission range is still elusive. Here, we combine discrete fiber simulations with continuum modeling to study the decay of displacements induced by a contractile cell in fibrous networks. We demonstrate that fiber nonlinearity and fiber reorientation both contribute to the strain-induced elastic anisotropy of the cell's local environment. This elastic anisotropy is a "lumped" parameter that governs the slow decay of displacements, and it depends on the magnitude of applied strain, either an external tension or an internal contraction, as a model of the cell. Furthermore, we show that accounting for artificially prescribed elastic anisotropy dictates the decay of displacements induced by a contracting cell. Our findings unify previous single effects into a mechanical theory that explains force transmission in fibrous networks. This work may provide insights into biological processes that involve communication of distant cells mediated by the ECM, such as those occurring in morphogenesis, wound healing, angiogenesis, and cancer metastasis. It may also provide design parameters for biomaterials to control force transmission between cells as a way to guide morphogenesis in tissue engineering.
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14
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Gomez D, Natan S, Shokef Y, Lesman A. Mechanical Interaction between Cells Facilitates Molecular Transport. ACTA ACUST UNITED AC 2019; 3:e1900192. [PMID: 32648678 DOI: 10.1002/adbi.201900192] [Citation(s) in RCA: 11] [Impact Index Per Article: 2.2] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/20/2019] [Revised: 09/24/2019] [Accepted: 09/25/2019] [Indexed: 01/06/2023]
Abstract
In vivo, eukaryotic cells are embedded in a matrix environment, where they grow and develop. Generally, this extracellular matrix (ECM) is an anisotropic fibrous structure, through which macromolecules and biochemical signaling molecules at the nanometer scale diffuse. The ECM is continuously remodeled by cells, via mechanical interactions, which lead to a potential link between biomechanical and biochemical cell-cell interactions. Here, it is studied how cell-induced forces applied on the ECM impact the biochemical transport of molecules between distant cells. It is experimentally observed that cells remodel the ECM by increasing fiber alignment and density of the matrix between them over time. Using random walk simulations on a 3D lattice, elongated fixed obstacles are implemented that mimic the fibrous ECM structure. Both diffusion of a tracer molecule and the mean first-passage time a molecule secreted from one cell takes to reach another cell are measured. The model predicts that cell-induced remodeling can lead to a dramatic speedup in the transport of molecules between cells. Fiber alignment and densification cause reduction of the transport dimensionality from a 3D to a much more rapid 1D process. Thus, a novel mechanism of mechano-biochemical feedback in the regulation of long-range cell-cell communication is suggested.
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Affiliation(s)
- David Gomez
- School of Mechanical Engineering, Tel Aviv University, Tel Aviv, 69978, Israel
| | - Sari Natan
- School of Mechanical Engineering, Tel Aviv University, Tel Aviv, 69978, Israel
| | - Yair Shokef
- School of Mechanical Engineering, Tel Aviv University, Tel Aviv, 69978, Israel.,Sackler Center for Computational Molecular and Materials Science, Tel Aviv University, Tel Aviv, 69978, Israel
| | - Ayelet Lesman
- School of Mechanical Engineering, Tel Aviv University, Tel Aviv, 69978, Israel
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15
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Proestaki M, Ogren A, Burkel B, Notbohm J. Modulus of Fibrous Collagen at the Length Scale of a Cell. EXPERIMENTAL MECHANICS 2019; 59:1323-1334. [PMID: 31680700 PMCID: PMC6824437 DOI: 10.1007/s11340-018-00453-4] [Citation(s) in RCA: 13] [Impact Index Per Article: 2.6] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/05/2023]
Abstract
The extracellular matrix provides macroscale structural support to tissues as well as microscale mechanical cues, like stiffness, to the resident cells. As those cues modulate gene expression, proliferation, differentiation, and motility, quantifying the stiffness that cells sense is crucial to understanding cell behavior. Whereas the macroscopic modulus of a collagen network can be measured in uniform extension or shear, quantifying the local stiffness sensed by a cell remains a challenge due to the inhomogeneous and nonlinear nature of the fiber network at the scale of the cell. To address this challenge, we designed an experimental method to measure the modulus of a network of collagen fibers at this scale. We used spherical particles of an active hydrogel (poly N-isopropylacrylamide) that contract when heated, thereby applying local forces to the collagen matrix and mimicking the contractile forces of a cell. After measuring the particles' bulk modulus and contraction in networks of collagen fibers, we applied a nonlinear model for fibrous materials to compute the modulus of the local region surrounding each particle. We found the modulus at this length scale to be highly heterogeneous, with modulus varying by a factor of 3. In addition, at different values of applied strain, we observed both strain stiffening and strain softening, indicating nonlinearity of the collagen network. Thus, this experimental method quantifies local mechanical properties in a fibrous network at the scale of a cell, while also accounting for inherent nonlinearity.
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Affiliation(s)
- M Proestaki
- Department of Engineering Physics, University of Wisconsin-Madison, Madison, WI
| | - A Ogren
- Department of Engineering Physics, University of Wisconsin-Madison, Madison, WI
| | - B Burkel
- Department of Engineering Physics, University of Wisconsin-Madison, Madison, WI
| | - J Notbohm
- Department of Engineering Physics, University of Wisconsin-Madison, Madison, WI
- University of Wisconsin Carbone Cancer Center, Madison, WI
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16
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Mann A, Sopher RS, Goren S, Shelah O, Tchaicheeyan O, Lesman A. Force chains in cell-cell mechanical communication. J R Soc Interface 2019; 16:20190348. [PMID: 31662075 DOI: 10.1098/rsif.2019.0348] [Citation(s) in RCA: 19] [Impact Index Per Article: 3.8] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/01/2023] Open
Abstract
Force chains (FCs) are a key determinant of the micromechanical properties and behaviour of heterogeneous materials, such as granular systems. However, less is known about FCs in fibrous materials, such as the networks composing the extracellular matrix (ECM) of biological systems. Using a finite-element computational model, we simulated the contraction of a single cell and two nearby cells embedded in two-dimensional fibrous elastic networks and analysed the tensile FCs that developed in the ECM. The role of ECM nonlinear elasticity on FC formation was evaluated by considering linear and nonlinear, i.e. exhibiting 'buckling' and/or 'strain-stiffening', stress-strain curves. The effect of the degree of cell contraction and network coordination value was assessed. We found that nonlinear elasticity of the ECM fibres influenced the structure of the FCs, facilitating the transition towards more distinct chains that were less branched and more radially oriented than the chains formed in linear elastic networks. When two neighbouring cells contract, a larger number of FCs bridged between the cells in nonlinear networks, and these chains had a larger effective rigidity than the chains that did not reach a neighbouring cell. These results suggest that FCs function as a route for mechanical communication between distant cells and highlight the contribution of ECM fibre nonlinear elasticity to the formation of FCs.
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Affiliation(s)
- Amots Mann
- School of Mechanical Engineering, Faculty of Engineering, Tel Aviv University, Tel Aviv, Israel
| | - Ran S Sopher
- School of Mechanical Engineering, Faculty of Engineering, Tel Aviv University, Tel Aviv, Israel
| | - Shahar Goren
- School of Mechanical Engineering, Faculty of Engineering, Tel Aviv University, Tel Aviv, Israel
| | - Ortal Shelah
- School of Mechanical Engineering, Faculty of Engineering, Tel Aviv University, Tel Aviv, Israel
| | - Oren Tchaicheeyan
- School of Mechanical Engineering, Faculty of Engineering, Tel Aviv University, Tel Aviv, Israel
| | - Ayelet Lesman
- School of Mechanical Engineering, Faculty of Engineering, Tel Aviv University, Tel Aviv, Israel
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17
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Nan H, Zheng Y, Lin YH, Chen S, Eddy CZ, Tian J, Xu W, Sun B, Jiao Y. Absorbing-active transition in a multi-cellular system regulated by a dynamic force network. SOFT MATTER 2019; 15:6938-6945. [PMID: 31432887 DOI: 10.1039/c9sm01244c] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/10/2023]
Abstract
Collective cell migration in 3D extracellular matrix (ECM) is crucial to many physiological and pathological processes. Migrating cells can generate active pulling forces via actin filament contraction, which are transmitted to the ECM fibers and lead to a dynamically evolving force network in the system. Here, we elucidate the role of this force network in regulating collective cell behaviors using a minimal active-particle-on-network (APN) model, in which active particles can pull the fibers and hop between neighboring nodes of the network following local durotaxis. Our model reveals a dynamic transition as the particle number density approaches a critical value, from an "absorbing" state containing isolated stationary small particle clusters, to an "active" state containing a single large cluster undergoing constant dynamic reorganization. This reorganization is dominated by a subset of highly dynamic "radical" particles in the cluster, whose number also exhibits a transition at the same critical density. The transition is underlaid by the percolation of "influence spheres" due to the particle pulling forces. Our results suggest a robust mechanism based on ECM-mediated mechanical coupling for collective cell behaviors in 3D ECM.
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Affiliation(s)
- Hanqing Nan
- Materials Science and Engineering, Arizona State University, Tempe, AZ 85287, USA.
| | - Yu Zheng
- Department of Physics, Arizona State University, Tempe, AZ 85287, USA
| | - Yiheng H Lin
- Materials Science and Engineering, Arizona State University, Tempe, AZ 85287, USA. and Shenzhen Middle School, Shenzhen 518001, P. R. China
| | - Shaohua Chen
- Department of Materials Engineering, KU Leuven, Kasteelpark Arenberg 44 Bus 2450, Leuven, Belgium
| | - Christopher Z Eddy
- Department of Physics, Oregon State University, Corvallis, OR 97331, USA.
| | - Jianxiang Tian
- Materials Science and Engineering, Arizona State University, Tempe, AZ 85287, USA. and Department of Physics, Qufu Normal University, Qufu 273165, P. R. China
| | - Wenxiang Xu
- Materials Science and Engineering, Arizona State University, Tempe, AZ 85287, USA. and College of Mechanics and Materials, Hohai University, Nanjing 211100, P. R. China.
| | - Bo Sun
- Department of Physics, Oregon State University, Corvallis, OR 97331, USA.
| | - Yang Jiao
- Materials Science and Engineering, Arizona State University, Tempe, AZ 85287, USA. and Department of Physics, Arizona State University, Tempe, AZ 85287, USA
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18
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Leiphart RJ, Chen D, Peredo AP, Loneker AE, Janmey PA. Mechanosensing at Cellular Interfaces. LANGMUIR : THE ACS JOURNAL OF SURFACES AND COLLOIDS 2019; 35:7509-7519. [PMID: 30346180 DOI: 10.1021/acs.langmuir.8b02841] [Citation(s) in RCA: 30] [Impact Index Per Article: 6.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/19/2023]
Abstract
At the plasma membrane interface, cells use various adhesions to sense their extracellular environment. These adhesions facilitate the transmission of mechanical signals that dictate cell behavior. This review discusses the mechanisms by which these mechanical signals are transduced through cell-matrix and cell-cell adhesions and how this mechanotransduction influences cell processes. Cell-matrix adhesions require the activation of and communication between various transmembrane protein complexes such as integrins. These links at the plasma membrane affect how a cell senses and responds to its matrix environment. Cells also communicate with each other through cell-cell adhesions, which further regulate cell behavior on a single- and multicellular scale. Coordination and competition between cell-cell and cell-matrix adhesions in multicellular aggregates can, to a significant extent, be modeled by differential adhesion analyses between the different interfaces even without knowing the details of cellular signaling. In addition, cell-matrix and cell-cell adhesions are connected by an intracellular cytoskeletal network that allows for direct communication between these distinct adhesions and activation of specific signaling pathways. Other membrane-embedded protein complexes, such as growth factor receptors and ion channels, play additional roles in mechanotransduction. Overall, these mechanoactive elements show the dynamic interplay between the cell, its matrix, and neighboring cells and how these relationships affect cellular function.
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Affiliation(s)
- Ryan J Leiphart
- Department of Bioengineering , University of Pennsylvania , 210 S 33rd St , Philadelphia , Pennsylvania 19104 , United States
- McKay Orthopedic Research Laboratory , University of Pennsylvania , Philadelphia , Pennsylvania , United States
| | - Dongning Chen
- Department of Bioengineering , University of Pennsylvania , 210 S 33rd St , Philadelphia , Pennsylvania 19104 , United States
- Center for Engineering Mechanobiology , University of Pennsylvania , Philadelphia , Pennsylvania , United States
| | - Ana P Peredo
- Department of Bioengineering , University of Pennsylvania , 210 S 33rd St , Philadelphia , Pennsylvania 19104 , United States
- McKay Orthopedic Research Laboratory , University of Pennsylvania , Philadelphia , Pennsylvania , United States
| | - Abigail E Loneker
- Department of Bioengineering , University of Pennsylvania , 210 S 33rd St , Philadelphia , Pennsylvania 19104 , United States
- Center for Engineering Mechanobiology , University of Pennsylvania , Philadelphia , Pennsylvania , United States
| | - Paul A Janmey
- Department of Bioengineering , University of Pennsylvania , 210 S 33rd St , Philadelphia , Pennsylvania 19104 , United States
- Institute for Medicine and Engineering, Department of Physiology , University of Pennsylvania , 3340 Smith Walk , Philadelphia , Pennsylvania 19104 , United States
- Center for Engineering Mechanobiology , University of Pennsylvania , Philadelphia , Pennsylvania , United States
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19
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Golkov R, Shokef Y. Elastic interactions between anisotropically contracting circular cells. Phys Rev E 2019; 99:032418. [PMID: 30999414 DOI: 10.1103/physreve.99.032418] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/14/2018] [Indexed: 12/14/2022]
Abstract
We study interactions between biological cells that apply anisotropic active mechanical forces on an elastic substrate. We model the cells as thin disks that along their perimeters apply radial, but angle-dependent forces on the substrate. We obtain analytical expressions for the elastic energy stored in the substrate as a function of the distance between the cells, the Fourier modes of applied forces, and their phase angles. We show how the relative phases of the forces applied by the cells can switch the interaction between attractive and repulsive, and relate our results to those for linear force dipoles. For long enough distances, the interaction energy decays in magnitude as a power law of the cell-cell distance with an integer exponent that generally increases with the Fourier modes of the applied forces.
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Affiliation(s)
- Roman Golkov
- School of Mechanical Engineering and The Sackler Center for Computational Molecular and Materials Science, Tel Aviv University, Tel Aviv 69978, Israel
| | - Yair Shokef
- School of Mechanical Engineering and The Sackler Center for Computational Molecular and Materials Science, Tel Aviv University, Tel Aviv 69978, Israel
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20
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Palmieri B, Scanlon C, Worroll D, Grant M, Lee J. Substrate mediated interaction between pairs of keratocytes: Multipole traction force models describe their migratory behavior. PLoS One 2019; 14:e0212162. [PMID: 30822310 PMCID: PMC6396918 DOI: 10.1371/journal.pone.0212162] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/12/2017] [Accepted: 01/29/2019] [Indexed: 11/18/2022] Open
Abstract
A series of traction force microscopy experiments involving pairs of keratocytes migrating on compliant substrates were analyzed. We observed several instances where keratocytes that are about to collide turn before they touch. We term this phenomenon collision avoidance behavior and we propose that the turning is caused by the substrate mediated elastic interactions between the cells. A multipole analysis of the cell traction reveals that the left-right symmetry of the keratocyte traction pattern is broken during collision avoidance events. The analysis further shows that the cell migration direction reorients before the principal traction dipoles as the cells turn. Linear elasticity theory is used to derive the cell-cell interaction energy between pairs of keratocytes. The traction force applied by each cell is modeled as a two points (dipole) or three points (tripod) force model. We show that both models predict that cells that are about to collide in a head-on manner will turn before touching. The tripod model is further able to account for the quadrupole components of the traction force profile that we observed experimentally. Also, the tripod model proposes a mechanism that may explain why cells tend to scatter with a finite angle after a collision avoidance event. A relationship between the scattering angle and the traction force quadrupole moment is also established. Dynamical simulations of migrating model cells are further used to explain the emergence of other cell pair trajectories that we observed experimentally.
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Affiliation(s)
- Benoit Palmieri
- Department of Physics, McGill University, Montréal, Québec, Canada
| | - Christine Scanlon
- Department of Molecular & Cell Biology, Storrs, CT, United States of America
| | - Daniel Worroll
- Department of Molecular & Cell Biology, Storrs, CT, United States of America
| | - Martin Grant
- Department of Physics, McGill University, Montréal, Québec, Canada
| | - Juliet Lee
- Department of Molecular & Cell Biology, Storrs, CT, United States of America
- * E-mail:
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21
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Ronceray P, Broedersz CP, Lenz M. Fiber plucking by molecular motors yields large emergent contractility in stiff biopolymer networks. SOFT MATTER 2019; 15:1481-1487. [PMID: 30608098 DOI: 10.1039/c8sm00979a] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/09/2023]
Abstract
The mechanical properties of the cell depend crucially on the tension of its cytoskeleton, a biopolymer network that is put under stress by active motor proteins. While the fibrous nature of the network is known to strongly affect the transmission of these forces to the cellular scale, our understanding of this process remains incomplete. Here we investigate the transmission of forces through the network at the individual filament level, and show that active forces can be geometrically amplified as a transverse motor-generated force "plucks" the fiber and induces a nonlinear tension. In stiff and densely connected networks, this tension results in large network-wide tensile stresses that far exceed the expectation drawn from a linear elastic theory. This amplification mechanism competes with a recently characterized network-level amplification due to fiber buckling, suggesting that that fiber networks provide several distinct pathways for living systems to amplify their molecular forces.
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Affiliation(s)
- Pierre Ronceray
- Princeton Center for Theoretical Science, Princeton University, Princeton, NJ 08544, USA
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22
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Fregin B, Czerwinski F, Biedenweg D, Girardo S, Gross S, Aurich K, Otto O. High-throughput single-cell rheology in complex samples by dynamic real-time deformability cytometry. Nat Commun 2019; 10:415. [PMID: 30679420 PMCID: PMC6346011 DOI: 10.1038/s41467-019-08370-3] [Citation(s) in RCA: 57] [Impact Index Per Article: 11.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/17/2018] [Accepted: 01/08/2019] [Indexed: 11/25/2022] Open
Abstract
In life sciences, the material properties of suspended cells have attained significance close to that of fluorescent markers but with the advantage of label-free and unbiased sample characterization. Until recently, cell rheological measurements were either limited by acquisition throughput, excessive post processing, or low-throughput real-time analysis. Real-time deformability cytometry expanded the application of mechanical cell assays to fast on-the-fly phenotyping of large sample sizes, but has been restricted to single material parameters as the Young's modulus. Here, we introduce dynamic real-time deformability cytometry for comprehensive cell rheological measurements at up to 100 cells per second. Utilizing Fourier decomposition, our microfluidic method is able to disentangle cell response to complex hydrodynamic stress distributions and to determine viscoelastic parameters independent of cell shape. We demonstrate the application of our technology for peripheral blood cells in whole blood samples including the discrimination of B- and CD4+ T-lymphocytes by cell rheological properties.
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Affiliation(s)
- Bob Fregin
- Zentrum für Innovationskompetenz: Humorale Immunreaktionen bei kardiovaskulären Erkrankungen, Universität Greifswald, Fleischmannstr. 42, 17489, Greifswald, Germany
| | - Fabian Czerwinski
- Zentrum für Innovationskompetenz: Humorale Immunreaktionen bei kardiovaskulären Erkrankungen, Universität Greifswald, Fleischmannstr. 42, 17489, Greifswald, Germany
| | - Doreen Biedenweg
- Universitätsmedizin Greifswald, Fleischmannstr. 8, 17489, Greifswald, Germany
| | - Salvatore Girardo
- Biotechnology Center, Center for Molecular and Cellular Bioengineering, Technische Universität Dresden, Tatzberg 47/49, 01307, Dresden, Germany
| | - Stefan Gross
- Universitätsmedizin Greifswald, Fleischmannstr. 8, 17489, Greifswald, Germany
- Deutsches Zentrum für Herz-Kreislauf-Forschung e.V., Standort Greifswald, Universitätsmedizin Greifswald, Fleischmannstr. 42, 17489, Greifswald, Germany
| | - Konstanze Aurich
- Universitätsmedizin Greifswald, Fleischmannstr. 8, 17489, Greifswald, Germany
| | - Oliver Otto
- Zentrum für Innovationskompetenz: Humorale Immunreaktionen bei kardiovaskulären Erkrankungen, Universität Greifswald, Fleischmannstr. 42, 17489, Greifswald, Germany.
- Deutsches Zentrum für Herz-Kreislauf-Forschung e.V., Standort Greifswald, Universitätsmedizin Greifswald, Fleischmannstr. 42, 17489, Greifswald, Germany.
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23
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Ronceray P, Broedersz CP, Lenz M. Stress-dependent amplification of active forces in nonlinear elastic media. SOFT MATTER 2019; 15:331-338. [PMID: 30556571 DOI: 10.1039/c8sm00949j] [Citation(s) in RCA: 10] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/09/2023]
Abstract
The production of mechanical stresses in living organisms largely relies on localized, force-generating active units embedded in filamentous matrices. Numerical simulations of discrete fiber networks with fixed boundaries have shown that buckling in the matrix dramatically amplifies the resulting active stresses. Here we extend this result to a continuum elastic medium prone to buckling subjected to an arbitrary external stress, and derive analytical expressions for the active, nonlinear constitutive relations characterizing the full active medium. Inserting these relations into popular "active gel" descriptions of living tissues and the cytoskeleton will enable investigations into nonlinear regimes previously inaccessible due to the phenomenological nature of these theories.
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Affiliation(s)
- Pierre Ronceray
- Princeton Center for Theoretical Science, Princeton University, Princeton, NJ 08544, USA
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24
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Nan H, Liang L, Chen G, Liu L, Liu R, Jiao Y. Realizations of highly heterogeneous collagen networks via stochastic reconstruction for micromechanical analysis of tumor cell invasion. Phys Rev E 2018; 97:033311. [PMID: 29776156 DOI: 10.1103/physreve.97.033311] [Citation(s) in RCA: 21] [Impact Index Per Article: 3.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/21/2017] [Indexed: 11/07/2022]
Abstract
Three-dimensional (3D) collective cell migration in a collagen-based extracellular matrix (ECM) is among one of the most significant topics in developmental biology, cancer progression, tissue regeneration, and immune response. Recent studies have suggested that collagen-fiber mediated force transmission in cellularized ECM plays an important role in stress homeostasis and regulation of collective cellular behaviors. Motivated by the recent in vitro observation that oriented collagen can significantly enhance the penetration of migrating breast cancer cells into dense Matrigel which mimics the intravasation process in vivo [Han et al. Proc. Natl. Acad. Sci. USA 113, 11208 (2016)PNASA60027-842410.1073/pnas.1610347113], we devise a procedure for generating realizations of highly heterogeneous 3D collagen networks with prescribed microstructural statistics via stochastic optimization. Specifically, a collagen network is represented via the graph (node-bond) model and the microstructural statistics considered include the cross-link (node) density, valence distribution, fiber (bond) length distribution, as well as fiber orientation distribution. An optimization problem is formulated in which the objective function is defined as the squared difference between a set of target microstructural statistics and the corresponding statistics for the simulated network. Simulated annealing is employed to solve the optimization problem by evolving an initial network via random perturbations to generate realizations of homogeneous networks with randomly oriented fibers, homogeneous networks with aligned fibers, heterogeneous networks with a continuous variation of fiber orientation along a prescribed direction, as well as a binary system containing a collagen region with aligned fibers and a dense Matrigel region with randomly oriented fibers. The generation and propagation of active forces in the simulated networks due to polarized contraction of an embedded ellipsoidal cell and a small group of cells are analyzed by considering a nonlinear fiber model incorporating strain hardening upon large stretching and buckling upon compression. Our analysis shows that oriented fibers can significantly enhance long-range force transmission in the network. Moreover, in the oriented-collagen-Matrigel system, the forces generated by a polarized cell in collagen can penetrate deeply into the Matrigel region. The stressed Matrigel fibers could provide contact guidance for the migrating cell cells, and thus enhance their penetration into Matrigel. This suggests a possible mechanism for the observed enhanced intravasation by oriented collagen.
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Affiliation(s)
- Hanqing Nan
- Materials Science and Engineering, Arizona State University, Tempe, Arizona 85287, USA
| | - Long Liang
- Department of Physics, Arizona State University, Tempe, Arizona 85287, USA
| | - Guo Chen
- College of Physics, Chongqing University, Chongqing 401331, China
| | - Liyu Liu
- College of Physics, Chongqing University, Chongqing 401331, China
| | - Ruchuan Liu
- College of Physics, Chongqing University, Chongqing 401331, China
| | - Yang Jiao
- Materials Science and Engineering, Arizona State University, Tempe, Arizona 85287, USA.,Department of Physics, Arizona State University, Tempe, Arizona 85287, USA
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25
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Sopher RS, Tokash H, Natan S, Sharabi M, Shelah O, Tchaicheeyan O, Lesman A. Nonlinear Elasticity of the ECM Fibers Facilitates Efficient Intercellular Communication. Biophys J 2018; 115:1357-1370. [PMID: 30217380 DOI: 10.1016/j.bpj.2018.07.036] [Citation(s) in RCA: 52] [Impact Index Per Article: 8.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/28/2018] [Revised: 07/09/2018] [Accepted: 07/25/2018] [Indexed: 11/18/2022] Open
Abstract
Biological cells embedded in fibrous matrices have been observed to form intercellular bands of dense and aligned fibers through which they mechanically interact over long distances. Such matrix-mediated cellular interactions have been shown to regulate various biological processes. This study aimed to explore the effects of elastic nonlinearity of the fibers contained in the extracellular matrix (ECM) on the transmission of mechanical loads between contracting cells. Based on our biological experiments, we developed a finite-element model of two contracting cells embedded within a fibrous network. The individual fibers were modeled as showing linear elasticity, compression microbuckling, tension stiffening, or both of the latter two. Fiber compression buckling resulted in smaller loads in the ECM, which were primarily directed toward the neighboring cell. These loads decreased with increasing cell-to-cell distance; when cells were >9 cell diameters apart, no such intercellular interaction was observed. Tension stiffening further contributed to directing the loads toward the neighboring cell, though to a smaller extent. The contraction of two neighboring cells resulted in mutual attraction forces, which were considerably increased by tension stiffening and decayed with increasing cell-to-cell distances. Nonlinear elasticity contributed also to the onset of force polarity on the cell boundaries, manifested by larger contractile forces pointing toward the neighboring cell. The density and alignment of the fibers within the intercellular band were greater when fibers buckled under compression, with tension stiffening further contributing to this structural remodeling. Although previous studies have established the role of the ECM nonlinear mechanical behavior in increasing the range of force transmission, our model demonstrates the contribution of nonlinear elasticity of biological gels to directional and efficient mechanical signal transfer between distant cells, and rehighlights the importance of using fibrous gels in experimental settings for facilitating intercellular communication. VIDEO ABSTRACT.
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Affiliation(s)
- Ran S Sopher
- School of Mechanical Engineering, Faculty of Engineering, Tel Aviv University, Tel Aviv, Israel
| | - Hanan Tokash
- School of Mechanical Engineering, Faculty of Engineering, Tel Aviv University, Tel Aviv, Israel
| | - Sari Natan
- School of Mechanical Engineering, Faculty of Engineering, Tel Aviv University, Tel Aviv, Israel
| | - Mirit Sharabi
- School of Mechanical Engineering, Faculty of Engineering, Tel Aviv University, Tel Aviv, Israel
| | - Ortal Shelah
- School of Mechanical Engineering, Faculty of Engineering, Tel Aviv University, Tel Aviv, Israel
| | - Oren Tchaicheeyan
- School of Mechanical Engineering, Faculty of Engineering, Tel Aviv University, Tel Aviv, Israel
| | - Ayelet Lesman
- School of Mechanical Engineering, Faculty of Engineering, Tel Aviv University, Tel Aviv, Israel.
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26
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Kang YG, Jang H, Yang TD, Notbohm J, Choi Y, Park Y, Kim BM. Quantification of focal adhesion dynamics of cell movement based on cell-induced collagen matrix deformation using second-harmonic generation microscopy. JOURNAL OF BIOMEDICAL OPTICS 2018; 23:1-8. [PMID: 29886577 DOI: 10.1117/1.jbo.23.6.065001] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/18/2018] [Accepted: 05/14/2018] [Indexed: 06/08/2023]
Abstract
Mechanical interactions of living cells with the surrounding environment via focal adhesion (FA) in three dimensions (3-D) play a key role in dynamic biological events, such as tissue regeneration, wound healing, and cancer invasion. Recently, several methods for observing 3-D cell-extracellular matrix (ECM) interactions have been reported, lacking solid and quantitative analysis on the dynamics of the physical interaction between the cell and the ECM. We measured the submicron displacements of ECM deformation in 3-D due to protrusion-retraction dynamics during cell migration, using second-harmonic generation without labeling the matrix structures. We then quantitatively analyzed the mechanical deformation between the ECM and the cells based on spatiotemporal volumetric correlations. The greatest deformations within the collagen matrix were found to occur at sites of colocalization of the FA site-related proteins vinculin and actin, which confirms that FA sites play a critical role in living cells within the ECM as a point for adhesion, traction, and migration. We believe that this modality can be used in studies of cell-ECM interaction during angiogenesis, wound healing, and metastasis.
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Affiliation(s)
- Yong Guk Kang
- Korea University, Department of Bio-convergence Engineering, Seoul, Republic of Korea
| | - Hwanseok Jang
- Korea University Medical Center, Department of Biomedical Science, College of Medicine, Seoul, Republic of Korea
| | - Taeseok Daniel Yang
- Korea University, Department of Bio-convergence Engineering, Seoul, Republic of Korea
| | - Jacob Notbohm
- University of Wisconsin-Madison, Department of Engineering Physics, Madison, Wisconsin, United States
| | - Youngwoon Choi
- Korea University, Department of Bio-convergence Engineering, Seoul, Republic of Korea
| | - Yongdoo Park
- Korea University Medical Center, Department of Biomedical Science, College of Medicine, Seoul, Republic of Korea
| | - Beop-Min Kim
- Korea University, Department of Bio-convergence Engineering, Seoul, Republic of Korea
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27
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Gnesotto FS, Mura F, Gladrow J, Broedersz CP. Broken detailed balance and non-equilibrium dynamics in living systems: a review. REPORTS ON PROGRESS IN PHYSICS. PHYSICAL SOCIETY (GREAT BRITAIN) 2018; 81:066601. [PMID: 29504517 DOI: 10.1088/1361-6633/aab3ed] [Citation(s) in RCA: 116] [Impact Index Per Article: 19.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/08/2023]
Abstract
Living systems operate far from thermodynamic equilibrium. Enzymatic activity can induce broken detailed balance at the molecular scale. This molecular scale breaking of detailed balance is crucial to achieve biological functions such as high-fidelity transcription and translation, sensing, adaptation, biochemical patterning, and force generation. While biological systems such as motor enzymes violate detailed balance at the molecular scale, it remains unclear how non-equilibrium dynamics manifests at the mesoscale in systems that are driven through the collective activity of many motors. Indeed, in several cellular systems the presence of non-equilibrium dynamics is not always evident at large scales. For example, in the cytoskeleton or in chromosomes one can observe stationary stochastic processes that appear at first glance thermally driven. This raises the question how non-equilibrium fluctuations can be discerned from thermal noise. We discuss approaches that have recently been developed to address this question, including methods based on measuring the extent to which the system violates the fluctuation-dissipation theorem. We also review applications of this approach to reconstituted cytoskeletal networks, the cytoplasm of living cells, and cell membranes. Furthermore, we discuss a more recent approach to detect actively driven dynamics, which is based on inferring broken detailed balance. This constitutes a non-invasive method that uses time-lapse microscopy data, and can be applied to a broad range of systems in cells and tissue. We discuss the ideas underlying this method and its application to several examples including flagella, primary cilia, and cytoskeletal networks. Finally, we briefly discuss recent developments in stochastic thermodynamics and non-equilibrium statistical mechanics, which offer new perspectives to understand the physics of living systems.
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Affiliation(s)
- F S Gnesotto
- Arnold-Sommerfeld-Center for Theoretical Physics and Center for NanoScience, Ludwig-Maximilians-Universität München, D-80333 München, Germany
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28
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Cell contraction induces long-ranged stress stiffening in the extracellular matrix. Proc Natl Acad Sci U S A 2018; 115:4075-4080. [PMID: 29618614 DOI: 10.1073/pnas.1722619115] [Citation(s) in RCA: 178] [Impact Index Per Article: 29.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/17/2023] Open
Abstract
Animal cells in tissues are supported by biopolymer matrices, which typically exhibit highly nonlinear mechanical properties. While the linear elasticity of the matrix can significantly impact cell mechanics and functionality, it remains largely unknown how cells, in turn, affect the nonlinear mechanics of their surrounding matrix. Here, we show that living contractile cells are able to generate a massive stiffness gradient in three distinct 3D extracellular matrix model systems: collagen, fibrin, and Matrigel. We decipher this remarkable behavior by introducing nonlinear stress inference microscopy (NSIM), a technique to infer stress fields in a 3D matrix from nonlinear microrheology measurements with optical tweezers. Using NSIM and simulations, we reveal large long-ranged cell-generated stresses capable of buckling filaments in the matrix. These stresses give rise to the large spatial extent of the observed cell-induced matrix stiffness gradient, which can provide a mechanism for mechanical communication between cells.
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29
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Grimmer P, Notbohm J. Displacement Propagation in Fibrous Networks Due to Local Contraction. J Biomech Eng 2018; 140:2666617. [DOI: 10.1115/1.4038744] [Citation(s) in RCA: 17] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/19/2017] [Indexed: 01/27/2023]
Abstract
The extracellular matrix provides macroscale structure to tissues and microscale guidance for cell contraction, adhesion, and migration. The matrix is composed of a network of fibers, which each deform by stretching, bending, and buckling. Whereas the mechanics has been well characterized in uniform shear and extension, the response to more general loading conditions remains less clear, because the associated displacement fields cannot be predicted a priori. Studies simulating contraction, such as due to a cell, have observed displacements that propagate over a long range, suggesting mechanisms such as reorientation of fibers toward directions of tensile force and nonlinearity due to buckling of fibers under compression. It remains unclear which of these two mechanisms produces the long-range displacements and how properties like fiber bending stiffness and fiber length affect the displacement field. Here, we simulate contraction of an inclusion within a fibrous network and fit the resulting radial displacements to ur ∼ r−n where the power n quantifies the decay of displacements over distance, and a value of n less than that predicted by classical linear elasticity indicates displacements that propagate over a long range. We observed displacements to propagate over a longer range for greater contraction of the inclusion, for networks having longer fibers, and for networks with lower fiber bending stiffness. Contraction of the inclusion also caused fibers to reorient into the radial direction, but, surprisingly, the reorientation was minimally affected by bending stiffness. We conclude that both reorientation and nonlinearity are responsible for the long-range displacements.
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Affiliation(s)
- Peter Grimmer
- Department of Engineering Physics, University of Wisconsin-Madison, 1500 Engineering Drive, Madison, WI 53706 e-mail:
| | - Jacob Notbohm
- Department of Engineering Physics, University of Wisconsin-Madison, 1500 Engineering Drive, Madison, WI 53706 e-mail:
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30
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Burkel B, Notbohm J. Mechanical response of collagen networks to nonuniform microscale loads. SOFT MATTER 2017; 13:5749-5758. [PMID: 28759060 DOI: 10.1039/c7sm00561j] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/28/2023]
Abstract
As force is applied to fibrous proteins such as collagen or fibrin, the fibers respond by bending, stretching, or buckling, which together bring about a nonlinear relationship between force and displacement. The nonlinearity is typically understood in terms of strain stiffening in uniform extension or shear, but there remains a critical lack of data on how fibrous materials respond to other more complicated loadings. Here we study the mechanics of collagen networks in response to nonuniform loads applied on the local scale of the fibers. For this, we use particles made of an active hydrogel that undergoes a temperature-induced phase transition causing a large decrease in volume. We embed these particles in networks of fibrous collagen and use them as microactuators to apply controlled microscale loading. The resulting fiber displacements propagate over a long range with radial displacements u scaling as r-n with n ≈ 1. By contrast, we find linear homogeneous materials have n ≈ 2, in agreement with classical linear elastic theory. Our experimental data supports the notion that the long range displacements result from buckling of fibers in compression and local straightening of fibers in tension, in agreement with previous studies. Surprisingly, global network anisotropy appears to have only a modest effect on the displacement propagation. These insights into the microscale mechanics demonstrate that the decay power n provides a useful metric to quantify the mechanics of fibrous materials. We therefore suggest it is a means to compare new theories with experimental data.
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Affiliation(s)
- Brian Burkel
- Department of Engineering Physics, University of Wisconsin-Madison, Madison, WI 53706, USA.
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31
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Freedman SL, Banerjee S, Hocky GM, Dinner AR. A Versatile Framework for Simulating the Dynamic Mechanical Structure of Cytoskeletal Networks. Biophys J 2017; 113:448-460. [PMID: 28746855 DOI: 10.1016/j.bpj.2017.06.003] [Citation(s) in RCA: 48] [Impact Index Per Article: 6.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/30/2016] [Revised: 05/26/2017] [Accepted: 06/02/2017] [Indexed: 01/07/2023] Open
Abstract
Computer simulations can aid in understanding how collective materials properties emerge from interactions between simple constituents. Here, we introduce a coarse-grained model that enables simulation of networks of actin filaments, myosin motors, and cross-linking proteins at biologically relevant time and length scales. We demonstrate that the model qualitatively and quantitatively captures a suite of trends observed experimentally, including the statistics of filament fluctuations, and mechanical responses to shear, motor motilities, and network rearrangements. We use the simulation to predict the viscoelastic scaling behavior of cross-linked actin networks, characterize the trajectories of actin in a myosin motility assay, and develop order parameters to measure contractility of a simulated actin network. The model can thus serve as a platform for interpretation and design of cytoskeletal materials experiments, as well as for further development of simulations incorporating active elements.
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Affiliation(s)
- Simon L Freedman
- Department of Physics, University of Chicago, Chicago, Illinois; James Franck Institute, University of Chicago, Chicago, Illinois
| | - Shiladitya Banerjee
- Department of Physics and Astronomy, University College London, London, United Kingdom; Institute for Physics of Living Systems, University College London, London, United Kingdom
| | - Glen M Hocky
- James Franck Institute, University of Chicago, Chicago, Illinois; Department of Chemistry, University of Chicago, Chicago, Illinois
| | - Aaron R Dinner
- James Franck Institute, University of Chicago, Chicago, Illinois; Department of Chemistry, University of Chicago, Chicago, Illinois; Institute for Biophysical Dynamics, University of Chicago, Chicago, Illinois.
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32
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Humphries DL, Grogan JA, Gaffney EA. Mechanical Cell-Cell Communication in Fibrous Networks: The Importance of Network Geometry. Bull Math Biol 2017; 79:498-524. [PMID: 28130739 PMCID: PMC5331102 DOI: 10.1007/s11538-016-0242-5] [Citation(s) in RCA: 21] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/30/2016] [Accepted: 12/14/2016] [Indexed: 01/24/2023]
Abstract
Cells contracting in extracellular matrix (ECM) can transmit stress over long distances, communicating their position and orientation to cells many tens of micrometres away. Such phenomena are not observed when cells are seeded on substrates with linear elastic properties, such as polyacrylamide (PA) gel. The ability for fibrous substrates to support far reaching stress and strain fields has implications for many physiological processes, while the mechanical properties of ECM are central to several pathological processes, including tumour invasion and fibrosis. Theoretical models have investigated the properties of ECM in a variety of network geometries. However, the effects of network architecture on mechanical cell-cell communication have received little attention. This work investigates the effects of geometry on network mechanics, and thus the ability for cells to communicate mechanically through different networks. Cell-derived displacement fields are quantified for various network geometries while controlling for network topology, cross-link density and micromechanical properties. We find that the heterogeneity of response, fibre alignment, and substrate displacement fields are sensitive to network choice. Further, we show that certain geometries support mechanical communication over longer distances than others. As such, we predict that the choice of network geometry is important in fundamental modelling of cell-cell interactions in fibrous substrates, as well as in experimental settings, where mechanical signalling at the cellular scale plays an important role. This work thus informs the construction of theoretical models for substrate mechanics and experimental explorations of mechanical cell-cell communication.
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Affiliation(s)
- D L Humphries
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, UK.
| | - J A Grogan
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, UK
| | - E A Gaffney
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, UK
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33
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Liang L, Jones C, Chen S, Sun B, Jiao Y. Heterogeneous force network in 3D cellularized collagen networks. Phys Biol 2016; 13:066001. [PMID: 27779119 DOI: 10.1088/1478-3975/13/6/066001] [Citation(s) in RCA: 40] [Impact Index Per Article: 5.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/27/2023]
Abstract
Collagen networks play an important role in coordinating and regulating collective cellular dynamics via a number of signaling pathways. Here, we investigate the transmission of forces generated by contractile cells in 3D collagen-I networks. Specifically, the graph (bond-node) representations of collagen networks with collagen concentrations of 1, 2 and 4 mg ml-1 are derived from confocal microscopy data and used to model the network microstructure. Cell contraction is modeled by applying correlated displacements at specific nodes of the network, representing the focal adhesion sites. A nonlinear elastic model is employed to characterize the mechanical behavior of individual fiber bundles including strain hardening during stretching and buckling under compression. A force-based relaxation method is employed to obtain equilibrium network configurations under cell contraction. We find that for all collagen concentrations, the majority of the forces are carried by a small number of heterogeneous force chains emitted from the contracting cells, which is qualitatively consistent with our experimental observations. The force chains consist of fiber segments that either possess a high degree of alignment before cell contraction or are aligned due to fiber reorientation induced by cell contraction. The decay of the forces along the force chains is significantly slower than the decay of radially averaged forces in the system, suggesting that the fibreous nature of biopolymer network structure can support long-range force transmission. The force chains emerge even at very small cell contractions, and the number of force chains increases with increasing cell contraction. At large cell contractions, the fibers close to the cell surface are in the nonlinear regime, and the nonlinear region is localized in a small neighborhood of the cell. In addition, the number of force chains increases with increasing collagen concentration, due to the larger number of focal adhesion sites in collagen networks with high concentrations.
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Affiliation(s)
- Long Liang
- Department of Physics, Arizona State University, Tempe, AZ, 85287, USA
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34
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Notbohm J, Lesman A, Rosakis P, Tirrell DA, Ravichandran G. Microbuckling of fibrin provides a mechanism for cell mechanosensing. J R Soc Interface 2016; 12:20150320. [PMID: 26040601 DOI: 10.1098/rsif.2015.0320] [Citation(s) in RCA: 55] [Impact Index Per Article: 6.9] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/04/2023] Open
Abstract
Biological cells sense and respond to mechanical forces, but how such a mechanosensing process takes place in a nonlinear inhomogeneous fibrous matrix remains unknown. We show that cells in a fibrous matrix induce deformation fields that propagate over a longer range than predicted by linear elasticity. Synthetic, linear elastic hydrogels used in many mechanotransduction studies fail to capture this effect. We develop a nonlinear microstructural finite-element model for a fibre network to simulate localized deformations induced by cells. The model captures measured cell-induced matrix displacements from experiments and identifies an important mechanism for long-range cell mechanosensing: loss of compression stiffness owing to microbuckling of individual fibres. We show evidence that cells sense each other through the formation of localized intercellular bands of tensile deformations caused by this mechanism.
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Affiliation(s)
- Jacob Notbohm
- Division of Engineering and Applied Science, California Institute of Technology, Pasadena, CA 91125, USA
| | - Ayelet Lesman
- Division of Chemistry and Chemical Engineering, California Institute of Technology, Pasadena, CA 91125, USA
| | - Phoebus Rosakis
- Department of Theoretical and Applied Mathematics, University of Crete, Heraklion 70013, Greece
| | - David A Tirrell
- Division of Chemistry and Chemical Engineering, California Institute of Technology, Pasadena, CA 91125, USA
| | - Guruswami Ravichandran
- Division of Engineering and Applied Science, California Institute of Technology, Pasadena, CA 91125, USA
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35
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Abstract
Large-scale force generation is essential for biological functions such as cell motility, embryonic development, and muscle contraction. In these processes, forces generated at the molecular level by motor proteins are transmitted by disordered fiber networks, resulting in large-scale active stresses. Although these fiber networks are well characterized macroscopically, this stress generation by microscopic active units is not well understood. Here we theoretically study force transmission in these networks. We find that collective fiber buckling in the vicinity of a local active unit results in a rectification of stress towards strongly amplified isotropic contraction. This stress amplification is reinforced by the networks' disordered nature, but saturates for high densities of active units. Our predictions are quantitatively consistent with experiments on reconstituted tissues and actomyosin networks and shed light on the role of the network microstructure in shaping active stresses in cells and tissue.
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36
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Licup AJ, Sharma A, MacKintosh FC. Elastic regimes of subisostatic athermal fiber networks. Phys Rev E 2016; 93:012407. [PMID: 26871101 DOI: 10.1103/physreve.93.012407] [Citation(s) in RCA: 35] [Impact Index Per Article: 4.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/01/2015] [Indexed: 11/07/2022]
Abstract
Athermal models of disordered fibrous networks are highly useful for studying the mechanics of elastic networks composed of stiff biopolymers. The underlying network architecture is a key aspect that can affect the elastic properties of these systems, which include rich linear and nonlinear elasticity. Existing computational approaches have focused on both lattice-based and off-lattice networks obtained from the random placement of rods. It is not obvious, a priori, whether the two architectures have fundamentally similar or different mechanics. If they are different, it is not clear which of these represents a better model for biological networks. Here, we show that both approaches are essentially equivalent for the same network connectivity, provided the networks are subisostatic with respect to central force interactions. Moreover, for a given subisostatic connectivity, we even find that lattice-based networks in both two and three dimensions exhibit nearly identical nonlinear elastic response. We provide a description of the linear mechanics for both architectures in terms of a scaling function. We also show that the nonlinear regime is dominated by fiber bending and that stiffening originates from the stabilization of subisostatic networks by stress. We propose a generalized relation for this regime in terms of the self-generated normal stresses that develop under deformation. Different network architectures have different susceptibilities to the normal stress but essentially exhibit the same nonlinear mechanics. Such a stiffening mechanism has been shown to successfully capture the nonlinear mechanics of collagen networks.
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Affiliation(s)
- A J Licup
- Department of Physics and Astronomy, VU University Amsterdam, 1081 HV Amsterdam, The Netherlands
| | - A Sharma
- Department of Physics and Astronomy, VU University Amsterdam, 1081 HV Amsterdam, The Netherlands
| | - F C MacKintosh
- Department of Physics and Astronomy, VU University Amsterdam, 1081 HV Amsterdam, The Netherlands
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37
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Xu X, Safran SA. Nonlinearities of biopolymer gels increase the range of force transmission. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 92:032728. [PMID: 26465519 DOI: 10.1103/physreve.92.032728] [Citation(s) in RCA: 34] [Impact Index Per Article: 3.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/21/2015] [Indexed: 06/05/2023]
Abstract
We present a model of biopolymer gels that includes two types of elastic nonlinearities, stiffening under extension and softening (due to buckling) under compression, to predict the elastic anisotropy induced by both external as well as internal (e.g., due to cell contractility) stresses in biopolymer gels. We show how the stretch-induced anisotropy and the strain-stiffening nonlinearity increase both the amplitude and power-law range of transmission of internal, contractile, cellular forces, and relate this to recent experiments.
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Affiliation(s)
- Xinpeng Xu
- Department of Materials and Interfaces, Weizmann Institute of Science, Rehovot 76100, Israel
| | - Samuel A Safran
- Department of Materials and Interfaces, Weizmann Institute of Science, Rehovot 76100, Israel
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38
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Feng J, Levine H, Mao X, Sander LM. Alignment and nonlinear elasticity in biopolymer gels. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 91:042710. [PMID: 25974530 DOI: 10.1103/physreve.91.042710] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/03/2014] [Indexed: 06/04/2023]
Abstract
We present a Landau-type theory for the nonlinear elasticity of biopolymer gels with a part of the order parameter describing induced nematic order of fibers in the gel. We attribute the nonlinear elastic behavior of these materials to fiber alignment induced by strain. We suggest an application to contact guidance of cell motility in tissue. We compare our theory to simulation of a disordered lattice model for biopolymers. We treat homogeneous deformations such as simple shear, hydrostatic expansion, and simple extension, and obtain good agreement between theory and simulation. We also consider a localized perturbation which is a simple model for a contracting cell in a medium.
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Affiliation(s)
- Jingchen Feng
- Bioengineering Department and Center for Theoretical Biological Physics, Rice University, Houston, Texas 77251-1892, USA
| | - Herbert Levine
- Bioengineering Department and Center for Theoretical Biological Physics, Rice University, Houston, Texas 77251-1892, USA
| | - Xiaoming Mao
- Department of Physics, University of Michigan, Ann Arbor, Michigan 48109-1040, USA
| | - Leonard M Sander
- Physics & Complex Systems, University of Michigan, Ann Arbor, Michigan 48109-1040, USA
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39
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Abstract
The first step in the spread of cancer is invasion by malignant cells of the normal tissue surrounding a tumor. There is considerable evidence both in vitro and in vivo that mechanical interactions with the tissue, in particular with the biopolymer network that makes up the extracellular matrix (ECM), are important factors in invasion. The interactions take two forms: (i) contractile cells on the surface of the tumor act on the nearby ECM and remodel it; in some cases, they align the fibers of the biopolymers; (ii) the aligned fibers can enhance invasion via contact guidance, the tendency of motile cells to follow alignment. Here, we give evidence, mainly for in vitro systems, that both effects are important. We discuss how alignment occurs in biopolymers such as collagen-I (a major component of the ECM). We propose a modeling framework for computing alignment and propose phenomenologic models for contact guidance. See all articles in this Cancer Research section, "Physics in Cancer Research."
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Affiliation(s)
- Leonard M Sander
- Physics and Complex Systems, University of Michigan, Ann Arbor Michigan
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40
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Ronceray P, Lenz M. Connecting local active forces to macroscopic stress in elastic media. SOFT MATTER 2015; 11:1597-1605. [PMID: 25594831 DOI: 10.1039/c4sm02526a] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/04/2023]
Abstract
In contrast with ordinary materials, living matter drives its own motion by generating active, out-of-equilibrium internal stresses. These stresses typically originate from localized active elements embedded in an elastic medium, such as molecular motors inside the cell or contractile cells in a tissue. While many large-scale phenomenological theories of such active media have been developed, a systematic understanding of the emergence of stress from the local force-generating elements is lacking. In this paper, we present a rigorous theoretical framework to study this relationship. We show that the medium's macroscopic active stress tensor is equal to the active elements' force dipole tensor per unit volume in both continuum and discrete linear homogeneous media of arbitrary geometries. This relationship is conserved on average in the presence of disorder, but can be violated in nonlinear elastic media. Such effects can lead to either a reinforcement or an attenuation of the active stresses, giving us a glimpse of the ways in which nature might harness microscopic forces to create active materials.
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Affiliation(s)
- Pierre Ronceray
- Univ. Paris-Sud, CNRS, LPTMS, UMR 8626, Orsay 91405, France.
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41
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Ben-Yaakov D, Golkov R, Shokef Y, Safran SA. Response of adherent cells to mechanical perturbations of the surrounding matrix. SOFT MATTER 2015; 11:1412-1424. [PMID: 25604950 DOI: 10.1039/c4sm01817f] [Citation(s) in RCA: 17] [Impact Index Per Article: 1.9] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/04/2023]
Abstract
We present a generic and unified theory to explain how cells respond to perturbations of their mechanical environment such as the presence of neighboring cells, slowly applied stretch, or gradients of matrix rigidity. Motivated by experiments, we calculate the local balance of forces that give rise to a tendency for the cell to locally move or reorient, with a focus on the contribution of feedback and homeostasis to cell contractility (manifested by a fixed displacement, strain or stress) that acts on the adhesions at the cell boundary. These forces can be either reinforced or diminished by elastic stresses due to mechanical perturbations of the matrix. Our model predicts these changes and how their balance with local protrusive forces that act on the cell's leading edge either increase or decrease the tendency of the cell to locally move (toward neighboring cells or rigidity gradients) or reorient (in the direction of slowly applied stretch or rigidity gradients).
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Affiliation(s)
- Dan Ben-Yaakov
- Dept. Materials and Interfaces, Weizmann Institute of Science, Rehovot, Israel.
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42
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Jansen KA, Bacabac RG, Piechocka IK, Koenderink GH. Cells actively stiffen fibrin networks by generating contractile stress. Biophys J 2014; 105:2240-51. [PMID: 24268136 DOI: 10.1016/j.bpj.2013.10.008] [Citation(s) in RCA: 104] [Impact Index Per Article: 10.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/15/2013] [Revised: 09/03/2013] [Accepted: 10/07/2013] [Indexed: 12/13/2022] Open
Abstract
During wound healing and angiogenesis, fibrin serves as a provisional extracellular matrix. We use a model system of fibroblasts embedded in fibrin gels to study how cell-mediated contraction may influence the macroscopic mechanical properties of their extracellular matrix during such processes. We demonstrate by macroscopic shear rheology that the cells increase the elastic modulus of the fibrin gels. Microscopy observations show that this stiffening sets in when the cells spread and apply traction forces on the fibrin fibers. We further show that the stiffening response mimics the effect of an external stress applied by mechanical shear. We propose that stiffening is a consequence of active myosin-driven cell contraction, which provokes a nonlinear elastic response of the fibrin matrix. Cell-induced stiffening is limited to a factor 3 even though fibrin gels can in principle stiffen much more before breaking. We discuss this observation in light of recent models of fibrin gel elasticity, and conclude that the fibroblasts pull out floppy modes, such as thermal bending undulations, from the fibrin network, but do not axially stretch the fibers. Our findings are relevant for understanding the role of matrix contraction by cells during wound healing and cancer development, and may provide design parameters for materials to guide morphogenesis in tissue engineering.
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Affiliation(s)
- Karin A Jansen
- Biological Soft Matter Group, FOM Institute AMOLF, Amsterdam, Netherlands
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43
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Wen Q, Janmey PA. Effects of non-linearity on cell-ECM interactions. Exp Cell Res 2013; 319:2481-9. [PMID: 23748051 PMCID: PMC3930572 DOI: 10.1016/j.yexcr.2013.05.017] [Citation(s) in RCA: 76] [Impact Index Per Article: 6.9] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/13/2013] [Accepted: 05/21/2013] [Indexed: 01/17/2023]
Abstract
Filamentous biopolymers such as F-actin, vimentin, fibrin and collagen that form networks within the cytoskeleton or the extracellular matrix have unusual rheological properties not present in most synthetic soft materials that are used as cell substrates or scaffolds for tissue engineering. Gels formed by purified filamentous biopolymers are often strain stiffening, with an elastic modulus that can increase an order of magnitude at moderate strains that are relevant to cell and tissue deformation in vivo. This review summarizes some experimental studies of non-linear rheology in biopolymer gels, discusses possible molecular mechanisms that account for strain stiffening, and explores the possible relevance of non-linear rheology to the interactions between cell and extracellular matrices.
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Affiliation(s)
- Qi Wen
- Department of Physics, Worcester Polytechnic Institute, 100 Institute Road, Worcester, MA 01609
| | - Paul A. Janmey
- Institute for Medicine and Engineering, University of Pennsylvania, 1010 Vagelos Laboratories, 3340 Smith Walk, Philadelphia, PA 19104
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44
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Sheinman M, Broedersz CP, MacKintosh FC. Actively stressed marginal networks. PHYSICAL REVIEW LETTERS 2012; 109:238101. [PMID: 23368268 DOI: 10.1103/physrevlett.109.238101] [Citation(s) in RCA: 33] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/27/2012] [Indexed: 06/01/2023]
Abstract
We study the effects of motor-generated stresses in disordered three-dimensional fiber networks using a combination of a mean-field theory, scaling analysis, and a computational model. We find that motor activity controls the elasticity in an anomalous fashion close to the point of marginal stability by coupling to critical network fluctuations. We also show that motor stresses can stabilize initially floppy networks, extending the range of critical behavior to a broad regime of network connectivities below the marginal point. Away from this regime, or at high stress, motors give rise to a linear increase in stiffness with stress. Finally, we demonstrate that our results are captured by a simple, constitutive scaling relation highlighting the important role of nonaffine strain fluctuations as a susceptibility to motor stress.
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Affiliation(s)
- M Sheinman
- Department of Physics and Astronomy, VU University, 1081 HV Amsterdam, The Netherlands
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