Nonlinear elasticity in biological gels


The mechanical properties of soft biological tissues are essential to their physiological function and cannot easily be duplicated by synthetic materials. Unlike simple polymer gels, many biological materials—including blood vessels1, mesentery tissue2, lung parenchyma3, cornea4 and blood clots5—stiffen as they are strained, thereby preventing large deformations that could threaten tissue integrity. The molecular structures and design principles responsible for this nonlinear elasticity are unknown. Here we report a molecular theory that accounts for strain-stiffening in a range of molecularly distinct gels formed from cytoskeletal and extracellular proteins and that reveals universal stress–strain relations at low to intermediate strains. The input to this theory is the force–extension curve for individual semi-flexible filaments and the assumptions that biological networks composed of these filaments are homogeneous, isotropic, and that they strain uniformly. This theory shows that systems of filamentous proteins arranged in an open crosslinked mesh invariably stiffen at low strains without requiring a specific architecture or multiple elements with different intrinsic stiffness.

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Figure 1: Neurofilament and fibrin protofibril networks.
Figure 2: Dynamic shear storage moduli measured at different strain amplitudes for a series of crosslinked biopolymer networks.
Figure 3: Scaled modulus–strain curves for various biopolymer networks compared to theory and the scaled force–extension relation for a semiflexible polymer.
Figure 4: Experimental data for fibrin protofilaments (dots) at various concentrations, and corresponding theoretical curves (solid lines) as computed from the extended theory including a stretch modulus.


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We are grateful to J.-F. Leterrier, S. Hvidt, P. Traub, J. Hartwig and E. Sawyer for collaboration in producing the protein networks and electron micrographs. This work was supported in part by the US-NIH and NSF/MRSEC programmes (P.A.J., T.C.L., C.S., J.J.P.) and by the Kavli Institute for Theoretical Physics at the University of California (F.C.M., P.A.J.), where some of this work was initiated.

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Correspondence to Cornelis Storm.

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Storm, C., Pastore, J., MacKintosh, F. et al. Nonlinear elasticity in biological gels. Nature 435, 191–194 (2005).

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