Skip to main content

Thank you for visiting You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

Optimal matrix rigidity for stress-fibre polarization in stem cells


The shape and differentiated state of many cell types are highly sensitive to the rigidity of the microenvironment. The physical mechanisms involved, however, are unknown. Here, we present a theoretical model and experiments demonstrating that the alignment of stress fibres within stem cells is a non-monotonic function of matrix rigidity. We treat the cell as an active elastic inclusion in a surrounding matrix, allowing the actomyosin forces to polarize in response to elastic stresses developed in the cell. The theory correctly predicts the monotonic increase of the cellular forces with the matrix rigidity and the alignment of stress fibres parallel to the long axis of cells. We show that the anisotropy of this alignment depends non-monotonically on matrix rigidity and demonstrate it experimentally by quantifying the orientational distribution of stress fibres in stem cells. These findings offer physical insight into the sensitivity of stem-cell differentiation to tissue elasticity and, more generally, introduce a cell-type-specific parameter for actomyosin polarizability.

This is a preview of subscription content, access via your institution

Access options

Rent or buy this article

Prices vary by article type



Prices may be subject to local taxes which are calculated during checkout

Figure 1: Actomyosin stress-fibre alignment in hMSCs sparsely plated on 2D substrates of different elasticities.
Figure 2: Cell adhesion and polarization represented by a 1D spring model.
Figure 3: Cell polarization as a function of the ratio of Young’s modulus of the matrix, Em, and the cell, Ec, in both our 2D and 3D models.
Figure 4: The effect of axial cell elongation on stress-fibre polarization and experimental values of the order parameter S for different elastic substrates.


  1. O’Neill, C., Jordan, P. & Ireland, G. Evidence for two distinct mechanisms of anchorage stimulation in freshly explanted and 3t3 swiss mouse fibroblasts. Cell 44, 489–496 (1986).

    Article  Google Scholar 

  2. Chen, C. S., Mrksich, M., Huang, S., Whitesides, G. M. & Ingber, D. E. Geometric control of cell life and death. Science 276, 1425–1428 (1997).

    Article  Google Scholar 

  3. McBeath, R., Pirone, D. M., Nelson, C. M., Bhadriraju, K. & Chen, C. S. Cell shape, cytoskeleton tension, and RohA regulate stem cell lineage commitment. Dev. Cell 6, 483–495 (2004).

    Article  Google Scholar 

  4. Engler, A. J. et al. Myotubes differentiate optimally on substrates with tissue-like stiffness: Pathological implications for soft or stiff microenvironments. J. Cell Biol. 166, 877–887 (2004).

    Article  Google Scholar 

  5. Yeung, T. et al. Effects of substrate stiffness on cell morphology, cytoskeletal structure, and adhesion. Cell Motil. Cytoskeleton 60, 24–34 (2005).

    Article  Google Scholar 

  6. Discher, D. E., Janmey, P. & Wang, Y. Tissue cells feel and respond to the stiffness of their substrate. Science 310, 1139–1143 (2005).

    Article  ADS  Google Scholar 

  7. Engler, A. J., Sen, S., Sweeney, H. L. & Discher, D. E. Matrix elasticity directs stem cell lineage specification. Cell 126, 677–689 (2006).

    Article  Google Scholar 

  8. Georges, P. C., Miller, W. J., Meaney, D. F., Sawyer, E. S. & Janmey, P. A. Matrices with compliance comparable to that of brain tissue select neuronal over glial growth in mixed cortical cultures. Biophys. J. 90, 3012–3018 (2006).

    Article  ADS  Google Scholar 

  9. Cai, Y. et al. Cytoskeletal coherence requires myosin-IIA contractility. J. Cell Sci. 123, 413–423 (2010).

    Article  Google Scholar 

  10. Bhattacharya, D., Talwar, S., Mazumder, A. & Shivashankar, G. V. Spatio-temporal plasticity in chromatin organization in mouse cell differentiation and during drosophila embryogenesis. Biophys. J. 96, 3832–3839 (2009).

    Article  ADS  Google Scholar 

  11. Wang, N., Tytell, J. D. & Ingber, D. E. Mechanotransduction at a distance: Mechanically coupling the extracellular matrix with the nucleus. Nat. Rev. Mol. Cell Biol. 10, 75–82 (2009).

    Article  Google Scholar 

  12. Bershadsky, A., Kozlov, M. & Geiger, B. Adhesion-mediated mechanosensitivity: A time to experiment, and a time to theorize. Curr. Opin. Cell Biol. 18, 472–481 (2006).

    Google Scholar 

  13. Iba, T. & Sumpio, B. Morphological response of human endothelial cells subjected to cyclic strain in vitro. Microvasc. Res. 42, 245–254 (1991).

    Article  Google Scholar 

  14. Rodriguez, J. P., Gonzalez, M., Rios, S. & Cambiazo, V. Cytoskeletal organization of human mesenchymal stem cells (msc) changes during their osteogenic differentiation. J. Cell. Biochem. 93, 721–731 (2004).

    Article  Google Scholar 

  15. Curtis, A., Aitchison, G. & Tsapikouni, T. Orthogonal (transverse) arrangements of actin in endothelia and fibroblasts. J. R. Soc. Interface 3, 753–756 (2006).

    Article  Google Scholar 

  16. Ghibaudo, M. et al. Traction forces and rigidity sensing regulate cell functions. Soft Matter 4, 1836–1843 (2008).

    Article  ADS  Google Scholar 

  17. Kumar, S. et al. Viscoelastic retraction of single living stress fibres and its impact on cell shape, cytoskeletal organization, and extracellular matrix mechanics. Biophys. J. 90, 3762–3773 (2006).

    Article  ADS  Google Scholar 

  18. Dubin-Thaler, B. J. et al. Quantification of cell edge velocities and traction forces reveals distinct motility modules during cell spreading. PLOS One 3, e3735 (2008).

    Article  ADS  Google Scholar 

  19. Griffin, M. A. et al. Patterning, prestress, and peeling dynamics of myocytes. Biophys. J. 86, 1209–1222 (2004).

    Article  ADS  Google Scholar 

  20. Wang, N. et al. Cell prestress. i. Stiffness and prestress are closely associated in adherent contractile cells. Am. J. Physiol. Cell Physiol. 282, C606–C616 (2002).

    Article  Google Scholar 

  21. Wang, N., Ostuni, E., Whitesides, G. M. & Ingber, D. E. Micropatterning tractional forces in living cells. Cell Motil. Cyto. 52, 97–106 (2002).

    Article  Google Scholar 

  22. Chicurel, M. E., Chen, C. S. & Ingber, D. E. Cellular control lies in the balance of forces. Curr. Opin. Cell Biol. 10, 232–239 (1998).

    Article  Google Scholar 

  23. Balaban, N. Q. et al. Force and focal adhesion assembly: A close relationship studied using elastic micropatterned substrates. Nature Cell Biol. 3, 466–472 (2001).

    Article  Google Scholar 

  24. Schwarz, U. S., Erdmann, T. & Bischofs, I. B. Focal adhesions as mechanosensors: The two-spring model. Biosystems 83, 225–232 (2006).

    Article  Google Scholar 

  25. Grinnell, F. Fibroblast-collagen-matrix contraction: Growth-factor signalling and mechanical loading. Trends Cell Biol. 10, 362–365 (2000).

    Article  Google Scholar 

  26. Eshelby, J. D. The determination of elastic field of an ellipsoidal inclusion, and related problems. Proc. R. Soc. A 241, 376–396 (1957).

    ADS  MathSciNet  MATH  Google Scholar 

  27. Mura, T. Micromechanics of Defects in Solids (Kluwer Academic, 1991).

    Google Scholar 

  28. Landau, L. D. & Lifshitz, E. M. Theory of Elasticity 3rd edn (Course of Theoretical Physics, Vol. 7, Reed Educational and Professional Publishing, 1986).

    MATH  Google Scholar 

  29. Jaswon, M. A. & Bhargava, R. D. Two-dimensional elastic inclusion problems. Proc. Camb. Phil. Soc. 57, 669–680 (1961).

    Article  ADS  MathSciNet  Google Scholar 

  30. Siems, R. Mechanical interactions of point defects. Phys. Status Solidi 30, 645–658 (1968).

    Article  Google Scholar 

  31. Schwarz, U. S. & Safran, S. A. Elastic interactions of cells. Phys. Rev. Lett. 88, 048102 (2002).

    Article  ADS  Google Scholar 

  32. Carlsson, A. E. Contractile stress generation by actomyosin gels. Phys. Rev. E 74, 051912 (2006).

    Article  ADS  Google Scholar 

  33. Riveline, D. et al. 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, 1175–1185 (2001).

    Article  Google Scholar 

  34. Schwarz, U. S. & Bischofs, I. B. Physical determinants of cell organization in soft media. Med. Eng. Phys. 27, 763–772 (2005).

    Article  Google Scholar 

  35. Pelham, R. J. & Wang, Y. L. Cell locomotion and focal adhesions are regulated by substrate flexibility. Proc. Natl Acad. Sci. USA 94, 13661–13665 (1997).

    Article  ADS  Google Scholar 

  36. Engler, A. J., Rehfeldt, F., Sen, S. & Discher, D. E. Microtissue Elasticity: Measurements by Atomic Force Microscopy and its Influence on Cell Differentiation Vol. 83, 521–545 (Academic, 2007).

    Google Scholar 

  37. Rasband, W. S. ImageJ US National Institute of Health, Bethesda, Maryland, USA, <>, 1997–2007.

  38. Haralick, R. & Shapiro, L. Computer and Robot Vision Vol. 1 (Addison-Wesley, 1992).

    Google Scholar 

  39. Otsu, N. Threshold selection method from grey-level histograms. IEEE Trans. Syst. Man Cybernetics 9, 62–66 (1979).

    Article  Google Scholar 

Download references


We thank R. De, R. Paul and N. Gov for many useful discussions. We are grateful to the Israel Science Foundation, the Clore Center for Biological Physics, the Schmidt Minerva Center and an EU Network grant for their support. F.R. gratefully acknowledges financial support through the Feodor Lynen fellowship of the Alexander von Humboldt Foundation. D.E.D. thanks NFS and NIH. A.E.X.B. was supported by a scholarship from the Natural Sciences and Engineering Research Council of Canada.

Author information

Authors and Affiliations



A.Z. and S.A.S. developed the theory. F.R., A.E.X.B. and D.E.D. designed the experiments; F.R. carried out the experiments; A.E.X.B. wrote the image analysis algorithm. All authors analysed the data and wrote the paper.

Corresponding author

Correspondence to A. Zemel.

Ethics declarations

Competing interests

The authors declare no competing financial interests.

Supplementary information

Supplementary Information

Supplementary Information (PDF 453 kb)

Rights and permissions

Reprints and permissions

About this article

Cite this article

Zemel, A., Rehfeldt, F., Brown, A. et al. Optimal matrix rigidity for stress-fibre polarization in stem cells. Nature Phys 6, 468–473 (2010).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:

This article is cited by


Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing