Computational analysis of viscoelastic properties of crosslinked actin networks

Abstract

Mechanical force plays an important role in the physiology of eukaryotic cells whose dominant structural constituent is the actin cytoskeleton composed mainly of actin and actin crosslinking proteins (ACPs). Thus, knowledge of rheological properties of actin networks is crucial for understanding the mechanics and processes of cells. We used Brownian dynamics simulations to study the viscoelasticity of crosslinked actin networks. Two methods were employed, bulk rheology and segment-tracking rheology, where the former measures the stress in response to an applied shear strain, and the latter analyzes thermal fluctuations of individual actin segments of the network. It was demonstrated that the storage shear modulus (G') increases more by the addition of ACPs that form orthogonal crosslinks than by those that form parallel bundles. In networks with orthogonal crosslinks, as crosslink density increases, the power law exponent of G' as a function of the oscillation frequency decreases from 0.75, which reflects the transverse thermal motion of actin filaments, to near zero at low frequency. Under increasing prestrain, the network becomes more elastic, and three regimes of behavior are observed, each dominated by different mechanisms: bending of actin filaments, bending of ACPs, and at the highest prestrain tested (55%), stretching of actin filaments and ACPs. In the last case, only a small portion of actin filaments connected via highly stressed ACPs support the strain. We thus introduce the concept of a 'supportive framework,' as a subset of the full network, which is responsible for high elasticity. Notably, entropic effects due to thermal fluctuations appear to be important only at relatively low prestrains and when the average crosslinking distance is comparable to or greater than the persistence length of the filament. Taken together, our results suggest that viscoelasticity of the actin network is attributable to different mechanisms depending on the amount of prestrain.

Figure 1. Two representative networks used in this study.

(A) A network bundled via ACP^B and (B) crosslinked via ACP^C, both of which consist of actin filaments of various lengths (cyan) and ACPs (red). For visualization, VMD was used . Two arms of each ACP drawn with partial red and cyan are connected to filaments, forming crosslinks. In (A), due to the small computational domain, ladder-like structures are predominant rather than long, thick bundles. The linear dimension of the simulation box is 2.8 µm. These two networks form the basis for all simulations; networks with low R were obtained from these by eliminating a portion of active ACPs. The inset of each plot shows the detailed geometry of bundled or crosslinked structures consisting of actin filaments and ACPs.

Figure 2. Comparison between computational and experimental results at <L _f> = 1.5 µm, R = 0.01, and C _A = 12.1 µM.

Solid symbols: G′, open symbols: G″. For simulation, bulk rheology (blue triangles) and segment-tracking rheology (red diamonds) were employed, while the experiment was conducted only using a bulk rheometer (black circles). Overlap occurs in the frequency range of 1–10 Hz.

Figure 3. Viscoelastic moduli of networks crosslinked by ACP^C.

(A) G′ and (B) G″. Open symbols: segment-tracking rheology, solid symbols: bulk rheology. R = 0.021 (black circles), 0.01 (red triangles), and 0 (blue diamonds). With more ACP^C, the magnitude of G′ increases, and its slope decreases. G″ is slightly larger for networks with higher R.

Figure 4. Viscoelastic moduli of networks bundled by ACP^B.

(A) G′ and (B) G″. Open symbols: segment-tracking rheology, solid symbols: bulk rheology. R = 0.04 (black circles), 0.02 (magenta triangles), 0.01 (blue inverted triangles), and 0 (green diamonds). Large discrepancies exist between results obtained by segment-tracking rheology and by bulk rheology with nonzero R due to heterogeneity of the bundled network.

Figure 5. Behaviors of prestrained networks.

(A) G′ (solid symbols) and G″ (open symbols) of networks with R = 0.021 at various prestrain: γ = 0.55 (black circles), 0.4 (magenta triangles), 0.2 (blue inverted triangles), and 0 (green diamonds). At high prestrain, G′ becomes nearly independent of frequency. G″ with high prestrain slightly increases at low frequency. (B) G′ at f _s = 3.16 Hz versus prestress, τ ₀. G′ begins to increase at about 0.1 Pa and follows a power law, G′∼τ ₀ ^0.85.

Figure 6. Importance and effects of extensional stiffness of actin filaments in prestrained networks.

(A) Color-map of the prestrained network with γ = 0.55 (red: highly stretched bonds, gray: intermediately stretched bonds, blue: least stretched bonds.). (B) Similar plot showing only actin filaments with bonds stretched by more than 0.5%. (C,D) Influence of the extensional stiffness of actin filaments, κ _s,A, on G′ and G″ for a network crosslinked by ACP^C (R = 0.021) at (C) γ = 0.55 and (D) γ = 0. Solid symbols: G′, and open symbols: G″ with κ _s,A = 1.69×10⁻² (black circles), 3.38×10⁻³ (red triangles), and 6.76×10⁻⁴ N/m (blue diamonds) (E) G′ at f _s = 3.16 Hz as a function of prestress, τ ₀, for κ _s,A = 6.764×10⁻⁴ (red triangles) and 0.01691 N/m (black circles). G′ of both cases remains nearly constant at low prestress, but starts to increase above ∼0.1 Pa. The behavior is similar for the two values of κ _s,A except that lower κ _s,A leads to a slight reduction in both the level and slope (∼0.7, dashed line) of G′ above the threshold stress level.

Figure 7. Effects of bending stiffness and thermal fluctuation on G′ and G″.

(A,B) Effects of bending stiffness of actin filaments, κ _b,A, on G′ (solid symbols) and G″ (open symbols) for a network crosslinked by ACP^C (R = 0.021) at (A) γ = 0 and (B) γ = 0.4. κ _b,A = 1.056×10⁻¹⁸ (black circles), 1.056×10⁻¹⁹ (red triangles), and 1.056×10⁻²⁰ Nm (blue diamonds). The changes in κ _b,A have large effects on G′ and G″ in (A) but not in (B) (C) Effects of thermal fluctuation (TF) of actin filaments on G′ at f _s = 10 Hz as a function of γ and l _p (calculated at 300 K). At high γ (≥0.4) or large l _p (∼20 µm), TF plays no significant role in G′. On the contrary, at low γ and l _p, G′ decreases without TF.

Figure 8. Effects of bending stiffnesses of ACP^C, κ _b,ACP,1 and κ _b,ACP,2, on G′ and G″.

The network is crosslinked by ACP^C with and without prestrain (R = 0.021). Solid symbols: G′, open symbols: G″. κ _b,ACP = control (black circles), 10-fold decrease (magenta triangles), 100-fold decrease (blue inverted triangles), and 0 (green diamonds) (A) γ = 0, (B) γ = 0.4, (C) γ = 0.55. The effects of changes in κ _b,ACP on G′ are the greatest in the network with γ = 0.4.

Figure 9. Relative decrease in G′ at f _s = 10 Hz due to 25-fold decrease of various stiffnesses at different prestrains.

Magenta triangles: κ _s,A/25, blue inverted triangles: κ _s,ACP/25, green diamonds: κ _b,ACP/25, black circles: κ _b,A/25 with thermal fluctuation, and cyan stars: κ _b,A/25 without thermal fluctuation. R = 0.021 was used. The influence of κ _s,A and κ _{s, ACP} increases at higher γ, and κ _b,ACP is significant at all prestrains. The effect of κ _b,A on G′ increases as γ decreases, and by comparing stars and circles, it can be inferred that thermal fluctuation plays an important role at very low γ when l _p is comparable to or less than l _c.

Figure 10. The supportive framework bearing most of stress.

(A) Network composed of filamentous actin connected via ACPs that support the highest 25% of ACP bending forces. In contrast to Figure 6B, there are filaments that are almost perpendicular to the diagonal direction on the x-z plane, which are not highly stretched yet transmit load by bending of ACPs connected to them. (B) Stress exerted by prestrained networks (γ = 0.4) consisting of a fraction of actin filaments and ACPs. The extent of ACP bending forces is employed as a criterion to retain elements. Each symbol corresponds to a different percentage ratio of the number of ACPs remaining in a rebuilt network to that in the original network: 100% (black circles), 75% (magenta triangles), 50% (blue inverted triangles), and 25% (green diamonds). The fraction of remaining actin segments is, respectively: 100%, 79%, 52%, and 28%. (inset) Orientation angles of actin segments projected onto the x-z plane for the network in Figure 10A. Segments oriented in the z direction have a value of 0°. Most actin segments in the reduced structure are oriented in the (+x)-(+z) direction (45°), but segments with other orientations are also important, presumably because they transmit stress through bending of the ACPs attached to them.

Figure 11. Schematic diagrams explaining the geometry of a network and the calculation of repulsive forces.

(A) A coarse-graining scheme using cylindrical segments with N _C = 5. Dashed lines show monomers and ACPs used to generate the network using the polymerization model . Once the network is formed, it is coarse-grained by replacing the original spheres by cylinders as shown. (B) A schematic diagram showing the distribution of the repulsive force acting on the point Y, onto two end points, α and β. The proportion of each force is determined by y, the distance between point α and point Y.