Mechanical forces play a vital role in the activities of cells and their interaction with biological and nonbiological material. Various experiments have successfully measured forces exerted by the cells when in contact with a substrate, but the intracellular contractile machinery leading to these actions is not entirely understood. Tan et al., (2003, “Cells Lying on a Bed of Microneedles: An Approach to Isolate Mechanical Force,” Proc. Natl. Acad. Sci. USA, 100(4), pp. 1484–1489) use a bed of PDMS posts as the substrate for cells and measure the localized mechanical forces exerted by the cell cytoskeleton on the posts. In live cell experiments for this setup, post deflections are measured, and from these results the forces applied by the cell are calculated. From such results, it is desirable to quantify the contractile tensions generated in the force-bearing elements corresponding to the stress fibers within the cell cytoskeleton that generate the loads applied to the posts. The purpose of the present article is to consider the cytoskeleton as a discrete network of force-bearing elements, and present a structural mechanics based methodology to estimate the configuration of the network, and the contractile tension in the corresponding stress fibers. The network of stress fibers is modeled as a structure of truss elements connected among the posts adhered to a single cell. In-plane force equilibrium among the network of stress fibers and the system of posts is utilized to calculate the tension forces in the network elements. A Moore-Penrose pseudo-inverse is used to solve the linear equations obtained from the mechanical equilibrium of the cell-posts system, thereby obtaining a least squares fit of the stress fiber tensions to the post deflections. The predicted network of force-bearing elements provides an approximated distribution of the prominent stress fibers connected among deflected posts, and the tensions in each fibril.

References

References
1.
Wang
,
N.
,
Butler
,
J. P.
, and
Ingber
,
D. E.
, 1993, “
Mechanotransduction Across the Cell Surface and Through the Cytoskeleton
,”
Science
,
260
(5111), pp.
1124
1127
.
2.
Alberts
,
B.
,
Johnson
,
A.
,
Lewis
,
J.
,
Raff
,
M.
,
Roberts
,
K.
, and
Watson
,
J. D.
, 2002,
Molecular Biology of the Cell
,
Garland Publishing
,
New York
.
3.
Choquet
,
D.
,
Felsenfeld
,
D. P.
, and
Sheetz
,
M. P.
, 1997, “
Extracellular Matrix Rigidity Causes Strengthening of Integrin-Cytoskeleton Linkages
,”
Cell
,
88
(
1
), pp.
39
48
.
4.
Balaban
,
N. Q.
,
Schwarz
,
U. S.
,
Riveline
,
D.
,
Goichberg
,
P.
,
Tzur
,
G.
,
Sabanay
,
I.
,
Mahalu
,
D.
,
Safran
,
S.
,
Bershadsky
,
A.
,
Addadi
,
L.
, and
Geiger
,
B.
, 2001, “
Force and Focal Adhesion Assembly: A Close Relationship Studied Using Elastic Micropatterned Substrates
,”
Nat. Cell Biol.
,
3
(
5
), pp.
466
472
.
5.
Burton
,
K.
, and
Taylor
,
D. L.
, 1997, “
Traction Forces of Cytokinesis Measured With Optically Modified Elastic Substrata
,”
Nature
,
385
(6615), pp.
450
454
.
6.
Chrzanowska-Wodnicka
,
M.
, and
Burridge
,
K.
, 1996, “
Rho-Stimulated Contractility Drives the Formation of Stress Fibers and Focal Adhesions
,”
J. Cell Biol.
,
133
(
6
), pp.
1403
1415
.
7.
Harris
,
A. K.
,
Wild
,
P.
, and
Stopak
,
D.
, 1980, “
Silicone Rubber Substrata: A New Wrinkle in the Study of Cell Locomotion
,”
Science
,
208
(4440), pp.
177
179
.
8.
Pelham
,
R. J.
, and
Wang
,
Y.-L.
, 1997, “
Cell Locomotion and Focal Adhesions Are Regulated by Substrate Flexibility
,”
Proc. Natl. Acad. Sci.
USA
,
94
(
25
), pp.
13661
13665
.
9.
Riveline
,
D.
,
Zamir
,
E.
,
Balaban
,
N. Q.
,
Schwarz
,
U. S.
,
Ishizaki
,
T.
,
Narumiya
,
S.
,
Kam
,
Z.
,
Geiger
,
B.
, and
Bershadsky
,
A. D.
, 2001, “
Focal Contacts as Mechanosensors: Externally Applied Local Mechanical Force Induces Growth of Focal Contacts by an Mdia1-Dependent and Rock-Independent Mechanism
,”
J. Cell Biol.
,
153
(
6
), pp.
1175
1186
.
10.
Dembo
,
M.
,
Oliver
,
T.
,
Ishihara
,
A.
, and
Jacobson
,
K.
, 1996, “
Imaging the Traction Stresses Exerted by Locomoting Cells With the Elastic Substratum Method
,”
Biophys. J.
,
70
(
4
), pp.
2008
2022
.
11.
Wang
,
N.
,
Naruse
,
K.
,
Stamenovic
,
D.
,
Fredberg
,
J. J.
,
Mijailovich
,
S. M.
,
Tolic-Norrelykke
,
I. M.
,
Polte
,
T.
,
Mannix
,
R.
, and
Ingber
,
D. E.
, 2001, “
Mechanical Behavior in Living Cells Consistent With the Tensegrity Model
,”
Proc. Natl. Acad. Sci. USA
,
98
(
14
), pp.
7765
7770
.
12.
Tan
,
J. L.
,
Tien
,
J.
,
Pirone
,
D. M.
,
Gray
,
D. S.
,
Bhadriraju
,
K.
, and
Chen
,
C. S.
, 2003, “
Cells Lying on a Bed of Microneedles: An Approach to Isolate Mechanical Force
,”
Proc. Natl. Acad. Sci. USA
,
100
(
4
), pp.
1484
1489
.
13.
Elson
,
E. L.
, 1988, “
Cellular Mechanics as an Indicator of Cytoskeletal Structure and Function
,”
Annu. Rev. Biophys. Bio.
,
17
(
1
), pp.
397
430
.
14.
Evans
,
E.
, and
Yeung
,
A.
, 1989, “
Apparent Viscosity and Cortical Tension of Blood Granulocytes Determined by Micropipet Aspiration
,”
Biophys. J.
,
56
(
1
), pp.
151
160
.
15.
Fung
,
Y. C.
, and
Liu
,
S. Q.
, 1993, “
Elementary Mechanics of the Endothelium of Blood Vessels
,”
J. Biomech. Eng.
,
115
(
1
), pp.
1
12
.
16.
Nelson
,
C. M.
,
Jean
,
R. P.
,
Tan
,
J. L.
,
Liu
,
W. F.
,
Sniadecki
,
N. J.
,
Spector
,
A. A.
,
Chen
,
C. S.
, and
Langer
,
R.
, 2005, “
Emergent Patterns of Growth Controlled by Multicellular Form and Mechanics
,”
Proc. Natl. Acad. Sci. USA
,
102
(
33
), pp.
11594
11599
.
17.
Ingber
,
D. E.
, 2003, “
Tensegrity I. Cell Structure and Hierarchical Systems Biology
,”
J. Cell Sci.
,
116
(
7
), pp.
1157
1173
.
18.
Ingber
,
D. E.
, 2003, “
Tensegrity II. How Structural Networks Influence Cellular Information Processing Networks
,”
J. Cell Sci.
,
116
(
8
), pp.
1397
1408
.
19.
Chen
,
C. S.
,
Alonso
,
J. L.
,
Ostuni
,
E.
,
Whitesides
,
G. M.
, and
Ingber
,
D. E.
, 2003, “
Cell Shape Provides Global Control of Focal Adhesion Assembly
,”
Biochem. Biophys. Res. Commun.
,
307
(
2
), pp.
355
361
.
20.
Stamenovic
,
D.
,
Fredberg
,
J. J.
,
Wang
,
N.
,
Butler
,
J. P.
, and
Ingber
,
D. E.
, 1996, “
A Microstructural Approach to Cytoskeletal Mechanics Based on Tensegrity
,”
J. Theor. Biol.
,
181
(
2
), pp.
125
136
.
21.
Satcher
,
R. L.
, Jr.,
, and
Dewey
,
C. F.
Jr.
, 1996, “
Theoretical Estimates of Mechanical Properties of the Endothelial Cell Cytoskeleton
,”
Biophys. J.
,
71
(
1
), pp.
109
118
.
22.
Mohrdieck
,
C.
,
Wanner
,
A.
,
Roos
,
W.
,
Roth
,
A.
,
Sackmann
,
E.
,
Spatz
,
J. P.
, and
Arzt
,
E.
, 2005, “
A Theoretical Description of Elastic Pillar Substrates in Biophysical Experiments
,”
ChemPhysChem
,
6
(
8
), pp.
1492
1498
.
23.
Deshpande
,
V. S.
,
McMeeking
,
R. M.
, and
Evans
,
A. G.
, 2007, “
A Model for the Contractility of the Cytoskeleton Including the Effects of Stress-Fibre Formation and Dissociation
,”
Proc. R. Soc. London, Ser. A-Math. Phy.
,
463
(2079), pp.
787
815
.
24.
Deshpande
,
V. S.
,
McMeeking
,
R. M.
, and
Evans
,
A. G.
, 2006, “
A Bio-Chemo-Mechanical Model for Cell Contractility
,”
Proc. Natl. Acad. Sci. USA
,
103
(
38
), pp.
14015
14020
.
25.
Deshpande
,
V. S.
,
Mrksich
,
M.
,
McMeeking
,
R. M.
, and
Evans
,
A. G.
, 2008, “
A Bio-Mechanical Model for Coupling Cell Contractility With Focal Adhesion Formation
,”
J. Mech. Phys. Solids
,
56
(
4
), pp.
1484
1510
.
26.
Wang
,
N.
, and
Suo
,
Z.
, 2005, “
Long-Distance Propagation of Forces in a Cell
,”
Biochem. Biophys. Res. Commun.
,
328
(
4
), pp.
1133
1138
.
27.
Meyer
,
C. D.
, 2001,
Matrix Analysis and Applied Linear Algebra
,
SIAM: Society for Industrial and Applied Mathematics
,
Philadelphia, PA
.
28.
Strang
,
G.
, 2003,
Introduction to Linear Algebra
,
Wellesley-Cambridge Press
,
Wellesley, MA
.
29.
MATLAB, 2007, Reference Guide, V. R2007a, The MathWorks, Inc., Natick, MA.
30.
Brangwynne
,
C. P.
,
Mackintosh
,
F. C.
,
Kumar
,
S.
,
Geisse
,
N. A.
,
Talbot
,
J.
,
Mahadevan
,
L.
,
Parker
,
K. K.
,
Ingber
,
D. E.
, and
Weitz
,
D. A.
, 2006, “
Microtubules Can Bear Enhanced Compressive Loads in Living Cells Because of Lateral Reinforcement
,”
J. Cell Biol.
,
173
(
5
), pp.
733
741
.
31.
Pathak
,
A.
, 2009, “
Computational Models for the Bio-Chemo-Mechanical Behavior of Cells in Diverse Extra-Cellular Settings
,” Ph.D. thesis, Department of Mechanical Engineering, University of California, Santa Barbara, CA.
32.
Hotulainen
,
P.
, and
Lappalainen
,
P.
, 2006, “
Stress Fibers Are Generated by Two Distinct Actin Assembly Mechanisms in Motile Cells
,”
J. Cell Biol.
,
173
(
3
), pp.
383
394
.
33.
Prass
,
M.
,
Jacobson
,
K.
,
Mogilner
,
A.
, and
Radmacher
,
M.
, 2006, “
Direct Measurement of the Lamellipodial Protrusive Force in a Migrating Cell
,”
J. Cell Biol.
,
174
(
6
), pp.
767
772
.
34.
Gardel
,
M. L.
,
Shin
,
J. H.
,
Mackintosh
,
F. C.
,
Mahadevan
,
L.
,
Matsudaira
,
P.
, and
Weitz
,
D. A.
, 2004, “
Elastic Behavior of Cross-Linked and Bundled Actin Networks
,”
Science
,
304
(
5675
), pp.
1301
1305
.
35.
Burridge
,
K.
, and
Chrzanowska-Wodnicka
,
M.
, 1996, “
Focal Adhesions, Contractility, and Signaling
,”
Ann. Rev. Cell Dev. Biol.
,
12
(
1
), pp.
463
519
.
36.
Beningo
,
K. A.
,
Dembo
,
M.
,
Kaverina
,
I.
,
Small
,
J. V.
, and
Wang
,
Y.-L.
, 2001, “
Nascent Focal Adhesions Are Responsible for the Generation of Strong Propulsive Forces in Migrating Fibroblasts
,”
J. Cell Biol.
,
153
(
4
), pp.
881
888
.
37.
Wei
,
Z.
,
Deshpande
,
V. S.
,
McMeeking
,
R. M.
, and
Evans
,
A. G.
, 2008, “
Analysis and Interpretation of Stress Fiber Organization in Cells Subject to Cyclic Stretch
,”
J. Biomech. Eng.
,
130
(
3
), p.
031009
.
38.
Pathak
,
A.
,
Deshpande
,
V. S.
,
McMeeking
,
R. M.
, and
Evans
,
A. G.
, 2008, “
The Simulation of Stress Fibre and Focal Adhesion Development in Cells on Patterned Substrates
,”
J. R. Soc. Interface.
,
5
(
22
), pp.
507
524
.
You do not currently have access to this content.