Figure 1.
Two representative networks used in this study.
(A) A network bundled via ACPB and (B) crosslinked via ACPC, both of which consist of actin filaments of various lengths (cyan) and ACPs (red). For visualization, VMD was used [53]. 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.
Table 1.
List of adjusted parameters.
Figure 2.
Comparison between computational and experimental results at <Lf> = 1.5 µm, R = 0.01, and CA = 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 ACPC.
(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 ACPC, 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 ACPB.
(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 fs = 3.16 Hz versus prestress, τ0. G′ begins to increase at about 0.1 Pa and follows a power law, G′∼τ00.85.
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, κs,A, on G′ and G″ for a network crosslinked by ACPC (R = 0.021) at (C) γ = 0.55 and (D) γ = 0. Solid symbols: G′, and open symbols: G″ with κs,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 fs = 3.16 Hz as a function of prestress, τ0, for κs,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 κs,A except that lower κs,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, κb,A, on G′ (solid symbols) and G″ (open symbols) for a network crosslinked by ACPC (R = 0.021) at (A) γ = 0 and (B) γ = 0.4. κb,A = 1.056×10−18 (black circles), 1.056×10−19 (red triangles), and 1.056×10−20 Nm (blue diamonds). The changes in κb,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 fs = 10 Hz as a function of γ and lp (calculated at 300 K). At high γ (≥0.4) or large lp (∼20 µm), TF plays no significant role in G′. On the contrary, at low γ and lp, G′ decreases without TF.
Figure 8.
Effects of bending stiffnesses of ACPC, κb,ACP,1 and κb,ACP,2, on G′ and G″.
The network is crosslinked by ACPC with and without prestrain (R = 0.021). Solid symbols: G′, open symbols: G″. κb,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 κb,ACP on G′ are the greatest in the network with γ = 0.4.
Figure 9.
Relative decrease in G′ at fs = 10 Hz due to 25-fold decrease of various stiffnesses at different prestrains.
Magenta triangles: κs,A/25, blue inverted triangles: κs,ACP/25, green diamonds: κb,ACP/25, black circles: κb,A/25 with thermal fluctuation, and cyan stars: κb,A/25 without thermal fluctuation. R = 0.021 was used. The influence of κs,A and κs, ACP increases at higher γ, and κb,ACP is significant at all prestrains. The effect of κb,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 lp is comparable to or less than lc.
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 NC = 5. Dashed lines show monomers and ACPs used to generate the network using the polymerization model [35]. 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.