Kinesin is a two-headed, ATP-dependent motor protein1,2 that moves along microtubules indiscrete steps3 of 8 nm. In vitro, single molecules produceprocessive movement4,5, motors typically take ∼100steps before releasing from a microtubule5,6,7 . A central question relates tomechanochemical coupling in this enzyme: how many molecules ofATP are consumed per step? For the actomyosin system,experimental approaches to this issue have generated considerablecontroversy8,9. Here we take advantage of theprocessivity of kinesin to determine the coupling ratio withoutrecourse to direct measurements of ATPase activity, which aresubject to large experimental uncertainties8,10,11,12. Beads carrying singlemolecules of kinesin moving on microtubules were tracked with highspatial and temporal resolution by interferometry3,13. Statistical analysis of theintervals between steps at limiting ATP, and studies offluctuations in motor speed as a function of ATPconcentration14,15, allow the coupling ratio to bedetermined. At near-zero load, kinesin moleculeshydrolyse a single ATP molecule per 8-nm advance. Thisfinding excludes various one-to-many andmany-to-one coupling schemes, analogous to thoseadvanced for myosin, and places severe constraints on models for movement.
Silica beads (0.5 µm diameter) were prepared with fewer than one molecule of kinesin, on average, bound nonspecifically to their surfaces. Beads were suspended in buffers containing variable amounts of ATP and introduced into a microscope flow cell. Individual diffusing beads were captured by an optical trap, deposited onto immobilized microtubules bound to the coverglass, and subsequent movements recorded with subnanometre resolution by optical-trapping interferometry3,13. To ensure that beads were propelled by single molecules, they were prepared using extremely low concentrations of kinesin protein, sufficiently diluted from stock to make it unlikely that more than one molecule was present on a bead5,13. In this regime, the fraction of beads moving as a function of kinesin concentration obeys Poisson statistics, confirming earlier findings4,5 (Fig. 1). To work at the lowest possible mechanical loads, the optical trap stiffness was set to just 7 fN nm−1, producing a mean applied force of <0.9 pN in the optical trap (<15% of kinesin stall force13).
Average rates of kinesin movement were measured over three decades of ATP concentration. At saturating ATP levels, speeds were statistically identical to those measured by video tracking13 under no load, confirming the near-zero load condition. Speed data were well fit by Michaelis–Menten kinetics, with an apparent Km for movement of 62 ± 5 µM and kcat of 680± 31 nm s−1, comparable to values found in microtubule gliding assays4 as well as solution studies of kinesin ATPase activity16,17,18. This implies a Hill coefficient of one, and excludes models for movement that depart significantly from Michaelis–Menten kinetics19. Because hydrolysis activity in solution16,17,18 and speed in vitro display the same functional dependence on ATP, the coupling ratio—defined as the number of ATP molecules hydrolysed per mechanical advance—must be independent of ATP concentration. At limiting concentrations of ATP, kcat/Km gives a velocity of 11 ± 1 nm s−1 µM−1. In this regime, kinesin molecules advanced in clear increments of 8 nm (Fig. 2a), implying a stepping rate of 1.4 ± 0.1 µM−1 s−1 (Fig. 1). Statistical analysis of records, based on power spectra derived from pairwise distance differences3, revealed no signals corresponding to steps of another size, although a small minority of steps of other sizes cannot be excluded (Fig. 2b,c). However, we did not confirm data from one study reporting roughly comparable numbers of 3-, 5- and 8-nm steps (ref. 20).
How long are the intervals between steps? Knowledge of this distribution at rate-limiting ATP may be used to determine the coupling. If kinesin molecules require just one ATP molecule per step, the distribution will be exponential, reflecting a solitary, rate-limiting biochemical reaction (binding of ATP). Conversely, a requirement for several (n) ATP molecules would lead to a distribution that is the convolution of n exponentials15. Pooling data from runs at 0.75, 1 and 2 µM ATP, during which steps could be identified in the records (Fig. 2), produced a distribution that was well fit by a single exponential (reduced χ12= 0.6) with a rate of 1.1 ± 0.1 µM−1 s−1 (Fig. 3a). This rate is in satisfactory agreement with the value derived independently from speed measurements (Fig. 1). In contrast, a convolution of two exponentials did not fit the data as well (χ22= 1.4, P(F = χ12/χ22) < 0.22), and yielded a stepping rate (0.8 ± 0.06 µM−1 s−1) less compatible with kcat/Km. The observed step distribution supports the view that kinesin proteins bind a single ATP molecule before advancing by 8 nm.
To extend and confirm this finding, but without the need to identify individual stepwise transitions, we performed a fluctuation analysis of kinesin movement. For single processive motors, fluctuations about the average speed reflect the underlying enzyme stochasticity14,15. Fluctuation analysis has been applied successfully to studies of the bacterial rotary motor21. Earlier work has shown that fluctuations can reveal the number of rate-limiting biochemical transitions per mechanical step14,15; for kinesin molecules moving at saturating levels of ATP, there are approximately two such rate-limiting events14.
Fluctuation analysis of processive motor proteins relies on the determination of the randomness parameter, r, a dimensionless measure of the temporal irregularity between steps14,15. A motor with r = 0 is perfectly clock-like, and takes an invariant time between steps. Conversely, a ‘Poisson motor’ with a single rate-limiting transition has r = 1. Additional rate-limiting transitions generally lead to 0< r < 1. For motors that step by a distance, d, whose positions are functions of time, x (t), the randomness parameter is defined as
where angle brackets denote the ensemble average. Both numerator and denominator in eqn (1)increase linearly with time (Fig. 2d), and their ratio approaches a constant whose reciprocal, r−1, supplies a continuous measure of the number of rate-limiting transitions per step. Computations of the randomness parameter are robust to sources of thermal and instrumentation noise14,15, thereby avoiding known difficulties associated with identifying individual steps against noisy backgrounds22. Nevertheless, the randomness parameter can be affected by factors that increase variance at long time scales, such as occasional backward steps, inactivated states, or steps of larger than normal size, which can all raise r beyond unity.
To ensure accurate determination of r, controls used both real and simulated data. We simulated runs at both saturating and limiting ATP conditions by generating stochastic ‘staircase’ records, consisting of 8-nm steps at either high or low average speeds (∼630 nm s−1, ∼10 nm s−1) corrupted by gaussian white noise with total power comparable to that of real thermal noise. The times between steps were described by either an exponential distribution or a convolution of two exponentials. Fluctuation analysis of such records demonstrated that r could be determined to better than 15%, and that the appropriate stepping statistics could thereby be distinguished (Table 1). To mimic data at low ATP, we fixed kinesin-coated beads on microtubules using 2 mM AMP-PNP, a non-hydrolysable ATP analogue. The microscope stage was then moved in nanometre-sized increments to generate simulated records with average speeds comparable to those in 1 µM ATP (∼10 nm s−1) (Fig. 1 ). Once again, the number of rate-limiting transitions per step could readily be distinguished from the value of r. Individual steps in the latter records could also be detected by low-pass filtering the data. As a final control, we reconstructed interval distributions from these records; these were consistently fit by the appropriate model with either one or two rate-limiting steps (data not shown).
From records of kinesin-driven movement, we computed r as a function of ATP over a thousandfold range of concentration (Fig. 3). The qualitative shape of this relation had been predicted from the catalytic pathway14,15. At high ATP levels, the curve is asymptotic to r ≈ 1/2. At intermediate ATP levels, the curve dips slightly, then rises at limiting ATP to a value near unity. The experimental value at saturating ATP confirms an earlier measurement14 and implies that each step involves a minimum of two rate-limiting transitions. Biochemical pathways derived from transient and steady-state kinetic studies also predict r ≈ 1/2 at high ATP levels, with the additional assumption that each mechanical step corresponds to one hydrolysis15,17,18. For ATP levels near Km, r drops as the rate of binding becomes comparably slow to other reactions. At limiting ATP, r rises through 1, reflecting a single rate-limiting transition occurring once per advance (ATP binding). The slight increase of r beyond unity reflects additional aspects of motor behaviour that tend to increase the variance.
Fluctuation analysis is robust to heterogeneity in the ATP binding rate (yielding heterogeneous mean speeds at limiting ATP), and controls allowed us to eliminate heterogeneity in bead size or in the stiffness of the bead–kinesin linkage as possible sources of additional variance (Table 1; Methods). Futile hydrolysis events that do not lead to a step can increase r, but not above unity14. Occasional pauses in movement, perhaps resulting from momentary sticking of beads to the coverglass or from motors transiently entering non-motile biochemical states, also increase variance. Sticking can be eliminated as an explanation because it would produce a concomitant reduction in brownian noise, and this was not observed. Transient inactive states causing r ≳ 1 at limiting ATP result in even further increases beyond this value of randomness at moderate to saturating levels of ATP, and are therefore inconsistent with the data. Two more possibilities merit examination: that ATP hydrolysis might occasionally produce backwards movement, a behaviour noted previously at higher external loads13 and also in this study; and that a single ATP hydrolysis might occasionally produce a double step of 16 nm, a transition that would appear indistinguishable from two 8-nm steps in rapid succession.
How frequent must such events be to give r = 1.2–1.5 at limiting ATP? We addressed this question by using analytical expressions for r subject to each of the candidate mechanisms (Methods). The observed randomness could be generated if either ∼16% of steps were 16 nm or ∼7% of steps were backwards. By visual examination of records, we scored 6–9% of all steps at low ATP as being backwards. However, using the identical approach, 4–5% of steps were spuriously identified as backwards in controls in which beads were attached in rigor and the stage stepped stochastically in one direction only. Neither mechanism (nor admixtures of the two) can therefore be eliminated, and either produces an acceptable, one-parameter fit to the data (Fig. 3b). However, the additional variance derives from a minority of events: 84–93% of 8-nm steps correspond to a single ATP hydrolysis. In contrast, if 16-nm steps per hydrolysis were the norm, this behaviour would be unmistakable in the records, and the value of r would be 2 (or more) at limiting ATP. Our data are equally incompatible with 8-nm steps arising from the hydrolysis of two ATP molecules, because neither mechanism is adequate to raise the limiting value of r at low ATP from 0.5 to the observed value: to do so would require that we fail to detect that in excess of ∼20% of steps went backwards, and no fraction whatsoever of 16-nm advances alone could raise r from 0.5 to beyond 1.
These findings place important constraints on theories of kinesin movement. The Michaelis–Menten dependence of kinesin velocity4 (Fig. 1) is consistent with a solitary ATP binding event per step, and is incompatible with a simultaneous requirement for two ATP molecules. However, Michaelis–Menten kinetics are equally consistent with kinetic schemes in which two ATP molecules bind sequentially to each of two heads. This can happen, in particular, whenever the two binding events are separated by a kinetically irreversible transition. Specific examples include ATP binding to each of two cooperative sites in an ordered sequence to produce a step, or ATP binding irreversibly in random order to each of two non-interacting sites. Such possibilities are excluded by our data. Furthermore, we can exclude models in which kinesin molecules hydrolyse multiple ATPs per 8-nm step23, and all models in which ATP hydrolysis results in an average advance in excess of 8–9 nm, (the so-called ‘many-to-one’ coupling scenarios9,11). Models that remain consistent with our study include those in which the centroid of the molecule advances in increments of 8 nm (refs 22, 24, 25), for example through alternating 16-nm steps by each of the two heads23,26,27, or those in which each ATP molecule produces a composite of two shorter ‘substeps’28. However, to be consistent with our findings, substeps must obey certain constraints. First, one of the substeps must be ATP-dependent, whereas the other must be ATP-independent. Second, the ATP-independent substep must be at least as fast as kcat (or faster), in view of the behaviour at saturating ATP levels. Putting in numbers, this implies the maximum duration between substeps must be under ∼15 ms at low loads, and so could not be resolved with instrumentation such as ours. If substeps were somehow to be detected under other conditions (for example, if 8-nm steps were actually a composite of 3- and 5-nm substeps that could be visualized at moderate load, separated by ∼50 ms and more20) this would imply that the ATP-independent substep was strongly load dependent, and that its rate slowed dramatically with increased load. However, one model that incorporates substeps predicts that the second substep should proceed rapidly, not slowly, under such loads23. Addressing this issue experimentally will require additional work at higher forces.
The size of the kinesin motor domain29 is quite small, 4.5 × 4.5 × 7.0 nm. A sequence of ∼40 amino acids beyond the carboxy end of the crystal structure is thought to form an α-helix, and may supply an additional ∼3 nm of leeway before the two kinesin heavy chains joint to form a coiled-coil tail. However, this segment is probably too short to constitute a ‘lever arm’ analogous to the structure proposed to be the site of displacement and force generation in myosin30. An 8-nm step therefore poses challenges for understanding the molecular basis of kinesin movement. We anticipate that further applications of fluctuation analysis will supply useful clues to unravelling the mechanochemistry. Moreover, the power of this approach to provide insight into single-molecule kinetics makes it applicable to other processive systems, such as nucleic acid enzymes, including polymerases, helicases, topoisomerases and ribosomes.
Assays. Motility assays were performed as previously described3,5,13, except that microtubules were made from bovine brain tubulin (Cytoskeleton), coverslips were coated with poly-L-lysine rather than silanized, KCl was replaced by K-acetate, and an oxygen scavenging system (250 µg ml−1 glucose oxidase, 30 µg ml−1 catalase, 4.5 mg ml−1 glucose; Sigma) was used. An ATP regenerating system stabilized low ATP levels5,13. Kinesin was purified from squid optic lobe as described1. As a precaution against variability and to ensure work in the single molecule regime, only data from assays in which fewer than half of all beads moved were analysed.
Calibration and analysis. Optical-trapping interferometry and calibrations were performed as described3,13. A trap stiffness of 7 fN nm−1 (average load <0.9 pN over 50–200 nm from the trap centre) was obtained using 12 mW illumination at the specimen plane from a Nd : YLF laser (1047 nm, Spectra Physics). Interferometer output signals were sampled at 2 kHz, anti-alias filtered at 1 kHz, digitized at 12 bits, and stored by computer. Analysis programs were written in LabView 4.01 (National Instruments). Velocities, fluctuations and pairwise distances were analysed as described previously3,14. Multiple runs over 50–200 nm from the trap centre were corrected to adjust for the kinesin-to-bead linkage compliance3,13, except for the example in Fig. 2a, which was selected for low noise3. Mean velocities at each ATP concentration, 〈v〉 = 〈x(t)〉/ t, were determined by line fits to the average pairwise separation versus time for each run; slopes from such fits were averaged over all runs. The variance, 〈x2(t)〉 − 〈 x(t)〉2, was computed as the mean-squared deviation from the trajectory 〈x( t)〉 = 〈v〉t, time-averaged for each run, then averaged over all runs. A line fit was made to this variance over the first 20 nm/〈v〉 time interval, after excluding the first 3.5 ms, because of the ∼1–3 ms correlation time for bead position3,14. Theory and simulations show that a linear increase of variancewith time (Fig. 2d) reflects a lack of any significant heterogeneity inmean velocity. The latter would produce a variance that grew quadratically with time. The standard error in variance also grew linearly with time, therebysupplying an error estimate for the slope. Randomness, r, was computed from the slope of the variance line fit divided by d〈v〉. Experimental and control runs at limiting ATP levels were smoothed over a 5-ms time window with a linear filter, then decimated at 5-ms intervals, a procedure that reduces computation time but does not affect r. Subject to the minimal assumption of exactly two rate-limiting transitions per step at saturating ATP, r as a function of ATP concentration is determined by two parameters: the apparent Michaelis–Menten constant, and the ATP binding rate, determined in Fig. 1. To ascertain the effect of backward movements or 16-nm steps on r, we used the following expressions: r = 1/(1 − 2 P−) + (r̃ − 1)(1 − 2P−), and r = (1 + 3P16)/(1 + P16) + (r̃ − 1)(1 + P16) , where P− and P16 are the probabilities of backwards and 16-nm events, respectively, and r̃ is the randomness in their absence (M.J.S. and S.M.B., unpublished). By using experimental values for Km and binding rate, the randomness data of Fig. 3were fit to each of these expressions using a single free parameter. For analysis of pairwise distances and times between advances, runs at 0.75, 1 and 2 µM ATP were median-filtered with a window of 20–75 ms. Step times were identified by eye using custom software that returned the interval selected by cursors superposed on a graphical display of displacement versus time. Events shorter than 0.5 µM s were excluded from analysis to avoid artefacts arising from the missing-event problem14.
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We thank S. Gross, W. Ryu, L. Satterwhite, M. Wang and especially K. Visscher for technical assistance and discussions; S. Gross for help in purifying kinesin; and P. Mitra and K. Svoboda for advice in the early stages of this project. This work was supported by a grant from NIGMS (S.M.B.) and predoctoral fellowships from NSF and American Heart Association (M.J.S.).
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Schnitzer, M., Block, S. Kinesin hydrolyses one ATP per 8-nm step. Nature 388, 386–390 (1997). https://doi.org/10.1038/41111
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