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Barrett JM, Malakoutian M, Fels S, Brown SHM, Oxland TR. Muscle short-range stiffness behaves like a maxwell element, not a spring: Implications for joint stability. PLoS One 2024; 19:e0307977. [PMID: 39141670 PMCID: PMC11324116 DOI: 10.1371/journal.pone.0307977] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/16/2024] [Accepted: 07/15/2024] [Indexed: 08/16/2024] Open
Abstract
INTRODUCTION Muscles play a critical role in supporting joints during activities of daily living, owing, in part, to the phenomenon of short-range stiffness. Briefly, when an active muscle is lengthened, bound cross-bridges are stretched, yielding forces greater than what is predicted from the force length relationship. For this reason, short-range stiffness has been proposed as an attractive mechanism for providing joint stability. However, there has yet to be a forward dynamic simulation employing a cross-bridge model, that demonstrates this stabilizing role. Therefore, the purpose of this investigation was to test whether Huxley-type muscle elements, which exhibit short-range stiffness, can stabilize a joint while at constant activation. METHODS We analyzed the stability of an inverted pendulum (moment of inertia: 2.7 kg m2) supported by Huxley-type muscle models that reproduce the short-range stiffness phenomenon. We calculated the muscle forces that would provide sufficient short-range stiffness to stabilize the system based in minimizing the potential energy. Simulations consisted of a 50 ms long, 5 Nm square-wave perturbation, with numerical simulations carried out in ArtiSynth. RESULTS Despite the initial analysis predicting shared activity of antagonist and agonist muscles to maintain stable equilibrium, the inverted pendulum model was not stable, and did not maintain an upright posture even with fully activated muscles. DISCUSSION & CONCLUSION Our simulations suggested that short-range stiffness cannot be solely responsible for joint stability, even for modest perturbations. We argue that short-range stiffness cannot achieve stability because its dynamics do not behave like a typical spring. Instead, an alternative conceptual model for short-range stiffness is that of a Maxwell element (spring and damper in series), which can be obtained as a first-order approximation to the Huxley model. We postulate that the damping that results from short-range stiffness slows down the mechanical response and allows the central nervous system time to react and stabilize the joint. We speculate that other mechanisms, like reflexes or residual force enhancement/depression, may also play a role in joint stability. Joint stability is due to a combination of factors, and further research is needed to fully understand this complex system.
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Affiliation(s)
- Jeff M. Barrett
- Department of Orthopaedics, The University of British Columbia, British Columbia, Canada
- ICORD Research Centre, The University of British Columbia, British Columbia, Canada
| | - Masoud Malakoutian
- ICORD Research Centre, The University of British Columbia, British Columbia, Canada
- Department of Mechanical Engineering, The University of British Columbia, British Columbia, Canada
| | - Sidney Fels
- Department of Electrical and Computer Engineering, The University of British Columbia, British Columbia, Canada
| | - Stephen H. M. Brown
- Department of Human Health and Nutritional Sciences, University of Guelph, Guelph, Canada
| | - Thomas R. Oxland
- Department of Orthopaedics, The University of British Columbia, British Columbia, Canada
- ICORD Research Centre, The University of British Columbia, British Columbia, Canada
- Department of Mechanical Engineering, The University of British Columbia, British Columbia, Canada
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2
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Torres-Sánchez A, Kerr Winter M, Salbreux G. Interacting active surfaces: A model for three-dimensional cell aggregates. PLoS Comput Biol 2022; 18:e1010762. [PMID: 36525467 PMCID: PMC9803321 DOI: 10.1371/journal.pcbi.1010762] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/23/2022] [Revised: 12/30/2022] [Accepted: 11/26/2022] [Indexed: 12/23/2022] Open
Abstract
We introduce a modelling and simulation framework for cell aggregates in three dimensions based on interacting active surfaces. Cell mechanics is captured by a physical description of the acto-myosin cortex that includes cortical flows, viscous forces, active tensions, and bending moments. Cells interact with each other via short-range forces capturing the effect of adhesion molecules. We discretise the model equations using a finite element method, and provide a parallel implementation in C++. We discuss examples of application of this framework to small and medium-sized aggregates: we consider the shape and dynamics of a cell doublet, a planar cell sheet, and a growing cell aggregate. This framework opens the door to the systematic exploration of the cell to tissue-scale mechanics of cell aggregates, which plays a key role in the morphogenesis of embryos and organoids.
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Affiliation(s)
| | - Max Kerr Winter
- Theoretical Physics of Biology laboratory, The Francis Crick Institute, London, United Kingdom
| | - Guillaume Salbreux
- Theoretical Physics of Biology laboratory, The Francis Crick Institute, London, United Kingdom
- Department of Genetics and Evolution, University of Geneva, Genève, Switzerland
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3
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Hanauer C, Bergeler S, Frey E, Broedersz CP. Theory of Active Intracellular Transport by DNA Relaying. PHYSICAL REVIEW LETTERS 2021; 127:138101. [PMID: 34623846 DOI: 10.1103/physrevlett.127.138101] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/08/2021] [Accepted: 08/12/2021] [Indexed: 06/13/2023]
Abstract
The spatiotemporal organization of bacterial cells is crucial for the active segregation of replicating chromosomes. In several species, including Caulobacter crescentus, the ATPase ParA binds to DNA and forms a gradient along the long cell axis. The ParB partition complex on the newly replicated chromosome translocates up this ParA gradient, thereby contributing to chromosome segregation. A DNA-relay mechanism-deriving from the elasticity of the fluctuating chromosome-has been proposed as the driving force for this cargo translocation, but a mechanistic theoretical description remains elusive. Here, we propose a minimal model to describe force generation by the DNA-relay mechanism over a broad range of operational conditions. Conceptually, we identify four distinct force-generation regimes characterized by their dependence on chromosome fluctuations. These relay force regimes arise from an interplay of the imposed ParA gradient, chromosome fluctuations, and an emergent friction force due to chromosome-cargo interactions.
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Affiliation(s)
- Christian Hanauer
- Arnold-Sommerfeld-Center for Theoretical Physics and Center for NanoScience, Ludwig-Maximilians-Universität München, D-80333 München, Germany
| | - Silke Bergeler
- Arnold-Sommerfeld-Center for Theoretical Physics and Center for NanoScience, Ludwig-Maximilians-Universität München, D-80333 München, Germany
| | - Erwin Frey
- Arnold-Sommerfeld-Center for Theoretical Physics and Center for NanoScience, Ludwig-Maximilians-Universität München, D-80333 München, Germany
| | - Chase P Broedersz
- Arnold-Sommerfeld-Center for Theoretical Physics and Center for NanoScience, Ludwig-Maximilians-Universität München, D-80333 München, Germany
- Department of Physics and Astronomy, Vrije Universiteit Amsterdam, 1081 HV Amsterdam, Netherlands
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4
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Irons L, Huang H, Owen MR, O'Dea RD, Meininger GA, Brook BS. Switching behaviour in vascular smooth muscle cell-matrix adhesion during oscillatory loading. J Theor Biol 2020; 502:110387. [PMID: 32603668 DOI: 10.1016/j.jtbi.2020.110387] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/16/2020] [Revised: 06/03/2020] [Accepted: 06/18/2020] [Indexed: 11/19/2022]
Abstract
Integrins regulate mechanotransduction between smooth muscle cells (SMCs) and the extracellular matrix (ECM). SMCs resident in the walls of airways or blood vessels are continuously exposed to dynamic mechanical forces due to breathing or pulsatile blood flow. However, the resulting effects of these forces on integrin dynamics and associated cell-matrix adhesion are not well understood. Here we present experimental results from atomic force microscopy (AFM) experiments, designed to study the integrin response to external oscillatory loading of varying amplitudes applied to live aortic SMCs, together with theoretical results from a mathematical model. In the AFM experiments, a fibronectin-coated probe was used cyclically to indent and retract from the surface of the cell. We observed a transition between states of firm adhesion and of complete detachment as the amplitude of oscillatory loading increased, revealed by qualitative changes in the force timecourses. Interestingly, for some of the SMCs in the experiments, switching behaviour between the two adhesion states is observed during single timecourses at intermediate amplitudes. We obtain two qualitatively similar adhesion states in the mathematical model, where we simulate the cell, integrins and ECM as an evolving system of springs, incorporating local integrin binding dynamics. In the mathematical model, we observe a region of bistability where both the firm adhesion and detachment states can occur depending on the initial adhesion state. The differences are seen to be a result of mechanical cooperativity of integrins and cell deformation. Switching behaviour is a phenomenon associated with bistability in a stochastic system, and bistability in our deterministic mathematical model provides a potential physical explanation for the experimental results. Physiologically, bistability provides a means for transient mechanical stimuli to induce long-term changes in adhesion dynamics-and thereby the cells' ability to transmit force-and we propose further experiments for testing this hypothesis.
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Affiliation(s)
- Linda Irons
- Centre for Mathematical Medicine and Biology, School of Mathematical Sciences, University of Nottingham, Nottingham, United Kingdom.
| | - Huang Huang
- Dalton Cardiovascular Research Center, Department of Medical Pharmacology and Physiology, University of Missouri, Columbia, MO, United States
| | - Markus R Owen
- Centre for Mathematical Medicine and Biology, School of Mathematical Sciences, University of Nottingham, Nottingham, United Kingdom
| | - Reuben D O'Dea
- Centre for Mathematical Medicine and Biology, School of Mathematical Sciences, University of Nottingham, Nottingham, United Kingdom
| | - Gerald A Meininger
- Dalton Cardiovascular Research Center, Department of Medical Pharmacology and Physiology, University of Missouri, Columbia, MO, United States
| | - Bindi S Brook
- Centre for Mathematical Medicine and Biology, School of Mathematical Sciences, University of Nottingham, Nottingham, United Kingdom
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5
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Stick-slip model for actin-driven cell protrusions, cell polarization, and crawling. Proc Natl Acad Sci U S A 2020; 117:24670-24678. [PMID: 32958682 DOI: 10.1073/pnas.2011785117] [Citation(s) in RCA: 21] [Impact Index Per Article: 5.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/09/2023] Open
Abstract
Cell crawling requires the generation of intracellular forces by the cytoskeleton and their transmission to an extracellular substrate through specific adhesion molecules. Crawling cells show many features of excitable systems, such as spontaneous symmetry breaking and crawling in the absence of external cues, and periodic and propagating waves of activity. Mechanical instabilities in the active cytoskeleton network and feedback loops in the biochemical network of activators and repressors of cytoskeleton dynamics have been invoked to explain these dynamical features. Here, I show that the interplay between the dynamics of cell-substrate adhesion and linear cellular mechanics is sufficient to reproduce many nonlinear dynamical patterns observed in spreading and crawling cells. Using an analytical formalism of the molecular clutch model of cell adhesion, regulated by local mechanical forces, I show that cellular traction forces exhibit stick-slip dynamics resulting in periodic waves of protrusion/retraction and propagating waves along the cell edge. This can explain spontaneous symmetry breaking and polarization of spreading cells, leading to steady crawling or bipedal motion, and bistability, where persistent cell motion requires a sufficiently strong transient external stimulus. The model also highlights the role of membrane tension in providing the long-range mechanical communication across the cell required for symmetry breaking.
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6
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Bressloff PC. Stochastic resetting and the mean-field dynamics of focal adhesions. Phys Rev E 2020; 102:022134. [PMID: 32942383 DOI: 10.1103/physreve.102.022134] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/20/2020] [Accepted: 08/12/2020] [Indexed: 11/07/2022]
Abstract
In this paper we investigate the effects of diffusion on the dynamics of a single focal adhesion at the leading edge of a crawling cell by considering a simplified model of sliding friction. Using a mean-field approximation, we derive an effective single-particle system that can be interpreted as an overdamped Brownian particle with spatially dependent stochastic resetting. We then use renewal and path-integral methods from the theory of stochastic resetting to calculate the mean sliding velocity under the combined action of diffusion, active forces, viscous drag, and elastic forces generated by the adhesive bonds. Our analysis suggests that the inclusion of diffusion can sharpen the response to changes in the effective stiffness of the adhesion bonds. This is consistent with the hypothesis that force fluctuations could play a role in mechanosensing of the local microenvironment.
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Affiliation(s)
- Paul C Bressloff
- Department of Mathematics, University of Utah Salt Lake City, Utah 84112, USA
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7
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Wierenga H, Wolde PRT. Diffusible Cross-linkers Cause Superexponential Friction Forces. PHYSICAL REVIEW LETTERS 2020; 125:078101. [PMID: 32857554 DOI: 10.1103/physrevlett.125.078101] [Citation(s) in RCA: 10] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/19/2019] [Accepted: 07/15/2020] [Indexed: 06/11/2023]
Abstract
The friction between cytoskeletal filaments is of central importance for the formation of cellular structures such as the mitotic spindle and the cytokinetic ring. This friction is caused by passive cross-linkers, yet the underlying mechanism and the dependence on cross-linker density are poorly understood. Here, we use theory and computer simulations to study the friction between two filaments that are cross-linked by passive proteins, which can hop between discrete binding sites while physically excluding each other. The simulations reveal that filaments move via rare discrete jumps, which are associated with free-energy barrier crossings. We identify the reaction coordinate that governs the relative microtubule movement and derive an exact analytical expression for the free-energy barrier and the friction coefficient. Our analysis not only elucidates the molecular mechanism underlying cross-linker-induced filament friction, but also predicts that the friction coefficient scales superexponentially with the density of cross-linkers.
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8
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Abstract
Integrins, and integrin-mediated adhesions, have long been recognized to provide the main molecular link attaching cells to the extracellular matrix (ECM) and to serve as bidirectional hubs transmitting signals between cells and their environment. Recent evidence has shown that their combined biochemical and mechanical properties also allow integrins to sense, respond to and interact with ECM of differing properties with exquisite specificity. Here, we review this work first by providing an overview of how integrin function is regulated from both a biochemical and a mechanical perspective, affecting integrin cell-surface availability, binding properties, activation or clustering. Then, we address how this biomechanical regulation allows integrins to respond to different ECM physicochemical properties and signals, such as rigidity, composition and spatial distribution. Finally, we discuss the importance of this sensing for major cell functions by taking cell migration and cancer as examples.
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9
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Mai MH, Camley BA. Hydrodynamic effects on the motility of crawling eukaryotic cells. SOFT MATTER 2020; 16:1349-1358. [PMID: 31934705 DOI: 10.1039/c9sm01797f] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/10/2023]
Abstract
Eukaryotic cell motility is crucial during development, wound healing, the immune response, and cancer metastasis. Some eukaryotic cells can swim, but cells more commonly adhere to and crawl along the extracellular matrix. We study the relationship between hydrodynamics and adhesion that describe whether a cell is swimming, crawling, or combining these motions. Our simple model of a cell, based on the three-sphere swimmer, is capable of both swimming and crawling. As cell-matrix adhesion strength increases, the influence of hydrodynamics on migration diminishes. Cells with significant adhesion can crawl with speeds much larger than their nonadherent, swimming counterparts. We predict that, while most eukaryotic cells are in the strong-adhesion limit, increasing environment viscosity or decreasing cell-matrix adhesion could lead to significant hydrodynamic effects even in crawling cells. Signatures of hydrodynamic effects include a dependence of cell speed on the presence of a nearby substrate or interactions between noncontacting cells. These signatures will be suppressed at large adhesion strengths, but even strongly adherent cells will generate relevant fluid flows that will advect nearby passive particles and swimmers.
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Affiliation(s)
- Melissa H Mai
- Department of Biophysics, Johns Hopkins University, Baltimore, Maryland, USA
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10
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Ouyang W, Ramakrishna SN, Rossi A, Urbakh M, Spencer ND, Arcifa A. Load and Velocity Dependence of Friction Mediated by Dynamics of Interfacial Contacts. PHYSICAL REVIEW LETTERS 2019; 123:116102. [PMID: 31573261 DOI: 10.1103/physrevlett.123.116102] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/17/2019] [Revised: 06/19/2019] [Indexed: 06/10/2023]
Abstract
Studying the frictional properties of interfaces with dynamic chemical bonds advances understanding of the mechanism underlying rate and state laws, and offers new pathways for the rational control of frictional response. In this work, we revisit the load dependence of interfacial chemical-bond-induced (ICBI) friction experimentally and find that the velocity dependence of friction can be reversed by changing the normal load. We propose a theoretical model, whose analytical solution allows us to interpret the experimental data on timescales and length scales that are relevant to experimental conditions. Our work provides a promising avenue for exploring the dynamics of ICBI friction.
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Affiliation(s)
- Wengen Ouyang
- School of Chemistry and The Sackler Center for Computational Molecular and Materials Science, Tel Aviv University, Tel Aviv 69978, Israel
| | - Shivaprakash N Ramakrishna
- Laboratory for Surface Science and Technology, Department of Materials, ETH Zurich, Vladimir-Prelog-Weg 5,CH-8093 Zurich, Switzerland
| | - Antonella Rossi
- Laboratory for Surface Science and Technology, Department of Materials, ETH Zurich, Vladimir-Prelog-Weg 5,CH-8093 Zurich, Switzerland
- Dipartimento di Scienze Chimiche e Geologiche, Università degli Studi di Cagliari, Cittadella Universitaria di Monserrato, I-09100 Cagliari, Italy
| | - Michael Urbakh
- School of Chemistry and The Sackler Center for Computational Molecular and Materials Science, Tel Aviv University, Tel Aviv 69978, Israel
| | - Nicholas D Spencer
- Laboratory for Surface Science and Technology, Department of Materials, ETH Zurich, Vladimir-Prelog-Weg 5,CH-8093 Zurich, Switzerland
| | - Andrea Arcifa
- Laboratory for Surface Science and Technology, Department of Materials, ETH Zurich, Vladimir-Prelog-Weg 5,CH-8093 Zurich, Switzerland
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11
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Uroz M, Garcia-Puig A, Tekeli I, Elosegui-Artola A, Abenza JF, Marín-Llauradó A, Pujals S, Conte V, Albertazzi L, Roca-Cusachs P, Raya Á, Trepat X. Traction forces at the cytokinetic ring regulate cell division and polyploidy in the migrating zebrafish epicardium. NATURE MATERIALS 2019; 18:1015-1023. [PMID: 31160803 DOI: 10.1038/s41563-019-0381-9] [Citation(s) in RCA: 24] [Impact Index Per Article: 4.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/24/2018] [Accepted: 04/18/2019] [Indexed: 06/09/2023]
Abstract
Epithelial repair and regeneration are driven by collective cell migration and division. Both cellular functions involve tightly controlled mechanical events, but how physical forces regulate cell division in migrating epithelia is largely unknown. Here we show that cells dividing in the migrating zebrafish epicardium exert large cell-extracellular matrix (ECM) forces during cytokinesis. These forces point towards the division axis and are exerted through focal adhesions that connect the cytokinetic ring to the underlying ECM. When subjected to high loading rates, these cytokinetic focal adhesions prevent closure of the contractile ring, leading to multi-nucleation through cytokinetic failure. By combining a clutch model with experiments on substrates of different rigidity, ECM composition and ligand density, we show that failed cytokinesis is triggered by adhesion reinforcement downstream of increased myosin density. The mechanical interaction between the cytokinetic ring and the ECM thus provides a mechanism for the regulation of cell division and polyploidy that may have implications in regeneration and cancer.
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Affiliation(s)
- Marina Uroz
- Institute for Bioengineering of Catalonia (IBEC), The Barcelona Institute for Science and Technology (BIST), Barcelona, Spain
| | - Anna Garcia-Puig
- Center of Regenerative Medicine in Barcelona (CMRB), Barcelona, Spain
- Centro de Investigación Biomédica en Red en Bioingeniería, Biomateriales y Nanomedicina (CIBER-BBN), Barcelona, Spain
| | - Isil Tekeli
- Center of Regenerative Medicine in Barcelona (CMRB), Barcelona, Spain
- Centro de Investigación Biomédica en Red en Bioingeniería, Biomateriales y Nanomedicina (CIBER-BBN), Barcelona, Spain
| | - Alberto Elosegui-Artola
- Institute for Bioengineering of Catalonia (IBEC), The Barcelona Institute for Science and Technology (BIST), Barcelona, Spain
| | - Juan F Abenza
- Institute for Bioengineering of Catalonia (IBEC), The Barcelona Institute for Science and Technology (BIST), Barcelona, Spain
| | - Ariadna Marín-Llauradó
- Institute for Bioengineering of Catalonia (IBEC), The Barcelona Institute for Science and Technology (BIST), Barcelona, Spain
| | - Silvia Pujals
- Institute for Bioengineering of Catalonia (IBEC), The Barcelona Institute for Science and Technology (BIST), Barcelona, Spain
| | - Vito Conte
- Institute for Bioengineering of Catalonia (IBEC), The Barcelona Institute for Science and Technology (BIST), Barcelona, Spain
- Department of Biomedical Engineering and the Institute for Complex Molecular Systems, Eindhoven University of Technology, Eindhoven, the Netherlands
| | - Lorenzo Albertazzi
- Institute for Bioengineering of Catalonia (IBEC), The Barcelona Institute for Science and Technology (BIST), Barcelona, Spain
- Department of Biomedical Engineering and the Institute for Complex Molecular Systems, Eindhoven University of Technology, Eindhoven, the Netherlands
| | - Pere Roca-Cusachs
- Institute for Bioengineering of Catalonia (IBEC), The Barcelona Institute for Science and Technology (BIST), Barcelona, Spain
- University of Barcelona, Barcelona, Spain
| | - Ángel Raya
- Center of Regenerative Medicine in Barcelona (CMRB), Barcelona, Spain
- Centro de Investigación Biomédica en Red en Bioingeniería, Biomateriales y Nanomedicina (CIBER-BBN), Barcelona, Spain
- Institució Catalana de Recerca i Estudis Avançats (ICREA), Barcelona, Spain
| | - Xavier Trepat
- Institute for Bioengineering of Catalonia (IBEC), The Barcelona Institute for Science and Technology (BIST), Barcelona, Spain.
- Centro de Investigación Biomédica en Red en Bioingeniería, Biomateriales y Nanomedicina (CIBER-BBN), Barcelona, Spain.
- University of Barcelona, Barcelona, Spain.
- Institució Catalana de Recerca i Estudis Avançats (ICREA), Barcelona, Spain.
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12
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Irons L, Owen MR, O'Dea RD, Brook BS. Effect of Loading History on Airway Smooth Muscle Cell-Matrix Adhesions. Biophys J 2019; 114:2679-2690. [PMID: 29874617 DOI: 10.1016/j.bpj.2018.04.026] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/27/2017] [Revised: 04/05/2018] [Accepted: 04/16/2018] [Indexed: 01/06/2023] Open
Abstract
Integrin-mediated adhesions between airway smooth muscle (ASM) cells and the extracellular matrix (ECM) regulate how contractile forces generated within the cell are transmitted to its external environment. Environmental cues are known to influence the formation, size, and survival of cell-matrix adhesions, but it is not yet known how they are affected by dynamic fluctuations associated with tidal breathing in the intact airway. Here, we develop two closely related theoretical models to study adhesion dynamics in response to oscillatory loading of the ECM, representing the dynamic environment of ASM cells in vivo. Using a discrete stochastic-elastic model, we simulate individual integrin binding and rupture events and observe two stable regimes in which either bond formation or bond rupture dominate, depending on the amplitude of the oscillatory loading. These regimes have either a high or low fraction of persistent adhesions, which could affect the level of strain transmission between contracted ASM cells and the airway tissue. For intermediate loading, we observe a region of bistability and hysteresis due to shared loading between existing bonds; the level of adhesion depends on the loading history. These findings are replicated in a related continuum model, which we use to investigate the effect of perturbations mimicking deep inspirations (DIs). Because of the bistability, a DI applied to the high adhesion state could either induce a permanent switch to a lower adhesion state or allow a return of the system to the high adhesion state. Transitions between states are further influenced by the frequency of oscillations, cytoskeletal or ECM stiffnesses, and binding affinities, which modify the magnitudes of the stable adhesion states as well as the region of bistability. These findings could explain (in part) the transient bronchodilatory effect of a DI observed in asthmatics compared to a more sustained effect in normal subjects.
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Affiliation(s)
- Linda Irons
- Centre for Mathematical Medicine and Biology, School of Mathematical Sciences, University of Nottingham, Nottingham, United Kingdom.
| | - Markus R Owen
- Centre for Mathematical Medicine and Biology, School of Mathematical Sciences, University of Nottingham, Nottingham, United Kingdom
| | - Reuben D O'Dea
- Centre for Mathematical Medicine and Biology, School of Mathematical Sciences, University of Nottingham, Nottingham, United Kingdom
| | - Bindi S Brook
- Centre for Mathematical Medicine and Biology, School of Mathematical Sciences, University of Nottingham, Nottingham, United Kingdom
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13
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Newhard CS, Walcott S, Swank DM. The load dependence of muscle's force-velocity curve is modulated by alternative myosin converter domains. Am J Physiol Cell Physiol 2019; 316:C844-C861. [PMID: 30865518 PMCID: PMC6620577 DOI: 10.1152/ajpcell.00494.2018] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/10/2018] [Revised: 02/15/2019] [Accepted: 03/05/2019] [Indexed: 01/07/2023]
Abstract
The hyperbolic shape of the muscle force-velocity relationship (FVR) is characteristic of all muscle fiber types. The degree of curvature of the hyperbola varies between muscle fiber types and is thought to be set by force-dependent properties of different myosin isoforms. However, the structural elements in myosin and the mechanism that determines force dependence are unresolved. We tested our hypothesis that the myosin converter domain plays a critical role in the force-velocity relationship (FVR) mechanism. Drosophila contains a single myosin heavy chain gene with five converters encoded by alternative exons. We measured FVR properties of Drosophila jump muscle fibers from five transgenic lines each expressing a single converter. Consistent with our hypothesis, we observed up to 2.4-fold alterations in FVR curvature. Maximum shortening velocity (v0) and optimal velocity for maximum power generation were also altered, but isometric tension and maximum power generation were unaltered. Converter 11a, normally found in the indirect flight muscle (IFM), imparted the highest FVR curvature and v0, whereas converter 11d, found in larval body wall muscle, imparted the most linear FVR and slowest v0. Jump distance strongly correlated with increasing FVR curvature and v0, meaning flies expressing the converter from the IFM jumped farther than flies expressing the native jump muscle converter. Fitting our data with Huxley's two-state model and a biophysically based four-state model suggest a testable hypothesis that the converter sets muscle type FVR curvature by influencing the detachment rate of negatively strained myosin via changes in the force dependence of product release.
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Affiliation(s)
- Christopher S Newhard
- Department of Biological Sciences, Department of Biomedical Engineering and Center for Biotechnology and Interdisciplinary Studies, Rensselaer Polytechnic Institute , Troy, New York
| | - Sam Walcott
- Department of Mathematics, University of California , Davis, California
| | - Douglas M Swank
- Department of Biological Sciences, Department of Biomedical Engineering and Center for Biotechnology and Interdisciplinary Studies, Rensselaer Polytechnic Institute , Troy, New York
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14
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Cao Y, Karmakar R, Ghabache E, Gutierrez E, Zhao Y, Groisman A, Levine H, Camley BA, Rappel WJ. Cell motility dependence on adhesive wetting. SOFT MATTER 2019; 15:2043-2050. [PMID: 30724956 DOI: 10.1039/c8sm01832d] [Citation(s) in RCA: 13] [Impact Index Per Article: 2.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/09/2023]
Abstract
Adhesive cell-substrate interactions are crucial for cell motility and are responsible for the necessary traction that propels cells. These interactions can also change the shape of the cell, analogous to liquid droplet wetting on adhesive substrates. To address how these shape changes affect cell migration and cell speed we model motility using deformable, 2D cross-sections of cells in which adhesion and frictional forces between cell and substrate can be varied separately. Our simulations show that increasing the adhesion results in increased spreading of cells and larger cell speeds. We propose an analytical model which shows that the cell speed is inversely proportional to an effective height of the cell and that increasing this height results in increased internal shear stress. The numerical and analytical results are confirmed in experiments on motile eukaryotic cells.
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Affiliation(s)
- Yuansheng Cao
- Department of Physics, University of California, San Diego, La Jolla, California 92093, USA.
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15
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Christensen A, West AKV, Wullkopf L, Terra Erler J, Oddershede LB, Mathiesen J. Friction-limited cell motility in confluent monolayer tissue. Phys Biol 2018; 15:066004. [DOI: 10.1088/1478-3975/aacedc] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/08/2023]
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16
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Camley BA. Collective gradient sensing and chemotaxis: modeling and recent developments. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2018; 30:223001. [PMID: 29644981 PMCID: PMC6252055 DOI: 10.1088/1361-648x/aabd9f] [Citation(s) in RCA: 26] [Impact Index Per Article: 4.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/05/2023]
Abstract
Cells measure a vast variety of signals, from their environment's stiffness to chemical concentrations and gradients; physical principles strongly limit how accurately they can do this. However, when many cells work together, they can cooperate to exceed the accuracy of any single cell. In this topical review, I will discuss the experimental evidence showing that cells collectively sense gradients of many signal types, and the models and physical principles involved. I also propose new routes by which experiments and theory can expand our understanding of these problems.
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Affiliation(s)
- Brian A Camley
- Departments of Physics & Astronomy and Biophysics, Johns Hopkins University, Baltimore, MD, United States of America
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17
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Elosegui-Artola A, Trepat X, Roca-Cusachs P. Control of Mechanotransduction by Molecular Clutch Dynamics. Trends Cell Biol 2018; 28:356-367. [PMID: 29496292 DOI: 10.1016/j.tcb.2018.01.008] [Citation(s) in RCA: 171] [Impact Index Per Article: 28.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/31/2017] [Revised: 01/24/2018] [Accepted: 01/30/2018] [Indexed: 02/09/2023]
Abstract
The linkage of cells to their microenvironment is mediated by a series of bonds that dynamically engage and disengage, in what has been conceptualized as the molecular clutch model. Whereas this model has long been employed to describe actin cytoskeleton and cell migration dynamics, it has recently been proposed to also explain mechanotransduction (i.e., the process by which cells convert mechanical signals from their environment into biochemical signals). Here we review the current understanding on how cell dynamics and mechanotransduction are driven by molecular clutch dynamics and its master regulator, the force loading rate. Throughout this Review, we place a specific emphasis on the quantitative prediction of cell response enabled by combined experimental and theoretical approaches.
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Affiliation(s)
- Alberto Elosegui-Artola
- Institute for Bioengineering of Catalonia (IBEC), The Barcelona Institute of Science and Technology (BIST), 08028 Barcelona, Spain.
| | - Xavier Trepat
- Institute for Bioengineering of Catalonia (IBEC), The Barcelona Institute of Science and Technology (BIST), 08028 Barcelona, Spain; University of Barcelona, 08028 Barcelona, Spain; Institució Catalana de Recerca i Estudis Avançats (ICREA), 08010 Barcelona, Spain; Centro de Investigación Biomédica en Red en Bioingeniería, Biomateriales y Nanomedicina, 28029 Madrid, Spain
| | - Pere Roca-Cusachs
- Institute for Bioengineering of Catalonia (IBEC), The Barcelona Institute of Science and Technology (BIST), 08028 Barcelona, Spain; University of Barcelona, 08028 Barcelona, Spain.
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18
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Xu GK, Feng XQ, Gao H. Orientations of Cells on Compliant Substrates under Biaxial Stretches: A Theoretical Study. Biophys J 2018; 114:701-710. [PMID: 29414715 PMCID: PMC5985023 DOI: 10.1016/j.bpj.2017.12.002] [Citation(s) in RCA: 24] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/11/2017] [Revised: 11/25/2017] [Accepted: 12/05/2017] [Indexed: 10/24/2022] Open
Abstract
Mechanical cues from the microenvironments play a regulating role in many physiological and pathological processes, such as stem cell differentiation and cancer cell metastasis. Experiments showed that cells adhered on a compliant substrate may change orientation with an externally applied strain in the substrate. By accounting for actin polymerization, actin retrograde flow, and integrin binding dynamics, here we develop a mechanism-based tensegrity model to study the orientations of polarized cells on a compliant substrate under biaxial stretches. We show that the cell can actively regulate its mechanical state by generating different traction force levels along its polarized direction. Under static or ultralow-frequency cyclic stretches, stretching a softer substrate leads to a higher increase in the traction force and induces a narrower distribution of cell alignment. Compared to static loadings, high-frequency cyclic loadings have a more significant influence on cell reorientation on a stiff substrate. In addition, the width of the cellular angular distribution scales inversely with the stretch amplitude under both static and cyclic stretches. Our results are in agreement with a wide range of experimental observations, and provide fundamental insights into the functioning of cellular mechanosensing systems.
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Affiliation(s)
- Guang-Kui Xu
- International Center for Applied Mechanics, State Key Laboratory for Strength and Vibration of Mechanical Structures, Xi'an Jiaotong University, Xi'an, China.
| | - Xi-Qiao Feng
- Institute of Biomechanics and Medical Engineering, Department of Engineering Mechanics, Tsinghua University, Beijing, China
| | - Huajian Gao
- School of Engineering, Brown University, Providence, Rhode Island
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19
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Miles CE, Keener JP. Bidirectionality from cargo thermal fluctuations in motor-mediated transport. J Theor Biol 2017; 424:37-48. [DOI: 10.1016/j.jtbi.2017.04.032] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/10/2016] [Revised: 04/26/2017] [Accepted: 04/29/2017] [Indexed: 11/29/2022]
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20
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Zhang S, Guy RD, Lasheras JC, Del Álamo JC. Self-organized mechano-chemical dynamics in amoeboid locomotion of Physarum fragments. JOURNAL OF PHYSICS D: APPLIED PHYSICS 2017; 50:204004. [PMID: 30906070 PMCID: PMC6430145 DOI: 10.1088/1361-6463/aa68be] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/28/2023]
Abstract
The aim of this work is to quantify the spatio-temporal dynamics of flow-driven amoeboid locomotion in small (~100 µm) fragments of the true slime mold Physarum polycephalum. In this model organism, cellular contraction drives intracellular flows, and these flows transport the chemical signals that regulate contraction in the first place. As a consequence of these non-linear interactions, a diversity of migratory behaviors can be observed in migrating Physarum fragments. To study these dynamics, we measure the spatio-temporal distributions of the velocities of the endoplasm and ectoplasm of each migrating fragment, the traction stresses it generates on the substratum, and the concentration of free intracellular calcium. Using these unprecedented experimental data, we classify migrating Physarum fragments according to their dynamics, finding that they often exhibit spontaneously coordinated waves of flow, contractility and chemical signaling. We show that Physarum fragments exhibiting symmetric spatio-temporal patterns of endoplasmic flow migrate significantly slower than fragments with asymmetric patterns. In addition, our joint measurements of ectoplasm velocity and traction stress at the substratum suggest that forward motion of the ectoplasm is enabled by a succession of stick-slip transitions, which we conjecture are also organized in the form of waves. Combining our experiments with a simplified convection-diffusion model, we show that the convective transport of calcium ions may be key for establishing and maintaining the spatiotemporal patterns of calcium concentration that regulate the generation of contractile forces.
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Affiliation(s)
- Shun Zhang
- Mechanical and Aerospace Engineering Department, University of California San Diego
| | - Robert D Guy
- Department of Mathematics, University of California Davis
| | - Juan C Lasheras
- Mechanical and Aerospace Engineering Department, University of California San Diego
- Department of Bioengineering, University of California San Diego
- Institute for Engineering in Medicine, University of California San Diego
| | - Juan C Del Álamo
- Mechanical and Aerospace Engineering Department, University of California San Diego
- Institute for Engineering in Medicine, University of California San Diego
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21
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Tao J, Li Y, Vig DK, Sun SX. Cell mechanics: a dialogue. REPORTS ON PROGRESS IN PHYSICS. PHYSICAL SOCIETY (GREAT BRITAIN) 2017; 80:036601. [PMID: 28129208 PMCID: PMC5518794 DOI: 10.1088/1361-6633/aa5282] [Citation(s) in RCA: 28] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
Abstract
Under the microscope, eukaryotic animal cells can adopt a variety of different shapes and sizes. These cells also move and deform, and the physical mechanisms driving these movements and shape changes are important in fundamental cell biology, tissue mechanics, as well as disease biology. This article reviews some of the basic mechanical concepts in cells, emphasizing continuum mechanics description of cytoskeletal networks and hydrodynamic flows across the cell membrane. We discuss how cells can generate movement and shape changes by controlling mass fluxes at the cell boundary. These mass fluxes can come from polymerization/depolymerization of actin cytoskeleton, as well as osmotic and hydraulic pressure-driven flow of water across the cell membrane. By combining hydraulic pressure control with force balance conditions at the cell surface, we discuss a quantitative mechanism of cell shape and volume control. The broad consequences of this model on cell mechanosensation and tissue mechanics are outlined.
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Affiliation(s)
- Jiaxiang Tao
- Departments of Mechanical Engineering, Johns Hopkins University, Baltimore MD, United States of America
- Physical Sciences in Oncology Center, Johns Hopkins University, Baltimore MD, United States of America
| | - Yizeng Li
- Departments of Mechanical Engineering, Johns Hopkins University, Baltimore MD, United States of America
- Institute of NanoBioTechnology, Johns Hopkins University, Baltimore MD, United States of America
| | - Dhruv K Vig
- Departments of Mechanical Engineering, Johns Hopkins University, Baltimore MD, United States of America
- Institute of NanoBioTechnology, Johns Hopkins University, Baltimore MD, United States of America
| | - Sean X Sun
- Departments of Mechanical Engineering, Johns Hopkins University, Baltimore MD, United States of America
- Biomedical Engineering, Johns Hopkins University, Baltimore MD, United States of America
- Physical Sciences in Oncology Center, Johns Hopkins University, Baltimore MD, United States of America
- Institute of NanoBioTechnology, Johns Hopkins University, Baltimore MD, United States of America
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22
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Lewis OL, Zhang S, Guy RD, del Álamo JC. Coordination of contractility, adhesion and flow in migrating Physarum amoebae. J R Soc Interface 2016; 12:rsif.2014.1359. [PMID: 25904525 DOI: 10.1098/rsif.2014.1359] [Citation(s) in RCA: 43] [Impact Index Per Article: 5.4] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022] Open
Abstract
This work examines the relationship between spatio-temporal coordination of intracellular flow and traction stress and the speed of amoeboid locomotion of microplasmodia of Physarum polycephalum. We simultaneously perform particle image velocimetry and traction stress microscopy to measure the velocity of cytoplasmic flow and the stresses applied to the substrate by migrating Physarum microamoebae. In parallel, we develop a mathematical model of a motile cell which includes forces from the viscous cytosol, a poro-elastic, contractile cytoskeleton and adhesive interactions with the substrate. Our experiments show that flow and traction stress exhibit back-to-front-directed waves with a distinct phase difference. The model demonstrates that the direction and speed of locomotion are determined by this coordination between contraction, flow and adhesion. Using the model, we identify forms of coordination that generate model predictions consistent with experiments. We demonstrate that this coordination produces near optimal migration speed and is insensitive to heterogeneity in substrate adhesiveness. While it is generally thought that amoeboid motility is robust to changes in extracellular geometry and the nature of extracellular adhesion, our results demonstrate that coordination of adhesive forces is essential to producing robust migration.
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Affiliation(s)
- Owen L Lewis
- Department of Mathematics, University of Utah, Salt Lake City, UT, USA
| | - Shun Zhang
- Mechanical and Aerospace Engineering Department, University of California San Diego, La Jolla, CA, USA
| | - Robert D Guy
- Department of Mathematics, University of California Davis, Davis, CA, USA
| | - Juan C del Álamo
- Mechanical and Aerospace Engineering Department, University of California San Diego, La Jolla, CA, USA
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23
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Schwarz US. Physical constraints for pathogen movement. Semin Cell Dev Biol 2015; 46:82-90. [DOI: 10.1016/j.semcdb.2015.09.025] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/29/2015] [Accepted: 09/29/2015] [Indexed: 10/22/2022]
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24
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Gralka M, Kroy K. Inelastic mechanics: A unifying principle in biomechanics. BIOCHIMICA ET BIOPHYSICA ACTA-MOLECULAR CELL RESEARCH 2015; 1853:3025-37. [PMID: 26151340 DOI: 10.1016/j.bbamcr.2015.06.017] [Citation(s) in RCA: 17] [Impact Index Per Article: 1.9] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/14/2015] [Revised: 05/13/2015] [Accepted: 06/26/2015] [Indexed: 01/16/2023]
Abstract
Many soft materials are classified as viscoelastic. They behave mechanically neither quite fluid-like nor quite solid-like - rather a bit of both. Biomaterials are often said to fall into this class. Here, we argue that this misses a crucial aspect, and that biomechanics is essentially damage mechanics, at heart. When deforming an animal cell or tissue, one can hardly avoid inducing the unfolding of protein domains, the unbinding of cytoskeletal crosslinkers, the breaking of weak sacrificial bonds, and the disruption of transient adhesions. We classify these activated structural changes as inelastic. They are often to a large degree reversible and are therefore not plastic in the proper sense, but they dissipate substantial amounts of elastic energy by structural damping. We review recent experiments involving biological materials on all scales, from single biopolymers over cells to model tissues, to illustrate the unifying power of this paradigm. A deliberately minimalistic yet phenomenologically very rich mathematical modeling framework for inelastic biomechanics is proposed. It transcends the conventional viscoelastic paradigm and suggests itself as a promising candidate for a unified description and interpretation of a wide range of experimental data. This article is part of a Special Issue entitled: Mechanobiology.
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Affiliation(s)
- Matti Gralka
- Institute for Theoretical Physics, University of Leipzig, Bruederstr. 16, 04103 Leipzig, Germany.
| | - Klaus Kroy
- Institute for Theoretical Physics, University of Leipzig, Bruederstr. 16, 04103 Leipzig, Germany.
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25
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Walcott S. Muscle activation described with a differential equation model for large ensembles of locally coupled molecular motors. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 90:042717. [PMID: 25375533 DOI: 10.1103/physreve.90.042717] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/27/2014] [Indexed: 06/04/2023]
Abstract
Molecular motors, by turning chemical energy into mechanical work, are responsible for active cellular processes. Often groups of these motors work together to perform their biological role. Motors in an ensemble are coupled and exhibit complex emergent behavior. Although large motor ensembles can be modeled with partial differential equations (PDEs) by assuming that molecules function independently of their neighbors, this assumption is violated when motors are coupled locally. It is therefore unclear how to describe the ensemble behavior of the locally coupled motors responsible for biological processes such as calcium-dependent skeletal muscle activation. Here we develop a theory to describe locally coupled motor ensembles and apply the theory to skeletal muscle activation. The central idea is that a muscle filament can be divided into two phases: an active and an inactive phase. Dynamic changes in the relative size of these phases are described by a set of linear ordinary differential equations (ODEs). As the dynamics of the active phase are described by PDEs, muscle activation is governed by a set of coupled ODEs and PDEs, building on previous PDE models. With comparison to Monte Carlo simulations, we demonstrate that the theory captures the behavior of locally coupled ensembles. The theory also plausibly describes and predicts muscle experiments from molecular to whole muscle scales, suggesting that a micro- to macroscale muscle model is within reach.
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Affiliation(s)
- Sam Walcott
- Department of Mathematics, University of California, Davis, Davis, California 95616, USA
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26
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Thøgersen K, Trømborg JK, Sveinsson HA, Malthe-Sørenssen A, Scheibert J. History-dependent friction and slow slip from time-dependent microscopic junction laws studied in a statistical framework. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 89:052401. [PMID: 25353806 DOI: 10.1103/physreve.89.052401] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/21/2014] [Indexed: 06/04/2023]
Abstract
To study how macroscopic friction phenomena originate from microscopic junction laws, we introduce a general statistical framework describing the collective behavior of a large number of individual microjunctions forming a macroscopic frictional interface. Each microjunction can switch in time between two states: a pinned state characterized by a displacement-dependent force and a slipping state characterized by a time-dependent force. Instead of tracking each microjunction individually, the state of the interface is described by two coupled distributions for (i) the stretching of pinned junctions and (ii) the time spent in the slipping state. This framework allows for a whole family of microjunction behavior laws, and we show how it represents an overarching structure for many existing models found in the friction literature. We then use this framework to pinpoint the effects of the time scale that controls the duration of the slipping state. First, we show that the model reproduces a series of friction phenomena already observed experimentally. The macroscopic steady-state friction force is velocity dependent, either monotonic (strengthening or weakening) or nonmonotonic (weakening-strengthening), depending on the microscopic behavior of individual junctions. In addition, slow slip, which has been reported in a wide variety of systems, spontaneously occurs in the model if the friction contribution from junctions in the slipping state is time weakening. Next, we show that the model predicts a nontrivial history dependence of the macroscopic static friction force. In particular, the static friction coefficient at the onset of sliding is shown to increase with increasing deceleration during the final phases of the preceding sliding event. We suggest that this form of history dependence of static friction should be investigated in experiments, and we provide the acceleration range in which this effect is expected to be experimentally observable.
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Affiliation(s)
- Kjetil Thøgersen
- Department of Physics, University of Oslo, Sem Sælands vei 24, NO-0316 Oslo, Norway
| | | | | | | | - Julien Scheibert
- Laboratoire de Tribologie et Dynamique des Systèmes, CNRS, Ecole Centrale de Lyon, 36, Avenue Guy de Collongue, 69134 Ecully Cedex, France
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27
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Stirbat TV, Tlili S, Houver T, Rieu JP, Barentin C, Delanoë-Ayari H. Multicellular aggregates: a model system for tissue rheology. THE EUROPEAN PHYSICAL JOURNAL. E, SOFT MATTER 2013; 36:84. [PMID: 23928795 DOI: 10.1140/epje/i2013-13084-1] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/15/2012] [Accepted: 06/21/2013] [Indexed: 06/02/2023]
Abstract
Morphogenetic processes involve cell flows. The mechanical response of a tissue to active forces is linked to its effective viscosity. In order to decouple this mechanical response from the complex genetic changes occurring in a developing organism, we perform rheometry experiments on multicellular aggregates, which are good models for tissues. We observe a cell softening behavior when submitting to stresses. As our technique is very sensitive, we were able to get access to the measurement of a yield point above which a creep regime is observed obtained for strains above 12%. To explain our rheological curves we propose a model for the cytoskeleton that we represent as a dynamic network of parallel springs, which will break under stress and reattach at null strain. Such a simple model is able to reproduce most of the important behavior of cells under strain. We highlight here the importance of considering cells as complex fluids whose properties will vary with time according to the history of applied stress.
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Affiliation(s)
- Tomita Vasilica Stirbat
- Institut Lumière Matière, UMR5306 Université de Lyon 1-CNRS, Université de Lyon, 69622 Villeurbanne cedex, France
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28
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Chaudhury MK, Goohpattader PS. Noise-activated dissociation of soft elastic contacts. THE EUROPEAN PHYSICAL JOURNAL. E, SOFT MATTER 2012; 35:9808. [PMID: 23239270 DOI: 10.1140/epje/i2012-12131-9] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/17/2012] [Revised: 11/16/2012] [Accepted: 11/20/2012] [Indexed: 06/01/2023]
Abstract
Adhesive forces are capable of deforming a soft elastic object when it comes in contact with a flat rigid substrate. The contact is in stable equilibrium if the total energy of the system arising from the elastic and surface forces exhibits a minimum at a zero or at a slightly negative load. However, as the system is continually unloaded, the energy barrier decreases and it eventually disappears, thus leading to a ballistic separation of the contact. While this type of contact splitting has received wide recognition, what has not been much appreciated with these types of soft adhesion problems is that rupture of a contact can also occur at any finite sub critical load in the presence of a noise. The soft contact problems are unique in that the noise can be athermal, whereas the metastable and stable states of the thermodynamic potential can arise from the competition of the elastic and the interfacial energies of the system. Analysis based on Kramers' theory and simulations based on Langevin dynamics show that the contact rupture dynamics is amenable to an Eyring's form of a force and noise-induced escape of a particle from a potential well that is generic to various types of colloidal and macromolecular processes. These ideas are useful in understanding the results of a recent experiment involving the noise-activated rolling dynamics of a rigid sphere on a surface, where it is pinned by soft micro-fibrillar contacts.
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Affiliation(s)
- M K Chaudhury
- Department of Chemical Engineering, Lehigh University, Bethlehem, PA 18015, USA.
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29
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Walcott S, Warshaw DM, Debold EP. Mechanical coupling between myosin molecules causes differences between ensemble and single-molecule measurements. Biophys J 2012; 103:501-510. [PMID: 22947866 PMCID: PMC3414898 DOI: 10.1016/j.bpj.2012.06.031] [Citation(s) in RCA: 77] [Impact Index Per Article: 6.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/13/2012] [Revised: 06/07/2012] [Accepted: 06/21/2012] [Indexed: 11/17/2022] Open
Abstract
In contracting muscle, individual myosin molecules function as part of a large ensemble, hydrolyzing ATP to power the relative sliding of actin filaments. The technological advances that have enabled direct observation and manipulation of single molecules, including recent experiments that have explored myosin's force-dependent properties, provide detailed insight into the kinetics of myosin's mechanochemical interaction with actin. However, it has been difficult to reconcile these single-molecule observations with the behavior of myosin in an ensemble. Here, using a combination of simulations and theory, we show that the kinetic mechanism derived from single-molecule experiments describes ensemble behavior; but the connection between single molecule and ensemble is complex. In particular, even in the absence of external force, internal forces generated between myosin molecules in a large ensemble accelerate ADP release and increase how far actin moves during a single myosin attachment. These myosin-induced changes in strong binding lifetime and attachment distance cause measurable properties, such as actin speed in the motility assay, to vary depending on the number of myosin molecules interacting with an actin filament. This ensemble-size effect challenges the simple detachment limited model of motility, because even when motility speed is limited by ADP release, increasing attachment rate can increase motility speed.
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Affiliation(s)
- Sam Walcott
- Department of Mathematics, University of California, Davis, California.
| | - David M Warshaw
- Department of Molecular Physiology and Biophysics, University of Vermont, Burlington, Vermont
| | - Edward P Debold
- Department of Kinesiology, University of Massachusetts, Amherst, Massachusetts
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30
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Schwarz US, Gardel ML. United we stand: integrating the actin cytoskeleton and cell-matrix adhesions in cellular mechanotransduction. J Cell Sci 2012; 125:3051-60. [PMID: 22797913 DOI: 10.1242/jcs.093716] [Citation(s) in RCA: 213] [Impact Index Per Article: 17.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/12/2022] Open
Abstract
Many essential cellular functions in health and disease are closely linked to the ability of cells to respond to mechanical forces. In the context of cell adhesion to the extracellular matrix, the forces that are generated within the actin cytoskeleton and transmitted through integrin-based focal adhesions are essential for the cellular response to environmental clues, such as the spatial distribution of adhesive ligands or matrix stiffness. Whereas substantial progress has been made in identifying mechanosensitive molecules that can transduce mechanical force into biochemical signals, much less is known about the nature of cytoskeletal force generation and transmission that regulates the magnitude, duration and spatial distribution of forces imposed on these mechanosensitive complexes. By focusing on cell-matrix adhesion to flat elastic substrates, on which traction forces can be measured with high temporal and spatial resolution, we discuss our current understanding of the physical mechanisms that integrate a large range of molecular mechanotransduction events on cellular scales. Physical limits of stability emerge as one important element of the cellular response that complements the structural changes affected by regulatory systems in response to mechanical processes.
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Affiliation(s)
- Ulrich S Schwarz
- BioQuant and Institute for Theoretical Physics, University of Heidelberg, Heidelberg, Germany.
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31
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Walcott S, Kim DH, Wirtz D, Sun SX. Nucleation and decay initiation are the stiffness-sensitive phases of focal adhesion maturation. Biophys J 2011; 101:2919-28. [PMID: 22208190 DOI: 10.1016/j.bpj.2011.11.010] [Citation(s) in RCA: 34] [Impact Index Per Article: 2.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/24/2011] [Revised: 11/07/2011] [Accepted: 11/10/2011] [Indexed: 01/09/2023] Open
Abstract
A cell plated on a two-dimensional substrate forms adhesions with that surface. These adhesions, which consist of aggregates of various proteins, are thought to be important in mechanosensation, the process by which the cell senses and responds to the mechanical properties of the substrate (e.g., stiffness). On the basis of experimental measurements, we model these proteins as idealized molecules that can bind to the substrate in a strain-dependent manner and can undergo a force-dependent state transition. The model forms molecular aggregates that are similar to adhesions. Substrate stiffness affects whether a simulated adhesion is initially formed and how long it grows, but not how that adhesion grows or shrinks. Our own experimental tests support these predictions, suggesting that the mechanosensitivity of adhesions is an emergent property of a simple molecular-mechanical system.
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Affiliation(s)
- Sam Walcott
- Department of Mathematics, University of California, Davis, California, USA
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32
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Guérin T, Prost J, Joanny JF. Dynamical behavior of molecular motor assemblies in the rigid and crossbridge models. THE EUROPEAN PHYSICAL JOURNAL. E, SOFT MATTER 2011; 34:60. [PMID: 21706282 DOI: 10.1140/epje/i2011-11060-5] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/10/2011] [Accepted: 05/09/2011] [Indexed: 05/31/2023]
Abstract
We present a detailed analysis of the dynamical instabilities appearing in two kinetic theories for the collective behavior of molecular motors: the rigid two-state model and the two-state crossbridge (or power-stroke) model with continuous binding sites. We calculate force-velocity relations, discuss their stability, plot a diagram that summarizes the oscillation regimes, identify the location of the Hopf bifurcation with a memory effect, discuss the oscillation frequency and make a link with single-molecule experiments. We show that the instabilities present in these models naturally translate into non-linearities in force-displacement relations, and at linear order give forces that are similar to the delayed stretch activation observed in oscillating muscles. We also find that instabilities can appear for both apparent load-decelerated and load-accelerated detachment rates in a 3-state crossbridge model.
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Affiliation(s)
- T Guérin
- Physicochimie Curie (Institut Curie/CNRS-UMR168/UPMC), Institut Curie, Centre de Recherche, Paris Cedex, France,
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33
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Braun OM, Peyrard M. Dependence of kinetic friction on velocity: master equation approach. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 83:046129. [PMID: 21599262 DOI: 10.1103/physreve.83.046129] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/05/2010] [Revised: 02/06/2011] [Indexed: 05/30/2023]
Abstract
We investigate the velocity dependence of kinetic friction with a model that makes minimal assumptions on the actual mechanism of friction so that it can be applied at many scales, provided the system involves multicontact friction. Using a recently developed master equation approach, we investigate the influence of two concurrent processes. First, at a nonzero temperature, thermal fluctuations allow an activated breaking of contacts that are still below the threshold. As a result, the friction force monotonically increases with velocity. Second, the aging of contacts leads to a decrease of the friction force with velocity. Aging effects include two aspects: the delay in contact formation and aging of a contact itself, i.e., the change of its characteristics with the duration of stationary contact. All these processes are considered simultaneously with the master equation approach, giving a complete dependence of the kinetic friction force on the driving velocity and system temperature, provided the interface parameters are known.
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Affiliation(s)
- O M Braun
- Institute of Physics, National Academy of Sciences of Ukraine, 46 Science Avenue, 03028 Kiev, Ukraine.
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34
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Abstract
When tissue cells are plated on a flexible substrate, durotaxis, the directed migration of cells toward mechanically stiff regions, has been observed. Environmental mechanical signals are not only important in cell migration but also seem to influence all aspects of cell differentiation and development, including the metastatic process in cancer cells. Based on a theoretical model suggesting that this mechanosensation has a mechanical basis, we introduce a simple model of a cell by considering the contraction of F-actin bundles containing myosin motors (stress fibers) mediated by the movement of adhesions. We show that, when presented with a linear stiffness gradient, this simple model exhibits durotaxis. Interestingly, since stress fibers do not form on soft surfaces and since adhesion sliding occurs very slowly on hard surfaces, the model predicts that the expected cell velocity reaches a maximum at an intermediate stiffness. This prediction can be experimentally tested. We therefore argue that stiffness-dependent cellular adaptations (mechanosensation) and durotaxis are intimately related and may share a mechanical basis. We therefore identify the essential physical ingredients, which combined with additional biochemical mechanisms can explain durotaxis and mechanosensation in cells.
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Affiliation(s)
- Ben Harland
- Department of Mechanical Engineering, Johns Hopkins University, Baltimore, MD 21218, USA
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35
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Walcott S, Sun SX. Active force generation in cross-linked filament bundles without motor proteins. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2010; 82:050901. [PMID: 21230430 DOI: 10.1103/physreve.82.050901] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/07/2010] [Revised: 11/09/2010] [Indexed: 05/30/2023]
Abstract
Cytoskeletal filaments often interact laterally through cross-linking proteins, contributing to passive cellular viscoelasticity and, perhaps surprisingly, active force generation. We present a theory, based on the formation and rupture of cross-linker bonds, that relates molecular properties of those interactions to the macroscale mechanics of filament bundles. Computing the force-velocity relation for such a bundle, we demonstrate significant contractile forces in the absence of molecular motors. This theory provides insight into cytokinesis, cytoskeletal mechanics, and stress-fiber contraction.
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Affiliation(s)
- Sam Walcott
- Department of Mechanical Engineering and Johns Hopkins Physical Science in Oncology Center, Johns Hopkins University, Baltimore, Maryland 21218, USA
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36
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Braun OM, Peyrard M. Master equation approach to friction at the mesoscale. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2010; 82:036117. [PMID: 21230149 DOI: 10.1103/physreve.82.036117] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/19/2010] [Revised: 08/25/2010] [Indexed: 05/30/2023]
Abstract
At the mesoscale friction occurs through the breaking and formation of local contacts. This is often described by the earthquakelike model which requires numerical studies. We show that this phenomenon can also be described by a master equation, which can be solved analytically in some cases and provides an efficient numerical solution for more general cases. We examine the effect of temperature and aging of the contacts and discuss the statistical properties of the contacts for different situations of friction and their implications, particularly regarding the existence of stick-slip.
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Affiliation(s)
- O M Braun
- Institute of Physics, National Academy of Sciences of Ukraine, 46 Science Avenue, 03028 Kiev, Ukraine.
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37
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Sabass B, Schwarz US. Modeling cytoskeletal flow over adhesion sites: competition between stochastic bond dynamics and intracellular relaxation. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2010; 22:194112. [PMID: 21386438 DOI: 10.1088/0953-8984/22/19/194112] [Citation(s) in RCA: 37] [Impact Index Per Article: 2.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/30/2023]
Abstract
In migrating cells, retrograde flow of the actin cytoskeleton is related to traction at adhesion sites located at the base of the lamellipodium. The coupling between the moving cytoskeleton and the stationary adhesions is mediated by the continuous association and dissociation of molecular bonds. We introduce a simple model for the competition between the stochastic dynamics of elastic bonds at the moving interface and relaxation within the moving actin cytoskeleton represented by an internal viscous friction coefficient. Using exact stochastic simulations and an analytical mean field theory, we show that the stochastic bond dynamics lead to biphasic friction laws as observed experimentally. At low internal dissipation, stochastic bond dynamics lead to a regime of irregular stick-and-slip motion. High internal dissipation effectively suppresses cooperative effects among bonds and hence stabilizes the adhesion.
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38
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A mechanical model of actin stress fiber formation and substrate elasticity sensing in adherent cells. Proc Natl Acad Sci U S A 2010; 107:7757-62. [PMID: 20385838 DOI: 10.1073/pnas.0912739107] [Citation(s) in RCA: 164] [Impact Index Per Article: 11.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/20/2022] Open
Abstract
Tissue cells sense and respond to the stiffness of the surface on which they adhere. Precisely how cells sense surface stiffness remains an open question, though various biochemical pathways are critical for a proper stiffness response. Here, based on a simple mechanochemical model of biological friction, we propose a model for cell mechanosensation as opposed to previous more biochemically based models. Our model of adhesion complexes predicts that these cell-surface interactions provide a viscous drag that increases with the elastic modulus of the surface. The force-velocity relation of myosin II implies that myosin generates greater force when the adhesion complexes slide slowly. Then, using a simple cytoskeleton model, we show that an external force applied to the cytoskeleton causes actin filaments to aggregate and orient parallel to the direction of force application. The greater the external force, the faster this aggregation occurs. As the steady-state probability of forming these bundles reflects a balance between the time scale of bundle formation and destruction (because of actin turnover), more bundles are formed when the cytoskeleton time-scale is small (i.e., on stiff surfaces), in agreement with experiment. As these large bundles of actin, called stress fibers, appear preferentially on stiff surfaces, our mechanical model provides a mechanism for stress fiber formation and stiffness sensing in cells adhered to a compliant surface.
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39
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Guérin T, Prost J, Martin P, Joanny JF. Coordination and collective properties of molecular motors: theory. Curr Opin Cell Biol 2010; 22:14-20. [PMID: 20074926 DOI: 10.1016/j.ceb.2009.12.012] [Citation(s) in RCA: 66] [Impact Index Per Article: 4.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/11/2009] [Revised: 11/25/2009] [Accepted: 12/15/2009] [Indexed: 10/20/2022]
Abstract
Many cellular processes require molecular motors to produce motion and forces. Single molecule experiments have led to a precise description of how a motor works. Under most physiological conditions, however, molecular motors operate in groups. Interactions between motors yield collective behaviors that cannot be explained only from single molecule properties. The aim of this paper is to review the various theoretical descriptions that explain the emergence of collective effects in molecular motor assemblies. These include bidirectional motion, hysteretic behavior, spontaneous oscillations, and self-organization into dynamical structures. We discuss motors acting on the cytoskeleton both in a prescribed geometry such as in muscles or flagella and in the cytoplasm.
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Affiliation(s)
- Thomas Guérin
- Laboratoire Physico-Chimie Curie, CNRS, Institut Curie, UPMC, 26 rue d'Ulm, F-75248 Paris Cedex 05, France
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