Elastic Behavior of Cross-Linked and Bundled Actin Networks

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Science  28 May 2004:
Vol. 304, Issue 5675, pp. 1301-1305
DOI: 10.1126/science.1095087
  • Fig. 1.

    (A) Schematic representation of the state diagram of cross-linked and bundled F-actin networks, showing the variations in the elasticity of the networks with changes in actin concentration, cA, or density of cross-linking R = cS/cA, where cS is the scruin concentration. By varying either cA or R, the elastic modulus, Go, can be varied by more than three orders of magnitude, from less than 0.1 Pa to 300 Pa. For large values of cA or R (indicated by the red plane), the network elasticity becomes nonlinear at very small deformations, and the elasticity becomes strain dependent and increases markedly upon increased strain. By contrast, at very low values of cA or R (indicated by the blue plane), the network mechanical response remains linear up to large strains, and no strain stiffening is observed. Transitions between these two regimes of elasticity can be made by changing either R or cA, as indicated by the dashed lines. (B to D) Confocal microscope images of actin networks, cA = 11.9 μM, labeled with Texas Red–phalloidin as a function of R. (B) R = 0, (C) R = 0.07, (D) R = 0.5. As R is increased for a fixed cA, actin filaments are cross-linked into tight bundles. The average bundle thickness is weakly proportional to R between R = 0.03 and R = 1. Because the total filament concentration remains constant, the average mesh size also increases. Bar (B), 3 μm. The bright circles in (B) to (D) are 1-μm-diameter colloidal particles.

  • Fig. 2.

    The elastic modulus, Go, and maximum strain, γmax, as a function of R and cA. (A) Go as a function of R for cA = 11.9 μM. The solid line indicates a scaling of GoR2. (B) γmax versus R for cA = 11.9 μM. The solid line indicates γmaxR–0.6. (Inset) Typical strain stiffening response for a composite network with cA = 11.9 μM and R = 0.03, indicating the onset of nonlinear elastic response, γcrit, and the maximum strain, γmax, before the network breaks. (C) Go as a function of cA for R = 0.03 (◼), 0.13 (△), and 0.3 (⚫). The solid line shows a scaling of Embedded Image and the error bars indicate our sample-to-sample reproducibility. (D) γmax as a function of c for the same values of R. The solid line shows a scaling of Embedded Image.

  • Fig. 3.

    A summary of the interpretation of results presented in this paper. (A) Networks are polymerized between two plates of area, A, and separation, h. By varying the filament or cross-link concentration, we vary the network microstructure; low filament and cross-link density are shown in the left cartoon, whereas high filament and cross-link density are shown in the right cartoon. (B) We measure the mechanical properties of the networks by applying a force, F, per unit area, A, or stress and measuring the deformation or strain γ ≡ x/h. The microscopic distribution of the strain differs in the networks. Our data show two distinct mechanical regimes, distinguished by their nonlinear response. One of these is consistent with macroscopic mechanical properties due to enthalpic filament bending, as expected for low cross-link or filament density. This leads to a highly inhomogeneous distribution of strain, as indicated by the red arrows. By contrast, the mechanical properties of dense networks are consistent with stretching of thermally induced filament fluctuations and an entropic elasticity. The resultant strain is uniform throughout the sample, as indicated by the red arrows. (C) The table summarizes the essential differences in the elasticity of entropic and enthalpic networks.

  • Fig. 4.

    The differential elastic modulus, K′, as a function of applied steady shear stress, σo, for R = 0.03 and several values of cA; the lines through the data indicate theoretical predictions for each concentration determined from the single-filament response. The values of cA are 29.4 μM(▲—), 21.4 μM (⚫ - ·), 11.9 μM (◼ - -), and 8.33 μM (♢ - ··). The stress stiffening response of a bundled network, R = 0.5, at cA = 7 μM is also shown (◯). The dotted line at the right indicates σo3/2o. (Inset) The data sets rescaled by σcrit and Go showing the universal form of the stress stiffening response; the rescaled theory is indicated by the solid line.

  • Fig. 5.

    The R-cA state diagram of actin:scruin networks detailing the tunability of Go; colors indicate a range from 0.03 Pa (purple) to 300 Pa (red), as shown by the legend. Symbols differentiate networks that stress stiffen (+) from those that do not (◯).

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  • Abstract
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    Elastic Behavior of Cross-Linked and Bundled Actin Networks
    M. L. Gardel, J. H. Shin, F. C. MacKintosh, L. Mahadevan, P. Matsudaira, D. A. Weitz

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