Microstructure Dynamics and Anomalous Diffusion in Entangled F-actin Networks

Plotted on a log-log scale, the ensemble averaged MSD is linear, suggesting that it scales as a power law in time. As the mesh size is increased relative to a constant particle size, it becomes easier for the particle to diffuse through the network, so the slope of the power law increases.

When the particle size is much larger than the mesh size, particle is tightly confined by the local elasticity of the network and its thermal motion can then be interpreted using microrheology to provides a direct measure of the plateau modulus, although the full frequency dependence of the bulk modulus is not captured by the thermal motion of a single particle due to heterogeneity in the network.

In an actin solution with the ratio of particle size to mesh size is 1.47, the MSD is approximately constant between 0.01 - 10 seconds, as shown by the open squares in the previous figure. This corresponds to an
elastic modulus of 2 Pa, consistent with measurements of bulk rheology.

See video of this motion here.

However, then the particle size is much smaller than the mesh size, the particle can diffuse freely and the diffusion coefficient D can be used to probe the viscosity of
the background fluid of the network.

In an actin solution where the ratio of particle size to mesh size is 0.79, the MSD evolves nearly linearly in time
between 0.01-10 seconds as shown by the open triangles in the previous figure. This corresponds to a solvent viscosity of 1.5 cP, slightly larger than the viscosity of pure water, presumably reflecting the effects of hydrodynamic interactions.

See video of this motion here.

When the particle radius is comparable to the mesh size, dramatically different results are observed. The MSD exhibits anomalous subdiffusion, increasing as a power law for large times. The motion of earlier lag times is also subdiffusive, albeit with a slightly larger exponent. Interpreting this anomalous diffusion using microrheological analysis yields a viscoelastic response whose magnitude and frequency dependence are in sharp disagreement with bulk measurements. Thus, the MSD does not provide an accurate or robust measure of the viscoelasticity.

Roughly 20% of the particles undergo constrained motion punctuated by large scale jumps as illustrated by the spatial and temporal trajectories above. The time scale of these jumps is very short compared to the residence time within the cages. This suggests that the particles randomly and rapidly jump between different microenvironments wherein they are constrained.

See video of this motion here.

Within these local environments, the MSD of jumping particles exhibits the same time dependence as the non-jumping particles of the majority of the particles. These are reflected by the red line and open circles in the figure to the right. Thus, the viscoelastic properties of these constraining microenvironments are the same as those of the rest of the network. However, when the MSD is ensemble averaged over all particles with the jumps included the power law temporal dependence is recovered as shown by the solid squares. Thus, the anomalous diffusion results from the dynamics of the jumps between local microenvironments.

In order to quantify these dynamics, the probability distribution of cage times within the local microenvironments is determined. For large cage times, this probability distribution function scales as a power law. To model this motion, we assume that the particle undergoes a random walk, but, unlike normal diffusive motion, the time scale of each step is chosen from a power law distribution. When the steps are chosen symmetrically, this model predicts that the MSD should also be subdiffusive.

We calculate the MSD of 0.25 um particles over a large range of actin concentrations and find that we can smoothly change the diffusive exponent by simply changing the ratio of the particle size to the mesh size as shown by the solid squares. By using several different sized tracer particles varying from 0.23 - 1.0 um, we find the diffusive exponent depends only on the ratio of the bead size to the mesh size rather than the actual bead radius or actin concentration. Over a very narrow range, we observe anomalous diffusion with the diffusive exponent decreasing rapidly. This steep decrease of the diffusive exponent in this range shows that even small variations in tracer diameter or mesh size can have large effects on the bead mobility in a network. These results strikingly demonstrate that the actin network is inhomogeneous and that the subdiffusive motion is trap-like; thus, interpreting the particle MSD within the framework of microrheology must be done with extreme care for these particles. Instead, the particle motion can reflect the dynamics of the network on length scales comparable to the particle size.

There exists an intermediate regime where the motion of particles in F-actin networks is subdiffusive, with the MSD scaling as a power law for large times. This results from taking an ensemble average over particles that jump infrequently between local microenvironments within the network. Within these local microenvironments, the particles measure a plateau elasticity that is consistent with bulk measurements. However, the jumping motion is driven by other mechanisms, so that the ensemble averaged MSD cannot be interpreted using microrheology. This suggests that particle motion under such conditions must be interpreted carefully, especially in materials where the structural length scale or dynamics are not well known.