Biopolymer Filament Entanglement Softens Then Hardens with Shear

It is unsatisfactory that regarding the problem of entangled macromolecules driven out of equilibrium, experimentally based understanding is usually inferred from the ensemble average of polydisperse samples. Here, confronting with single-molecule imaging this common but poorly understood situation, over a wide range of shear rate we use single-molecule fluorescence imaging to track alignment and stretching of entangled aqueous filamentous actin filaments in a homebuilt rheo-microscope. With increasing shear rate, tube “softening” is followed by “hardening.” Physically, this means that dynamical localization first weakens from molecular alignment, then strengthens from filament stretching, even for semiflexible biopolymers shorter than their persistence length.

Figure 1

In situ single-molecule imaging of entangled F-actin under shear. (a) “Tube” trajectories, each of a different single filament, after removing the mass center velocity, compared in the rest state, in the tube-softening regime (shear rate $0.1{\mathrm{s}}^{-1}$) and in the tube-hardening regime (shear rate $1{\mathrm{s}}^{-1}$). (b) As a function of logarithmic shear rate, this graph shows the mean filament length $⟨L⟩$ and the orientation angle $⟨|\theta |⟩$ with respect to shear, where $⟨⟩$ denotes the ensemble average of chains with different $L$. (c) The distribution of $L$ in the rest state, showing standard deviation σ, normalized to the peak value. (d) The distribution of $|\theta |$ at shear rate $1{\mathrm{s}}^{-1}$, normalized to the peak value.

Figure 2

Correlation between filament alignment and shear thinning. (a) Plotted against logarithmic shear rate, the orientation parameter of F-actin filaments shows strongest dependence for the subpopulation of shortest filaments (squares) and weakest dependence for the subpopulation of longest filaments (diamonds). The middle population (filled circles) and the mean of all populations (open circles) are indistinguishable and indicated by the dotted line. (b) Logarithmic viscosity, normalized to the value measured at $0.001{\mathrm{s}}^{-1}$, is plotted against mean orientation parameter $⟨P_2⟩$ and compared to experimental data for rigid rods [33] and flexible chains [32] as indicated.

Figure 3

Tube nonlinearities. (a) Distribution of stretch ($l/L$), normalized to the peak value, for entangled F-actin filaments at rest (black), at $0.001{\mathrm{s}}^{-1}$ (red), and at $1{\mathrm{s}}^{-1}$ (blue). (b) Stretch ($l/L$) plotted against logarithmic shear rate for the subpopulations of shortest, medium-length, and longest filaments (black, red, and blue, respectively). (c) Probability distribution of mean tube size of medium-length filaments in the rest state (gray), at $0.01{\mathrm{s}}^{-1}$ (red), and at $1{\mathrm{s}}^{-1}$ (blue). (d) Mean tube diameter of medium-length filaments plotted against the logarithmic product of shear rate and Rouse relaxation time, normalized to the mean tube diameter in the rest state. The two dotted lines compare to theoretical predictions [12, 14] for “tube softening” and “tube hardening.” The inset shows schematically the definition of tube diameter $d_T$.