The Kinesin-8 Kip3 Switches Protofilaments in a Sideward Random Walk Asymmetrically Biased by Force

Abstract

Molecular motors translocate along cytoskeletal filaments, as in the case of kinesin motors on microtubules. Although conventional kinesin-1 tracks a single microtubule protofilament, other kinesins, akin to dyneins, switch protofilaments. However, the molecular trajectory—whether protofilament switching occurs in a directed or stochastic manner—is unclear. Here, we used high-resolution optical tweezers to track the path of single budding yeast kinesin-8, Kip3, motor proteins. Under applied sideward loads, we found that individual motors stepped sideward in both directions, with and against loads, with a broad distribution in measured step sizes. Interestingly, the force response depended on the direction. Based on a statistical analysis and simulations accounting for the geometry, we propose a diffusive sideward stepping motion of Kip3 on the microtubule lattice, asymmetrically biased by force. This finding is consistent with previous multimotor gliding assays and sheds light on the molecular switching mechanism. For kinesin-8, the diffusive switching mechanism may enable the motor to bypass obstacles and reach the microtubule end for length regulation. For other motors, such a mechanism may have implications for torque generation around the filament axis.

Introduction

Translocation of motor proteins along cytoskeletal filaments fulfills diverse cellular functions (1). For example, the dimeric kinesin-1 transports cargo by taking 8 nm steps in a hand-over-hand fashion along microtubules (2–5). Microtubules consist of circularly arranged protein chains, so-called protofilaments, assembled of α/β-tubulin dimers (1). Kinesin-1 tracks a single protofilament (6–8), seldom switching between them (9,10). Sideward motion of other motors has been detected via rotations of filaments driven by multiple motors in gliding assays and by off-axis movement of motor-attached microspheres or quantum dots used as tracking probes. Probe and microtubule rotations imply torque generation for all cytoskeletal motors: myosin (11), dynein (9,12–17) and kinesin (kinesin-1 monomers (18) and dimers (10), kinesin-2 (19,20), kinesin-5 (21), kinesin-8 (16,22), and kinesin-14 (23)). Occasional directed sideward steps—as suggested for kinesin-8—may explain microtubule rotations of motor-ensemble gliding assays (22) or the spiralling motion of multimotor-coated microspheres around microtubules (20). However, motors may also randomly switch protofilaments with a bias toward one direction, resulting in the same net ensemble motion. Although data on single cytoplasmic dyneins suggest a diffusive, i.e., undirected, sideward stepping mechanism (15–17), the switching mechanism of single kinesin motors remains poorly understood. In both cases, how sideward stepping may depend on load is unclear.

To determine the switching mechanism for a kinesin motor, we investigated the translocation of the budding yeast kinesin-8, Kip3. Instead of transporting cargo, Kip3 depolymerizes microtubules and thereby regulates their dynamics and length (24–26). To do so, the motor needs to reach the microtubule end and thus, ideally, should have two features: 1) it should be highly processive—taking many steps without dissociating from the microtubule; and 2) it should be able to bypass obstacles. Whereas the high processivity of Kip3, with an average run length of up to 12 μm (24,27), has been shown to be due to a weakly bound slip state (28,29), in addition to a second microtubule binding site at the motor’s tail (27), the obstacle-bypassing capability has only been proposed (22). To bypass an obstacle, the motor should have the ability to switch microtubule protofilaments, as suggested by microtubule rotations seen in motor-ensemble gliding assays (22). Here, we probed how protofilament switching occurs on the molecular level by applying alternating sideward loads on single, microsphere-coupled Kip3 motors using optical tweezers. In addition, we simulated the motion with force-dependent sideward stepping rates. Both the data and simulation are consistent with a random sideward walk asymmetrically biased by force. The ability to sidestep suggests that Kip3 is well-suited to bypass obstacles on microtubules.

Materials and Methods

Microtubule preparation

Porcine tubulin was polymerized in BRB80 buffer (80 mM PIPES, 1 mM EGTA, and 1 mM MgCl₂, pH 6.9) with 5% dimethyl sulfoxide, 4 mM MgCl₂, and 1 mM GTP for 1 h at 37°C. Afterward, microtubules were diluted with BRB80 containing 10 μM taxol (BRB80T), spun down in a Beckman airfuge (Beckman Coulter, Krefeld, Germany), and resuspended in BRB80T. Unless noted otherwise, all chemicals are from Sigma-Aldrich (St. Louis, MO). Microtubules were visualized with differential interference contrast employing a light-emitting diode (30).

Microsphere preparation

Motors were bound to microspheres via antibodies and a flexible polymer spacer. Carboxylated polystyrene microspheres (mean diameter 0.59 μm, Bangs Lab, Fishers, IN) were coated covalently with a 3 kDa polyethylene glycol linker and a green fluorescent protein (GFP) antibody as described previously (28,31). The GFP antibody was expressed and purified in the protein expression facility of the Max Planck Institute for Molecular Cell Biology and Genetics (Dresden, Germany).

Sample preparation and assay

Experiments were performed in flow cells constructed with silanized, hydrophobic coverslips, as described in our previous work (32). The motor proteins budding yeast Kip3 (His₆-Kip3-eGFP) and rat kinesin-1 (His₆-rkin430-eGFP) were expressed and purified according to methods described in Jannasch et al. (28) and Rogers et al. (33). The motility buffer for Kip3 stepping assays is BRB80 supplemented with 1 mM ATP, casein, taxol, and an antifading mix (28). Motility buffer for assays with nonmotile Kip3 contained 1 mM adenosine 5′-(β,γ-imido)triphosphate (AMPPNP) (Jena Bioscience, Jena, Germany) instead of ATP; motility buffers for kinesin-1 assays had 1–2 μM ATP. Polyethylene glycol microspheres were mixed with kinesins in motility buffer to a motor/microsphere ratio where every third microsphere showed motility, implying single-molecule conditions with 95% confidence (34). The channels of flow cells were washed with BRB80, filled and incubated for 20 min successively with anti β-tubulin I (monoclonal antibody SAP.4G5 from Sigma-Aldrich in BRB80), Pluronic F-127 (1% in BRB80) and microtubules in BRB80T. Finally, the kinesin-microsphere mix was flowed in.

Optical tweezers setup

Measurements were performed in a single-beam optical tweezers setup as described previously (29,35,36). This setup is equipped with a millikelvin precision temperature control, a lateral-force feedback using piezo tilt mirrors and an axial force feedback using the sample stage with a feedback rate of 1 kHz. The trapping objective temperature was 29.2°C. The optical trap is calibrated by analysis of the height-dependent power-spectrum density, as described previously (35,37).

Applying sideward loads with optical tweezers

The force clamp had a sampling rate of 4 kHz and a feedback rate of 1 kHz. The feedback parameters were tuned by using AMPPNP-bound Kip3 under single-molecule conditions and simulating 8 nm steps by moving the sample stage with a stepping rate of 5 Hz (corresponding to a velocity of 40 nm/s). In stepping assays, sideward loads of 0.5 pN were applied with a trap stiffness of 0.02–0.03 pN/nm and alternated every 1.75–20 s. No load was applied along the microtubule axis. In addition, sideward loads of 0.25, 1, and 2.1 pN were applied for an alternating time of 5 s. The range of the force feedback was 3.5 μm, limiting our ability to determine the run length of the motor as a function of sideward load force. Data for forces and microsphere positions were smoothed with a running median filter for visualization. No data points from transients between the alternating loads were used for the analysis. As a measure for the overall sideward motion during alternating times, the y-position signal was fitted by a line; the difference between the end and starting point of this fit has been taken as sideward displacement, $\text{Δ}y$. Means and variances were calculated; variances were averaged between different experiments. We preferentially used microtubules that were parallel to the flow-cell channel, coinciding with the x-axis of the detector and the differential interference contrast camera image. The mean angle relative to this axis of all microtubules used in this work was −2 ± 7° (mean ± SD). The microtubule angle was determined via image analysis with a precision of <1°. Note that the force-feedback automatically tracked the microtubule axis. The recorded data were rotated by this measured angle. Occasionally, this rotation angle was fine-adjusted in the MATLAB analysis to minimize any overall trend in the y-position signal. We determined the zero-position of the y-axis with a precision of ≈10 nm, corresponding approximately to the root mean-square noise on the position traces. For a typical 50-s-long trace, this precision resulted in a systematic deviation of (10/50 = 0.2) nm/s. For an alternating time of 20 s, the systematic deviation would be ∼20 s × 0.2 nm/s = 4 nm. Because we averaged data obtained from many different microtubules, the mean of the systematic deviations should be zero. The error on the rotation angle increased the variance of $\text{Δ}y$—for the 20 s example, it would be 16 nm², which is approximately equal to our measurement precision. For shorter alternating times, the effect is even smaller.

To ensure the functionality of the motors, we measured the speed of Kip3-coated microspheres by video tracking (30) with the optical trap turned off. We determined the speed by linear fits to the tracked x position. The speed was 40 ± 2 nm/s (N = 52, mean ± SE unless noted otherwise), consistent with previous reports (24,28,29). With the trap turned on and no load applied in any direction (zero-force feedback), the speed was 39 ± 2 nm/s (N = 90, zero-force data point in Fig. 1 C), ensuring that the trap and feedback did not affect the motor.

Figure 1.

Sideward loads slow down Kip3 motors. (A) Schematic side view of the stepping assay. A Kip3-coated microsphere is trapped with optical tweezers and placed on an immobilized microtubule. (B) Top view of a motor translocating toward the microtubule plus end (x-axis, black path) with no force applied in the direction of the microtubule axis. The motor is subjected to alternating, constant sideward loads of $F_y={\kappa }_y\text{Δ}{y}_{\text{trap}}$ perpendicular to the microtubule axis (in the y-direction), where Δy_trap is the microsphere displacement from the trap center and ${\kappa }_y$ is the trap stiffness in the y-direction. Schematics are not to scale. (C) Forward speed as a function of the absolute value of sideward (open and solid black circles for left and right, respectively, lower axis) and upward load (open red diamonds, upper axis). Lines represent linear fits to the data. Dashed and solid black lines indicate sideward loads with slopes of −8.6 ± 0.6 nm s⁻¹ pN⁻¹ (mean ± SE, N = 10) and −7.3 ±0.6 nm s⁻¹ pN⁻¹ (N = 9) for leftward and rightward loads, respectively. The red dotted line indicates upward loads with a slope of −4.1 ± 1.2 nm s⁻¹ pN⁻¹ (N = 5). The vertical blue line indicates the mean detachment force and the blue shaded region its standard deviation. To see this figure in color, go online.

Results

Kip3 slowed down under off-axis loads

To measure sideward motion, we used optical tweezers (35–38) to precisely track the forward and sideward motion of single, microsphere-coupled Kip3 motors subjected to sideward loads (Fig. 1, A and B; see Materials and Methods). Kip3-coated microspheres were trapped and placed on taxol-stabilized microtubules. These microtubules generally have 14 protofilaments and a mean supertwist of ≈8 μm (6). Using a force feedback, we applied no load along the microtubule axis and different—on average constant—sideward loads perpendicular to the microtubule axis. We define the motion along the axis of the microtubule to be in the x direction and off-axis, sideward motion to be in the y direction—left and right with respect to the forward direction, corresponding to positive and negative y-values, respectively (Fig. 1 B). Before investigating the sideward motion, we asked how sideward loads affect the forward translocation. During single Kip3 runs, we successively increased the sideward load, with no load applied along the microtubule axis. We found that the motor slowed down and eventually detached at a mean force of 2.9 ± 0.2 pN (N = 31) averaged over left and right (Fig. 1 C). Detachment forces and slopes did not significantly depend on the pulling direction. Based on the extrapolated fits, motors stalled at a sideward load of ∼4–5 pN. Note that due to the geometry, the sideward load (F_y) also causes an upward load (F_z) on the motor. We address the issue of the geometry below. The above results show that sideward loads moderately slowed down Kip3 compared to its ≈1 pN stall force on backward loads (28). However, in contrast to kinesin-1 (39), no significant asymmetry was observed with respect to the sideward pulling direction.

Single Kip3 motors switched protofilaments

To precisely measure the sideward motion of Kip3, we applied sideward loads of 0.5 pN perpendicular to the microtubule axis in the y-direction with no load along the microtubule axis (Figs. 1 B and 2 A). We regularly changed the direction of the load from left to right with different alternating times and recorded the x and y positions of the trapped microsphere as a function of time (see Materials and Methods). At 0.5 pN sideward load, the forward speed was 31.5 ± 0.6 nm/s (N = 400), corresponding to a forward stepping rate of $k_f\approx$ 4 s⁻¹ for 8-nm steps (28) (Fig. 1 C and Fig. S1 in the Supporting Material). In the sideward direction, there were large transient displacements, Δy_L, upon changing the load direction (Fig. 2 B), which we attribute to the lever of the microsphere, motor, and linker. After these transient displacements, small sideward displacements occurred during the constant-load time. We observed discrete sideward steps (Fig. 2 C, left). Our step finder (29) confidently detected a broad distribution of steps with and against load down to ≈3 nm and as large as 30 nm (see Fig. 2 C, right), with a mean dwell time of ≈1 s (Fig. S2). To check for an asymmetry in the mean dwell times, in analogy to limping (3), we separately calculated the dwell times with or against load for left and right, respectively. We did not find a significant difference. Surprisingly, the largest detected step sizes are larger than the largest projected distance between protofilaments of ≈6 nm and even larger than the ≈25 nm diameter of a microtubule. We attribute the large microsphere steps to 1) the geometry (see below), 2) the possibility of two fast subsequent steps, and 3) the root mean-square noise on the traces.

Figure 2.

Kip3-coupled microspheres moved sideward with and against sideward loads. (A) Sideward load force as a function of time (gray trace, raw data with 4 kHz; black trace, median filtered to 8 Hz). Zero load was applied along the microtubule axis (light green trace, raw data with 4 kHz; dark green trace, median filtered to 8 Hz). (B) Sideward position, y, perpendicular to the microtubule axis as a function of time. Δy_L indicates the mean distance between the leftward- and rightward-pulled microsphere. (Insets) Linear fits (red lines) to the sideward motion resulted in the sideward displacement values, Δy, indicated in red. (C) (Left) Representative traces for leftward (positive y) and rightward displacement, plotted together with detected steps (red lines). Traces are offset for clarity. (Right) Histogram of detected step sizes for steps in the direction of or against the applied sideward load. (D) Histograms of sideward displacement, Δy, for leftward and rightward loads of 0.5 pN and an alternating time of 5 s (red bars). Gaussian fit (red line) and the mean (± SE) values are indicated. To see this figure in color, go online.

Because we could not detect all steps, and because the ones we could detect had a large, continuous variation in step size, we measured the sideward displacement, Δy, during the alternating time by a linear fit to the position traces after the transient displacement (Fig. 2 B, insets). For this fit-based sideward displacement, we also measured a broad distribution of both positive and negative sideward displacements independent of the loading direction, with displacements again exceeding the microtubule diameter (Fig. 2 D, red bars). Interestingly, the mean values of the distribution significantly differed from zero: with a leftward and rightward load, there was a small mean displacement to the left and right, respectively, with a larger absolute value for the rightward displacement. Together, these results suggest that Kip3 switched protofilaments in both directions and that the switching was asymmetrically biased by the loading force.

The variance of the sideward displacements increased with time

To determine whether protofilament switching was due to a directed or a random process, we varied the alternating time and determined the mean and variance of the sideward displacement distribution (see histograms in Fig. S3). The absolute mean sideward displacements for left and right (Fig. 3, open and solid red circles, respectively) first increased for alternating times <∼10 s and then, within the sampling error, leveled off or slightly decreased (Fig. 3 A). The initial increase was significantly larger than control measurements using 1) Kip3 in the presence of the nonhydrolyzable ATP analog AMPPNP (Fig. 3 A, blue square, and Fig. S4, A, C, and D); 2) kinesin-1 (Fig. 3, open and solid black diamonds for left and right displacements, respectively, and Fig. S5); or 3) paused motors (Fig. S4, B and E–H). The controls using kinesin-1 also showed no dependence on the alternating time. Thus, for Kip3, the increase in the mean sideward displacement supports the notion that force biases the sideward stepping motion in the direction of the applied load.

Figure 3.

Mean sideward displacement and variance increased with alternating time. Mean sideward displacement (A) (error bars indicate the mean ± SE) and variance (B) as a function of alternating time with a sideward load of 0.5 pN for Kip3 (open circles, leftward displacement; solid circles, rightward displacement), AMPPNP-bound Kip3 (blue square), and rkin430 (open diamonds, leftward displacement; solid diamonds, rightward displacement). Dashed and solid lines are the best-fit simulations for leftward and rightward displacement, respectively (red, Kip3; black, rkin430). A linear fit (gray dashed line), $2Dt+2{\epsilon }^2$, to the first four variance data points, including both directions for Kip3, resulted in a sideward diffusion coefficient, discussed in the text, and a measurement precision of $\epsilon \approx 3$ nm. To see this figure in color, go online.

For Kip3, the variances of the distributions increased with time while also leveling off for longer times (Fig. 3 B). Compared to the Kip3 data, the kinesin-1 data showed a much smaller increase (the initial slope is sevenfold smaller). For Kip3, the initial linear increase of the variance is reminiscent of a diffusive process. A linear fit (Fig. 3 B, dashed gray line) resulted in a sideward diffusion coefficient of D = 20 ± 2 nm² s⁻¹. We can use this diffusion coefficient to estimate the mean sideward stepping rate. To this end, we estimated the average projected distance between protofilaments on the top half of the microtubule to be $\langle \text{Δ}p\rangle \approx 4$ nm (Fig. S6 A). With this estimate, the sideward stepping rate was $k_{\text{Δ}p}=D/{\langle \text{Δ}p\rangle }^2\approx 1.3$ s⁻¹. Sideward movement due to the microtubule supertwist results in net sideward stepping rates that are much smaller compared to this estimate. Therefore, the supertwist should not have a large influence on our measurements (see also Section S11 in the Supporting Material). Taken together, the variance data for short alternating times are consistent with a random sideward walk. Yet this simple analysis cannot account for the data at long alternating times and do not explain the broad distribution of sideward displacements and step sizes.

In addition to the alternating time, we varied the sideward load for an alternating time of 5 s (Fig. S7). The measurement revealed an increasing absolute mean sideward displacement with higher loads in both directions (Fig. S7 A). The variance decreased up to 0.5 pN (Fig. S7 B). For forces >0.5 pN, the variance was larger. Based on video images, we attribute this larger variance to insufficiently immobilized microtubules. In extreme cases, which were not used for data anlysis, microtubules were visibly displaced in the lateral direction for forces >0.5 pN.

A simulation accounting for the geometry supports an asymmetrically biased diffusion mechanism

To gain a deeper understanding of the observed data, we considered the experimental geometry to scale. In the above estimate for the sideward stepping rate, we tacitly assumed that the projected distance between protofilaments corresponds to the measured microsphere displacements. This assumption does not hold on close inspection of the geometry drawn to scale in Fig. 4 A. The microsphere with radius R is held by the optical trap, which pulls taut the linker of length L between the microtubule of radius r and the microsphere. We assume that the microsphere does not change its distance, h, to the surface during the stepping motion of the motors. This assumption is supported by surface-force measurements (35) and the vertical displacements during the alternating time, which on average do not significantly differ from zero (Fig. 4 A, lower inset, and Fig. S8). With this geometry, the lateral microsphere center position, y, can be calculated from the angular motor position, ϕ, on the microtubule (for an analytic expression, and for other details of the geometry, see Sections S7 and S10, respectively, in the Supporting Material).

Figure 4.

Geometry to scale and simulated traces. (A) Geometry of microsphere (gray), linker, and microtubule (red) with R = 295 nm, L = 34 nm (see Sections S7 and S10 in the Supporting Material and Fig. S9), h = 20 nm (35), and r = 12.5 nm, drawn to scale. Different-colored lines (black, green, and blue) correspond to different angular motor positions. (Upper inset) Magnification of microsphere center position. (Lower inset) Histogram of vertical displacement, $\text{Δ}z$, with mean ± SE and Gaussian fit for an alternating time of 2.5 s. (B) Microsphere position, y, as a function of angular motor position, ϕ, with a 360/13 ≈ 28° grid to illustrate the various sideward step sizes between protofilaments. Colored circles (black, green, and blue) indicate the angular positions sketched in (A). (C) Plot of the tangential (red solid line) and normal forces (green dashed line) acting on Kip3 as a function of angular position, ϕ, relative to the applied sideward load (ocher dotted line). At the maximum microsphere position corresponding to the angle ϕ_max ≈ 62°, the tangential force is zero. (D) Typical simulated traces of forward (left) and sideward (right, red line, left axis) motion of a Kip3-coupled microsphere along with the angular motor position (right, green line, right axis) as a function of time. (Inset) Magnification of forward steps showing small forward displacements at the times of sideward steps. To see this figure in color, go online.

Fig. 4 B shows the nonlinear dependence of the microsphere position, y(ϕ), on the angular motor position, ϕ. This function and the corresponding microsphere displacements upon sideward stepping are consistent with the large sideward displacements seen in the experiment (Fig. 2, C and D). For the angular range shown in Fig. 4 B, the difference between the maximum and minimum positions is Δy ≈ 60 nm. Thus, sideward microsphere displacements of up to 60 nm are possible, even though the motor has only moved roughly the projected distance corresponding to the microtubule diameter of 25 nm. The geometry results in an amplified microsphere displacement compared to the motor displacement. For single steps, the microsphere displacements can continuously vary between −30 nm and +40 nm (Fig. S6 B), consistent with the measured broad step-size distribution (Fig. 2 C).

The function y(ϕ) is not monotonic, but has a maximum. This maximum results in a counterintuitive phenomenon: if in a gedanken experiment the motor starts at an angular position close to the maximum (Fig. 4, A and B, green lines and circle, respectively) and takes two clockwise angular steps to the right (Fig. 4, A and B, blue lines and circle, respectively), the microsphere center position also moves to the right (Fig. 4 A, upper inset). However, if the motor takes two counterclockwise angular steps to the left (Fig. 4, A and B, black lines and circle, respectively), the microsphere center position moves to the right, as in the case of clockwise steps, not to the left. Thus, the microsphere movement may reflect neither the angular directionality of the motor nor the projected distance between protofilaments. For the same reason, the directionality and magnitude of the force acting on the motor may differ from the applied load (Fig. 4 C). Depending on the angular position of the motor, the force tangential to the microtubule cross section—corresponding to the sideward force in the reference frame of the motor (Fig. 4 C, red solid line, and Fig. S9)—can be more than twice the sideward load applied with the optical tweezers (Fig. 4 C, ocher dotted line) and also of opposite direction. Note that at the maximal displacement of the microsphere, the tangential force is zero and the normal force (Fig. 4 C, green dashed line) reaches a maximum. Using the relationship between microsphere position and angular motor position, we can reestimate the sideward stepping rate. Based on the expected mean sideward stepping distance measured via the microsphere, ≈8.9 ± 0.4 nm (Fig. S6), the rate is $k_{\text{Δ}y}=\left(0.30\pm 0.06\right)$ s⁻¹, which is around four times lower than the estimate based on the projected filament distance. Thus, the geometry leads to counterintuitive movement of the microsphere and does not allow for an analytical solution to describe our data.

To quantitatively describe all of our data, we simulated the sideward stepping motion of the kinesin motors accounting for the geometry (see Section S11 in the Supporting Material for details). Because force biased the sideward motion, we simulated the motor translocation along the microtubule with Arrhenius-type, force-dependent sideward stepping rates,

$$k_{\text{l,r}}\left(F_{\text{tang}}\right)=k_{\text{l,r}}^0\text{exp}\left[\pm F_{\text{tang}}\times x_{\text{l,r}}^{†}\text{/}\left(k_{\text{B}}T\right)\right],$$ (1)

where $k_{\text{l}}^0$ and $k_{\text{r}}^0$ are the zero-force sideward stepping rates toward the left and right (+ and − in the exponential, respectively), $x_{\text{l,r}}^{†}$ are the distances to the transition states for the respective directions, $F_{\text{tang}}$ is the tangential force acting on the motor, $k_{\text{B}}$ is the Boltzmann constant, and T is the absolute temperature. In addition, we accounted for the microtubule supertwist, even though it had little influence on the results of the simulation (see Section S11 in the Supporting Material and Fig. S11). Furthermore, because the distance to the directly adjacent tubulin dimers is shortest (1,22), we assumed that a sideward step is to one of the neighboring tubulin dimers (not diagonally to the front left or front right). The motor translocation was simulated and analyzed in the same manner in which the experiments were performed. We simulated 200 traces consisting of the same number of alternating cycles as we acquired during the experiments for >100,000 sets of the sideward stepping parameters ($k_{\text{l}}^0,k_{\text{r}}^0,x_{\text{l}}^{†},x_{\text{r}}^{†}$) as a function of alternating time. Then, we calculated the simulated mean and variance of the sideward-displacement histograms. To account for the kinesin-1 variance increase, we added a global best-fit linear increase to both the simulated kinesin-1 and Kip3 variance. Subsequently, we calculated the mean-square deviation of the simulated values from the experimental data normalized by the experimental error bars and number of degrees of freedom. In this manner, we determined a reduced ${\chi }_{\text{red}}^2$-value for each set of simulated parameters.

The best-fit kinetic parameters corresponding to the simulation set with the smallest ${\chi }_{\text{red}}^2$-value averaged over a simulation with and without variable supertwist are given in Table 1 (see Sections S10 and S11 in the Supporting Material). Within the sample error, the zero-force sideward stepping rates toward the left and right of $k_0\approx 0.3$ s⁻¹ did not differ. The distance to the transition states of 3.6 ± 0.8 nm and 2.5 ± 1.0 nm (mean ± SD of N = 95 steps) for left and right, respectively, depended on the direction. Thus, in the absence of force, the simulation results support a purely diffusive sideward motion with a sideward stepping time of $\tau ={\left(2k_0\right)}^{-1}\approx 1.7$ s. A directed process, i.e., with one of the rates being zero, does not describe our data well (see Figs. S11 and S12). The mean and variance of the simulated sideward displacements using the best-fit parameters are plotted in Fig. 3 (dotted and solid lines, respectively); an exemplary stepping trace is plotted in Fig. 4 D. The simulated trace shows the counterintuitive effect that the sideward displacement, y, may be positive, close to zero, or negative, even though the angular position always changes in the same direction with the same magnitude (see angular steps at ≈2 s, 3 s, and 5 s in Fig. 4 D). Overall, the simulation fits the experimental data very well, supporting our diffusive, asymmetrically force-biased stepping model.

Table 1.

Best-fit simulation parameters

$x_{\text{l}}^{†}$	$x_{\text{r}}^{†}$	$k_{\text{l}}^0$	$k_{\text{r}}^0$
3.6 ± 0.8 nm	2.5 ± 1.0 nm	0.31 ± 0.03 s−1	0.28 ± 0.04 s−1

Values are expressed as the mean ± SD (N = 95).

Discussion

Our experiments show that 1) Kip3 motor-attached microspheres moved on average sideward, in the direction of load; 2) the variance of the sideward displacement distribution increased with increasing sideward pulling time; and 3) detected individual sideward steps had a broad distribution both in the direction of applied load and against it. Although the latter two points are indicative of a diffusive process, the geometry may also be the cause for apparent bidirectional microsphere steps for a unidirectional, i.e., directed angular motion of the motors. However, such unidirectional motion is inconsistent with the load-induced sideward motion and steric hindrance. For a unidirectional angular motion, the motor would have to pass through underneath the microtubule, which is sterically impossible due to the attached microsphere.

Why do the mean and variance of the sideward displacement distributions saturate or even decrease for long alternating times? The geometry of the experiment explained the broad distribution in sideward step sizes and sideward distances. Moreover, because of the geometry, in addition to the force-dependent sideward stepping rates, we expect that for long alternating times the motor should on average localize to the protofilament oriented at the angle ϕ_max, for which the tangential force is zero (Fig. 4 C). For deviations away from this angular position, the tangential force exponentially increases the counteracting sideward stepping rate while exponentially decreasing the rate in the direction pointing away from ϕ_max. Therefore, after a transient displacement, the microsphere position should fluctuate around the position y(ϕ_max). For long alternating times, the transient displacement to reach ϕ_max contributes little to the linear-fit-based sideward displacement (as defined in Fig. 2 B), resulting in an overall mean sideward displacement, Δy, approaching zero. Since the force dependence of the leftward rate is larger, we expect that the mean angular position ϕ_max is reached faster for leftward displacement compared to rightward displacement. Therefore, the transient displacement is shorter and contributes less to the mean sideward displacement. The smaller contribution results in an overall smaller absolute mean sideward displacement to the left compared to the right. Thus, the dynamics of the system causes a larger mean displacement to the right, even though the leftward stepping rate is more sensitive to force. For long alternating times, we expect the variance to approach a constant value resulting from the fluctuations around the mean angular position. Taken together, our model is consistent with all of our experimental observations.

We assumed that the normal force did not affect the sideward stepping. However, we observed that the sideward load slowed down the forward motion. We attribute this slowdown to the normal force acting on the motor. To support this notion, we measured the forward speed while pulling upward on the motor (Fig. 1 C, red open diamonds, top axis). We scaled the upward-force axis relative to the sideward-force axis by dividing the latter by cos(ϕ_max), which corresponds to the normal load force expected according to our model under sideward pulling conditions after the transient displacement. With such a scaling, the data overlap: within the margin of error, the decrease in forward speed upon upward loading suggests that the normal load is the key parameter that slows down the forward motion of the motor. Although we could not measure any limping, we would expect the motor to limp with increasing sideward and normal loads.

For a single motor, our model suggests a purely diffusive sideward motion in the absence of loads, with about every ninth step ($\left[2k_0+k_f\right]/\left(2k_0\right)=\left[0.6+5\right]/0.6\approx 9$) of the motor being a sideward step in a random direction. However, microtubule rotations observed in gliding assays suggested a leftward bias of the steps (22). According to our model, this leftward bias is due to the different force dependence of the sideward stepping rates (Table 1). The molecular origin of this difference may be due to the asymmetric structure of the kinesin head with respect to the neck linker (22). In the gliding assays, multiple motors operate together. Because motors do not step in synchrony, more advanced motors should apply loads on lagging ones. Since motors are attached to various angular positions on the microtubule, these loads have components both in the direction of the microtubule axis and perpendicular to it, causing tangential forces on the motor. Because of the asymmetric force bias in the sideward stepping rates, the ensemble of motors should on average step to the left, consistent with the gliding assays (see Fig. S13 and Section S12 in the Supporting Material). A similar collective effect has been observed for dynein (17), suggesting that dynein may also show a difference in the force dependence of the sideward stepping rates. Kip3 has a weakly bound slip state (28). The motor switches to this short-lived state at a frequency comparable to the sideward stepping rate. Whether sideward stepping is related to this state is unclear at the moment. A weakly bound state may give the motor a longer reach to binding sites on neighboring protofilaments (22). Whether sideward stepping is coupled to ATP hydrolysis, and how the sideward stepping rate depends on the ATP concentration, is also unclear. We tested trapping assays at reduced ATP concentrations. However, a quantitative analysis of these assays turned out to be nearly impossible due to very low motor speeds and a seemingly reduced stall force. If sideward stepping is coupled to ATP hydrolysis, we would expect a small zero-force asymmetry in the sideward stepping rates, which we could not determine within our error margins. Such an asymmetry should arise because of the helical microtubule geometry and the asymmetry of the motor (22).

What biological relevance does the diffusive sideward stepping mechanism have for Kip3? For axonemal dynein motors, off-axis movement—causing microtubule rotations and, thus, torque—may be important for the three-dimensional motion of the flagellar beat (12,13); for cytoplasmic dynein, sideward steps may be an essential biological requirement such that heads can pass each other, obstacles, or counter-propagating kinesin motors (15–17,40). For kinesin motors, the ability to bypass obstacles is also an essential property for cargo transport (41–43). How torque generation (12,13,18–21) on the cargo, i.e., a rotation of the cargo around the filament axis, induced by sideward stepping influences cargo transport remains to be seen. Kip3 does not transport cargo but must reach the microtubule end for length regulation. Therefore, being able to bypass obstacles on both sides seems to be the most efficient way to achieve this. The asymmetric force bias may not have a biological function for Kip3.

Author Contributions

M.B. and E.S. designed the research, M.B. performed all measurements, and E.B. programmed the simulation. M.B. and E.S. analyzed the data, developed the model, and wrote the manuscript.

Supporting Citations

Reference (44) appears in the Supporting Material.