# Difference between revisions of "Microrheology of entangled F-actin solutions"

Entry by Helen Wu, AP225 Fall 2010

## Reference

M. L. Gardel, M. T. Valentine, J. C. Crocker, A. R. Bausch, D. A. Weitz, Physical Review Letters, 91, 158302 (2003).

## Keywords

microrheology, biopolymer, semiflexible networks

## Overview

The dynamics of networks of semiflexible polymers such as filamentous actin (F-actin) are affected by many characteristic length scales and frequency scales. The polymers become entanled at very low volume fractions and the resulting networks have large elastic moduli and long relaxation times compared to flexible polymers. Since the volume fraction is so low, individual fibers are sterically hindered. However, instead of a constant elastic modulus (over various frequencies), bulk rheological measurements show a monotonically increasing $G'(\omega)$ that approaches a plateau asymptotically. Thus, the authors determined the frequency and length scale dependencies in order to understand the system. They used one-particle (1P) and two-particle (2P) microrheology to accomplish this.

## Results and discussion

Figure 1. Comparison of 1P (filled symbols) and 2P (open symbols) MSDs.
Figure 3. Comparison between elastic modulus G' (closed symbols) and loss modulus G" (open symbols) from 1P (squares) and 2P (circles) data. Conventional rheometer data (triangles) is also represented. (a) 1.0mg/ml F-actin, (b) 0.3mg/ml F-actin.

Using the data obtained during experiments, the authors calculated the one-dimensional ensemble averaged mean-squared displacement (1P MSD) and scaled it by the particles' radii for the size-dependent viscous drag. Figure 1 shows that there was little change in this value over time for particles 0.32$\mu$m and greater, but at 0.23$\mu$m, the value increases. This was found to be due to the fact that 0.23$\mu$m particles were traveling through the network whereas the larger particles were trapped.

The 2P MSD gave information about dynamics at larger length scales than the radius. It represents the one-particle motion from long-wavelength modes. Assuing the material was incompressible, the scaling factor should be 2/radius.

The 1P and 2P MSDs are very different until about $\tau$=10s, where they converge (the right edge of Figure 1).

The generalized Stokes-Einstein relation was used to approximate the bulk elastic modulus G'($\omega$) and viscous modulus G"($\omega$). Figure 3 shows that these approximations were close to the measured bulk values. 2P microrheology measures a viscoelastic response and indicates that at low frequencies (<0.1rad/s), the elastic modulus dominates. However, at intermediate frequencies (<30rad/s), longitudinal fluctuations of the filaments affect the bulk response.

Looking at the 1P microrheology with the generalized Stokes-Einstein relation produced information on the origins of the viscoelasticity observed using 2P microrheology. Since 1P microrheology significanly underestimates viscoelasticity in the bulk material, particles are again permeating through the network. 1P viscoelasticity seems to be independent of both frequency and particle size. Thus, the authors suggest that the differences between 1P and 2P microrheology comes from coupling between particles.

Entanglements also affect the bulk viscoelasticity. They determine the plateau elasticity $G_0 ~ \rho k_b T/l_e$, which contains terms for the filament density $\rho$. 1P and 2P microrheology were shown to both effectively measure the low frequency plateau of the modulus due to entanglement, so local heterogeneities had little to no effect on the measurements.

1P microrheology may be applied to in vitro or in vivo measurements because crosslinking proteins reduce the importance of longitudinal fluctuations, which 2P microrheology can account for.

## Experimental Setup

Actin was polymerized in glass sample chambers and then imaged with CCD cameras.