Abstract. Microtubules are long, proteinaceous filaments that perform structural functions in eukaryotic cells by defining cellular shape and serving as tracks for intracellular motor proteins. We report the first accurate measurements of the flexural rigidity of microtubules. By analyzing the thermally driven fluctuations in their shape, we estimated the mean flexural rigidity of taxol-stabilized microtubules to be 2.2 x 10 -23 Nm 2 (with 6.4% uncertainty) for seven unlabeled microtubules and 2.1 x 10 -23 Nm 2 (with 4.7% uncertainty) for eight rhodamine-labeled microtubules. These values are similar to earlier, less precise estimates of microtubule bending stiffness obtained by modeling flagellar motion. A similar analysis on seven rhodaminephalloidin-labeled actin filaments gave a flexural rigidity of Z3 x 10 -26 Nm 2 (with 6% uncertainty), consistent with previously reported results. The flexural rigidity of these microtubules corresponds to a persistence length of 5,200 #m showing that a microtubule is rigid over cellular dimensions. By contrast, the persistence length of an actin filament is only ,,,,17.7 #m, perhaps explaining why actin filaments within cells are usually cross-linked into bundles. The greater flexural rigidity of a microtubule compared to an actin filament mainly derives from the formers larger cross-section. If tubulin were homogeneous and isotropic, then the microtubule's Young's modulus would be -1.2 GPa, similar to Plexiglas and rigid plastics. Microtubules are expected to be almost inextensible: the compliance of cells is due primarily to filament bending or sliding between filaments rather than the stretching of the filaments themselves.

We describe a high-resolution, high-bandwidth technique for determining the local viscoelasticity of soft materials such as polymer gels. Loss and storage shear moduli are determined from the power spectra of thermal fluctuations of embedded micron-sized probe particles, observed with an interferometric microscope. This provides a passive, small-amplitude measurement of rheological properties over a much broader frequency range than previously accessible to microrheology. We study both F-actin biopolymer solutions and polyacrylamide (PAAm) gels, as model semiflexible and flexible systems, respectively. We observe high-frequency ω 3/4 scaling of the shear modulus in F-actin solutions, in contrast to ω 1/2 scaling for PAAm.PACS numbers: 61.30. Cz, 64.70.Md, 83.70.Jr The local analysis of viscoelasticity can explore the small-scale structure of complex fluids. It can also potentially measure bulk viscoelastic quantities in small samples. Experiments have been done to characterize the local viscoelasticity of materials since at least the 1920s, when magnetic particles in gelatin were manipulated by field gradients [1]. Crick used a similar technique to study living cells [2]. With the recent advent of methods for force generation, detection, and manipulation of particles on sub-micrometer scales, experimental possibilities have expanded greatly, and interest in microrheology has grown substantially [3][4][5][6][7][8]. Here we report an optical technique for high resolution and high bandwidth observations of the thermal fluctuations of particles embedded in soft materials. Using dispersion relations from linear response theory, the frequency-dependent loss and elastic storage shear moduli are both calculated from the fluctuation power spectra, which we measure with a resolution of 2Åfrom 0.1 Hz to 20 kHz. This frequency range exceeds that of video-based microrheological experiments (although large-volume rheometers can reach frequencies up to 500 kHz [9]) and allowed us to observe high-frequency ω 3/4 scaling of the shear modulus for entangled, semiflexible solutions. These dynamics differ fundamentally from those of flexible polymer systems.We have studied F-actin solutions as a model semiflexible polymer, and polyacrylamide (PAAm) gels as a flexible polymer control. Actin is one of the primary components of the cytoskeleton of plant and animal cells and is largely responsible for the viscoelastic response of cells [10]. Actin is a particularly accessible model system because individual filaments can be hundreds of microns in length. Viscoelastic properties of entangled F-actin solutions in vitro have been measured using conventional macroscopic rheology [11][12][13][14][15]. Actin has also been the subject of recent microrheological studies [3][4][5][6].The relationship between thermal motion and hydrodynamic response is well-known in the context of Brownian motion in simple viscous fluids. Less obviously, the bulk viscoelastic properties of complex fluids can also be determined from thermal motion. Such a passi...

The lateral position of an optically trapped object in a microscope can be monitored with a quadrant photodiode to within nanometers or better by measurement of intensity shifts in the back focal plane of the lens that is collimating the outgoing laser light. This detection is largely independent of the position of the trap in the field of view. We provide a model for the essential mechanism of this type of detection, giving a simple, closed-form analytic solution with simplifying assumptions. We identify intensity shifts as first-order far-field interference between the outgoing laser beam and scattered light from the trapped particle, where the latter is phase advanced owing to the Gouy phase anomaly. This interference also reflects momentum transfer to the particle, giving the spring constant of the trap. Our response formula is compared with the results of experiments.

We construct a model for the dynamic shear modulus G() of entangled or crosslinked networks of semiflexible polymer that can account for the high-frequency scaling behavior, G()ϳ 3/4 , that has recently been observed in solutions of the biopolymer F-actin. As we argue, this behavior should not be unique to F-actin, but rather should be a clear characteristic of semiflexible polymers in general. We also report molecular dynamics simulations that support the single filament response that is the basis of our model for the network shear modulus. ͓S1063-651X͑98͒51908-3͔

In an optical tweezers experiment intense laser light is tightly focused to intensities of MW/cm(2) in order to apply forces to submicron particles or to measure mechanical properties of macromolecules. It is important to quantify potentially harmful or misleading heating effects due to the high light intensities in biophysical experiments. We present a model that incorporates the geometry of the experiment in a physically correct manner, including heat generation by light absorption in the neighborhood of the focus, balanced by outward heat flow, and heat sinking by the glass surfaces of the sample chamber. This is in contrast to the earlier simple models assuming heat generation in the trapped particle only. We find that in the most common experimental circumstances, using micron-sized polystyrene or silica beads, absorption of the laser light in the solvent around the trapped particle, not in the particle itself, is the most important contribution to heating. To validate our model we measured the spectrum of the Brownian motion of trapped beads in water and in glycerol as a function of the trapping laser intensity. Heating both increases the thermal motion of the bead and decreases the viscosity of the medium. We measured that the temperature in the focus increased by 34.2 +/- 0.1 K/W with 1064-nm laser light for 2200-nm-diameter polystyrene beads in glycerol, 43.8 +/- 2.2 K/W for 840-nm polystyrene beads in glycerol, 41.1 +/- 0.7 K/W for 502-nm polystyrene beads in glycerol, and 7.7 +/- 1.2 K/W for 500-nm silica beads and 8.1 +/- 2.1 K/W for 444-nm silica beads in water. Furthermore, we observed that in glycerol the heating effect increased when the bead was trapped further away from the cover glass/glycerol interface as predicted by the model. We show that even though the heating effect in water is rather small it can have non-negligible effects on trap calibration in typical biophysical experimental circumstances and should be taken into consideration when laser powers of more than 100 mW are used.