Branching influences force-velocity curves and length fluctuations in actin networks

We investigate collective dynamics of branched actin networks growing against a rigid movable wall constrained by a resistive force. Computing the force velocity relations, we show that the stall force of such networks depends not only on the average number of filaments touching the wall, but also on the amount of fluctuation of the leading edge of the network. These differences arise due to differences in the network architecture, namely, distance between two adjacent branching points and the initial distance of the starting filament from the wall, with their relative magnitudes influencing the nature of the force velocity curves (convex versus concave). We also show that the introduction of branching results in nonmonotonic diffusion constant, a quantity that measures the growth in length fluctuation of the leading edge of the network, as a function of externally applied force. Together our results demonstrate how the collective dynamics of a branched network differs from that of a parallel filament network.

Figure 1

(a) Schematic of an actin network with branching. We start from a single filament (n1) positioned at an initial distance of $D_{\mathrm{wall}}$ from a rigid movable wall constrained by a resistive force $f$. The back side of the filament is attached to a fixed wall. The network is allowed to grow via force-dependent polymerization, depolymerization, and branching. Branching occurs, stochastically, at an angle of ${70}^{\circ }$ from the parent filament when it grows to a length of $D_{\mathrm{br}}$ from the previous branching point. n1, n2, n3, and n4 represent the four different filaments in the network. $x_{\mathrm{old}}$ and $x_{\mathrm{new}}$ represent the wall positions before polymerization (old) and after polymerization (new), respectively. (b) While polymerization of a horizontal filament displaces the wall by a distance of 1 subunit $\left(\delta \right)$, an angled filament will displace the wall by a distance of $\delta cos\theta$.

Figure 2

(a) Force-velocity relationship of actin networks for different values of branching distance $(7\delta \le D_{\mathrm{br}}\le 100\delta$; from right to left), fixing $D_{\mathrm{wall}}=10\delta$. Curve $N_{\mathrm{fl}}=1$ (the left most) is the force-velocity relation for a single filament without any branching. Increasing $D_{\mathrm{br}}$ is seen to change the nature of the force-velocity curve. Stall forces for $D_{\mathrm{br}}=12\delta$ and $D_{\mathrm{br}}=15\delta$ etc. differ due to dependence of stall forces on wall fluctuation (see text). (b) Fluctuation of the wall (leading edge) for two different forces. The horizontal line (blue, solid) corresponds to $D_{\mathrm{br}}=15\delta$. When $f=1.97f_0 >f_1^s$(stall force of one filament), the wall fluctuates wildly and crosses the branching distance of $15\delta$ very often leading to branching and growth (red curve, solid line). However, when $f=3.29f_0$ (green curve, solid line with open circle), the fluctuation never crossed the horizontal line (blue, solid line), and hence no branching and growth was observed at $f=3.29f_0$. (c) Force-velocity relationship of actin networks when the root filament is oriented at an angle ${20}^{\circ }$ clockwise with the horizontal axis for different $D_{\mathrm{br}}$ values. Here we have $D_{\mathrm{wall}}=10\delta$. (d) Force-velocity profile of actin networks starting with two horizontal root filaments for varying branching distances $\left(7\delta \le D_{\mathrm{br}}\le 100\delta \right)$, keeping $D_{\mathrm{wall}}=10\delta$. In all the force-velocity curves here, the standard error is smaller than the size of the data points shown.

Figure 3

Number of filament tips within one monomer distance away from the wall $\left(N_{\mathrm{t}}\right)$ at the end of the simulation $\left(t=1800{k_{\mathrm{off}}}^{-1}\right)$, as a function of the external force applied, for different branching distances $\left(7\delta \le D_{\mathrm{br}}\le 100\delta \right)$. $D_{\mathrm{wall}}$ is fixed at $10\delta$. The standard error here is smaller than the size of the data points shown.

Figure 4

(a) Diffusion coefficient of the leading edge of network $\left(D\right)$ as a function of force for different branching distances $D_{\mathrm{br}}(7\delta \le D_{\mathrm{br}}\le 100\delta )$ exhibits nonmonotonic behavior. The standard error here is smaller than the size of the data points shown. (b) Three possible configurations of the branched actin network with $N_{\mathrm{t}}=3$. The force generated for each case will be different (as discussed in the text) and contributes to the nonmonotonic dependence of $D$ on $f$. Arrows show the direction of growth of filaments.

Figure 5

Two-state theory to understand the diffusion coefficient: (a) example configurations of state $s$, where the wall dynamics is dominated by “straight” (horizontal) filaments, and state $a$ where the wall dynamics is dominated by filaments at an angle from the horizontal. (b) $D\left(f\right)$ showing a single peak when Eq. (5) is plotted against force assuming $v_s=6.44exp(-f/f_0),v_a=6.44exp(-fcos70/f_0),f_{sa}=exp(f/1000)$, and $f_{sa}=1.0$. (c) $D\left(f\right)$ showing two peaks when the same parameters as in (b) were taken except that the velocity of the angled state is changed to have a different concave-like curvature near $f=0$; e.g., here we took, for simplicity, $v_a=6.44exp(-0.4f^2)$. Other functions with concave-like curvature gave similar results.

Figure 6

(a) Force-velocity relationship of actin networks for varying maximum number of filaments $N_{\mathrm{fl}} \left(10\le N_{\mathrm{fl}}\le 600\right)$. Both $D_{\mathrm{br}}$ and $D_{\mathrm{wall}}$ were kept fixed at $10\delta$. Increasing $N_{\mathrm{fl}}$ is seen to change the nature of the force-velocity curve from convex $\left(N_{\mathrm{fl}}\le 40\right)$ to concave $\left(N_{\mathrm{fl}}>100\right)$. (b) Number of filament tips within one monomer distance away from the wall $\left(N_{\mathrm{t}}\right)$ at the end of the simulation $\left(t=1800{k_{\mathrm{off}}}^{-1}\right)$, as a function of the external force applied, for different values of $N_{\mathrm{fl}}$. (c) Diffusion coefficient of network edge $\left(D\right)$ vs force for different values of $N_{\mathrm{fl}}$. (d) Comparison of force-velocity relationship of a bundle of parallel filaments and branched actin network having number of filaments, $N_{\mathrm{fl}}=200$. Here we have four different conditions: (1) $Br1$ is the force-velocity curve for branched network with $D_{\mathrm{br}}=10\delta$. Here we take $D_{\mathrm{wall}}=10\delta$ and $N_{\mathrm{fl}}=200$. (2) $Br2$ is the force-velocity curve for branched network with $D_{\mathrm{br}}=3\delta ,D_{\mathrm{wall}}=10\delta$, and $N_{\mathrm{fl}}=200$. (3) $St1$ is the force-velocity curve for 200 straight parallel filaments with $D_{\mathrm{wall}}=0,D_{\mathrm{br}}=0$, and $N_{\mathrm{fl}}=200$. (4) $St2$ is the force-velocity curve for 200 straight parallel filaments with $D_{\mathrm{wall}}=10\delta ,D_{\mathrm{br}}=0$, and $N_{\mathrm{fl}}=200$. (e) Force-velocity relationship of actin networks when the growth starts with two root horizontal filaments(perpendicular to the wall) for varying maximum number of filaments $N_{\mathrm{fl}} \left(10\le N_{\mathrm{fl}}\le 600\right)$. Both $D_{\mathrm{br}}$ and $D_{\mathrm{wall}}$ were kept fixed at $10\delta$. The standard error here is smaller than the size of the data points shown.

Figure 7

Summary of results obtained in this study.

Figure 8

(a) Force-velocity relation for a single (red [line with filled circle], left) and two filaments [brown (line with filled square), right], perpendicular to the movable wall. (b) Force-velocity relationship of a two-filament system, when one of them is growing perpendicular to the movable wall, and the second one is growing at an angle of ${70}^{\circ }$ with the perpendicular filament. Schematic of Fig. 8 would be similar to Fig. 1. Note that the stall force for this case $\approx 7.31f_0$.