3084
Langmuir 2005, 21, 3084-3089
Probe Surface Chemistry Dependence and Local Polymer Network Structure in F-Actin Microrheology Byeong Seok Chae and Eric M. Furst* Department of Chemical Engineering, Colburn Laboratory, University of Delaware, Newark, Delaware 19716 Received July 28, 2004 We investigate the dependence of F-actin microrheology on probe surface chemistry using diffusing wave spectroscopy. Polystyrene probe particles exhibit subdiffusive mean-squared displacements, where 〈∆r2(t)〉 ∼ t0.77(0.03 consistent with previous experiments and theory. However, polystyrene probes preadsorbed with bovine serum albumin (BSA) interact weakly with the surrounding polymer network and exhibit a scaling exponent similar to pure diffusion 〈∆r2(t)〉 ∼ t, which decreases as particle size and actin concentration increases. Using models of particle diffusion in locally heterogeneous viscoelastic microenvironments, we find that the microrheological response of BSA-treated particles is consistent with the formation of a polymer-depleted shell surrounding the probes. The shell thickness scales with particle size but not polymer concentration. These results suggest that the depletion is caused by exclusion or orientation of actin filaments near probes due to their long length and rigidity.
Introduction Microrheology measures the response of forced or thermally driven colloidal probe particles embedded in a polymer network. In the latter case, the viscoelastic modulus of the network is related to the particle displacement through a generalized Stokes-Einstein relationship (GSER),
〈∆r˜ 2(s)〉 )
kBT 6πasG ˜ (s)
(1)
where a is the particle radius, kBT is the thermal energy, and 〈∆r˜ 2(s)〉 and G ˜ (s) are the Laplace transforms of the mean-squared displacement (MSD) and viscoelastic modulus, respectively.1 Because of its advantages, such as small sample sizes, minimal mechanical perturbation, and a temporal range that spans many decades (10-6 to >100 s), microrheology has emerged as an important method for studying polymer network structure and response. In particular, it has been widely used to characterize the mechanics and rheology of individual, living cells2,3 and the properties of reconstituted cytoskeletal networks.4,5,6 However, despite many instances where microrheology has been shown to agree with bulk rheological measurements,1,7,8 there are notable circumstances in which it provides anomalous results.9,10 For instance, it was recently demonstrated that the measured low-frequency * Corresponding author. Electronic address: furst@ che.Udel.edu. Tel: +1 (302) 831-0102, Fax: +1 (302) 831-1048. (1) Mason, T. G.; Weitz, D. A. Phys. Rev. Lett. 1995, 74, 1250-1253. (2) Valberg, P.; Albertini, D. J. Cell Biol. 1985, 101, 130-140. (3) Fabry, B.; Maksym, G.; Butler, J.; Glogauer, M.; Navajas, D.; Fredberg, J. Phys. Rev. Lett. 2001, 87, 148102. (4) Gittes, F.; Schnurr, B.; Olmsted, P.; MacKintosh, F.; Schmidt, C. Phys. Rev. Lett. 1997, 79, 3286-3289. (5) Gisler, T.; Weitz, D. A. Phys. Rev. Lett. 1999, 82, 1606-1609. (6) Le Goff, L.; Hallatschek, O.; Frey, E.; Amblard, F. Phys. Rev. Lett. 2002, 89, 258101. (7) Mason, T. G.; Ganesan, K.; van Zanten, J. H.; Wirtz, D.; Kuo, S. C. Phys. Rev. Lett. 1997, 79, 3282-3285. (8) Mason, T. G.; Gang, H.; Weitz, D. A. J. Opt. Soc. Am. 1997, 14, 139-149. (9) Lu, Q.; Solomon, M. Phys. Rev. E 2002, 66, 061504. (10) McGrath, J. L.; Hartwig, J. H.; Kuo, S. C. Biophys. J. 2000, 79, 3258-3266.
plateau modulus of entangled filamentous actin (F-actin), a helical protein filament and principal component of the cytoskeleton, depends strongly on the surface chemistry of the probe particles.10 This apparent modulus varies by an order of magnitude depending on the affinity between the particles and actin filaments, and may limit the application of microrheology both in vivo and in vitro. However, the mechanism by which surface chemistry of the probe particle affects the microrheological response remains to be understood. Here, we report high-frequency microrheology experiments to further understand the dependence of F-actin microrheology on probe particle surface chemistry. F-actin is of interest not only for its central role in cellular organization, intracellular transport, and signaling pathways11 but also because it is a model semiflexible polymer, with an average diameter of 7 nm, persistence length Lp ≈ 15 µm and contour length, L, of several to tens of micrometers.12,13,6 A sequence of image stills shows the thermal fluctuations of a single actin filament in Figure 1 under quiescent conditions. Note that, because Lp ≈ L, the filament maintains an extended conformation with small, thermally driven bends. This places F-actin in an intermediate region between coil-like (L . Lp) and rodlike (L , Lp) behavior.14 F-actin has been the subject of a large number of experimental and theoretical studies which have played a critical role in developing the understanding of the structure and rheology of semiflexible polymer networks for both cell biology and technological applications.4-6,15-20 (11) Bray, D. Cell Movements; Garland: New York, 2001. (12) Ott, A.; Magnasco, M.; Simon, A.; Libchaber, A. Phys. Rev. E 1993, 48, R1642-1645. (13) Isambert, H.; Venier, P.; Maggs, A.; Fattoum, A.; Kassab, R.; Pantoloni, D.; Carlier, M. J. Biol. Chem. 1995, 270, 11437-11444. (14) Morse, D. Macromolelcules 1998, 31, 7030-7043. (15) Ka¨s, J.; Strey, H.; Sackmann, E. Nature 1994, 368, 226-9. (16) Wilhelm, J.; Frey, E. Phys. Rev. Lett. 1996, 77, 2581-2584. (17) Amblard, F.; Maggs, A. C.; Yurke, B.; Pargellis, A. N.; Leibler, S. Phys. Rev. Lett. 1996, 77, 4470-4473. (18) Gittes, F.; MacKintosh, F. Phys. Rev. E 1998, 58, R1241-1244. (19) Palmer, A.; Mason, T. G.; Xu, J.; Kuo, S. C.; Wirtz, D. Biophys. J. 1999, 76, 1063-1071. (20) Everaers, R.; Ju¨licher, F.; Ajdari, A.; Maggs, A. C. Phys. Rev. Lett. 1999, 82, 3717-3720.
10.1021/la0480890 CCC: $30.25 © 2005 American Chemical Society Published on Web 03/01/2005
F-Actin Microrheology
Figure 1. A sequence of image stills showing the thermal fluctuations of a single actin filament fluorescently labeled with rhodamine-phalloidin under quiescent conditions. Actin is a semiflexible polymer with persistence length Lp ≈ 15 µm. The scalebar is 10 µm. Filaments are suspended in a buffer solution (25 mM imidazole, 2 mM MgCl2, 0.5 mM ATP, 50 mM KCl, 0.5 mM DTT, 0.01% NaN3) and are visualized using a cooled (-20 °C) intensified CCD camera (Lhesa). The sample cell is a 1 µm thick BSA-coated chamber, which confines filaments in the focal plane of the microscope.
The experiments discussed in this paper provide an opportunity to investigate interactions and fluid structure in colloid-polymer solutions where the polymer end-toend length is greater than the particle size. While interactions in colloid-polymer solutions have long been recognized as a critical component toward understanding and controlling the phase behavior, rheology, stability, and vitrification of these systems, the interactions, structure, and dynamics investigated here are expected to be significantly more complex than those that give rise to depletion attractions between colloids in the presence of dilute polymer additives.21 The long length of polymer relative to the particle size is more closely related to the “protein limit” of colloid-polymer solutions, which has been of recent practical and theoretical interest.22,23,24 A general understanding of these phenomena could have broad implications for protein crystallization and filled polymer systems of industrial importance, including nanocomposites and high-performance coatings. We begin by first discussing the experimental aspects of the present work followed by a presentation of our results that demonstrate distinct microrheological responses depending on the surface chemistry of the probe particles. Polystyrene (PS) particles, as well as identical beads coated with bovine serum albumin (BSA), serve as probes with two distinct surface chemistries that have been shown to give low-frequency modulus amplitudes that differ by a factor of ∼4.10 This difference correlated positively with the degree of binding between F-actin and the particles, with BSA reducing particle-actin adhesion over untreated probes. To elucidate the underlying mechanisms of this behavior, we evaluate the response in terms of recently proposed models of the motion of particles in a heterogeneous microenvironment. Experimental Section Frozen monomeric actin (G-actin) from rabbit skeletal muscle (Cytoskeleton) is stored at -80 °C. Prior to experiments, we thaw the actin rapidly and dilute in G-buffer (5 mM Tris-HCl pH 8.0, (21) Gast, A.; Hall, C.; Russel, W. J. Colloid Interface Sci. 1983, 96, 251-267. (22) Kulkarni, A.; Zukoski, C. J. Crystal Growth 2001, 232, 156164. (23) Fuchs, M.; Schweizer, K. S. J. Phys.: Condens. Matter 2002, 14, R239-R269. (24) Chen, Y.-L.; Schweizer, K. S. Langmuir 2002, 18, 7354-7363.
Langmuir, Vol. 21, No. 7, 2005 3085 0.2 mM ATP, 0.2 mM CaCl2, 0.5 mM dithiothreitol (DTT)). The G-actin is dialyzed overnight against at least 100 volumes of fresh G-buffer at 4 °C. Sample-to-sample variation of the actin microrheological response is negligible. Monodisperse polymer latex particles in water (0.109, 0.356, 1.053, and 3.156 µm polystyrene (PS) microspheres, Polysciences, Inc.) are used as microrheological probes. The particles are synthesized via a surfactant-free polymerization and are chargestabilized by sulfonate groups. Identical beads coated with bovine serum albumin (PS-BSA) are prepared by first incubating the particles in a 200 mg/mL BSA solution for 24 h. Just prior to their use, all particles are washed in G-buffer with 3-5 successive centrifugation and redispersion steps to remove residual impurities and unadsorbed BSA. The beads are dispersed in G-actin (volume fraction φ ) 0.01) and polymerized into F-actin by raising the KCl concentration of each sample to 50 mM and MgCl2 to 1 mM using a 10× solution in G-buffer. Samples are polymerized in plastic cuvettes (path length l ) 0.4 cm) and are carefully sealed to avoid contamination. The final F-actin concentration is 0.63 mg/mL. For 0.356, 1.053, and 3.156 µm PS-BSA, the final concentrations are varied from 0.32, 0.63, and 0.8 mg/mL, with a corresponding network mesh size of ξ ≈ 530, 380, and 335 nm, determined by the relation ξ ) 0.3/xcA µm, where cA is the actin concentration in mg/mL.25 The ensemble-average mean-squared displacement 〈∆r2(t)〉 of particles embedded in an F-actin sample is measured by diffusing wave spectroscopy (DWS) using a plane-wave source from an Ar+ laser (λ ) 514 nm).26 A single-mode fiber optic coupled to an aspheric lens collects transmitted light after passing through a cross-polarizer to ensure that it is depolarized and highly multiply scattered. The fiber optic is then split to two PMT detectors operating in a pseudo-cross correlation mode to reduce spurious noise and afterpulsing. The time-averaged intensity autocorrelation 〈I(0)I(t)〉/〈I〉2 is calculated using a commercial high-speed correlator (Brookhaven Instruments, BI-9000) and is converted to the field autocorrelation function g(1)(t) via the Siegert relation 〈I(0)I(t)〉/〈I〉2 ) 1 + β|g1(t)2|, where the intercept value β accounts for optical properties, including the laser coherence and numerical aperture of the optical fiber. The mean-squared displacement is found by numerically inverting g(1)(t) for the transmission geometry26 g(1)(t) )
(l*l +34)xk 〈∆r (t)〉 4 4 l l 1 + k 〈∆r (t)〉 sinh[( )xk 〈∆r (t)〉] + k 〈∆r (t)〉 cosh[( )xk 〈∆r (t)〉] 9 l* 3x l* 2 0
2 0
2
2 0
2
2
2 0
2
2 0
2
(2)
where k0 ) 2πn/λ is the incident wavevector for a laser with vacuum wavelength λ in a medium of refractive index n. We measure the photon mean-free path, l*, for each sample by comparing the average transmitted light intensity against a sample with a known value of l*. Typical values are 0.6, 0.2, and 0.15 µm for 3.156, 1.053, and 0.356 µm diameter particles, respectively. To ensure that particles have not aggregated during sample preparation, we compare l* with a sample of identical particles dispersed in water at the same volume fraction. PSBSA particles are also observed under a microscope to verify that they do not aggregate.
Results The microrheology results are summarized in Figure 2, which shows the mean-squared displacement scaled with particle radius 〈∆r2(t)〉a versus time for the actin concentration 0.63 mg/mL. When plotted in this manner, the curves for PS particles of 0.356, 1.053, and 3.156 µm diameter collapse onto a single master curve, as expected for generalized Stokes-Einstein behavior. In addition, 〈∆r2(t)〉 is subdiffusive, scaling as t0.77(0.03. This time (25) Schmidt, C. F.; Ba¨rmann, M.; Isenberg, G.; Sackmann, E. Macromolecules 1989, 22, 3638-3649. (26) Pine, D. J.; Weitz, D. A.; Zhu, J. X.; Herbolzheimer, E. J. Phys. (France) 1990, 51, 2101.
3086
Langmuir, Vol. 21, No. 7, 2005
Figure 2. Mean squared displacements normalized with particle size for (a) PS and (b) PS-BSA in F-actin (0.63 mg/mL) for particle sizes 2a ) 0.109 (squares), 0.356 (circle), 1.053 (up triangle), and 3.156 (down triangle).
dependence is in very good agreement with the expected scaling 〈∆r2(t)〉 ∼ t3/4, which reflects the Rouse-like dynamics of semiflexible polymers.4,5,14,17,27 However, for probe particles smaller than ξ (2a )100 nm), we find that 〈∆r2(t)〉 exhibits the same scaling dependence but does not exhibit the particle size dependence expected from the GSER. Instead, the mobility is lower than expected. In startling contrast to PS particles, for surface-modified PS-BSA particles, the Brownian motion appears almost entirely independent of the actin network on the short time scales probed by DWS, as shown in Figure 2b. Instead of the distinct t3/4 scaling, the mean-squared displacement scaling is similar to pure self-diffusion, 〈∆r2(t)〉 ∼ t. However, the response still obeys the GSER. Scaling 〈∆r2(t)〉 for PS-BSA probes with a collapses the data. The mobility of the particles on these short times is approximately a factor of two to three times lower than in free solvent. It is interesting to note that at long time scales for the largest probe particles (2a ) 3.156 µm), 〈∆r2(t)〉 begins to show a subdiffusive character. Finally, we show the dependence of the PS-BSA particle response on particle size and F-actin concentration in Figure 3 for 0.356, 1.053, and 3.156 µm particles in 0.32 and 0.80 mg/mL F-actin (Figure 3a and b, respectively.) Increasing the particle size or F-actin concentration reduces the scaling exponent of the mean-squared displacement, with the effect more pronounced at higher concentrations. For 3.156 µm particles at 0.8 mg/mL, the scaling exponent is lowest at 〈∆r2(t)〉 ∼ t0.86(0.04. Discussion The contrast in behavior between PS and PS-BSA particles in the high-frequency regimes provides new (27) Morse, D. Macromolecules 1998, 31, 7044-7076.
Chae and Furst
Figure 3. Mean squared displacement of PS-BSA particles with diameters (a) 2a ) 1.053 and (b) 3.156 µm in F-actin concentrations of 0.32, 0.63, and 0.8 mg/mL (square, circle, and triangle, respectively) The lines have slopes of 1 for comparison.
Figure 4. Mean squared displacement of PS-BSA (circle) and PS (squares). DWS data (filled) using 2a ) 1.053 µm particles in 0.63 mg/mL F-actin shows excellent agreement with measurements using laser tracking microrheology (open) obtained from McGrath et al.10 for 2a ) 0.914 µm in 1.0 mg/mL F-actin when adjusted for concentration and particle size.
insight into the probe surface chemistry dependence of F-actin microrheology reported earlier.10 Most startling is the scaling of 〈∆r2(t)〉, which increases significantly for PS-BSA probe particles at short times. However, results for both surface chemistries are in excellent quantitative agreement with previous reports of F-actin microrheology. For instance, Figure 4 shows that the magnitude and scaling of 〈∆r2(t)〉 matches remarkably well for both PSBSA and PS particles when compared to the single-particle laser tracking experiments of McGrath et al.10 We account for small differences in the particle size and F-actin concentration by scaling the mean-squared displacement by the concentration dependence of the modulus and
F-Actin Microrheology
particle radius c1.8a.28 Second, microrheological experiments using PS particles reproduce results previously reported for high-frequency F-actin measurements.5,19,29 The PS particles appear strongly coupled to the actin filaments and limited in their motion by the polymer dynamics, as shown in Figure 2a. The distinct t3/4 scaling is characteristic of the dominant bending fluctuations of actin filaments at short time scales, in good agreement with theoretical models describing the dynamics of semiflexible polymers.14,18,27 It is notable, however, that PS probe particles smaller than ξ (2a ) 0.109 µm) show the same scaling behavior with less amplitude than what we expect from the GSER. These results indicate that binding of the PS to F-actin filaments for particles with 2a < ξ causes a violation of the assumptions of continuity in the GSER. Recently, Wong and co-workers found that smaller probes exhibit infrequent “hops” in F-actin solutions.30 Our experimental time scales (10-6-10-3 s) are short compared to video microscopy (10-2-101 s), which was used to observe these dynamics. This indicates that particles smaller than ξ percolate through the network through a series of unbinding and binding events in addition to local polymer rearrangements. We expect that the motion of particles on time scales shorter than the unbinding time are dominated by fluctuations of the actin filaments, thus implying that the unbinding time scale is greater than 10-3 s. Examining the response of PS-BSA particles, we find that the mean-squared displacements shown in Figures 2 and 3 indicate that particles are less sensitive to the surrounding polymer network, yet are confined on long times, as if diffusing in a small cavity. In the remainder of this paper, we focus on elucidating the mechanism that gives rise to this unique response. A typical analysis of microrheological data is based on the assumption that the medium is locally uniform on length scales of the probe size. A situation in which this assumption could break down is if the polymer network exhibits microheterogeneity in the vicinity of the particle, either due to variations of the polymer medium on length scales equal to or larger than the probe or due to perturbations caused by the presence of the particle itself. We hypothesize that the BSA-coating of the microrheological probes results in a new local network structure by reducing the enthalpic driving force of filament adsorption to the particle surfaces. In the absence of these interactions, several effects could determine the structure of the polymer network in the vicinity of the probe particle. First, the length and stiffness of F-actin could lead to a layer depleted of polymer due to the exclusion of filament orientations near the particle, as suggested earlier by Morse.27 In this case, the polymer segment density surrounding the particle would be similar to c(r) ≈ c0(r2 - a2)1/2/r for infinitely long filaments, where c0 is the segment density in the bulk.27 Note that this depleted layer thickness scales with the particle size. Second, the local orientational order profile of the network may change, resulting in the preferential alignment of filaments parallel to the probe particle surface.31 Finally, entropic depletion may play a role, again resulting in a reduced segment density near the particle. This has been observed (28) Gardel, M.; Valentine, M.; Crocker, J.; Bausch, A.; Weitz, D. Phys. Rev. Lett. 2003, 91, 158302. (29) Mason, T. G.; Gisler, T.; Kroy, K.; Frey, E.; Weitz, D. A. J. Rheol. 2000, 44, 917-928. (30) Wong, I.; Gardel, M.; Reichman, D.; Weeks, E.; Valentine, M.; Bausch, A.; Weitz, D. A. Phys. Rev. Lett. 2004, 92, 178101. (31) Groh, B.; Dietrich, S. Phys. Rev. E 1999, 59, 4216-4228.
Langmuir, Vol. 21, No. 7, 2005 3087
Figure 5. The local polymer structure surrounding BSA-coated probe particles is modeled using using the local composite model of Levine and Lubensky.33 The particle, represented by the inner gray circle, is surrounded by a shell of thickness ∆ ) b - a with modulus G*loc. The local modulus differs from the bulk modulus G*bulk.
for entangled semiflexible polymers in the coil-like limit, L . Lp. For instance, recent one- and two-point microrheology of bacteriophage lambda DNA (λ-DNA) solutions show that a depleted layer forms around embedded particles on the order of the network correlation length.32 In light of these possible local perturbations, our aim is to characterize and understand the microenvironment of the PS-BSA probe particles. We take advantage of the fact that the long-range hydrodynamic disturbance caused by the particle effectively “scatters” off of the surrounding network, and thus its motion probes the local structure and response of confined regions of the polymer.33 This novel application of microrheology has recently been termed “rheological microscopy”.32 A significant advantage of DWS microrheology in F-actin is that the local polymer structure should not change substantially over the time scales probed, which are shorter than the characteristic time scales of filament entanglement relaxation.14 The theory of Levine and Lubensky considers the response of a particle in a locally heterogeneous viscoelastic medium,33 which we use to extract a length scale of the structural disturbance surrounding the colloidal probes without specifying its exact nature. We divide the local region into inner and outer volumes surrounding each probe particle, as shown in Figure 5. The inner region, which extends to a radius b, results in a shell around the particle of thickness ∆ ) b - a. This shell has a viscoelastic modulus of G*loc(ω), which differs from the surrounding bulk material G*bulk(ω). In one-point microrheology, the probe response is influenced by both G*loc(ω) and G*bulk(ω), and the apparent modulus, calculated from 〈∆r2(t)〉, is given by
G*1(ω) ) 2G*bulk(ω)[κ′′ - 2β5κ′] 4β6κ′2 - 9β5κκ′ + 10β3κκ′ - 9βκ′2 - 15βκ′ + 2κκ′′
(3)
where β ) a/b, κ ) G*bulk(ω)/G*loc(ω), κ′ ) κ - 1, and κ′′ ) 3 + 2κ.34 The storage G′1(ω) and loss G′′1(ω) moduli are calculated from the mean squared displacements by35
|G1(ω)| )
kBT πa〈∆r2(1/ω)〉Γ[1 + R(ω)]ω)1/t
(4)
R(ω) ) |[∂ln〈∆r2(ω)〉/∂ ln ω]|ω)1/t
(5)
G1′(ω) ) |G1(ω)| cos[πR(ω)/2]
(6)
G1′′(ω) ) |G1(ω)| sin[πR(ω)/2]
(7)
where
3088
Langmuir, Vol. 21, No. 7, 2005
Chae and Furst
Figure 6. Viscoelastic moduli using 2a ) 1.053 (circles) and 3.156 µm (triangles) PS-BSA particles in 0.32 mg/mL F-actin. (a) One-point storage (solid symbols) and loss (open symbols) moduli derived from the mean-squared displacements. (b) The modlui from one-point microrheology are collapsed onto the storage (solid line) and loss (dashed line) moduli extrapolated from low-frequency two-point microrheology28 using eq 3 with ∆ as an adjustable parameter.
Figure 7. Viscoelastic moduli using 2a ) 0.356 (squares), 1.053 (circles), and 3.156 µm (triangles) PS-BSA particles in 0.63 mg/mL F-actin. (a) One-point storage (solid symbols) and loss (open symbols) moduli derived from the mean-squared displacements. (b) The modlui from one-point microrheology are collapsed onto the storage (solid line) and loss (dashed line) moduli extrapolated from low-frequency two-point microrheology28 using eq 3 with ∆ as an adjustable parameter.
The gamma function, Γ, results in at most a 12% correction.35 G*1(ω) values are shown in Figures 6a, 7a, and 8a for F-actin concentrations of 0.32, 0.63 and 0.80 mg/mL, respectively. Equation 3 enables the calculation of the one-point microrheological response if G*loc(ω), G*bulk(ω), and the shell thickness, ∆, are known. Alternatively, G*1(ω), G*loc(ω), and G*bulk(ω) can be used to find the length scale of the heterogeneous structure. Using ∆ as an adjustable parameter, G*1(ω) is collapsed onto G*bulk(ω), assuming that the shell is composed of a viscous fluid, G*loc(ω) ) iωη0 with η0 ) 1.0 mPa‚s, as shown in Figures 6b, 7b, and 8b. Normally, two-point microrheology, which is based on the correlated displacements of particle pairs, is used to approximate the bulk rheological data over the same frequency range as the one-point microrheology. However, corresponding two-point measurements do not exist for F-actin at the high frequencies probed by DWS. Instead, we must approximate G*bulk(ω) by extrapolating two-point microrheology data available at lower frequencies,28 by assuming the scaling behavior G*(ω) ∼ ω3/4. This is a good assumption above the frequency cutoff for entanglements, since the rheology should be dominated by the Rouse-like modes of the semiflexible filaments. The extrapolated data is represented by the solid (G′bulk) and dashed (G′′bulk) lines in Figures 6-8.
At the two lowest F-actin concentrations investigated, 0.32 and 0.63 mg/mL, we find that eq 3 collapses the data for multiple particle sizes onto the estimated bulk moduli. At higher actin concentrations, reasonable agreement is observed. Significantly, we find little dependence of the shell thickness, ∆, on the actin concentration, as shown in the inset of Figure 9. This is in stark contrast to the local structure which forms around particles in semidilute λ-DNA solutions,32 where ∆ is similar to the polymer correlation length, which in turn decreases with increasing polymer concentration. Instead, Figure 9 shows that ∆ for PS-BSA probes in F-actin is proportional to the probe particle size. Returning to the possible mechanisms of local structure in colloid-polymer solutions, the above results for PSBSA particles appear consistent with the hypothesis put forth by Morse,27 that in the absence of filament adsorption, the exclusion of orientations near the probes results in a region depleted of polymer when L g Lp and L g a. The local orientational order parameter of filaments may play a role as well. While both F-actin and DNA solutions exhibit similar evidence of microheterogeneity surrounding embedded probe particles, the characteristics are distinct. Presumably, this discrepancy arises from the fact that λ-DNA, with contour length L ) 16.5 µm, persistence length Lp ≈ 50 nm and radius of gyration Rg ≈ 500 nm, is a semiflexible polymer in the coil-like limit, in contrast to the more rodlike behavior of F-actin. Therefore, we expect local interactions in F-actin solutions (i.e., excluded orientations of filaments near the particle surface) to be transmitted over longer distances than in entangled DNA solutions. However, the collective structure of the suspen-
(32) Chen, D. T.; Weeks, E. R.; Crocker, J. C.; Islam, M. F.; Verma, R.; Gruber, J.; Levine, A. J.; Lubensky, T. C.; Yodh, A. G. Phys. Rev. Lett. 2003, 90, 108301. (33) Levine, A.; Lubensky, T. Phys. Rev. E 2001, 63, 041510. (34) Levine, A.; Lubensky, T. Phys. Rev. E 2001, 65, 011501. (35) Mason, T. G. Rheol. Acta 2000, 39, 371-378.
F-Actin Microrheology
Langmuir, Vol. 21, No. 7, 2005 3089
Figure 9. Shell thickness (∆) extracted from one-point microrheology using PS-BSA probes as a function of particle radius in 0.32 (squares), 0.63 (circles), and 0.80 mg/mL (triangles) entangled F-actin solutions. Inset: shell thickness as a function of actin concentration for 2a ) 1.053 (circles) and 3.056 µm (triangles).
Figure 8. Viscoelastic moduli using 2a ) 1.053 (circles) and 3.156 µm (triangles) PS-BSA particles in 0.80 mg/mL F-actin. (a) One-point storage (solid symbols) and loss (open symbols) moduli derived from the mean-squared displacements. (b) The modlui from one-point microrheology are collapsed onto the storage (solid line) and loss (dashed line) moduli extrapolated from low-frequency two-point microrheology28 using eq 3 with ∆ as an adjustable parameter.
sion is likely to be more complex than this since we have only considered the polymer-colloid interactions and have neglected colloid-colloid correlations. In recent theoretical studies of dense solutions of ideal rodlike polymers and colloids, colloid-colloid correlations are found to “imprint” their structure on the surrounding polymer network.24 Further modeling and experimental work will help clarify the origin of this intriguing local structure and its dependence on the relative magnitudes of the polymer length, persistence length, particle size, and particle and polymer concentrations. Conclusions High-frequency microrheology using DWS has enabled us to elucidate the mechanisms underlying the probe
surface chemistry dependence of F-actin microrheology. Particles which are pretreated to limit filament adsorption exhibited less subdiffusive behavior than untreated particles, reflecting a change in the microenvironment of the probes. Using theoretical models of probe diffusion in heterogeneous environments, we found the length scale of the local polymer network perturbation surrounding the particles. The magnitude and dependence on particle size of this layer are consistent with a depletion layer caused by the exclusion of actin filaments due to their long length and rigidity. This is an interesting contrast with work using semiflexible polymers in the coil-like limit (DNA), in which the depleted layer was observed to scale with the correlation length of the network, and thus the polymer concentration. Although the basic hydrodynamic models used here provide critical insight into the length scale of the structure underlying probe surface-chemistry dependence of F-actin microrheology, further experimental and theoretical work will enable us to more fully understand the polymer-colloid, polymer-polymer, and colloid-colloid correlations in these filled semiflexible polymer solutions.23,24 Acknowledgment. We thank L. Le Goff, A. Levine, and D. Morse for helpful discussions. This work was supported by ARL (CMR-18), NIH (5-P20RR15588), and the University of Delaware Research Foundation. Funding from the NSF (CTS-0238689 and DBI-0304051) is also gratefully acknowledged. LA0480890
