Biomechanics and Modeling in Mechanobiology

, Volume 9, Issue 5, pp 563–572 | Cite as

Power-law rheology analysis of cells undergoing micropipette aspiration

  • E. H. Zhou
  • S. T. Quek
  • C. T. Lim
Original Paper


Accurate quantification of the mechanical properties of living cells requires the combined use of experimental techniques and theoretical models. In this paper, we investigate the viscoelastic response of suspended NIH 3T3 fibroblasts undergoing micropipette aspiration using power-law rheology model. As an important first step, we examine the pipette size effect on cell deformation and find that pipettes larger than ~7 μm are more suitable for bulk rheological measurements than smaller ones and the cell can be treated as effectively continuum. When the large pipettes are used to apply a constant pressure to a cell, the creep deformation is better fitted with the power-law rheology model than with the liquid drop or spring-dashpot models; magnetic twisting cytometry measurement on the rounded cell confirms the power-law behavior. This finding is further extended to suspended cells treated with drugs targeting their cytoskeleton. As such, our results suggest that the application of relatively large pipettes can provide more effective assessment of the bulk material properties as well as support application of power-law rheology to cells in suspension.


Cell mechanics Cytoskeleton Deformability Soft glassy rheology Viscoelasticity Optical magnetic twisting cytometry 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Supplementary material

10237_2010_197_MOESM1_ESM.doc (352 kb)
ESM 1 (DOC 352 kb)


  1. An SS, Fabry B, Trepat X, Wang N, Fredberg JJ (2006) Do biophysical properties of the airway smooth muscle in culture predict airway hyperresponsiveness. Am J Respir Cell Mol Biol 35(1): 55–64CrossRefGoogle Scholar
  2. Baaijens FPT, Trickey WR, Laursen TA, Guilak F (2005) Large deformation finite element analysis of micropipette aspiration to determine the mechanical properties of the chondrocyte. Ann Biomed Eng 33(4): 492–499CrossRefGoogle Scholar
  3. Bausch AR, Ziemann F, Boulbitch AA, Jacobson K, Sackmann E (1998) Local measurements of viscoelastic parameters of adherent cell surfaces by magnetic bead microrheometry. Biophys J 75(4): 2038–2049CrossRefGoogle Scholar
  4. Burridge K, Kelly T, Mangeat P (1982) Nonerythrocyte spectrins: actin-membrane attachment proteins occurring in many cell types. J Cell Biol 95(2 Pt 1): 478–486CrossRefGoogle Scholar
  5. Bursac P, Fabry B, Trepat X, Lenormand G, Butler JP, Wang N, Fredberg JJ, An SS (2007) Cytoskeleton dynamics: fluctuations within the network. Biochem Biophys Res Commun 355(2): 324–330CrossRefGoogle Scholar
  6. Charras GT, Yarrow JC, Horton MA, Mahadevan L, Mitchison TJ (2005) Non-equilibration of hydrostatic pressure in blebbing cells. Nature 435(7040): 365–369CrossRefGoogle Scholar
  7. Dahl KN, Engler AJ, Pajerowski JD, Discher DE (2005) Power-law rheology of isolated nuclei with deformation mapping of nuclear substructures. Biophys J 89(4): 2855–2864CrossRefGoogle Scholar
  8. Danowski BA (1989) Fibroblast contractility and actin organization are stimulated by microtubule inhibitors. J Cell Sci 93(Pt 2): 255–266Google Scholar
  9. Desprat N, Richert A, Simeon J, Asnacios A (2005) Creep function of a single living cell. Biophys J 88(3): 2224–2233CrossRefGoogle Scholar
  10. Drury JL, Dembo M (2001) Aspiration of human neutrophils: effects of shear thinning and cortical dissipation. Biophys J 81(6): 3166–3177CrossRefGoogle Scholar
  11. Elson EL (1988) Cellular mechanics as an indicator of cytoskeletal structure and function. Annu Rev Biophys Biophys Chem 17: 397–430CrossRefGoogle Scholar
  12. Fabry B, Maksym GN, Butler JP, Glogauer M, Navajas D, Fredberg JJ (2001a) Scaling the microrheology of living cells. Phys Rev Lett 87(14): 148102CrossRefGoogle Scholar
  13. Fabry B, Maksym GN, Shore SA, Moore PE, Panettieri RA Jr, Butler JP, Fredberg JJ (2001b) Selected contribution: time course and heterogeneity of contractile responses in cultured human airway smooth muscle cells. J Appl Physiol 91(2): 986–994Google Scholar
  14. Fabry B, Maksym GN, Butler JP, Glogauer M, Navajas D, Taback NA, Millet EJ, Fredberg JJ (2003) Time scale and other invariants of integrative mechanical behavior in living cells. Phys Rev E Stat Nonlin Soft Matter Phys 68(4 Pt 1): 041914Google Scholar
  15. Fernandez P, Pullarkat PA, Ott A (2006) A master relation defines the nonlinear viscoelasticity of single fibroblasts. Biophys J 90(10): 3796–3805CrossRefGoogle Scholar
  16. Flugge W (1967) Viscoelasticity. Blaisdell Publishing Company, WalthamGoogle Scholar
  17. Guck J, Schinkinger S, Lincoln B, Wottawah F, Ebert S, Romeyke M, Lenz D, Erickson HM, Ananthakrishnan R, Mitchell D, Kas J, Ulvick S, Bilby C (2005) Optical deformability as an inherent cell marker for testing malignant transformation and metastatic competence. Biophys J 88(5): 3689–3698CrossRefGoogle Scholar
  18. Guilak F, Erickson GR, Ting-Beall HP (2002) The effects of osmotic stress on the viscoelastic and physical properties of articular chondrocytes. Biophys J 82(2): 720–727CrossRefGoogle Scholar
  19. Herant M, Marganski WA, Dembo M (2003) The mechanics of neutrophils: synthetic modeling of three experiments. Biophys J 84(5 SU): 3389–3413CrossRefGoogle Scholar
  20. Hochmuth RM (2000) Micropipette aspiration of living cells. J Biomech 33(1): 15–22CrossRefGoogle Scholar
  21. Hoffman BD, Massiera G, Van Citters KM, Crocker JC (2006) The consensus mechanics of cultured mammalian cells. Proc Natl Acad Sci USA 103(27): 10259–10264CrossRefGoogle Scholar
  22. Jones WR, Ting-Beall PH, Lee GM, Kelley SS, Hochmuth RM, Guilak F (1999) Alterations in the Young’s modulus and volumetric properties of chondrocytes isolated from normal and osteoarthritic human cartilage. J Biomech 32(2): 119–127CrossRefGoogle Scholar
  23. Lam J, Herant M, Dembo M, Heinrich V (2009) Baseline mechanical characterization of J774 macrophages. Biophys J 96(1): 248–254CrossRefGoogle Scholar
  24. Lee GY, Lim CT (2007) Biomechanics approaches to studying human diseases. Trends Biotechnol 25(3): 111–118CrossRefMathSciNetGoogle Scholar
  25. Lenormand G, Millet E, Fabry B, Butler JP, Fredberg JJ (2004) Linearity and time-scale invariance of the creep function in living cells. J R Soc Interface 1: 91–97CrossRefGoogle Scholar
  26. Lomakina EB, Spillmann CM, King MR, Waugh RE (2004) Rheological analysis and measurement of neutrophil indentation. Biophys J 87(6): 4246–4258CrossRefGoogle Scholar
  27. Mahaffy RE, Park S, Gerde E, Kas J, Shih CK (2004) Quantitative analysis of the viscoelastic properties of thin regions of fibroblasts using atomic force microscopy. Biophys J 86(3): 1777–1793CrossRefGoogle Scholar
  28. Mijailovich SM, Kojic M, Zivkovic M, Fabry B, Fredberg JJ (2002) A finite element model of cell deformation during magnetic bead twisting. J Appl Physiol 93(4): 1429–1436Google Scholar
  29. Mitchison TJ, Charras GT, Mahadevan L (2008) Implications of a poroelastic cytoplasm for the dynamics of animal cell shape. Semin Cell Dev Biol 19(3): 215–223CrossRefGoogle Scholar
  30. Mizutani T, Haga H, Kawabata K (2004) Cellular stiffness response to external deformation: tensional homeostasis in a single fibroblast. Cell Motil Cytoskeleton 59(4): 242–248CrossRefGoogle Scholar
  31. Nash GB, Obrien E, Gordonsmith EC, Dormandy JA (1989) Abnormalities in the mechanical-properties of red blood-cells caused by plasmodium-falciparum. Blood 74(2): 855–861Google Scholar
  32. Needham D, Hochmuth RM (1990) Rapid flow of passive neutrophils into a 4 microns pipet and measurement of cytoplasmic viscosity. J Biomech Eng 112(3): 269–276CrossRefGoogle Scholar
  33. Price LS, Leng J, Schwartz MA, Bokoch GM (1998) Activation of Rac and Cdc42 by integrins mediates cell spreading. Mol Biol Cell 9(7): 1863–1871Google Scholar
  34. Roca-Cusachs P, Almendros I, Sunyer R, Gavara N, Farre R, Navajas D (2006) Rheology of passive and adhesion-activated neutrophils probed by atomic force microscopy. Biophys J 91(9): 3508–3518CrossRefGoogle Scholar
  35. Rosenbluth MJ, Lam WA, Fletcher DA (2006) Force microscopy of nonadherent cells: a comparison of leukemia cell deformability. Biophys J 90(8): 2994–3003CrossRefGoogle Scholar
  36. Sato M, Theret DP, Wheeler LT, Ohshima N, Nerem RM (1990) Application of the micropipette technique to the measurement of cultured porcine aortic endothelial cell viscoelastic properties. J Biomech Eng 112(3): 263–268CrossRefGoogle Scholar
  37. Schmid-Schonbein GW, Sung KL, Tozeren H, Skalak R, Chien S (1981) Passive mechanical properties of human leukocytes. Biophys J 36(1): 243–256CrossRefGoogle Scholar
  38. Semmrich C, Storz T, Glaser J, Merkel R, Bausch AR, Kroy K (2007) Glass transition and rheological redundancy in F-actin solutions. Proc Natl Acad Sci USA 104(51): 20199–20203CrossRefGoogle Scholar
  39. Smith BA, Tolloczko B, Martin JG, Grutter P (2005) Probing the viscoelastic behavior of cultured airway smooth muscle cells with atomic force microscopy: stiffening induced by contractile agonist. Biophys J 88(4): 2994–3007CrossRefGoogle Scholar
  40. Stamenovic D, Suki B, Fabry B, Wang N, Fredberg JJ (2004) Rheology of airway smooth muscle cells is associated with cytoskeletal contractile stress. J Appl Physiol 96(5): 1600–1605CrossRefGoogle Scholar
  41. Sunyer R, Trepat X, Fredberg JJ, Farre R, Navajas D (2009) The temperature dependence of cell mechanics measured by atomic force microscopy. Phys Biol 6(2): 25009CrossRefGoogle Scholar
  42. Suresh S, Spatz J, Mills JP, Micoulet A, Dao M, Lim CT, Beil M, Seufferlein T (2005) Connections between single-cell biomechanics and human disease states: gastrointestinal cancer and malaria. Acta Biomaterialia 1: 15–30CrossRefGoogle Scholar
  43. Theret DP, Levesque MJ, Sato M, Nerem RM, Wheeler LT (1988) The application of a homogeneous half-space model in the analysis of endothelial cell micropipette measurements. J Biomech Eng 110(3): 190–199CrossRefGoogle Scholar
  44. Trepat X, Deng L, An SS, Navajas D, Tschumperlin DJ, Gerthoffer WT, Butler JP, Fredberg JJ (2007) Universal physical responses to stretch in the living cell. Nature 447(7144): 592–595CrossRefGoogle Scholar
  45. Trepat X, Lenormand G, Fredberg JJ (2008) Universality in cell mechanics. Soft Matter 4: 1750–1759CrossRefGoogle Scholar
  46. Trickey WR, Lee GM, Guilak F (2000) Viscoelastic properties of chondrocytes from normal and osteoarthritic human cartilage. J Orthop Res 18(6): 891–898CrossRefGoogle Scholar
  47. Tsai MA, Frank RS, Waugh RE (1993) Passive mechanical behavior of human neutrophils: power-law fluid. Biophys J 65(5): 2078–2088CrossRefGoogle Scholar
  48. Tsai MA, Waugh RE, Keng PC (1998) Passive mechanical behavior of human neutrophils: effects of colchicine and paclitaxel. Biophys J 74(6): 3282–3291CrossRefGoogle Scholar
  49. Wang N, Tolic-Norrelykke IM, Chen J, Mijailovich SM, Butler JP, Fredberg JJ, Stamenovic D (2002) Cell prestress. I. Stiffness and prestress are closely associated in adherent contractile cells. Am J Physiol Cell Physiol 282(3): C606–C616Google Scholar
  50. Wottawah F, Schinkinger S, Lincoln B, Ananthakrishnan R, Romeyke M, Guck J, Kas J (2005) Optical rheology of biological cells. Phys Rev Lett 94(9): 098103CrossRefGoogle Scholar
  51. Yanai M, Butler JP, Suzuki T, Sasaki H, Higuchi H (2004) Regional rheological differences in locomoting neutrophils. Am J Physiol Cell Physiol 287(3): C603–C611CrossRefGoogle Scholar
  52. Yeung A, Evans E (1989) Cortical shell-liquid core model for passive flow of liquid-like spherical cells into micropipets. Biophys J 56(1): 139–149CrossRefGoogle Scholar
  53. Zhou EH (2006) PhD thesis: experimental and numerical studies on the viscoelastic behavior of living cells. Department of Civil Engineering. National University of Singapore, SingaporeGoogle Scholar
  54. Zhou EH, Lim CT, Quek ST (2005) Finite element simulation of the micropipette aspiration of a viscoelastic cell undergoing large deformation. Mech Adv Mater Struct 12(6): 501–512CrossRefGoogle Scholar
  55. Zhou EH, Quek ST, Lim CT (2010) Finite element simulation of micropipette aspiration based on power-law rheology (in preparation)Google Scholar
  56. Zhou EH, Trepat X, Park CY, Lenormand G, Oliver MN, Mijailovich SM, Hardin C, Weitz DA, Butler JP, Fredberg JJ (2009) Universal behavior of the osmotically compressed cell and its analogy to the colloidal glass transition. Proc Natl Acad Sci USA 106(26): 10632–10637CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Nano Biomechanics Laboratory, Division of Bioengineering and Department of Mechanical EngineeringNational University of SingaporeSingaporeSingapore
  2. 2.Department of Civil EngineeringNational University of SingaporeSingaporeSingapore
  3. 3.Department of Environmental HealthHarvard School of Public HealthBostonUSA

Personalised recommendations