Abstract
In an attempt to understand the role of structural rearrangement onto the cell response during imposed cyclic stresses, we simulated numerically the frequency-dependent behavior of a viscoelastic tensegrity structure (VTS model) made of 24 elastic cables and 6 rigid bars. The VTS computational model was based on the nonsmooth contact dynamics (NSCD) method in which the constitutive elements of the tensegrity structure are considered as a set of material points that mutually interact. Low amplitude oscillatory loading conditions were applied and the frequency response of the overall structure was studied in terms of frequency dependence of mechanical properties. The latter were normalized by the homogeneous properties of constitutive elements in order to capture the essential feature of spatial rearrangement. The results reveal a specific frequency-dependent contribution of elastic and viscous effects which is responsible for significant changes in the VTS model dynamical properties. The mechanism behind is related to the variable contribution of spatial rearrangement of VTS elements which is decreased from low to high frequency as dominant effects are transferred from mainly elastic to mainly viscous. More precisely, the elasticity modulus increases with frequency while the viscosity modulus decreases, each evolution corresponding to a specific power-law dependency. The satisfactorily agreement found between present numerical results and the literature data issued from in vitro cell experiments suggests that the frequency-dependent mechanism of spatial rearrangement presently described could play a significant and predictable role during oscillatory cell dynamics.
1.
Pavalko
, F. M.
, Chen
, N. X.
, Turner
, C. H.
, Burr
, D. B.
, Atkinson
, S.
, Hsieh
, Y. F.
, Qiu
, J.
, and Duncan
, R. L.
, 1998, “Fluid Shear-Induced Mechanical Signaling in MC3T3-E1 Osteoblasts Requires Cytoskeleton-Integrin Interactions
,” Am. J. Physiol.
0002-9513, 275
, pp. C1591
–C1601
.2.
Planus
, E.
, Galiacy
, S.
, Matthay
, M.
, Laurent
, V.
, Gavrilovic
, J.
, Murphy
, G.
, Clérici
, C.
, Isabey
, D.
, Lafuma
, C.
, and d’Ortho
, M. P.
, 1999, “Role of Collagenase in Mediating In Vitro Alveolar Epithelial Wound Repair
,” J. Opt. Soc. Am. B
0740-3224, 112
(2
), pp. 243
–252
.3.
Bereiter-Hahn
, J.
, 1994, “Functional Morphology and Biomechanics
,” Verh. Dtsch. Zool. Ges.
0070-4342, 87
(2
), pp. 129
–145
.4.
Elson
, E. L.
, 1988, “Cellular Mechanism as an Indicator of CSK Structure and Function
,” Annu. Rev. Biophys. Biophys. Chem.
0883-9182, 17
, pp. 397
–430
.5.
Vico
, L.
, Lafage-Proust
, M. H.
, and Alexandre
, C.
, 1998, “Effects of Gravitational Changes on the Bone System In Vitro and In Vivo
,” Bone (N.Y.)
8756-3282, 22
(5
), pp. 95S
–100S
.6.
Ingber
, D. E.
, 2000, “Opposing Views on Tensegrity as a Structural Framework for Understanding Cell Mechanics
,” J. Appl. Physiol.
8750-7587 89
(4
), pp. 1663
–1670
.7.
Laurent
, V. M.
, Cañadas
, P.
, Fodil
, R.
, Planus
, E.
, Asnacios
, A.
, Wendling
, S.
, and Isabey
, D.
, 2002, “Tensegrity Behaviour of Cortical and Cytosolic Cytoskeletal Components in Twisted Living Adherent Cells
,” Acta Biotheor.
0001-5342, 50
(4
), pp. 331
–356
.8.
Forgacs
, G.
, 1995, “On the Possible Role of Cytoskeletal Filamentous Networks in Intracellular Signaling: An Approach Based on Percolation
,” J. Cell. Sci.
0021-9533, 108
(6
), pp. 2131
–2143
.9.
Choquet
, D.
, Felsenfeld
, D. P.
, and Sheetz
, M. P.
, 1997, “Extracellular Matrix Rigidity Causes Strengthening of Integrin-Cytoskeleton Linkages
,” Cell
0092-8674, 88
(1
), pp. 39
–48
.10.
Davies
, P. F.
, 1995, “Flow-Mediated Endothelial Mechanotransduction
,” Physiol. Rev.
0031-9333, 75
(3
), pp. 519
–560
.11.
Davies
, P. F.
, Barbee
, K. A.
, Volin
, M. V.
, Robotewskyj
, A.
, Chen
, J.
, Joseph
, L.
, Griem
, M. L.
, Wernick
, M. N.
, Jacobs
, E.
, Polacek
, D. C.
, DePaola
, N.
, and Barakat
, A. L.
, 1997, “Spatial Relationships in Early Signaling Events of Flow-Mediated Endothelial Mechanotransduction
,” Annu. Rev. Physiol.
0066-4278, 59
, pp. 527
–549
.12.
Thoumine
, O.
, and Ott
, A.
, 1997, “Time Scale Dependent Viscoelastic and Contractile Regimes in Fibroblasts Probed by Microplate Manipulation
,” J. Cell. Sci.
0021-9533, 110
, pp. 2109
–2116
.13.
Dennerll
, T. J.
, Buxbaum
, R. E.
, and Heidemann
, S. R.
, 1988, “Tension and Compression in the Cytoskeleton of PC-12 Neurites II: Quantitative Measurements
,” J. Cell Biol.
0021-9525, 107
(2
), pp. 665
–664
.14.
Heidemann
, S. R.
, and Buxbaum
, R. E.
, 1990, “Tension as a Regulator and Integrator of Axonal Growth
,” Cell Motil. Cytoskeleton
0886-1544, 17
, pp. 6
–10
.15.
Heidemann
, S. R.
, Kaech
, S.
, Buxbaum
, R. E.
, and Matus
, A.
, 1999, “Direct Observations of the Mechanical Behaviors of the Cytoskeleton in Living Fibroblasts
,” J. Cell Biol.
0021-9525, 145
(1
), pp. 109
–122
.16.
Maniotis
, A. J.
, Chen
, C. S.
, and Ingber
, D. E.
, 1997, “Demonstration of Mechanical Connections between Integrins, Cytoskeletal Filaments and Nucleoplasm that Stabilize Nuclear Structure
,” Proc. Natl. Acad. Sci. U.S.A.
0027-8424, 94
, pp. 849
–854
.17.
Wang
, N.
, Butler
, J.
, and Ingber
, D.
, 1993, “Mechanotransduction Across the Cell Surface and Through the Cytoskeleton
,” Science
0036-8075, 260
, pp. 1124
–1127
.18.
Wendling
, S.
, Planus
, E.
, Laurent
, V. M.
, Barbe
, L.
, Mary
, A.
, Oddou
, C.
, and Isabey
, D.
, 2000, “Role of Cellular Tone and Microenvironmental Conditions on Cytoskeleton Stiffness Assessed by Tensegrity Model
,” Eur. Phys. J.: Appl. Phys.
1286-0042, 9
, pp. 51
–62
.19.
Wendling
, S.
, Oddou
, C.
, and Isabey
, D.
, 1999, “Stiffening Response of a Cellular Tensegrity Model
,” J. Theor. Biol.
0022-5193, 196
(3
), pp. 309
–325
.20.
Wang
, N.
, Tolic-Norrelykke
, I. M.
, Chen
, J.
, Mijailovich
, S. M.
, Butler
, J. P.
, Fredberg
, J. J.
, and Stamenovic
, D.
, 2002, “Cell Prestress I. Stiffness and Prestress are closely associated in Adherent Contractile Cells
,” Am. J. Physiol.: Cell Physiol.
0363-6143, 282
, pp. 606
–616
.21.
Stamenovic
, D.
, Liang
, Z.
, Chen
, J.
, and Wang
, N.
, 2002a, “Effect of the Cytoskeletal Prestress on the Mechanical Impedance of Cultured Airway Smooth Muscle Cells
,” J. Appl. Physiol.
8750-7587, 92
(4
), pp. 1443
–1450
.22.
Stamenovic
, D.
, Mijailovich
, S. M.
, Tolic-Norrelykke
, I. M.
, Chen
, J.
, and Wang
, N.
, 2002b, “Cell Prestress II. Contribution of Microtubules
,” Am. J. Physiol.: Cell Physiol.
0363-6143, 282
, pp. 617
–624
.23.
Sultan
, C.
, Stamenovic
, D.
, and Ingber
, D. E.
, 2004, “A Computational Tensegrity Model Predicts Dynamic Rheological Behaviors in Living Cells
,” Ann. Biomed. Eng.
0090-6964, 32
(4
), pp. 520
–530
.24.
Laurent
, V. M.
, Fodil
, R.
, Canadas
, P.
, Fereol
, S.
, Louis
, B.
, Planus
, E.
, and Isabey
, D.
, 2003, “Partitioning of Cortical and Deep Cytoskeleton Responses From Transient Magnetic Bead Twisting
,” Ann. Biomed. Eng.
0090-6964, 31
(10
), pp. 1263
–1278
.25.
Baush
, A. R.
, Moller
, W.
, and Sackman
, E.
, 1999, “Measurements of Local Viscoelasticity and Forces in Living Cells by Magnetic Tweezers
,” Biophys. J.
0006-3495, 76
, pp. 573
–579
.26.
Mathur
, A. B.
, Truskey
, G. A.
, and Reichert
, W. M.
, 2000, “Atomic Force and Total Reflection Fluorescence Microscopy for the Study of Force Transmission in Endothelial Cells
,” Biophys. J.
0006-3495, 78
, pp. 1725
–1735
.27.
Satcher
, R. L.
, and Dewey
, C. F.
, 1996, “Theoretical Estimates of Mechanical Properties of the Endothelial Cell Cytoskeleton
,” Biophys. J.
0006-3495, 71
, pp. 109
–118
.28.
Thoumine
, O.
, and Ott
, A.
, 1997, “Comparison of the Mechanical Properties of Normal and Transformed Fibroblasts
,” Biorheology
0006-355X, 34
(4/5
), pp. 309
–326
.29.
Fabry
, B.
, Maksym
, G. N.
, Butler
, J. P.
, Glogauer
, M.
, Navajas
, D.
, and Fredberg
, J. J.
, 2001, “Scaling the Microrheology of Living Cells
,” Phys. Rev. Lett.
0031-9007, 87
(14
), pp. 148102
–148105
.30.
Maksym
, G. N.
, Fabry
, B.
, Butler
, J. P.
, Navajas
, D.
, Tschumperlin
, D. J.
, Laporte
, J. D.
, and Fredberg
, J. J.
, 2000, “Mechanical Properties of Cultured Human Airway Smooth Muscle Cells From 0.05 to 0.4Hz
,” J. Appl. Physiol.
8750-7587, 89
, pp. 1619
–1632
.31.
Wu
, H. W.
, Kuhn
, T.
, and Moy
, V. T.
, 1998, “Mechanical Properties of L929 Cells Measured by Atomic Force Microscopy: Effects of Anticytoskeletal Drugs and Membrane Crosslinking
,” Scanning
0161-0457, 20
, pp. 389
–397
.32.
Yamada
, S.
, Wirtz
, D.
, and Kuo
, S. C.
, 2000, “Mechanics of Living Cells Measured by Laser Tracking Microrheology
,” Biophys. J.
0006-3495, 78
, pp. 1736
–1747
.33.
Stamenovic
, D.
, Ingber
, D. E.
, Wang
, N.
, and Fredberg
, J. J.
, 1996, “A Microstructural Approach to Cytoskeletal Mechanics Based on Tensegrity
,” J. Theor. Biol.
0022-5193, 181
, pp. 125
–136
.34.
Stamenovic
, D.
, and Coughlin
, M. F.
, 1999, “The Role of Prestress and Architecture of the Cytoskeleton and Deformability of Cytoskeletal Filaments in Mechanics of Adherent Cells: A Quantitative Analysis
,” J. Theor. Biol.
0022-5193, 201
, pp. 63
–74
.35.
Volokh
, K. Y.
, Vilnay
, O.
, and Belsky
, M.
, 2000, “Tensegrity Architecture Explains Linear Stiffening and Predicts Softening of Living Cells
,” J. Biomech.
0021-9290, 33
, pp. 1543
–1549
.36.
Cañadas
, P.
, Laurent
, V. M.
, Oddou
, C.
, Isabey
, D.
, and Wendling
, S.
, 2002, “A Cellular Tensegrity Model to Analyse the Structural Viscoelasticity of the Cytoskeleton
,” J. Theor. Biol.
0022-5193, 218
, pp. 155
–173
.37.
Ingber
, D.
, 2003, “Tensegrity I. Cell Structure and Hierarchical Systems Biology
,” J. Cell. Sci.
0021-9533, 116
, pp. 1157
–1173
.38.
Ohayon
, J.
, Tracqui
, P.
, Fodil
, R.
, Féréol
, S.
, Laurent
, V. M.
, Planus
, E.
, and Isabey
, D.
, 2004, “Analysis of Nonlinear Responses of Adherent Epithelial Cells Probed by Magnetic Bead Twisting: A Finite Element Model Based on An Homogenization Approach
,” J. Biomech. Eng.
0148-0731, 126
(6
), pp. 685
–698
.39.
Cañadas
, P.
, Laurent
, V. M.
, Chabrand
, P.
, Isabey
, D.
, and Wendling-Mansuy
, S.
, 2003, “Mechanisms Governing the Visco-Elastic Responses of Living Cells Assessed by Foam and Tensegrity Models
,” Med. Biol. Eng. Comput.
0140-0118, 41
(6
), pp. 733
–739
.40.
Mohri
, F.
, and Motro
, R.
, 1993, “Static and Kinematic Determination of Generalized Space Reticulated Systems
,” Struct. Eng. Rev.
0952-5807, 5
(3
), pp. 231
–237
.41.
Pugh
, A.
, 1976, Introduction to Tensegrity
, University of California Press
, Berkeley and Los Angeles, CA.42.
Wendling
, S.
, Cañadas
, P.
, Oddou
, C.
, and Meunier
, A.
, 2002, “Interrelations Between Elastic Energy and Strain in a Tensegrity Model; Contribution to the Analysis on the Mechanical Response in Living Cells
,” Comput. Methods Biomech. Biomed. Eng.
1025-5842, 5
(1
), pp. 1
–6
.43.
Jean
, M.
, 1999, “The Non-Smooth Contact Dynamics Method
,” Appl. Mech. Eng.
1425-1655, 177
, pp. 235
–257
.44.
Wendling
, S.
, Cañadas
, P.
, and Chabrand
, P.
, 2003, “Toward a Generalized Tensegrity Model Describing the Mechanical Behaviour of the Cytoskeleton Structure
,” Comput. Methods Biomech. Biomed. Eng.
1025-5842, 1
, pp. 1
–8
.45.
Wendling
, S.
, Oddou
, C.
, and Isabey
, D.
, 2000, “Approche Structurale de la Mécanique du Cytosquelette: Solide Alvéolaire vs Modèle de Tenségrité
,” C. R. Acad. Sci., Ser. IIb Mec. Phys. Astron.
1287-4620, 328
, pp. 97
–104
.46.
Rubin
, C.
, Turner
, A. S.
, Mallinckrodt
, C.
, Jerome
, C.
, Mcleod
, K.
, and Bain
, S.
, 2002, “Mechanical Strain Induced Noninvasively in the High-Frequency Domain, is Anabolic
,” Bone (N.Y.)
8756-3282, 30
(3
), pp. 445
–452
.47.
Alcaraz
, J.
, Buscemi
, L.
, Grabulosa
, M.
, Trepat
, X.
, Fabry
, B.
, Farre
, R.
, and Navajas
, D.
, 2003, “Microrheology of Human Lung Epithelial Cells Measured by Atomic Force Microscopy
,” Biophys. J.
0006-3495, 84
, pp. 2071
–2079
.48.
Balland
, M.
, Richert
, A.
, and Gallet
, F.
, 2005, “The Dissipative Contribution of Myosin II in the Cytoskeleton Dynamics of Myoblasts
,” Eur. Biophys. J.
0175-7571, 34
(3
), pp. 255
–261
.49.
Puig-de-Morales
, M.
, Millet
, E. J.
, Fabry
, B.
, Navajas
, D.
, Wang
, N.
, Butler
, J. P.
, and Fredberg
, J. J.
, 2004, “Cytoskeletal Mechanics in Adherent Human Airway Smooth Muscle Cells: Probe Specificity and Scaling of Protein-Protein Dynamics
,” Am. J. Physiol.: Cell Physiol.
0363-6143 287
, pp. 643
–654
.50.
Palmer
, A.
, Mason
, T. G.
, Xu
, J.
, Kuo
, S. C.
, and Wirtz
, D.
, 1999, “Diffusing Wave Spectroscopy Microrheology of Actin Filament Networks
,” Biophys. J.
0006-3495, 76
, pp. 1063
–1071
.51.
Fabry
, B.
, Maksym
, G. N.
, Butler
, J. P.
, Glogauer
, M.
, Navajas
, D.
, Taback
, N. A.
, Millet
, E. J.
, and Fredberg
, J. J.
, 2003, “Time Scale and Other Invariants of Integrative Mechanical Behavior in Living Cells
,” Phys. Rev. E
1063-651X, 68
(4
), 041914
, pp. 1–18.52.
Fredberg
, J. J.
, and Stamenovic
, D.
, 1989, “On the Imperfect Elasticity of Lung Tissue
,” J. Appl. Physiol.
8750-7587, 67
(6
), pp. 2408
–2419
.53.
Cambou
, B.
, and Jean
, M.
, 2001, Micromécanique des Matériaux Granulaires
, Hermes Science Publication
, Paris, France.Copyright © 2006
by American Society of Mechanical Engineers
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