Skip to main content
SpringerLink
Log in

Matrix Strains Induced by Cells: Computing How Far Cells Can Feel

  • Published:
Cellular and Molecular Bioengineering Aims and scope Submit manuscript

Abstract

Many tissue cells exert contractile forces that mechanically couples them to elastic matrices and that influence cell adhesion, cytoskeletal organization, and even cell differentiation. However, strains within the depths of matrices are often unclear and are likely relevant not only to the fact that some matrices such as so-called basement membranes are thin relative to cell dimensions but also to defining how far cells can ‘feel’. Here we briefly present experimental results for cell spreading on thin, ligand-coated gels and for prestress in stem cells in relation to gel stiffness. We then introduce a finite element computation in which a cell is placed on an elastic matrix, while matrix elasticity and thickness are varied in order to compute and compare elastostatic deformations within the matrix. We focus on the response at the cell-matrix interface because this is the proximal location of likely tactile sensors, including focal adhesions and membrane channels. Average interfacial strains between cell and matrix show large deviations only when soft matrices are a fraction of the height and width of a cell, proving consistent with experiments. Three-dimensional (3D) cell morphologies that model stem cell-derived neurons, myoblasts, and osteoblasts show that a cylinder-shaped myoblast induces the highest strains, consistent with the prominent contractility of muscle. Groups of such cells show a weak crosstalk in matrix strains, but the cells must be much closer than a cell-width. Cells thus feel on length scales closer to that of adhesions than on cellular scales or higher.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Figure 1
Figure 2
Figure 3
Figure 4
Figure 5
Figure 6
Figure 7

Similar content being viewed by others

References

  1. Balaban NQ, Schwarz US, Riveline D, et al. Force and focal adhesion assembly: a close relationship studied using elastic micropatterned substrates. Nat Cell Biol. 2001;3(5):466–472.

    Article  Google Scholar 

  2. Beningo KA, Dembo M, Kaverina I, Small JV, Wang Y-l. Nascent Focal Adhesions Are Responsible for the Generation of Strong Propulsive Forces in Migrating Fibroblasts. J. Cell Biol. 2001;153(4):881–888.

    Article  Google Scholar 

  3. Bischofs IB, Safran SA, Schwarz US. Elastic interactions of active cells with soft materials. PRE. 2004;69(2).

    Google Scholar 

  4. Bischofs IB, Schwarz US. Cell organization in soft media due to active mechanosensing. PNAS. 2003;100(16):9274–9279.

    Article  Google Scholar 

  5. Butler JP, Tolic-Norrelykke IM, Fabry B, Fredberg JJ. Traction fields, moments, and strain energy that cells exert on their surroundings. Am J Physiol Cell Physiol. 2002;282(3):C595–605.

    Google Scholar 

  6. Cai Y, Biais N, Giannone G, et al. Nonmuscle myosin IIA-dependent force inhibits cell spreading and drives F-actin flow. Biophys. J. 2006;91(10):3907–20.

    Article  Google Scholar 

  7. Charras GT, Horton MA. Determination of cellular strains by combined atomic force microscopy and finite element modeling. Biophys. J. 2002;83(2):858–79.

    Article  Google Scholar 

  8. Cukierman E, Pankov R, Stevens DR, Yamada KM. Taking Cell-Matrix Adhesions to the Third Dimension. Science. 2001;294(5547):1708–1712.

    Article  Google Scholar 

  9. Dembo M, Wang Y-L. Stresses at the Cell-to-Substrate Interface during Locomotion of Fibroblasts. Biophys. J. 1999;76(4):2307–2316.

    Article  Google Scholar 

  10. DiMilla PA, Barbee K, Lauffenburger DA. Mathematical model for the effects of adhesion and mechanics on cell migration speed. Biophys. J. 1991;60(1):15–37.

    Article  Google Scholar 

  11. Engler AJ, Bacakova L, Newman C, Hategan A, Griffin M, Discher DE. Substrate Compliance versus Ligand Density in Cell on Gel Responses. Biophys. J. 2004;86(1):617–628.

    Article  Google Scholar 

  12. Engler AJ, Griffin MA, Sen S, Bonnemann CG, Sweeney HL, Discher DE. Myotubes differentiate optimally on substrates with tissue-like stiffness: pathological implications for soft or stiff microenvironments. J. Cell Biol. 2004;166(6):877–887.

    Article  Google Scholar 

  13. Engler AJ, Richert R, Wong JY, Picart C, Discher DE. Surface probe measurements of the elasticity of sectioned tissue, thin gels and polyelectrolyte multilayer films: Correlations between substrate stiffness and cell adhesion. J. Surface Science. 2004;570:142–154.

    Article  Google Scholar 

  14. Engler AJ, Sen S, Sweeney HL, Discher DE. Matrix elasticity directs stem cell lineage specification. Cell. 2006;126(4):677–89.

    Article  Google Scholar 

  15. Flanagan LA JY, Marg B, Osterfield M, Janmey PA. Neurite branching on deformable substrates. Neuroreport. 2002;13(18):2411–5.

    Article  Google Scholar 

  16. A.J. García Reyes CD. Bio-adhesive Surfaces to Promote Osteoblast Differentiation and Bone Formation Journal of Dental Research. 2005;84(5):407–413.

    Article  Google Scholar 

  17. Griffin MA, Sen S, Sweeney HL, Discher DE. Adhesion-contractile balance in myocyte differentiation. J. Cell Sci. 2004;117(24):5855–5863.

    Article  Google Scholar 

  18. Guilak F MV. The mechanical environment of the chondrocyte: a biphasic finite element model of cell-matrix interactions in articular cartilage. J Biomechanics. 2000;33(12):1663–73.

    Article  Google Scholar 

  19. Guilak F, Erickson GR, Ting-Beall HP. The Effects of Osmotic Stress on the Viscoelastic and Physical Properties of Articular Chondrocytes. Biophys. J. 2002;82(2):720–727.

    Article  Google Scholar 

  20. Harris AK Stopak D, Wild P. Fibroblast traction as a mechanism for collagen morphogenesis. Nature. 1981;290(5803):249–51.

    Article  Google Scholar 

  21. Harris AK WP, Stopak D. Silicone rubber substrata: a new wrinkle in the study of cell locomotion. Science. 1980;208(4440):177–9.

    Article  Google Scholar 

  22. Karcher H, Lammerding J, Huang H, Lee RT, Kamm RD, Kaazempur-Mofrad MR. A three-dimensional viscoelastic model for cell deformation with experimental verification. Biophys. J. 2003;85(5):3336–49.

    Article  Google Scholar 

  23. Khatiwala CB, Peyton SR, Putnam AJ. Intrinsic mechanical properties of the extracellular matrix affect the behavior of pre-osteoblastic MC3T3-E1 cells. Am. J. Physiol. Cell Physiol. 2006;290(6):C1640–50.

    Article  Google Scholar 

  24. Kumar S, Maxwell IZ, Heisterkamp A, et al. Viscoelastic Retraction of Single Living Stress Fibers and Its Impact on Cell Shape, Cytoskeletal Organization, and Extracellular Matrix Mechanics. Biophys. J. 2006;90(10):3762–3773.

    Article  Google Scholar 

  25. Maloney J, Walton EB, Bruce CM, Van Vliet KJ. Influence of finite thickness and stiffness on cellular adhesion-induced deformation of compliant substrata. Phys. Rev. E 2008;78(041923):1–15.

    Google Scholar 

  26. McBeath R, Pirone DM, Nelson CM, Bhadriraju K, Chen CS. Cell shape, cytoskeletal tension, and RhoA regulate stem cell lineage commitment. Dev Cell. 2004;6(4):483–95.

    Article  Google Scholar 

  27. McGarry JG, Prendergast PJ. A three-dimensional finite element model of an adherent eukaryotic cell. European Cells and Materials. 2004;7:27–34.

    Google Scholar 

  28. Merkel R, Kirchgessner N, Cesa CM, Hoffmann B. Cell Force Microscopy on Elastic Layers of Finite Thickness. Biophys. J. 2007;93(9):3314–3323.

    Article  Google Scholar 

  29. Paszek MJ, Zahir N, Johnson KR, et al. Tensional homeostasis and the malignant phenotype. Cancer Cell. 2005;8(3):241–54.

    Article  Google Scholar 

  30. Pelham RJ, Jr., Wang Y-l. Cell locomotion and focal adhesions are regulated by substrate flexibility. PNAS. 1997;94(25):13661–13665.

    Article  Google Scholar 

  31. Peyton SR, Putnam AJ. Extracellular matrix rigidity governs smooth muscle cell motility in a biphasic fashion. J. Cell Physiol. 2005;204(1):198–209.

    Article  Google Scholar 

  32. Rajagopalan P, Marganski WA, Brown XQ, Wong JY. Direct comparison of the spread area, contractility, and migration of balb/c 3T3 fibroblasts adhered to fibronectin- and RGD-modified substrata. Biophys. J. 2004;87(4):2818–27.

    Article  Google Scholar 

  33. Raucher D, Sheetz MP. Characteristics of a Membrane Reservoir Buffering Membrane Tension. Biophys. J. 1999;77(4):1992–2002.

    Article  Google Scholar 

  34. Schwarz US, Balaban NQ, Riveline D, Bershadsky A, Geiger B, Safran SA. Calculation of Forces at Focal Adhesions from Elastic Substrate Data: The Effect of Localized Force and the Need for Regularization. Biophys. J. 2002;83(3):1380–1394.

    Article  Google Scholar 

  35. Schwarz US, Safran SA. Elastic Interactions of Cells. PRL. 2002;88(4), 048102.

    Article  Google Scholar 

  36. Stedman HH, Sweeney HL, Shrager JB, et al. The mdx mouse diaphragm reproduces the degenerative changes of Duchenne muscular dystrophy. Nature. 1991;352(6335):536–9.

    Article  Google Scholar 

  37. Stenmark KR, Mecham RP. Cellular and molecular mechanisms of pulmonary vascular remodeling. Ann. Rev. Physiol. 1997;59(1):89–144.

    Article  Google Scholar 

  38. Wang HB, Dembo M, Wang YL. Substrate flexibility regulates growth and apoptosis of normal but not transformed cells. Am. J. Physiol. Cell Physiol. 2000;279(5):C1345–50.

    Google Scholar 

  39. Wang N, Tolic-Norrelykke IM, Chen J, et al. Cell prestress I Stiffness and prestress are closely associated in adherent contractile cells. Am J Physiol Cell Physiol. 2002;282(3):C606–616.

    Google Scholar 

  40. Weber KT, Sun Y, Campbell SE, et al. Chronic mineralocorticoid excess and cardiovascular remodeling. Steroids. 1995;60(1):125–32.

    Article  Google Scholar 

  41. Yeung T, Georges PC, Flanagan LA, et al. Effects of substrate stiffness on cell morphology, cytoskeletal structure, and adhesion. Cell Motil. Cytoskeleton. 2005;60(1):24–34.

    Article  Google Scholar 

  42. Zaman MH, Trapani LM, Sieminski AL, et al. Migration of tumor cells in 3D matrices is governed by matrix stiffness along with cell-matrix adhesion and proteolysis. Proc. Natl. Acad. Sci. U S A. 2006;103(29):10889–94.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Biophysical Engineering Lab, University of Pennsylvania, Philadelphia, PA, 19104, USA

    Shamik Sen & Dennis E. Discher

  2. Laboratory for Stem Cell Biology & Bioengineering, University of California, San Diego, CA, 92093, USA

    Adam J. Engler

Authors
  1. Shamik Sen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Adam J. Engler
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Dennis E. Discher
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dennis E. Discher.

Electronic supplementary material

Below are the links to the electronic supplementary material.

Figure S1

Cell displacements on thick matrices. Similar to trends observed with 〈ε∞〉, 〈u ∞〉 scales with E gel in a power-law dependent manner, with stem cells being maximally mechanosensitive. (a) Lateral propagation of displacements for a stem cell on gels of different stiffness. The common, characteristic decay length is ~0.25 R cell. (DOCX 67 kb)

Figure S2

Depth sensing: gel strain distributions. Individual interfacial strain components (〈ε rr 〉, 〈ε zz 〉, 〈ε rz 〉, 〈ε θθ 〉) plotted versus gel thickness for different values of E gel exhibit different transition regimes. (DOCX 300 kb)

Figure S3

Comparison of prestress distributions (soft gel). Uniform prestress distribution, used in this paper is compared with edge prestress and interfacial prestress to study differences in the displacement and strain maps. In comparison to edge prestress, where peak displacements and strain compare well with those obtained with uniform prestress, interfacial prestress produces very low displacement and strains. (DOCX 109 kb)

Rights and permissions

Reprints and permissions

About this article

Cite this article

Sen, S., Engler, A.J. & Discher, D.E. Matrix Strains Induced by Cells: Computing How Far Cells Can Feel. Cel. Mol. Bioeng. 2, 39–48 (2009). https://doi.org/10.1007/s12195-009-0052-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12195-009-0052-z

Keywords

Search

Navigation