An improved optical tweezers assay for measuring the force generation of single kinesin molecules

Abstract

Numerous microtubule-associated molecular motors, including several kinesins and cytoplasmic dynein, produce opposing forces that regulate spindle and chromosome positioning during mitosis. The motility and force generation of these motors are therefore critical to normal cell division, and dysfunction of these processes may contribute to human disease. Optical tweezers provide a powerful method for studying the nanometer motility and piconewton force generation of single motor proteins in vitro. Using kinesin-1 as a prototype, we present a set of step-by-step, optimized protocols for expressing a kinesin construct (K560-GFP) in Escherichia coli, purifying it, and studying its force generation in an optical tweezers microscope. We also provide detailed instructions on proper alignment and calibration of an optical trapping microscope. These methods provide a foundation for a variety of similar experiments.

Fig. 1

Optical tweezers assay for kinesin motility and force production (not to scale). A polystyrene microsphere covalently bound to anti-GFP antibodies binds a single kinesin K560-GFP dimer, and is trapped by a near-infrared optical trapping beam focused via a high numerical aperture microscope objective lens. The trap holds the microsphere directly above a MT that is covalently linked to the glass surface of the cover slip. When the kinesin binds to and moves along the MT in the presence of ATP, it pulls the attached microsphere with it. The trap resists this motion, exerting a force F = −k×Δx on the microsphere-motor complex, where k is the trap stiffness (force per displacement) and Δx is the distance from the trapping beam longitudinal (z) axis (”trap center”) to the center of the microsphere. The detection beam—which overlaps the trapping beam, but does not contribute to trapping—is used to measure Δx via back focal plane interferometry. This figure was prepared with VMD [164] (using PDB entries 3KIN, 1GFL, and 1IGT) and the Persistence of Vision Raytracer (POV-Ray www.povray.org)

Fig. 2

Protocol summary. Each pathway (a–d) summarizes the major steps in preparing reagents and instrumentation for the final assay: (a, b) purifying kinesin and attaching it to optical trapping microspheres bound to antibodies; (c) preparation of a slide chamber with immobilized, Cy3-labeled MTs (see Chap. 9, this issue); (d) aligning and calibrating the optical tweezers instrument. These tools are combined to precisely measure force production and motility of single kinesin molecules

Fig. 3

Force-fluorescence microscope (see text for description). A aperture; AD aperture diaphragm; AOD acousto-optic deflector; BD beam dump; BPF band-pass filter; C beam collimator; CCD charge-coupled device (for bright-field detection); CL condenser/collection lens; CPS coarse-positioning stage; DM dichroic mirror; EMCCD electron-multiplying CCD (for fluorescence detection); F single-mode, polarization-maintaining optical fiber; FD field diaphragm; IP image plane; L lens; L* lens mounted on a translation stage for fine focus adjustment; LED light-emitting diode; M mirror; MB microscope body; ND neutral density filter; NPS nanopositioning stage; OI optical isolator; OL objective lens; PBS polarizing beam splitter; PM polychroic mirror; Pzt-M piezo-driven mirror mount; QPD quadrant photodiode; RL relay lens; S shutter; TL tube lens; TS translation stage; WP half-wave plate

Fig. 4

Force-fluorescence trapping microscope. Note the optical pathway elevated on an optical breadboard and enclosed in an airtight box. The friction break [165] consists of an optical post or other rigid element bolted firmly to the table and pressed against the microscope fine-focus knob to prevent drift of the objective over time. The region demarcated by the red dashed line (back focal plane imaging arm) is shown in greater detail in Fig. 12. Refer also to Fig. 3

Fig. 5

K560 purification. The full-length motor is first purified from other proteins and incomplete fragments by nickel-nitrilotriacetic acid (Ni-NTA) chromatography, isolating only those proteins with the polyhistidine (His₆) tag (marked by the asterisk; note the inset photograph showing the elution with intense green color due to GFP). Next, His₆-containing fragments (degradation products) and motors incompetent to bind MTs are removed by MT binding and sedimentation in the presence of AMP-PNP. Motors unable to release MTs in response to saturating concentrations of ATP are then removed, and the purified, functional K560 is aliquotted, flash-frozen in liquid nitrogen, and stored

Fig. 6

Coupling antibodies to optical trapping microspheres. First, a microsphere bearing surface carboxyl groups reacts (a) with the carbodiimide EDAC (1-ethyl-3-(3-dimethylaminopropyl)carbodiimide) to form an unstable O-acylisourea intermediate. This short-lived species is reactive toward primary amines and thus capable of forming stable, covalent amide bonds with lysyl residues on the antibody surface (b) and yielding an isourea byproduct (not shown). However, this O-acylisourea is also readily and rapidly hydrolyzed, yielding the original carboxyl group (c). Adding NHSS (3-sulfo-N-hydroxysulfosuccinimide) to the reaction greatly enhances microsphere–antibody coupling efficiency by forming a relatively water-stable, but amine-reactive NHSS ester (d) that can likewise bind antibodies (e). BSA (not shown) is added with the antibody and also binds via its surface lysines, thus blocking the remainder of the microsphere surface. After antibodies and BSA are bound, any residual NHSS esters are removed by adding hydroxylamine (f), which reacts with the ester to form hydroxamic acid

Fig. 7

Pyranose 2-oxidase (P2O) oxygen scavenging system, with comparison to glucose oxidase (GO). Both flavoenzymes contain flavin adenine dinucleotide (FAD) prosthetic groups that are reduced by glucose, thereby generating FADH₂ and removing two hydrogens from the sugar. Whereas GO removes hydrogens from the C1 position (green), P2O acts on C2 (magenta), yielding d-glucono-δ-lactone and 2-keto-d-glucose, respectively. Whereas 2-keto-d-glucose is stable, in the presence of water, d-glucono-δ-lactone hydrolyzes to gluconic acid, thereby acidifying the reaction solution. The reduced enzymes (FADH₂ forms) are oxidized by O₂, regenerating the original enzymes and producing H₂O₂, which is converted by catalase to H₂O and O₂. In the net reaction, for each O₂ removed, two glucose molecules are consumed, and for GO, two gluconic acid molecules are generated. Although β-d-glucose, the substrate for GO, is shown, P2O has no anomeric preference and can also catalyze the oxidation of α-d-glucose [43, 111]

Fig. 8

Thermal equilibration of the optical tweezers laser pathway. When the optical trapping laser is initially powered on, the trap center is typically offset from its previous stable position (left-hand image; the cross to the upper left of the bead is the stable position to which the bead moves after the optical elements in the pathway expand due to heating by the trapping laser). The new position is typically very close (tens of nanometers) to the stable position during the preceding use of the instrument (right-hand image). The graph shows a representative example of microsphere movement over time as the optical pathway thermally equilibrates. The position traces were obtained by tracking the trapped microsphere in images acquired every second for 60 min, using a simple centroid-finding algorithm. The paths shown are followed fairly consistently each time the instrument is powered on before experiments. The instrument requires approximately 45–60 min to fully stabilize

Fig. 9

Important optical relationships for focusing a collimated beam. BFP, back focal plane; FFP, front focal plane. Solid rays represent a beam perfectly aligned with and centered on the optical axis. (a) Pure translation of the beam in the back aperture leads to tilting of the beam as it focuses in the FFP, but does not change the position of the focus. This induces asymmetry in intensity pattern of the retroreflected beam. (b) Pure tilting of the beam in the BFP leads to translation of the focus in the FFP, essentially without affecting the beam angle as it approaches the FFP. This shifts the position of the intensity pattern of the retroreflected beam, with minimal effects on the intensity distribution

Fig. 10

Coarse alignment using the retroreflected trapping and detection beams. (a) Diagram of retroreflected beam detection. The input beam (trapping or detection, solid rays in the diagram) reflects off a dichroic mirror with high reflectivity at the laser wavelength and passes through the microscope objective lens to be focused on the cover slip. When the focused beam strikes the interface between the cover slip and the aqueous solution of the sample chamber, a small proportion of the intensity is reflected back into the objective, traveling a reverse path through the system (dashed rays). Despite the high reflectivity of the dichroic mirror, a small fraction of this light is transmitted to the imaging optics to form an intensity pattern on the CCD. The size and shape of the pattern depend on the distance δz between the glass–solution interface and the focal plane of the objective (δz is negative when the focal plane is below the interface). (b) Images of the retroreflected trapping beam intensity pattern for various values of δz (adjusted by moving the nanopositioning stage holding the slide chamber), for well-aligned (top row) and poorly aligned (bottom row) beams. The well-aligned beam passes directly through the center of the objective, parallel with the longitudinal axis of the lens (optical axis), forming a symmetrical pattern at each stage position. The poorly aligned beam, which may be displaced, tilted, or both relative to the well aligned beam, forms an asymmetrical pattern that changes (and may shift position in the image) depending on δz. Note that when δz = 0 (near perfect focusing on the glass–solution interface), both patterns look very similar, and it is only when the beams are “defocused” that the differences become apparent. After proper alignment of the (c) trapping beam and (d) detection beam, the two patterns appear much more symmetrical and are concentric with each other (e, composite image: detection beam pseudocolored green, and trapping beam magenta). Scale bars in (c–e) are 2.25 µm

Fig. 11

Refinement of trapping beam alignment. (a) The trap holds the microsphere above the cover slip with a distance Z_B between the glass and the surface of the microsphere (left, not to scale; Z_B is measured as the axial stage position minus the height of the bottom surface of the bead at its normal trapped position). As the nanopositioning stage moves the cover slip upward, the magnitude of Z_B progressively decreases. Once the cover slip moves high enough, it displaces the bead axially from its equilibrium trapped position (right, at this point, Z_B is negative). (b) Following coarse adjustment using retroreflections, with Z_B ≈ 0.5 µm, the microsphere is well centered, and the reflected laser light forms a fairly symmetrical concentric pattern around it (left). With the bright-field illumination turned off and the CCD gain increased, it is easier to view the reflected light (right). (c) After moving the stage upward (Z_B ≈ −0.75 µm), the microsphere position deviates laterally (left) and the reflected light pattern is asymmetrical and off-center (right), indicating a slight misalignment of the laser beam. (d) After readjustment, the microsphere is centered (left) and the reflection pattern is more symmetrical and centered. (e) Following multiple rounds of minor adjustments, the microsphere remains well centered and the retroreflection symmetrical, over a wide range of stage displacements

Fig. 12

Back focal plane detection of microsphere position. (a) Optical configuration for back focal plane detection. The bright-field illumination (gray, exiting the field diaphragm (FD) from above) and detection beam (yellow) propagate in opposite directions through the system. After exiting the objective lens on the inverted microscope, the detection beam enters the slide chamber fixed to the nanopositioning stage (NPS), interacts with the trapped microsphere, and is collected by the condenser lens (CL), which is confocal with the objective. The detection beam is then redirected toward the quadrant photodiode (QPD) detector by a short-pass dichroic mirror (DM) that reflects the 830-nm detection beam, but transmits the 470-nm bright-field illumination. The trapping beam, which follows the same path, is blocked by an 830-nm band-pass filter (F). The relay lens (RL) is positioned such that it images the aperture diaphragm/back focal plane (AD/BFP) of the condenser lens onto the QPD (in this case, the lens is placed three focal lengths from the AD/BFP and 1.5 focal lengths from the QPD, respectively, to achieve a magnification of ½). A support post (SP) with vibration-dampening foam on top helps stabilize the QPD detection arm and eliminate unwanted movements. (b) The nature of the interaction of the detection beam with the bead depends somewhat on its size. Very small (<1 µm) particles act as scattering point sources. In this case, the pattern in the back focal plane arises due to interference between the unscattered portion of the detection beam (which essentially propagates without interacting with the particle) and the light scattered by the particle (left, solid and dashed lines represent optical wave fronts). Larger particles may significantly alter the path of the detection beam via refraction of the light as it passes through the microsphere (right), causing the entire pattern in the back focal plane to shift. (c) Examples of the patterns observed by replacing the QPD with a CCD camera. The dark regions with sharp edges in each corner are images of the aperture diaphragm (located at the back focal plane of the condenser and visible here because it has been partially closed). The left and right columns correspond to 500- and 920-nm-diameter beads, respectively. Approximate bead–trap separations for each set of images are given on the right side. Note that the larger particle has a more pronounced effect on the overall beam position. (d) The voltage signals from the four quadrants of the QPD (labeled V_A, V_B, V_C, and V_D, respectively) are used to calculate response signals in three dimensions (normalized by the total voltage for the x and y directions). The response signals each have similar shapes and are linear with displacement near the center of the detection beam (solid black line, slope = 1/β). In this region, the QPD response X_Vnorm can be directly converted to displacement ΔX = β_x X_Vnorm (and identically for Δ Y)

Fig. 13

Alignment of the trapping and detection beams by sweeping the trapping beam in a triangle-wave pattern. (a) A trapped bead is swept transversely across the detection beam (along either the x or y axis) by using the AOD to move the trap position in a triangle-wave pattern. This modulates the normalized QPD position signals, X_Vnorm and Y_Vnorm. The objective of adjustment is to position the detection beam so that the response in each channel is symmetrical about the origin. (b) After coarse alignment of the trap and detection beams, the QPD position signal (red trace, V_norm) for each channel is very nearly zero. However, when the trap position is modulated in a triangle-wave pattern (solid black line), it is clear that the response is not symmetrical about the origin (but rather around another position, in this case one slightly negative), indicating the detection beam and trapping beam do not overlap (step 1, the dashed lines through 0 V_norm and ±0.4 V_norm can be used as landmarks to help guide the eye). The center position of the sweep pattern is moved toward the origin using mirror M1 (see Fig. 3), with the consequence that the symmetry of the response is lost (step 2). Adjusting M2 restores the symmetry, while reversing some of the movement toward the origin (step 3). However, the net result is that the response is symmetrical and that the center of the pattern is closer to the origin. Performing steps 2 and 3 repeatedly yields the desired response (step 4). This process is repeated iteratively for each channel, until both exhibit symmetrical responses centered at the origin

Fig. 14

Identification and correction of erroneous QPD rotation. After aligning the instrument as precisely as possible, when the trap sweeps a bead back and forth across the detection beam in the in one axis (y in this example), the QPD signal (X_Vnorm) for the non-sweeping axis exhibits substantial displacements if the QPD is rotated relative to the axes of the microscope (left). This example is exaggerated for clarity, and in practice, the “crosstalk” between channels is usually much subtler. Correcting the QPD rotation decreases excursion in the non-sweeping axis, and when the axes are aligned (right), the signal for the non-sweeping axis is virtually unaffected even by large displacements in the orthogonal direction. For counterclockwise (as judged when looking toward the face of the QPD detector) rotations relative to the microscope axes, y sweeping produces deflections of the same sign in X_Vnorm and Y_Vnorm (as shown in the figure), whereas x sweeping produces deflections of the opposite sign in X_Vnorm and Y_Vnorm. For clockwise rotations, the situation is reversed

Fig. 15

Determination of bead–cover slip separation. (a) The stage is swept over a ±1,200 nm range in the axial (z) direction. First, a bead strongly stuck to the cover slip is carefully centered on the detection beam so that the x and y QPD signals are zero. Sweeping the bead through the trapping beam induces the characteristic response in the QPD sum signal (solid black line). If the same experiment is performed with a trapped bead (green dots), essentially no response is produced until the cover slip makes contact with the bead surface and pushes it out of the trap (at which point it follows the same trajectory and induces the same QPD signal as the stuck bead). Note that while the position (in nm) at which the contact point occurs depends on the distance between the cover slip and bead, the QPD voltage at which this happens depends on the position of the detection beam focus relative to the trapped bead (i.e., the trapping beam focus). In this example, the focus of the detection beam is slightly below the trapped bead, since the two curves overlap beyond the central voltage value for the stuck bead. (b) To find the bead–cover slip separation during an experiment, the position of the inflection point (where the sum signal for the trapped bead turns upward and follows the path of the signal for the stuck bead) is determined by finding the intersection of a line fit to the initial segment and a third-order polynomial fit to the peak, in this case 65 nm. Note that this value must be corrected to account for the focal shift (see Fig. 16), so that the true initial separation was actually ~0.82 × 65 = 53 nm. The periodic modulation observed in the sum signal as the cover slip comes very close to the bead is expected behavior and arises because the surfaces of the bead and cover slip essentially form a miniature Fabry–Pérot interferometer [62], as shown in the inset. Light backscattered/reflected from the bead (red dashed rays) reflects off the cover slip and interferes with forward-propagating (unscattered/forward-scattered) light (solid black rays) at the detector. The constructive/destructive nature of the interaction depends on the bead–cover slip distance and oscillates with a spatial frequency dependent on the wavelength of the detection beam

Fig. 16

The focal shift and its effects on bead–cover slip separation. (a) Ray diagram for a collimated beam focused by an oil immersion microscope objective. Although the indices of refraction are closely matched between the objective, immersion oil, and cover slip (n_oil ≈ n_glass ≈ 1.51), the refractive index of the sample buffer is significantly smaller (n_buffer ≈ n_water = 1.33). This causes the rays to refract at the interface, bringing the actual focus position (Z_AFP, measured relative to the cover slip surface) closer to the objective than the nominal focus position (Z_NFP) expected if the refractive indices were equal (dashed rays). Z_AFP depends on the indices of refraction and on the distance the beam travels in each medium (glass/oil vs. buffer), which is determined by the separation between the cover slip and objective. A change ΔZ in this separation (due to movement of the objective or cover slip) induces a corresponding change ΔZ_AFP in the actual focus position, which is linear with ΔZ over several micrometers. The proportionality constant relating ΔZ_AFP to ΔZ is referred to as the focal shift, FS (which is defined differently by some authors, as the difference [Z_NFP − Z_AFP]). Using high-NA oil-immersion objectives and aqueous buffer, a full electromagnetic treatment of the problem gives FS ≈ 0.75. (b) In optical tweezers, the figure of interest is not the focal shift per se, but rather the effective focal shift (EFS), defined analogously, though in reference to the trapped bead position, h, rather than to the focus itself. EFS ≈ 0.82 is slightly greater than FS for aqueous solutions, but still significantly different from unity. This must be accounted for when moving the objective or the nanopositioning stage holding the cover slip in the axial direction. Moving the stage toward the trapped bead by 100 nm decreases the separation not by 100 nm, but rather by ~82 nm. Truly reducing the bead–cover slip separation by 100 nm requires moving the stage by ~122 nm instead

Fig. 17

QPD position calibration. The trap is used to step a bead across the detection beam separately in each axis by a known distance, in this case ±250 nm (the inset in the upper left shows the sweep in the y axis). For the central region (approximately ±100 nm), the response is linear (the lower right inset shows this region for the y axis), but exhibits curvature for larger displacements (see Fig. 13). At each step (every 5 nm in this example), many samples of the QPD response signal are recorded and averaged (weaker traps require more samples because the bead is less confined; in this example, with k ≈ 0.06 pN/nm, 100 samples were averaged at each point). These values are then plotted against the position and fit with a third-degree polynomial. The coefficients from this fit are then used to determine the bead position from the QPD signal during experiments

Fig. 18

Mapping of bead diffusion and determination of the trap spring constant. (a) Y position trace for a trapped bead over a period of 20 s (dashed lines are ±standard deviation, σ = 7.6 nm). The variance of the positions is σ² = 57.4 nm², which for a value of k_BT = 4.116 pN nm (T = 25 °C) yields k_y = 0.072 pN/nm by the equipartition method. (b) For the same experiment as in (a), plotting a two-dimensional histogram of bead positions provides a useful method for determining the shape of the trapping potential (limited to the central region in which the bead can diffuse) and identifying any irregularities or asymmetries. (c) The data from (a) are plotted in a histogram (50 equally spaced bins of ~1.3 nm width) and fit with a Gaussian function P (Y), also yielding σ = 7.6 nm (agreement between σ calculated “blind” and by fitting shows that the data are Gaussian distributed and do not require fitting to determine σ). (d) Given the (non-normalized) probability distribution in (c), it is possible to calculate the potential U (Y), which, assuming a harmonic potential, can be fit by the parabola ½k_y Y² + b, where b is a constant. This procedure yields k_y = 0.071 pN/nm, in good agreement with the equipartition method. The good fit of the parabola to the calculated potential shows that the trap potential is indeed harmonic (like a Hookean spring) near the center

Fig. 19

Power spectrum analysis for optical trap calibration. For (a – c) the separation Z_B between the bead and cover slip is ~70 nm, and the sample rate for the data is 65,536 Hz (2¹⁶ Hz). (a) Log–log plot of the x and y power spectral densities for a trapped bead, with Lorentzian fit to the frequencies above 100 Hz (x, dashed line; y, solid line; note that the x power spectrum and dashed-line fit are only partially visible because they almost exactly overlap the y spectrum). The inset shows the entire spectrum, which exhibits non-Lorentzian character at very low frequencies (this is attributable to slow, long-term drift and is minimized by enclosing the optical pathway and otherwise stabilizing the instrument). The parameters extracted from the fit (for x and y, respectively) are corner frequencies (f_c) 683 and 685 Hz and effective drag coefficients 2.23 × γ_water and 2.22 × γ_water, yielding spring constants k_x = k_y = 0.074 pN/nm (compared to k_x = 0.074 pN/nm and k_y = 0.072 pN/nm calculated by the equipartition method). (b) Power spectrum analysis for forced oscillations (sinusoidal nanopositioning stage movement at f_drive = 32 Hz) of the same bead as in (a). In this example, the stage oscillates in the y direction with 98 nm measured amplitude (see Note 75), giving rise to a sharp peak at f_drive only in the y channel (note that the spatial units here are V²_norm rather than nm²). The amplitude of this peak is used to calculate β (the V_norm-to-nm conversion factor for the linear region of the QPD response signal). A Lorentzian fit (black line) to the block-averaged y data above 100 Hz (white trace) yields f_c = 696 Hz, β_y = 474 nm/V_norm, γ_y = 2.22 × γ_water, and k_y = 0.073 pN/nm, in very good agreement with the standard power spectrum analysis in (a) and with the AOD-based QPD calibration (β_y = 478 nm/V_norm; not shown). The x and y power spectra apparently overlap somewhat less than in (a); however, this is attributable to the small difference in β between the two channels (β_x = 434 nm/V_norm), rather than a difference in physical motion. (c) For frequencies much greater than f_c, the product of frequency squared times the power spectral density approaches a constant value C, such that γ = k_BT/[C (π β)²]. Thus, if γ is known, C can be used to find β and vice versa. Given β from the AOD-based calibration of the QPD response, we use this phenomenon to generate an initial guess for γ in our Lorentzian fits of the power spectra in (a) and (b). The data shown here (from the same underlying position data as the power spectrum in (a)) give C_x = 0.122 V²_norm Hz and C_y = 0.102 V²_norm Hz, yielding γ_x = 2.31 × γ_water and γ_y = 2.22 × γ_water, in relatively good agreement with the values used in the final fit. (d) The drag γ experienced by a sphere in close proximity to a surface is significantly larger than that experienced in bulk solution, γ₀, and is approximated by Faxén’s law. The ratio γ/γ₀ is plotted for different bead diameters as a function of the bead-surface–cover slip surface separation, Z_B

Fig. 20

Viscous drag determination of trap stiffness and mapping of trap potential. (a) With a bead in the trap, the stage is moved in the y direction at a constant velocity, v_stage. This pulls the solution in the chamber past the bead, thus inducing a viscous drag force γv_stage that pulls the bead away from the trap center and is balanced by the restoring force of the trap, k_yΔy. (b) Example data, showing bead displacements during constant-velocity stage movement, interspersed with periods of pausing. For this experiment, the 920-nm-diameter bead was approximately 500 nm above the cover slip, the stage velocity was 0.3 mm/s, and k_y was measured to be 0.076 pN/nm via equipartition and power spectrum, with γ ≈ 1.5 γ_water. The predicted Δy for these parameters is ~46 nm, in very good agreement with the 43 nm measured (red dashed lines; see (c)). (c) Measurement of Δy. The positions from the data in (b) are plotted in a histogram (100 equally spaced bins), with a major peak at 0 nm due to the pauses between movement. The inset shows fits to the two side peaks, with data from ±20 nm discarded, yielding means of ±43 nm. (d) Repeating these processes at many different velocities allows the mapping of the trapping potential in regions far from the center, in order to test the validity of trap linearity in these regions, and provides a more robust estimation for k_y, as determined from the slope of a line fit to the data. During the course of the experiment, k_y was measured to be ~0.070 pN/nm on average (via equipartition and power spectrum), with γ ≈ 1.5 γ_water. The fit is consistent with a linear force–displacement relationship for displacements of at least 150 nm, and the calculated slope of 1.7 × 10⁻⁴ s yields k_y = 0.068 pN/nm, in reasonably good agreement with the standard methods

Fig. 21

Linear dependence of lateral trap stiffness (spring constant) on the trapping laser power demonstrated for the x dimension. Laser power was measured at the entrance pupil of the objective lens, using a weakly focusing lens to ensure complete collection of the entire beam. Each measurement of the spring constant was obtained as an average between the equipartition and power spectrum analyses. Each point represents the mean of five different measurements taken over several minutes (error bars are ± standard deviation), and the line is a least-squares fit (weighted by the inverse variance of each point) through the origin; slope = (1.90 ± 0.02) × 10³ pN nm⁻¹ mW⁻¹. Identical analysis for the y dimension (not shown) essentially overlaps with the data here, with the fit yielding an identical slope

Fig. 22

K560 motility and force generation in the optical tweezers. The force/motion trace shows repeated force generation events in the trap in the y direction (k_y = 0.07 pN/nm). The motor repeatedly stalls at ~6 pN backward load. As it approaches stalling, the velocity slows considerably, and ~8-nm steps are clearly resolvable. Occasionally, the motor detaches from the MT before stalling (asterisk). Although the bead displacement is almost entirely monotonically increasing, the motor occasionally takes backward steps when it stalls (double asterisk)