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Emergence of tissue-like mechanics from fibrous networks confined by close-packed cells


The viscoelasticity of the crosslinked semiflexible polymer networks—such as the internal cytoskeleton and the extracellular matrix—that provide shape and mechanical resistance against deformation is assumed to dominate tissue mechanics. However, the mechanical responses of soft tissues and semiflexible polymer gels differ in many respects. Tissues stiffen in compression but not in extension1,2,3,4,5, whereas semiflexible polymer networks soften in compression and stiffen in extension6,7. In shear deformation, semiflexible polymer gels stiffen with increasing strain, but tissues do not1,2,3,4,5,6,7,8. Here we use multiple experimental systems and a theoretical model to show that a combination of nonlinear polymer network elasticity and particle (cell) inclusions is essential to mimic tissue mechanics that cannot be reproduced by either biopolymer networks or colloidal particle systems alone. Tissue rheology emerges from an interplay between strain-stiffening polymer networks and volume-conserving cells within them. Polymer networks that soften in compression but stiffen in extension can be converted to materials that stiffen in compression but not in extension by including within the network either cells or inert particles to restrict the relaxation modes of the fibrous networks that surround them. Particle inclusions also suppress stiffening in shear deformation; when the particle volume fraction is low, they have little effect on the elasticity of the polymer networks. However, as the particles become more closely packed, the material switches from compression softening to compression stiffening. The emergence of an elastic response in these composite materials has implications for how tissue stiffness is altered in disease and can lead to cellular dysfunction9,10,11. Additionally, the findings could be used in the design of biomaterials with physiologically relevant mechanical properties.

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Fig. 1: Multiaxial rheological behaviour of adipose tissue, reconstituted ECM networks and blood clots.
Fig. 2: Effect of dense cell packing on multiaxial mechanics.
Fig. 3: Multiaxial mechanics of networks with embedded particles.
Fig. 4: Theoretical model of fibre networks with volume-conserving inclusions.

Data availability

The data that support the findings of this study are available from the corresponding authors upon reasonable request.

Code availability

The computational code developed in this work is included as Supplementary Code. The source code is covered under the GNU general public license, version 2.0 (GPL-2.0).


  1. Mihai, L. A., Chin, L., Janmey, P. A. & Goriely, A. A comparison of hyperelastic constitutive models applicable to brain and fat tissues. J. R. Soc. Interface 12, 20150486 (2015).

    Article  Google Scholar 

  2. Perepelyuk, M. et al. Normal and fibrotic rat livers demonstrate shear strain softening and compression stiffening: a model for soft tissue mechanics. PLoS ONE 11, e0146588 (2016).

    Article  Google Scholar 

  3. Pogoda, K. et al. Compression stiffening of brain and its effect on mechanosensing by glioma cells. New J. Phys. 16, 075002 (2014).

    Article  ADS  Google Scholar 

  4. Mihai, L. A., Budday, S., Holzapfel, G. A., Kuhl, E. & Goriely, A. A family of hyperelastic models for human brain tissue. J. Mech. Phys. Solids 106, 60–79 (2017).

    Article  ADS  MathSciNet  Google Scholar 

  5. Budday, S. et al. Mechanical characterization of human brain tissue. Acta Biomater. 48, 319–340 (2017).

    Article  CAS  Google Scholar 

  6. Vahabi, M. et al. Elasticity of fibrous networks under uniaxial prestress. Soft Matter 12, 5050–5060 (2016).

    Article  ADS  CAS  Google Scholar 

  7. van Oosten, A. S. et al. Uncoupling shear and uniaxial elastic moduli of semiflexible biopolymer networks: compression-softening and stretch-stiffening. Sci. Rep. 6, 19270 (2016).

    Article  ADS  Google Scholar 

  8. Shah, J. V. & Janmey, P. A. Strain hardening of fibrin gels and plasma clots. Rheol. Acta 36, 262–268 (1997).

    Article  CAS  Google Scholar 

  9. Boucher, Y., Baxter, L. T. & Jain, R. K. Interstitial pressure gradients in tissue-isolated and subcutaneous tumors: implications for therapy. Cancer Res. 50, 4478–4484 (1990).

    CAS  PubMed  Google Scholar 

  10. Wyss, H. M. et al. Biophysical properties of normal and diseased renal glomeruli. Am. J. Physiol. Cell Physiol. 300, C397–C405 (2011).

    Article  CAS  Google Scholar 

  11. Millonig, G. et al. Liver stiffness is directly influenced by central venous pressure. J. Hepatol. 52, 206–210 (2010).

    Article  Google Scholar 

  12. Park, J. A. et al. Unjamming and cell shape in the asthmatic airway epithelium. Nat. Mater. 14, 1040–1048 (2015).

    Article  ADS  CAS  Google Scholar 

  13. Angelini, T. E. et al. Glass-like dynamics of collective cell migration. Proc. Natl Acad. Sci. USA 108, 4714–4719 (2011).

    Article  ADS  CAS  Google Scholar 

  14. Bi, D., Yang, X., Marchetti, M. C. & Manning, M. L. Motility-driven glass and jamming transitions in biological tissues. Phys. Rev. X 6, 021011 (2016).

    PubMed  PubMed Central  Google Scholar 

  15. Schötz, E.-M., Lanio, M., Talbot, J. A. & Manning, M. L. Glassy dynamics in three-dimensional embryonic tissues. J. R. Soc. Interface 10, 20130726 (2013).

    Article  Google Scholar 

  16. Bernal, J. & Mason, J. Packing of spheres: coordination of randomly packed spheres. Nature 188, 910–911 (1960).

    Article  ADS  Google Scholar 

  17. Wang, H., Abhilash, A., Chen, C. S., Wells, R. G. & Shenoy, V. B. Long-range force transmission in fibrous matrices enabled by tension-driven alignment of fibers. Biophys. J. 107, 2592–2603 (2014).

    Article  ADS  CAS  Google Scholar 

  18. Freed, A. D. & Doehring, T. C. Elastic model for crimped collagen fibrils. J. Biomech. Eng. 127, 587–593 (2005).

    Article  Google Scholar 

  19. Berryman, J. G. Random close packing of hard spheres and disks. Phys. Rev. A 27, 1053–1061 (1983).

    Article  ADS  CAS  Google Scholar 

  20. Scott, G. & Kilgour, D. The density of random close packing of spheres. J. Phys. D 2, 863–866 (1969).

    Article  ADS  Google Scholar 

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We acknowledge R. Wells and M. Perepelyuk for collection of rat blood, S. Diamond and R. Li for surplus blood products and D. Iwamoto for reading the manuscript. This work was supported by NIH R01GM09697, NIH U54-CA193417, EB017753 and NSF-DMR-1120901 (to P.A.J., L.C., A.E.P., K.P., K.C. and A.S.G.v.O.), by the NSF Center for Engineering Mechanobiology (CMMI-154857) through grants NSF MRSEC/DMR-1720530 R01CA232256 and U01CA202177 (X.C. and V.B.S.), by a Fulbright Science and Technology Award (A.S.G.v.O.) and by Prins Bernhard Cultuurfonds-Kuitse Fonds (A.S.G.v.O.). K.P. acknowledges partial support from the National Science Center, Poland under grant number UMO2017/26/D/ST4/00997 and from the Polish-American Fulbright Commission.

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Nature thanks Jasna Brujic, Ellen Kuhl and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Authors and Affiliations



A.S.G.v.O. and P.A.J. designed the experiments. A.S.G.v.O. performed the experiments that gave the data presented in Figs. 1b–g (except the adipose tissue data), 2b, d, 3a, c, f and Supplementary Figs. 2–9, 12–16, 18–20. L.C. performed the experiments that gave the results shown in Figs. 1b, c (adipose tissue data), 2a, g and Supplementary Figs. 1, 7–9, 11. K.P. obtained the data in Fig. 2f, h and Supplementary Figs. 10, 11. P.A.J., K.C. and A.E.P. provided the results in Figs. 2e, 3e, f and Supplementary Figs. 10, 11, 14, 17. V.B.S. and X.C. designed the computational model. X.C. generated the computational data. All authors contributed to the manuscript preparation.

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Correspondence to Vivek B. Shenoy or Paul A. Janmey.

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van Oosten, A.S.G., Chen, X., Chin, L. et al. Emergence of tissue-like mechanics from fibrous networks confined by close-packed cells. Nature 573, 96–101 (2019).

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