F‑Actin Fragmentation Induces Distinct Mechanisms of Stress Relaxation in the Actin Cytoskeleton

The diverse mechanical properties of the F-actin cytoskeleton mediate essential physical behaviors of the cells, including cell division, migration, and shape change. These properties include strain stiffening and stress relaxation that limit cell shape change and determine its rate. To date, stress relaxation has been mainly attributed to the transient nature of cross-linkers that connect F-actins. By contrast, the potential impacts of rich F-actin dynamics to the stress relaxation have been neglected in most previous studies. Thus, in this study, we use a novel computational model to demonstrate that F-actin severing arising from compression-induced filament buckling coordinates with cross-linker unbinding, leading to very distinct modes of stress relaxation. Furthermore, we establish the conditions under which the F-actin severing dominates the mechanical response, providing additional mechanistic insight into the viscoelasticity of the F-actin cytoskeleton

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

<p>(A) Color-map of the prestrained network with <i>γ</i> = 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, <i>κ</i>_s,A, on <i>G</i>′ and <i>G</i>″ for a network crosslinked by ACP^C (<i>R</i> = 0.021) at (C) <i>γ</i> = 0.55 and (D) <i>γ</i> = 0. Solid symbols: <i>G</i>′, and open symbols: <i>G</i>″ with <i>κ</i>_s,A = 1.69×10⁻² (black circles), 3.38×10⁻³ (red triangles), and 6.76×10⁻⁴ N/m (blue diamonds) (E) <i>G</i>′ at <i>f</i>_s = 3.16 Hz as a function of prestress, <i>τ</i>₀, for <i>κ</i>_s,A = 6.764×10⁻⁴ (red triangles) and 0.01691 N/m (black circles). <i>G</i>′ 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 <i>κ</i>_s,A except that lower <i>κ</i>_s,A leads to a slight reduction in both the level and slope (∼0.7, dashed line) of <i>G</i>′ above the threshold stress level.</p

Two representative networks used in this study.

<p>(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 <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1000439#pcbi.1000439-Humphrey1" target="_blank">[53]</a>. 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 <i>R</i> 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.</p

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

<p>(A) A coarse-graining scheme using cylindrical segments with <i>N</i>_C = 5. Dashed lines show monomers and ACPs used to generate the network using the polymerization model <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1000439#pcbi.1000439-Kim1" target="_blank">[35]</a>. 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, <i>α</i> and <i>β</i>. The proportion of each force is determined by <i>y</i>, the distance between point <i>α</i> and point Y.</p

Behaviors of prestrained networks.

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

Viscoelastic moduli of networks crosslinked by ACP^C.

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

Viscoelastic moduli of networks bundled by ACP^B.

<p>(A) <i>G</i>′ and (B) <i>G</i>″. Open symbols: segment-tracking rheology, solid symbols: bulk rheology. <i>R</i> = 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 <i>R</i> due to heterogeneity of the bundled network.</p

Relative decrease in <i>G</i>′ at <i>f</i>_s = 10 Hz due to 25-fold decrease of various stiffnesses at different prestrains.

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

The supportive framework bearing most of stress.

<p>(A) Network composed of filamentous actin connected via ACPs that support the highest 25% of ACP bending forces. In contrast to <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1000439#pcbi-1000439-g006" target="_blank">Figure 6B</a>, 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 (<i>γ</i> = 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%. (<i>inset</i>) 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.</p

Effects of bending stiffness and thermal fluctuation on <i>G</i>′ and <i>G</i>″.

<p>(A,B) Effects of bending stiffness of actin filaments, <i>κ</i>_b,A, on <i>G</i>′ (solid symbols) and <i>G</i>″ (open symbols) for a network crosslinked by ACP^C (<i>R</i> = 0.021) at (A) <i>γ</i> = 0 and (B) <i>γ</i> = 0.4. <i>κ</i>_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 <i>κ</i>_b,A have large effects on <i>G</i>′ and <i>G</i>″ in (A) but not in (B) (C) Effects of thermal fluctuation (TF) of actin filaments on <i>G</i>′ at <i>f</i>_s = 10 Hz as a function of <i>γ</i> and <i>l</i>_p (calculated at 300 K). At high <i>γ</i> (≥0.4) or large <i>l</i>_p (∼20 µm), TF plays no significant role in <i>G</i>′. On the contrary, at low <i>γ</i> and <i>l</i>_p, <i>G</i>′ decreases without TF.</p