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.
This content is only available via PDF.
You do not currently have access to this content.