Actin filament length tunes elasticity of flexibly cross-linked actin networks

Abstract

Networks of the cytoskeletal biopolymer actin cross-linked by the compliant protein filamin form soft gels that stiffen dramatically under shear stress. We demonstrate that the elasticity of these networks shows a strong dependence on the mean length of the actin polymers, unlike networks with small, rigid cross-links. This behavior is in agreement with a model of rigid filaments connected by multiple flexible linkers.

Figure 1

(A) Schematic of network of stiff polymers of mean length L connected by flexible cross-links. (B and C) Schematic of stiff polymer and attached cross-links in a network before (B) and after (C) shear.

Figure 2

Mean F-actin length L as a function of the molar ratio of actin/gelsolin, R_G⁻¹. L decreases from its unregulated value (dashed line) as gelsolin is added. (Solid line) Linear scaling.

Figure 3

Microstructure of filamin-gelsolin-F-actin networks. (A and B) For R_F ≲ 0.01, networks are a homogeneous mesh of F-actin. (C and D) Large bundles are present at high R_F. (A) Confocal image, R_F = 0.002, R_G = 0 (L = 15 μm). (B) Confocal image, R_F = 0.01, R_G = 1:370 (L = 1 μm). (C) Confocal image, R_F = 0.01, R_G = 0. (Inset) TEM image. (D) Confocal image, R_F = 0.04, R_G = 1:370. (Inset) Confocal image at lower magnification to show network connectivity. Scale bars are 10 μm for confocal images and 0.5 μm for TEM image.

Figure 4

Linear viscoelasticity of cross-linked F-actin networks. Elastic moduli G′ (solid) and viscous moduli G″ (open). Filamin cross-linked networks are soft, viscoelastic solids that become stiffer with increasing R_F or L: (A) L = 15 μm with various R_F and (C) R_F = 0.01 with various L (in μm). Rigidly cross-linked networks become stiffer with increasing L and significantly stiffer and more solidlike with increasing R_B: (B) L = 15 μm with various R_B, and (D) R_B = 0.01 with various L.

Figure 5

Dependence on L of the linear elastic modulus measured at a frequency of 0.1 Hz, G_o. (A) For filamin networks G_o increases stronger than linearly with L. R_F = 0.001 (open), 0.002 (light shaded), 0.005 (medium shaded), and 0.01 (solid). (B) Rigidly cross-linked networks show qualitatively different behavior. R_B = 0.01. (C) G_o for networks cross-linked with filamin at different R_F collapse onto a single curve when plotted versus R_FL² with nearly linear scaling above R_FL² = 0.1. Shaded bar represents the range of moduli measured for F-actin solutions with 2 < L < 7 μm.

Figure 6

Nonlinear stiffening in strain ramps with a rate of 0.1/s. The derivative of the stress-strain curve, K, normalized by its initial value, K_o, as a function of strain. (A) Filamin with R_F = 0.01 and L = 15 (short dash), 10 (long dash), 7 (dash-dot), 5 (solid), and 2 (dot) μm. For the network with L = 15 μm, K/K_o = 1 at small strains before beginning to stiffen above γ_c = 0.06. Networks with shorter filaments initially display weakening behavior, where K/K_o decreases below 1, due to their lower network connectivity, but stiffen at higher strains where the slope of the curve becomes positive. The value γ_c increases with decreasing L. (B) Rigid cross-links with R_B = 0.01 and L = 10 (dash), 5 (solid), and 2 (dot) μm. Networks with long filaments display stiffening behavior that is independent of L, while networks with short filaments display weakening behavior. (Insets) Same data plotted versus stress.

Figure 7

Dependence of the critical and maximum strains on L. Filamin with R_F = 0.01 (solid). Rigid cross-links with R_B = 0.01 (open). (A) The value γ_c of the filamin networks increases with increasing L⁻¹. In contrast, γ_c for rigidly cross-linked networks is independent of L, with mean value denoted (solid line). (B) The value γ_m versus L⁻¹.

Figure 8

Nonlinear stiffening in prestress measurements. (A) Filamin networks with R_F = 0.003 and L = 1 (squares), 5 (triangles), 15 μm (solid circles); R_F = 0.005 with L = 15 μm (diamonds); and R_F = 0.01 with L = 15 μm (open circles, bundled network). K′ is independent of prestress, σ_o, for small prestresses before beginning to increase with σ_o at a critical stress, σ_c. Networks with higher R_F and L stiffen more and support larger stresses before breaking. (Line) Linear scaling predicted by the model. (B) Rigidly cross-linked network with L = 15 μm and R_B = 0.0003 (diamonds), 0.001 (inverted triangles), 0.003 (triangles), 0.03 (circles), and 0.3 (squares). These networks also display stiffening behavior, but the maximum stiffness and stress are roughly the same for every sample. (Line) The σ_o^3/2 scaling predicted by the affine thermal model.

Figure 9

Rescaled nonlinear stiffening of nonbundled networks with filamin (solid) or rigid cross-links (light shaded) at various values of R_F or R_B, L, and c_A. Filamin data is consistent with model of rigid rods connected by flexible linkers (20) (medium shaded line), while rigid cross-linking data is consistent with model of semiflexible filaments connected by rigid linkers (5) (dark shaded line).

Figure 10

Scaling of the maximum stress, σ_m, with L for nonbundled filamin networks. (A) The value σ_m increases with L. From prestress measurements (circles) or 0.1/s strain ramps (squares): R_F = 0.001 (open), 0.003 (light shaded), 0.005 (medium shaded), and 0.01 (solid). (B) The value σ_m for samples of different compositions collapse onto a single curve when plotted versus R_FL. Above R_FL = 0.01, the data scale roughly linearly with R_FL. L = 15 (circles), 10 (diamonds), 7 (inverted triangles), 5 (triangles), 2 (pentagons), and 1 (squares) μm. (Inset) The value σ_m grows as the logarithm of loading rate in stress ramps for networks with L = 15 μm, R_F = 0.005.