Analysis of Nonlinear Responses of Adherent Epithelial Cells Probed by Magnetic Bead Twisting: A Finite Element Model Based on a Homogenization Approach

[+] Author and Article Information
Jacques Ohayon, Philippe Tracqui

Laboratoire TIMC-IMAG, Equipe DynaCell, CNRS UMR 5525, Institut de l’Ingénierie et de l’Information de Santé, Faculté de Médecine, 38706 La Tronche Cedex, France

Redouane Fodil, Sophie Féréol, Valérie M. Laurent

INSERM, UMR 492, Physiopathologie et Thérapeutique Respiratoires, Faculté de Médecine, 8, rue du Général Sarrail, 94010 Créteil cedex, France

Emmanuelle Planus

Laboratoire TIMC-IMAG, Equipe DynaCell, CNRS UMR 5525, Institut del’ Ingénierie et de l’Information de Santé Faculté de Médecine, 38706 La Tronche, Cedex France INSERM, UMR 492, Physiopathologie et Thérapeutique Respiratoires, Faculté de Médecine, 8, rue du Général Sarrail, 94010 Créteil cedex, France

Daniel Isabey

INSERM, UMR 492 Physiopathologie et Thérapeutique Respiratoires, Faculté de Médecine, 8, rue du Général Sarrail, 94010 Créteil cedex, France

J Biomech Eng 126(6), 685-698 (Feb 04, 2005) (14 pages) doi:10.1115/1.1824136 History: Received October 09, 2003; Revised July 26, 2004; Online February 04, 2005
Copyright © 2004 by ASME
Your Session has timed out. Please sign back in to continue.


Davies,  P. F., Robotewskyj,  A., and Griem,  M. L., 1993, “Endothelial Cell Adhesion in Real Time. Measurements In Vitro by Tandem Scanning Confocal Image Analysis,” J. Clin. Invest., 91(6), pp. 2640–2652.
Forgacs,  G., 1995, “Biological Specificity and Measurable Physical Properties of Cell Surface Receptors and Their Possible Role in Signal Transduction Through the Cytoskeleton,” Biochem. Cell Biol., 73(7–8), pp. 317–326.
Ingber,  D. E., 1997, “Tensegrity: The Architectural Basis of Cellular Mechanotransduction,” Annu. Rev. Physiol., 59, pp. 575–599.
Hamill,  O. P., and Martinac,  B., 2001, “Molecular Basis of Mechanotransduction in Living Cells,” Physiol. Rev., 81(2), pp. 685–740.
Ricci,  D., Tedesco,  M., and Grattarola,  M., 1997, “Mechanical and Morphological Properties of Living 3T6 Cells Probed Via Scanning Force Microscopy,” Microsc. Res. Tech., 36, pp. 165–171.
Pourati,  J., Maniotis,  A., Spiegel,  D., Schaffer,  J. L., Butler,  J. P., Fredberg,  J. J., Ingber,  D. E., Stamenovic,  D., and Wang,  N., 1998, “Is Cytoskeletal Tension a Major Determinant of Cell Deformability in Adherent Endothelial Cells?” Am. J. Physiol., 274(5 Pt 1), pp. C1283–C1289.
Ingber,  D. E., with Heidemann,  S. R., Lamoureux,  P., and Buxbaum,  R. E., 2000, “Opposing Views on Tensegrity as a Structural Framework for Understanding Cell Mechanics,” J. Appl. Physiol., 89(4), pp. 1663–1670.
Rahman,  A., Tseng,  Y., and Wirtz,  D., 2002, “Micromechanical Coupling Between Cell Surface Receptors and RGD Peptides,” Biochem. Biophys. Res. Commun., 296(3), pp. 771–778.
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.
Sato,  M., Ohshima,  N., and Nerem,  R. M., 1996, “Viscoelastic Properties of Cultured Porcine Aortic Endothelial Cells Exposed to Shear Stress,” J. Biomech., 29(4), pp. 461–467.
Thoumine,  O., and Ott,  A., 1997, “Time Scale Dependent Viscoelastic and Contractile Regimes in Fibroblasts Probed by Microplate Manipulation,” J. Cell. Sci., 110(Pt 17), pp. 2109–2116.
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.
Wang,  N., and Ingber,  D. E., 1994, “Control of Cytoskeletal Mechanics by Extracellular Matrix, Cell Shape, and Mechanical Tension,” Biophys. J., 66(6), pp. 2181–2189.
Wang,  N., and Ingber,  D. E., 1995, “Probing Transmembrane Mechanical Coupling and Cytomechanics Using Magnetic Twisting Cytometry,” Biochem. Cell Biol., 73(7–8), pp. 327–335.
Wang,  N., Planus,  E., Pouchelet,  M., Fredberg,  J. J., and Barlovatz-Meimon,  G., 1995, “Urokinase Receptor Mediates Mechanical Force Transfer Across the Cell Surface,” Am. J. Physiol., 268(4 Pt 1), C1062–C1066.
Wendling,  S., Planus,  E., Laurent,  V., Barbe,  L., Mary,  A., Oddou,  C., and Isabey,  D., 2000, “Role of Cellular Tone and Microenvironment on Cytoskeleton Stiffness Predicted by Tensegrity Model,” Eur. Phys. J.: Appl. Phys., 9, pp. 51–62.
Laurent,  V. M., Henon,  S., Planus,  E., Fodil,  R., Balland,  M., Isabey,  D., and Gallet,  F., 2002, “Assessment of Mechanical Properties of Adherent Living Cells by Bead Micromanipulation: Comparison of Magnetic Twisting Cytometry vs Optical Tweezers,” J. Biomech. Eng., 124(4), pp. 408–421.
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., 31(10), 1263–1278.
Sato,  M., Theret,  D. P., Wheeler,  L. T., Ohshima,  N., and Nerem,  R. M., 1990, “Application of the Micropipette Technique to the Measurement of Cultured Porcine Aortic Endothelial Cell Viscoelastic Properties,” J. Biomech. Eng., 112(3), pp. 263–268.
Hubmayr,  R. D., Shore,  S. A., Fredberg,  J. J., Planus,  E., Panettieri,  R. A., Moller,  W., Heyder,  J., and Wang,  N., 1996, “Pharmacological Activation Changes Stiffness of Cultured Human Airway Smooth Muscle Cells,” Am. J. Physiol., 271(5 Pt 1), pp. C1660–C1668.
Potard,  U. S., Butler,  J. P., and Wang,  N., 1997, “Cytoskeletal Mechanics in Confluent Epithelial Cells Probed Through Integrins and E-Cadherins,” Am. J. Physiol., 272(5 Pt 1), pp. C1654–C1663.
Fabry,  B., Maksym,  G., Hubmayr,  R., Butler,  J., and Fredberg,  J., 1999, “Implications of Heterogeneous Bead Behavior on Cell Mechanical Properties Measured With Magnetic Twisting Cytometry,” J. Magn. Magn. Mater., 194, pp. 120–125.
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., 87(14), pp. 1481–1502.
Fabry,  B., Maksym,  G. N., Shore,  S. A., Moore,  P. E., Panettieri,  R. A., Butler,  J. P., and Fredberg,  J. J., 2001, “Selected Contribution: Time Course and Heterogeneity of Contractile Responses in Cultured Human Airway Smooth Muscle Cells,” J. Appl. Physiol., 91, pp. 986–994.
Thoumine,  O., Ott,  A., Cardoso,  O., and Meister,  J. J., 1999, “Microplates: A New Tool for Manipulation and Mechanical Perturbation of Individual Cells,” J. Biochem. Biophys. Methods, 39(1–2), pp. 47–62.
Svoboda,  K., and Block,  S. M., 1994, “Biological Applications of Optical Forces,” Annu. Rev. Biophys. Biomol. Struct., 23, pp. 247–285.
Sheetz,  M. P., 1998, “Laser Tweezers in Cell Biology. Introduction,” Methods Cell Biol., 55, pp. xi–xii.
Henon,  S., Lenormand,  G., Richert,  A., and Gallet,  F., 1999, “A New Determination of the Shear Modulus of the Human Erythrocyte Membrane Using Optical Tweezers,” Biophys. J., 76(2), pp. 1145–1151.
Lenormand,  G., Henon,  S., Richert,  A., Simeon,  J., and Gallet,  F., 2001, “Direct Measurement of the Area Expansion and Shear Moduli of the Human Red Blood Cell Membrane Skeleton,” Biophys. J., 81(1), pp. 43–56.
Bausch,  A. R., Ziemann,  F., Boulbitch,  A. A., Jacobson,  K., and Sackmann,  E., 1998, “Local Measurements of Viscoelastic Parameters of Adherent Cell Surfaces by Magnetic Bead Microrheometry,” Biophys. J., 75(4), pp. 2038–2049.
Bausch,  A. R., Möller,  W., and Sackmann,  E., 1999, “Measurement of Local Viscoelasticity and Forces in Living Cells by Magnetic Tweezers,” Biophys. J., 76, pp. 573–579.
Shroff,  S. G., Saner,  D. R., and Lal,  R., 1995, “Dynamic Micromechanical Properties of Cultured Rat Atrial Myocytes Measured by Atomic Force Microscopy,” Am. J. Physiol., 269(1 Pt 1), pp. C286–C292.
Dimitriadis,  E. K., Horkay,  F., Maresca,  J., Kachar,  B., and Chadwick,  R. S., 2002, “Determination of Elastic Moduli of Thin Layers of Soft Material Using the Atomic Force Microscope,” Biophys. J., 82(5), pp. 2798–2810.
Bausch,  A. R., Hellerer,  U., Essler,  M., Aepfelbacher,  M., and Sackmann,  E., 2001, “Rapid Stiffening of Integrin Receptor–Actin Linkages in Endothelial Cells Stimulated With Thrombin: A Magnetic Bead Microrheology Study,” Biophys. J., 80(6), pp. 2649–2657.
Laurent,  V. M., Planus,  E., Fodil,  R., and Isabey,  D., 2003, “Mechanical Assessment by Magnetocytometry of the Cytosolic and Cortical Cytoskeletal Compartments in Adherent Epithelial Cells,” Biorheology, 40(1–3), pp. 235–240.
Stamenovic,  D., Ingber,  D. E., Wang,  N., and Fredberg,  J. J., 1996, “A Microstructural Approach to Cytoskeletal Mechanics Based on Tensegrity,” J. Theor. Biol., 181, pp. 125–136.
Wendling,  S., Oddou,  C., and Isabey,  D., 1999, “Stiffening Response of a Cellular Tensegrity Model,” J. Theor. Biol., 196(3), pp. 309–325.
Stamenovic,  D., and Coughlin,  M. F., 2000, “A Quantitative Model of Cellular Elasticity Based on Tensegrity,” J. Biomech. Eng., 122(1), pp. 39–43.
Canadas,  P., Laurent,  V. M., Oddou,  C., Isabey,  D., and Wendling,  S., 2002, “A Cellular Tensegrity Model to Analyze the Structural Viscoelasticity of the Cytoskeleton,” J. Theor. Biol., 218(2), pp. 155–173.
Caille,  N., Thoumine,  O., Tardy,  Y., and Meister,  J. J., 2002, “Contribution of the Nucleus to the Mechanical Properties of Endothelial Cells,” J. Biomech., 35(2), pp. 177–187.
Charras,  G. T., and Horton,  M. A., 2002, “Determination of Cellular Strains by Combined Atomic Force Microscopy and Finite Element Modeling,” Biophys. J., 83(2), pp. 858–879.
Mijailovich,  S. M., Kojic,  M., Zivkovic,  M., Fabry,  B., and Fredberg,  J. J., 2002, “A Finite Element Model of Cell Deformation During Magnetic Bead Twisting,” J. Appl. Physiol., 93(4), pp. 1429–1436.
Bornert, M., Bretheau, T., and Gilormini, P., 2001, Homogénéisation en Mécanique des Matériaux, Hermes Science Publications, Paris.
Fodil,  R., Laurent,  V., Planus,  E., and Isabey,  D., 2003, “Characterization of Cytoskeleton Mechanical Properties and 3D-Actin Structure in Twisted Adherent Epithelial Cells,” Biorheology, 40(1–3), pp. 241–245.
Schneider,  S. W., Pagel,  P., Rotsch,  C., Danker,  T., Oberleithner,  H., Radmacher,  M., and Schwab,  A., 2000, “Volume Dynamics in Migrating Epithelial Cells Measured With Atomic Force Microscopy,” Pfluegers Arch., 439(3), pp. 297–303.
Holzapfel, G. A., 2001, Nonlinear Solid Mechanics, Wiley, New York, p. 455.
Costa,  K. D., and Yin,  F. C. P., 1999, “Analysis of Indentation: Implications for Measuring Mechanical Properties With Atomic Force Microscopy,” J. Biomech. Eng., 121, pp. 462–469.
Yeoh,  O. H., 1990, “Characterization of Elastic Properties of Carbon-Black-Filled Rubber Vulcaizates,” Rubber Chem. Technol., 63, pp. 792–805.
Shin,  D., and Athanasiou,  K., 1999, “Cytoindentation for Obtaining Cell Biomechanical Properties,” J. Orthop. Res., 17, pp. 880–890.
Kamm,  R. D., McVittie,  A. K., and Bathe,  M., 2000, “On the Role of Continuum Models in Mechanobiology,” ASME International Congress—Mechanics in Biology,242, pp. 1–9.
Mathur,  A. B., Collinsworth,  A. M., Reichert,  W. M., Kraus,  W. E., and Truskey,  G. A., 2001, “Endothelial, Cardiac Muscle and Skeletal Muscle Exhibit Different Viscous and Elastic Properties as Determined by Atomic Force Microscopy,” J. Biomech., 34, pp. 1545–1553.
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, 68(4 Pt 1), p. 041914.
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 0.4 Hz,” J. Appl. Physiol., 89(4), pp. 1619–1632.
Doornaert,  B., Leblond,  V., Planus,  E., Galiacy,  S., Laurent,  V. M., Gras,  G., Isabey,  D., and Lafuma,  C., 2003, “Time Course of Actin Cytoskeleton Stiffness and Matrix Adhesion Molecules in Human Bronchial Epithelial Cell Cultures,” Exp. Cell Res., 287, pp. 199–208.
Wozniak,  M., Fausto,  A., Carron,  C. P., Meyer,  D. M., and Hruska,  K. A., 2000, “Mechanically Strained Cells of the Osteoblast Lineage Organize Their Extracellular Matrix Through Unique Sites of Alphavbeta3-Integrin Expression,” J. Bone Miner. Res., 15(9), pp. 1731–1745.
Mathur,  A. B., Truskey,  G., and Reichert,  W. M., 2000, “Atomic Force and Total Internal Reflection Fluorescence Microscopy for the Study of Force Transmission in Endothelial Cells,” Biophys. J., 78(4), pp. 1725–1735.
Balaban,  N. Q., Schwarz,  U. S., Riveline,  D., Goichberg,  P., Tzur,  G., Sabanay,  I., Mahalu,  D., Safran,  S., Bershadsky,  A., Addadi,  L. , 2001, “Force and Focal Adhesion Assembly: A Close Relationship Studied Using Elastic Micropatterned Substrates,” Nat. Cell Biol., 3(5), pp. 466–472.
Schwarz,  U. S., Balaban,  N. Q., Riveline,  D., Bershadsky,  A., Geiger,  B., and Safran,  S. A., 2002, “Calculation of Forces at Focal Adhesions From Elastic Substrate Data: The Effect of Localized Force and the Need for Regularization,” Biophys. J., 83(3), pp. 1380–1394.
Hu,  S., Chen,  J., Fabry,  B., Numaguchi,  Y., Gouldstone,  A., Ingber,  D. E., Fredberg,  J. J., Butler,  J. P., and Wang,  N., 2003, “Intracellular Stress Tomography Reveals Stress Focusing and Structural Anisotropy in Cytoskeleton of Living Cells,” Am. J. Physiol. Cell Physiol.,285(5), pp. C1082–C1090.
Pommerenke,  H., Schreiber,  E., Durr,  F., Nebe,  B., Hahnel,  C., Moller,  W., and Rychly,  J., 1996, “Stimulation of Integrin Receptors Using a Magnetic Drag Force Device Induces an Intracellular Free Calcium Response,” Eur. J. Cell Biol., 70(2), pp. 157–164.


Grahic Jump Location
Representative cross section of cell–bead geometry (bead #25, Table 1) in a plane running through the center of the bead and perpendicular to the substrate. This cross section was obtained from a 3D reconstruction of confocal images performed after staining F-actin with fluorescent phalloidin. The bead-embedding half-angle α is estimated as the mean of the two bead-embedding half-angles α1 and α2, and hu is the measured cell thickness under the bead. Scaling is given by the bead diameter, i.e., 3.2 μm.
Grahic Jump Location
Finite element mesh of the half-representative cell volume element (RCVE) measuring a/2×a×h. The bead surface and the portion of the cell surface (area bounded by white lines) that may come into contact with the bead during bead rotation have been meshed with contact elements. The arrow indicates the orientation of the torque applied at the center of the bead. Free-boundary conditions are applied to the RCVE surfaces, except for the displacement conditions indicated in the figure.
Grahic Jump Location
Values of cell material parameters obtained from the MTC data. (A) The RCVE homogeneous model response (solid line) gives a very good fit to the experimental data (open circles ±SD) for the two optimal material parameters a1=2.25 Pa and a2=50 Pa. The mean response in the 25 individual simulations performed with the geometrical features of each bead listed in Table 1 is superimposed (full triangles ±SD). (B) Representative curve of the Young modulus Ecell derived from the two-parameter Yeoh strain-energy function and computed from Eq. (B1) in Appendix B with a1=2.25 Pa and a2=50 Pa.
Grahic Jump Location
3D color maps showing the deformed cell shapes and the spatial distributions of the Lagrangian strains near the bead. These numerical results were obtained when a torque T equal to 1010.5 pN⋅μm was applied to the bead, inducing a bead rotation θ of 29.6° and a translation Δy along the Oy axis of 0.77 μm. The bead was removed to visualize more clearly the spatial strain distributions on the cell–bead contact area. The Lagrangian strain fields shown in the figure are (A) the normal strain Eyy; (B) the normal strain Ezz; (C) the shear strain Eyz, and (D) the effective strain eeff. The Std set of parameter values was used for these simulations.
Grahic Jump Location
2D color maps showing the spatial distribution of the secant modulus Es (in Pa) in the plane of symmetry (see Fig. 1) and in the vicinity of the bead. Increasing values of the applied torque T are considered: (A) T=31.7 pN⋅μm, bead rotation θ=5.2°, and translation along the Oy axis Δy=−0.11 μm; (B) T=87.7 pN⋅μm, θ=9.6°, and Δy=−0.22 μm; (C) T=385 pN⋅μm, θ=19.6°, and Δy=−0.49 μm; and (D) T=1010.5 pN⋅μm, θ=29.6°, and Δy=−0.77 μm. The Std set of parameter values was used for these simulations. Propagation of the strain from the bead contact area to the substrate induced a dramatic increase in the secant modulus from (A) to (D). The inserts correspond to the same figure but with the color code rescaled between 0 and 100 Pa.
Grahic Jump Location
Influence of the Yeoh strain-energy function parameters a1 and a2 (in Pa) on the bead rotation amplitude θ (A), (C) and on the associated normalized translation Δy*y/R (B), (D). The Std curve is the control simulation obtained with the Std set of parameter values. In each simulation, except for the parameter under study, the parameter values were those of the Std set.
Grahic Jump Location
Influence of the geometrical parameters R (A), (B), α (C), (D), and h (E), (F) on the bead rotation amplitude θ and on the associated normalized translation Δy*y/R. The response curves simulated with the Std set of parameter values is given in each case as the reference (Std curves). In each simulation, except for the parameter under study, the parameter values were those of the Std set.
Grahic Jump Location
Influence on the cell mechanical response of two schematic cell adhesion patterns, PF1 and PF2. Top left: Top view of the two models PF1 and PF2. The white areas correspond to absence of cell adhesion. The arrows indicate the orientation of the torque applied at the center of the bead. Bottom left: Cross-section view of the two models PF1 and PF2. The 2D color maps show the heterogeneous spatial distribution of the effective strain fields eeff in the plane of symmetry when the bead rotation is equal to 30°. Right: Computed cell response obtained for each cell adhesion pattern, compared to the control simulation (full cell adhesion). All computations were done using the Std set of parameter values.
Grahic Jump Location
Variation in normalized apparent cell stiffness [Gθ/Ecell, with Gθ=T/(6Vbθ), where Vb is the bead volume] with increasing values of the bead-immersion half-angle α and in the special case of a semi-infinite medium. The numerical results given by the linear approximation of the hyperelastic model are compared to the analytical expression proposed by Laurent et al. [Gθ/Ecell=sin3(α)/6; see Ref. 17] and to the numerical results reported by Mijailovitch et al. 42. The geometrical parameter values were chosen in agreement with those of Mijailovitch et al., i.e., R=2.25 μm and h=20 μm.
Grahic Jump Location
(A) Mechanical response when considering the cell as a linear elastic material with increasing Young modulus (E=6a1) values from 13.5 to 135 Pa. In each simulation, except for the parameter under study, the parameter values were those of the Std set. (B) Nonlinear effect induced by the arc cosine function on the apparent mechanical response when considering the cell as a linear elastic material with increasing Young modulus (E=6a1) values from 30 to 240 Pa. For each value of E, we calculated the torque-apparent bead rotation response (i.e., T–θ* curves) according to the procedure described in the text. Note that open circles (±SD) are the experimental MTC data and that all simulations were performed using a neo-Hookean strain-energy function.



Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In