Metastable intermediates in the condensation of semiflexible polymers.

Motivated by results from an earlier Brownian dynamics simulation for the collapse of a single, stiff polymer in a poor solvent [B. Schnurr, F. C. MacKintosh, and D. R. M. Williams, Europhys. Lett. 51, 279 (2000)] we calculate the conformational energies of the intermediate (racquet) states suggested by the simulations. In the absence of thermal fluctuations (at zero temperature) the annealed shapes of these intermediates are well-defined in certain limits, with their major structural elements given by a particular case of Euler's elastica. In appropriate units, a diagram emerges that displays the relative stability of all states, tori, and racquets. We conclude that, in marked contrast to the collapse of flexible polymers, the condensation of semiflexible or stiff polymers generically proceeds via a cascade through metastable intermediates, the racquets, towards a ground state, the torus or ring, as seen in the dynamical simulations.

Two-dimensional tracking of ncd motility by back focal plane interferometry.

A technique for detecting the displacement of micron-sized optically trapped probes using far-field interference is introduced, theoretically explained, and used to study the motility of the ncd motor protein. Bead motions in the focal plane relative to the optical trap were detected by measuring laser intensity shifts in the back-focal plane of the microscope condenser by projection on a quadrant diode. This detection method is two-dimensional, largely independent of the position of the trap in the field of view and has approximately 10-micros time resolution. The high resolution makes it possible to apply spectral analysis to measure dynamic parameters such as local viscosity and attachment compliance. A simple quantitative theory for back-focal-plane detection was derived that shows that the laser intensity shifts are caused primarily by a far-field interference effect. The theory predicts the detector response to bead displacement, without adjustable parameters, with good accuracy. To demonstrate the potential of the method, the ATP-dependent motility of ncd, a kinesin-related motor protein, was observed with an in vitro bead assay. A fusion protein consisting of truncated ncd (amino acids 195-685) fused with glutathione-S-transferase was adsorbed to silica beads, and the axial and lateral motions of the beads along the microtubule surface were observed with high spatial and temporal resolution. The average axial velocity of the ncd-coated beads was 230 +/- 30 nm/s (average +/- SD). Spectral analysis of bead motion showed the increase in viscous drag near the surface; we also found that any elastic constraints of the moving motors are much smaller than the constraints due to binding in the presence of the nonhydrolyzable nucleotide adenylylimidodiphosphate.

Interference model for back-focal-plane displacement detection in optical tweezers.

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.

Directional loading of the kinesin motor molecule as it buckles a microtubule.

Single kinesin motor molecules were observed to buckle the microtubules along which they moved in a modified in vitro gliding assay. In this assay a central portion of the microtubule was clamped to the glass substrate via biotin-streptavidin bonds, while the plus end of the microtubule was free to interact with motors adsorbed at low density to the substrate. A statistical analysis of the length of microtubules buckled by single motors showed a decreasing probability of buckling for loads greater than 4-6 pN parallel to the filament. This is consistent with kinesin stalling forces found in other experiments. A detailed analysis of some buckling events allowed us to estimate both the magnitude and direction of the loading force as it developed a perpendicular component tending to pull the motor away from the microtubule. We also estimated the motor speed as a function of this changing vector force. The kinesin motors consistently reached unexpectedly high speeds as the force became nonparallel to the direction of motor movement. Our results suggest that a perpendicular component of load does not hinder the kinesin motor, but on the contrary causes the motor to move faster against a given parallel load. Because the perpendicular force component speeds up the motor but does no net work, perpendicular force acts as a mechanical catalyst for the reaction. A simple explanation is that there is a spatial motion of the kinesin molecule during its cycle that is rate-limiting under load; mechanical catalysis results if this motion is oriented away from the surface of the microtubule.

The force exerted by a single kinesin molecule against a viscous load.

Kinesin is a motor protein that uses the energy derived from the hydrolysis of ATP to power the transport of organelles along microtubules. To probe the mechanism of this chemical-to-mechanical energy transduction reaction, the movement of microtubules across glass surfaces coated with kinesin was perturbed by raising the viscosity of the buffer solution. When the viscosity of the solution used in the low density motility assay was increased approximately 100-fold through addition of polysaccharides and polypeptides, the longer microtubules, which experienced a larger drag force from the fluid, moved more slowly than the shorter ones. The speed of movement of a microtubule depended linearly on the drag force loading the motor. At the lowest kinesin density, where dilution experiments indicated that the movement was caused by a single kinesin molecule, extrapolation of the linear relationship yielded a maximum time-averaged drag force of 4.2 +/- 0.5 pN per motor (mean +/- experimental SE). The magnitude of the force argues against one type of "ratchet" model in which the motor is hypothesized to rectify the diffusion of the microtubule; at high viscosity, diffusion is too slow to account for the observed speeds. On the other hand, our data are consistent with models in which force is a consequence of strain developed in an elastic element within the motor; these models include a different "ratchet" model (of the type proposed by A. F. Huxley in 1957) as well as "power-stroke" models.

Flexural rigidity of microtubules and actin filaments measured from thermal fluctuations in shape.

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) Nm2 (with 6.4% uncertainty) for seven unlabeled microtubules and 2.1 x 10(-23) Nm2 (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 rhodamine-phalloidin-labeled actin filaments gave a flexural rigidity of 7.3 x 10(-26) Nm2 (with 6% uncertainty), consistent with previously reported results. The flexural rigidity of these microtubules corresponds to a persistence length of 5,200 microns showing that a microtubule is rigid over cellular dimensions. By contrast, the persistence length of an actin filament is only approximately 17.7 microns, 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 former's larger cross-section. If tubulin were homogeneous and isotropic, then the microtubule's Young's modulus would be approximately 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.

Estimating mean particle volume and number from random sections by sampling profile boundaries.

We describe some new shape-independent stereological estimates of particle mean volume and surface area. Finding volumes or surface areas of cell nuclei, from electron micrographs of random thin sections, is a central problem of biological stereology. The well-known point-sampled intercept (PSI) method samples profile interiors to find the volume-weighted mean volume. This can be used in place of the true mean volume, but to do so introduces bias when volumes vary a great deal, as they do in fixed specimens. Jensen and Gundersen quite recently extended the PSI estimator to provide particle surface area, with no bias in the case of uniform surface areas. Here we extend the PSI volume estimator in a different way, sampling profile boundaries rather than their interiors. We obtain a boundary-sampled intercept (BSI) volume estimator, simpler than the PSI surface area estimator, but also unbiased for uniform surface areas. Both of these estimators are attractive, for example, in measuring and counting cell nuclei, where membrane surface area varies less than volume. Furthermore, they have no shape bias whatsoever. This paper also examines the general relationship between boundary- and area-sampled estimates, and we clarify the formal connection between our volume estimator and the PSI surface area estimator. We also calculate and compare their theoretical efficiencies.