| Microrheology of entangled F-actin solutions |
| Introduction |
The cytoskeleton is composed of dense actin filament networks that regulate important cellular processes such as cell shape, motility and division. The mechanical properties of these networks control proper biological function, but these mechanical properties are difficult to measure in vivo since the behavior of the cytoskeletal microstructure is not well understood. This has motivated an extensive effort to measure the mechanical properties and microstructure of reconstituted F-actin networks in vitro. F-actin is a semiflexible polymer which is characterized by a persistence length of 10-20 um, roughly three orders of magnitude larger than the filament diameter of 7 nm. This large aspect ratio allows semi-dilute solutions of actin to form entangled networks with an average mesh size of order 100 nm to 1 um. Thus, solutions of entangled F-actin are an ideal model system to study the unique and unresolved dynamics and mechanical properties of semiflexible polymer networks and to investigate the implications of these mechanisms for the behavior of the cytoskeleton. |
| Introduction | |
F-actin is a
semi-flexible polymer found in the cytoskeleton with a diameter of 5
nm and a persistence length of ~15 um. The persistence length,
lp, is a measure of polymer rigidity and is the length scale
over which the polymer loses orientational order. A
flexible polymer has a persistence length of order a few
monomers (~ nm) and a rigid rod has an infinite persistence
length. Semiflexible polymers have intermediate persistence
lengths and this new length scale is important in the thermally driven
polymer dynamics. In vivo, actin filament contour lengths, l,
are typically of order 1 micron and individual filaments are fairly
rigid. However, in vitro, filament lengths can be > 10 um and
actin filaments are entangled polymer solutions at semi-dilute actin
concentrations, cA.
These entangled semi-flexible polymer solutions exhibit a rich
frequency-dependent mechanical response that depends on cA,
l, and lp.
Thus, the characteristic lengths scales of F-actin solutions range
from 0.2 - 17 microns. |
cartoon of actin filament with relevent dimensions
T.E.M. of entangled F-actin |
| One- and Two-Particle Microrheology | |
We utilize multiple particle tracking to examine the dynamics of probe
particles. A typical movie is shown here. Microrheology
utilizes the thermally driven motion of micron-sized tracer particles
to make precise, local measurements of viscosity and mechanical properties
at the length scale of the tracer particle. The particle trajectories
can be captured using optical techniques and analyzed using video
tracking software in IDL. Approximately one hundred particles
are imaged over a period of half an hour, yielding good statistics
for calculating an ensemble averaged mean squared displacements (MSD).
Even though this is an ensemble average, we often refer to the MSD as
a one-particle (1P) measurement because it reflects the local
properties of each tracer. Alternatively, we can also examine
the correlated motion between pairs of particles to measure the
distinct or
two-particle (2P) MSD. This correlated motion is a direct
measure of the correlated motion at long wavelengths. |
 |
| 1P and 2P MSD show different magnitude and time dependence | |
The 1P and 2P MSD for
several differently sized particles in 0.9 mg/mL F-actin (mesh size of
0.32 mm) are shown, normalized by
the bead radius. The smallest particle sizes with a radius, a,
of 0.23 mm, are observed to
percolate through the network. Their motion is described here.
In contrast, the 0.32 and 0.42 mm
beads are constrained at times larger than 0.3 sec.
Differences observed in the 1P motion are eliminated with a 2P
analysis. Furthermore, in contrast to the 1P MSD, the 2P MSD is
weakly time dependent, growing as a weak power law in time, ta,
where a ~ 0.3-0.4. At long
times, the magnitude of the constrained 1P and 2P MSDs converge.
|
 |
| 2P Microrheology Measure Bulk Properties | |
We interpret the 1P and 2P MSD using the generalized Stokes-Einstein
relation as a frequency dependent elastic modulus, G', and
loss modulus, G''. As seen in the figure to the right,
the 2P measurement (red) is in much better agreement with both the
frequency dependence and magnitude of the bulk measurements (black).
The data to the right is measured with 1 mm
carboxylate-modified particles. Both of these measurements
show a weak frequency dependence of G' and G'' over
the measured frequency range and G' and G'' are
similar in magnitude. At the loweet frequencies, G'
begins to dominate the response and becomes independent of
frequency. In contrast, the 1P rheology is sensitive to the
size of the particle and does not reflect either the frequency
dependence or magnitude of the viscoelastic response.
Surprisingly, we find that for the constrained particles (i.e. at
higher concentratations as in (a)), the magnitude of G'
measured with 1P matches that for 2P at the lowest frequencies. |
 |
| What does 1P measure? | |
 |
We examine the value of G' measured with 1P for constrained particles over a large range of actin concentrations and particle radii ranging from 0.23- 1 mm. We find that the magnitude of G' measured at low frequencies with 1P is independent of bead size. Furthermore, we find that the magnitude of G' measured with both 1P and 2P is similar in magnitude and that G' ~ cAb, where b~1.8. This scaling is similar to that measured with bulk rheology. Thus, we conclude that 1P does measure the correct low frequency elastic modulus, even though the frequency dependence and magnitude at higher frequencies is in stark disagreement with either 2P or bulk measurements. |
| What does this all mean? |
We are also trying to figure
this out and you can read more about our current efforts
here.
However, there are a few essential results of this study.
1) Interpretations of
multiple particle tracking experiments must be done with great care.
The local viscoelasticity measured with this technique can be in stark
disagreement with bulk measurements. This is especially important
in biological materials that are structured on micron length scales.
2) It appears that the
discrepancy between 1P and 2P is frequency dependent. The elastic
modulus measured from 1P at low frequencies is in good agreement with
those from a 2P/bulk measurement. 3) Furthermore, the fact that 2P is an accurate
measure of bulk rheology indicates there is a critical length scale
between 1- 5 mm that determines the
onset of continuum elastic response. Is the 1P measurement
an accurate measure of elasticity at 1 mm
length scales? We are currently trying to figure out how we can
exploit microrheology techniques to examine the microscopic origins of
the bulk viscoelastic response of biomaterials. |
| References |
"Anomalous Diffusion Probes Microstructure Dynamics of Entangled F-actin
Networks" I.Y. Wong, M.L. Gardel, D.R. Reichman, E.R. Weeks, M.T. Valentine, A.R. Bausch and D.A. Weitz, Phys. Rev. Lett. 92 178101 (2004).
"Microrheology of entangled F-actin solutions" M.L. Gardel, M.T. Valentine, J.C. Crocker, A.R. Bausch, and D. A. Weitz, Phys. Rev. Lett. 91 158302 (2003).
"Microrheology" M.L. Gardel, M.T. Valentine, and D. A. Weitz, In: Microscale Diagnostic Techniques K. Breuer (Ed.) Springer Verlag 2002. (in press).
"Microrheology of polyethylene oxide using diffusing wave spectroscopy
and single scattering" Bivash R. Dasgupta, Shang-You Tee, John C. Crocker, B.J. Frisken, and D.A. Weitz, PRE 65 051505 (2002). |
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