Interplay between Brownian motion and cross-linking controls bundling dynamics in actin networks

Abstract

Morphology changes in cross-linked actin networks are important in cell motility, division, and cargo transport. Here, we study the transition from a weakly cross-linked network of actin filaments to a heavily cross-linked network of actin bundles through microscopic Brownian dynamics simulations. We show that this transition occurs in two stages: first, a composite bundle network of small and highly aligned bundles evolves from cross-linking of individual filaments and, second, small bundles coalesce into the clustered bundle state. We demonstrate that Brownian motion speeds up the first stage of this process at a faster rate than the second. We quantify the time to reach the composite bundle state and show that it strongly increases as the mesh size increases only when the concentration of cross-links is small and that it remains roughly constant if we decrease the relative ratio of cross-linkers as we increase the actin concentration. Finally, we examine the dependence of the bundling timescale on filament length, finding that shorter filaments bundle faster because they diffuse faster.

Figure 1

Bundling dynamics on small and large scales. Top: a small-scale bundling process with three filaments and snapshots taken at times t = 0, 2, 4, 6, 8, and 10 s. Bottom: snapshots of the bundling process taken (from left to right and top down) at t = 5, 10, 20, and 40 s for semiflexible fibers with stiffness $\kappa =0.07$ pN⋅μm² are shown. Fibers in the same bundle are colored with the same color. The two networks at the middle are before the coalescence transition time ${\tau }_c\approx 16$ s, whereas the two networks at the bottom are after the coalescence time. To see this figure in color, go online.

Figure 2

Statistics for the bundling process with filaments of varying stiffness. We compare the base parameters ($\kappa =0.07$ pN⋅μm², blue) with the systems with smaller bending stiffness ($\kappa =0.007$, orange) and rigid fibers ($\kappa \to \infty$, yellow). After ${\tau }_c\approx 16$ s, the bundling dynamics for the less stiff fibers are significantly faster. Fibers with similar bending stiffness to actin are well approximated by rigid fibers. Error bars are the error in the mean over five trials. To see this figure in color, go online.

Figure 3

Statistics for the bundling process with and without thermal fluctuations. The blue lines show the results without thermal movement, whereas the orange lines show the results with translational and rotational diffusion. Here, we use $\text{Δ}t={10}^{-4}$; we have verified that the statistical noise exceeds the time-stepping error for this time-step size. The peak in the bundle density occurs at ${\tau }_c\approx 16$ s for systems without diffusion, although for systems with diffusion it occurs at ${\tau }_c\approx 4$ s. Error bars are the error in the mean over five trials. To see this figure in color, go online.

Figure 4

Bundling timescales for a range of initial mesh sizes ${\ell }_m$ and binding rate $k_{\text{on}}$. The first three frames show the trajectory of the bundle density for the different mesh sizes, where blue lines denote our base value of $k_{\text{on}}=k_{\text{on}}^{\left(0\right)}=5$$(\mu$m⋅s$)$, orange lines denote $k_{\text{on}}/k_{\text{on}}^{\left(0\right)}=1/4$, and yellow lines denote $k_{\text{on}}/k_{\text{on}}^{\left(0\right)}=4$. Error bars are the error in the mean over five trials. The bottom right frame shows the dependence of the critical bundling time ${\tau }_c$ on ${\ell }_m$ and $k_{\text{on}}$. To see this figure in color, go online.

Figure 5

Snapshots of the network at $t={\tau }_c$ with initial mesh size ${\ell }_m=0.4$ μm and varying CL concentration. The networks contain $F=400$ filaments of length $L=1$ in a domain of size $L_d=4$ with $k_{\text{on}}/k_{\text{on}}^{\left(0\right)}=1/4$ (left, ${\tau }_c\approx 17$), $k_{\text{on}}/k_{\text{on}}^{\left(0\right)}=1$ (middle, ${\tau }_c\approx 5$), and $k_{\text{on}}/k_{\text{on}}^{\left(0\right)}=4$ (right, ${\tau }_c\approx 2.5$). A smaller $k_{\text{on}}$ (smaller CL concentration) gives fewer but larger bundles at $t={\tau }_c$, as well as a smaller percentage of fibers in bundles. To see this figure in color, go online.

Figure 6

Effect of changing filament length for rigid fibers with and without Brownian motion, with constant initial mesh size ${\ell }_m=0.2$ μm. (a) Without fiber diffusion, we show the statistics for filaments of length $L=0.5$ μm (orange) and $L=1$ μm (blue), where we observe dynamics occurring on a similar timescale, especially in the initial stage ($t\le {\tau }_c\approx 20$) of bundling. (b) When we add fiber diffusion, the bundling process for $L=0.5$ μm (purple) is significantly faster than $L=1$ μm (yellow), because filaments can diffuse faster. Error bars are the error in the mean over five trials. To see this figure in color, go online.

Figure 7

Steady-state morphologies for systems with turnover. We introduce filament turnover with mean filament lifetime ${\tau }_f$ (see (3) for implementation details) and observe the steady-state bundle density (left) and percentage of fibers in bundles (right) for ${\tau }_f/{\tau }_c=1/2$ (black), 1 (green), and 2 (red). Note that using a constant ${\tau }_f/{\tau }_c$ in the two systems ensures ${\tau }_f^{\left(\text{B}\right)}/{\tau }_f^{\left(\text{NB}\right)}={\tau }_c^{\left(\text{B}\right)}/{\tau }_c^{\left(\text{NB}\right)}$. Using both non-Brownian (lighter colors, ${\tau }_c\approx 16$ s) and Brownian (darker colors, ${\tau }_c\approx 4$ s) filaments, we show that the steady-state bundling statistics are roughly the same when ${\tau }_f/{\tau }_c$ is matched. Error bars are the error in the mean over five trials. To see this figure in color, go online.