Tube Width Fluctuations in F-Actin Solutions

We determine the statistics of the local tube width in F-actin solutions, beyond the usually reported mean value. Our experimental observations are explained by a segment fluid theory based on the binary collision approximation. In this systematic generalization of the standard mean-field approach, effective polymer segments interact via a potential representing the topological constraints. The analytically predicted universal tube width distribution with a stretched tail is in good agreement with the data.

Figure 1

(a) Superimposed confocal microscopy images of a fluorescent actin filament in a background solution and a spline representing the tube backbone; scale bar, $5\mu \mathrm{m}$; (b) rectified image; (c) tube radius profile $R(s)$ determined as standard deviation from Gaussian fits to the transverse intensity profile; (d) normalized tube radius distribution $P(R)$ obtained from cumulative contour lengths of 536, 804, 301, 225, and $116\mu \mathrm{m}$ for actin concentrations $c=0.2$, 0.4, 0.6, 0.8, $1.0\mathrm{mg}/\mathrm{ml}$, respectively. Solid lines represent a global fit by Eq. (3) with $\rho =5.95\mu {\mathrm{m}}^{-2}c[\mathrm{mg}/\mathrm{ml}]$ and $L=0.91L_e^{\infty }$, including corrections accounting for the line spread function [21].

Figure 2

(a) A pair of colliding filaments is assigned a “distance” $x$ and “topology” $\sigma =\pm$ as exemplified for $\sigma =+$ with $x>0$ (left) and $x<0$ (right). A reflection at the horizontal midplane amounts to $x\to -x$, $\sigma \to -\sigma$. (b) Overlap region (shaded) for two segments of length $L$ with orientations $\mathbf{u}$, ${\mathbf{u}}^{\prime }$ enclosing an arbitrary angle.

Figure 3

Dimensionless tube radius distribution $\tilde{P}(\tilde{R})$ ($\tilde{P}d\tilde{R}=PdR)$ with scaling variable $\tilde{R}\equiv L^{-3/8}{\overline{R}}^{-3/4}{l}_p^{1/8}R$ for reduced concentrations $\rho L\overline{R}=0.5$, 1, 2 (bottom to top): Holtzmark distribution, Eq. (2) (solid line); gamma distribution approximation, Eq. (3) (dashed line); numerical integration (dotted line). A bimodal structure develops if the theory is pushed towards the (unphysical) limit $\rho L\overline{R}\to 0$. Inset: Rescaled distribution as a function of $R/\overline{R}$ in semilogarithmic representation: experimental data for $c=0.2$ (circles), 0.4 (squares), 0.6 (diamonds), 0.8 (upright triangles), and $1.0\mathrm{mg}/\mathrm{ml}$ (downward-facing triangles), analytical approximation from Eq. (3) as in Fig. 1 (dashed line), and self-consistent theory described in the main text with a slightly renormalized value of the segment length $L=1.62L_e$ (solid line).

Figure 4

Left: Autocovariance function (ACF) of the tube radius profile averaged over 24, 37, 18, 13, and 9 filaments for $c=0.2$, 0.4, 0.6, 0.8, and $1.0\mathrm{mg}/\mathrm{ml}$, respectively (symbols as in Fig. 3, inset), with the abscissas rescaled by their initial slope ${\lambda }^{-1}$ (obtained by quadratic extrapolation for a varying fit interval $s=s_0,\dots ,s_{\max }$, $s_{\max }\to 0$ [21]). Because of the finite optical resolution of the microscope, data points below $s_0=230\mathrm{nm}$ were not considered. Right: $\lambda$ versus actin concentration $c$ with best power-law fit (exponent $-0.44\pm 0.09$).