Confining Potential when a Biopolymer Filament Reptates

Using single-molecule fluorescence imaging, we track Brownian motion perpendicular to the contour of tightly entangled F-actin filaments and extract the confining potential. The chain localization presents a small-displacement Hookean regime followed by a large amplitude regime where the effective restoring force is independent of displacement. The implied heterogeneity characterized by a distribution of tube width is modeled.

Figure 1

The main idea of this experiment. (a) Schematic representation of a chain constrained into a tubelike region by an effective harmonic potential, the collective topological effect of surrounding chains. Minima of this potential define the primitive path. As discussed in the text, the radius of the tube is irregular and polydisperse. (b) A representative image showing two labeled segments of a long actin filament. Analyzing the raw data using a two-dimensional Gaussian spreading function allows one to determine the actual position of segments with subpixel resolution ($\sim 10\mathrm{nm}$ in this set of experiments), indicated by the crosses. (c) An overview of the variety of curvilinearlike diffusive motions of labeled actin segments in entangled F-actin networks. Colors denote time lapse of the trajectory. The F-actin concentration is $0.7\mathrm{mg}\mathrm{\text{−}}{\mathrm{mL}}^{-1}$. Temporal step is 0.5 sec. The dimension of the view is $50\mu \mathrm{m}\times 50\mu \mathrm{m}$.

Figure 2

Histograms of RMS transverse fluctuations (distribution of “second moments”) of single time traces along filaments at actin concentrations of 1 (a) and $0.2\mathrm{mg}{\mathrm{ml}}^{-1}$ (b). The solid lines compare to the expected distribution predicted by the ensemble-averaged model presented in the text, assuming that each filament feels a homogenous environment. The discrepancy between these two distributions demonstrates that the heterogeneity quantified here occurs on length scales smaller than the filament length.

Figure 3

Illustrative transverse fluctuation distributions. (a) Transverse distance probability distributions, plotted against distance normal to the primitive path, for an ensemble of hundreds of filaments (circles) and for the trajectory of a single filament (crosses). Actin concentration is $0.5\mathrm{mg}\mathrm{\text{−}}{\mathrm{mL}}^{-1}$. The solid line is a fit using Eq. (1) with $\alpha =1$. The dashed line is the corresponding Gaussian fit to the center part of the distribution. (b) Effective restoring forces, inferred from a Boltzmann distribution law as explained in the text, plotted against transverse distance. The actin concentrations are 1 (crosses), 0.5 (triangles), and $0.2\mathrm{mg}{\mathrm{ml}}^{-1}$ (circles).

Figure 4

The tube radius distribution. (a) The distribution of tube radius, ${\overline{R}}_e{P}_{R_e}({\tilde{R}}_e)=\frac{5\pi }{12}{\tilde{R}}_e^{2/3}\exp (-\frac{\pi }{4}{\tilde{R}}_e^{5/3})$, is plotted against the dimensionless ${\tilde{R}}_e$ defined in the text, on linear scales. The shadowed region shows the needed lower cutoff determined by fitting the data. (b) The distribution of tube radius plotted on semilogarithmic scales rescaled by mean tube radius, taking lower cutoff into consideration so that the average tube radius $⟨R_e⟩=1.6{\overline{R}}_e$. (c) Logarithmic plot of ${\overline{R}}_e$ versus actin concentration. The error bars show the standard deviation from many experiments. The solid line has slope $-3/5$.