Fig 1.
(A) A branched network is created by self-assembly and interactions of F-actin (cyan), Arp2/3 complex (green), ACP (yellow), and motor (red), all of which are simplified via cylindrical segments. (B) An underlying substrate is simplified into a triangulated mesh with the chain length of 50 nm (gray). The endpoints of actin segments can transiently form an elastic link (yellow) to the substrate. (C) Description of different regions for dynamic events in the computational domain (5 µm × 2.5 µm). Actin polymerization occurs only in the assembly region (y = 2.125 - 2.5 µm), whereas actin depolymerization takes place only in the disassembly region (y = 0 - 0.375 µm). Motors are initially located near the -y boundary (y = 0 - 0.1875 µm). The links between the substrate and the network can form only in the focal adhesion region (y = 0.325 - 0.675 µm). We use F-actins located near the +y boundary for the calculation of retrograde flow speed. The periodic boundary condition is applied in the x direction, whereas repulsive and sticky boundary conditions are applied to the +y and -y boundaries, respectively.
Fig 2.
The dynamic steady state of the branched network under the reference condition.
(A) Snapshots of networks taken at 0 s, 150 s, and 300 s. The network was homogeneous and maintained continuity regardless of time points. (B) Snapshot showing links (green circles) formed in the focal adhesion region (y = 0.81 - 1.69 µm) and forces acting on the underlying substrate via color scaling (red: high force, white: low force) taken at 300 s. (C) Time evolution of the number of the links formed between the substrate and the network. (D) The total substrate force exerted by the network via the links as a function of time. (E) Time evolution of retrograde flow speed. (F) Kymograph of flow speed as a function of y position and time. Quantities shown in (C-F) change more at t < 100 s but exhibit smaller fluctuations around average values after 100 s. (G) Kymograph of actin density as a function of y position and time. Low actin density at y < 0.1 μm between 50 s and 100 s is indicative of the lack of overlap between F-actins and motors. Thus, it took more time for motors to bind and walk along F-actins to recover the retrograde flow, which explains large fluctuations shown in (C-F). Under this reference condition, key parameter values are k+,A = 12 µM-1s-1, RACP = 0.04, RArp2/3 = 0.01, k-,A = 6 s-1, k0,sev = 10-45 s-1, λsev = 1.0 deg-1, RM = 0.004, k20 = 17 s-1, and AFA = 0.35.
Fig 3.
Intermediate actin depolymerization rate led to a continuous flow with uniform network morphology.
(A, B) Snapshots of branched networks taken at ~150 s with a lower (0 s-1) or higher (10 s-1) value for actin depolymerization rate (k-,A) relative to that of the reference condition, 6 s-1. Network contraction without actin depolymerization led to loss of connection between the network and the leading edge, whereas too fast actin depolymerization resulted in loss of connection between the network and the motors. (C, D) Kymographs of actin density as a function of y position and time with different k-,A. (E) Heterogeneity of the network quantified as a coefficient of variation in actin density in the y direction. A lower value is indicative of a more homogeneous network. The network without actin depolymerization (k-,A = 0 s-1) showed very high heterogeneity which is attributed to the severe network accumulation shown in (A). (F) Retrograde flow speed for different values of k-,A. With intermediate values of k-,A, the network was more homogenous, and the flow speed was comparable with experimental observations.
Fig 4.
Intermediate F-actin severing activity is necessary to maintain a dynamic steady state.
(A, B) Snapshots of the branched networks taken at ~150 s with a lower (0 s-1) or higher (10-25 s-1) value of severing rate constant (k0,sev) relative to that of the reference condition, 10-45 s-1. In the absence of severing, a bundle structure was formed by the initial accumulation of F-actins. Due to the delayed disassembly of the bundle, motors remained trapped within the bundle until ~60 s and could not induce further network contraction. Once the bundle was finally disassembled by depolymerization, motors were unable to find F-actins to bind, leading to disconnection between the network and the motors. With fast severing activity, the filaments were immediately disassembled before motors could induce contraction, and the network eventually lost connection to the motors. (C, D) Kymographs of actin density as a function of y position and time with different k0,sev. (E) Heterogeneity of the network quantified as a coefficient of variation in actin density in the y direction. (F) Retrograde flow speed for different k0,sev. With intermediate values of k0,sev, network was more homogeneous, and the flow was faster with smaller fluctuations.
Fig 5.
An increase in motor density makes the retrograde flow faster by enhancing contractile forces.
(A, B) Snapshots of the branched network taken at ~150 s with a lower (0.001) or higher (0.04) value of motor density (RM) relative to that of the reference condition, 0.004. (C, D) Kymographs of actin density as a function of y position and time with different values of RM. The case with insufficient motors did not show a noticeable flow because it could not overcome frictional forces from FAs. As RM increased, more F-actins were accumulated as a bundle near the -y boundary. (E) Heterogeneity of the network quantified as a coefficient of variation in actin density in the y direction. The network was most homogeneous with intermediate values of RM. (F) Actin retrograde flow speed depending on RM. With higher RM, the flow tended to be faster. (G) Total force acting on the entire substrate via elastic links with different RM. The total force showed a clear plateau at high RM.
Fig 6.
Focal adhesions (FAs) hinder the actin retrograde flow by exerting frictional forces.
(A, B) Snapshots of the branched network taken at ~150 s with smaller (0) or larger (0.5) size of the FA region (AFA) relative to that of the reference condition, 0.35. (C, D) Kymographs of actin concentration depending on time and y position with AFA. Without the FA region, the accumulation of F-actins into a bundle was noticeable, whereas large FA region led to discontinuity between the network and the motors. (E) Heterogeneity of the network quantified as a coefficient of variation in actin density in the y direction. (F) Actin retrograde flow speed depending on AFA. The flow speed tended to be smaller with larger FA region due to higher frictional forces. (G) Total force acting on the substrate by the network with different AFA. At AFA ≤ 0.35, the total substrate force was proportional to AFA due to the formation of more links, but the case with AFA = 0.5 showed a lower total force because of the discontinuity between the network and the motors occurring at later times.
Fig 7.
Different modes of dynamic steady states exist.
(A) Two ways to rescue the case showing a negligible flow with RM = 0.0015, AFA = 0.175, and k-,A = 6 s-1. Since contractile forces were insufficient to overcome the frictional forces in this case, it was able to be rescued by either increasing RM to 0.004 or reducing AFA to 0. (B) Rescuing a case exhibiting a minimal flow with RM = 0.004, AFA = 0.15, and k-,A = 10 s-1. Since actin depolymerization was too fast compared to flow speed, this case was able to be rescued by enhancing flow speed, either by increasing RM to 0.01 or decreasing AFA to 0.05. (C, D) Kymographs of actin density as a function of y position and time for cases shown in (A) and (B), respectively.