Figure 2. Rotation of V₁.

(a–e) Typical time courses of rotation with a 40-nm bead (or duplex). (a) Rotation at 4 mM ATP captured at 8,000 fps; (b) 200 μM ATP at 2,000 fps; (c) 40 μM ATP at 250 fps; (d) 4 μM ATP at 4,000 fps and (e) 2 mM ATP at 4,000 fps, obtained from the same molecule as in d after medium exchange. Trajectories of the bead centroid (axis divisions: 11.1 nm) and histograms of angular positions, both for the indicated portion of the records, are shown in the upper and lower insets, respectively. (f–i) Histograms of dwell times between 120° steps. (f) Dwell times at 4 mM ATP with 125 μs bin size obtained from 6 molecules observed at 8,000 fps; (g) 200 μM ATP, 250 μs bin size, 6 molecules at 8,000 fps; (h) 40 μM ATP, 1 ms bin size, 6 molecules at 4,000 fps; (i) 4 μM ATP, 4 ms bin, 15 molecules at 2,000 fps. Orange curves show fit with the sequential two-reaction scheme with rates k_a and k_b: constant·(exp(−k_at) − exp(−k_bt)). At 4 mM ATP, the two rates turned out to be indistinguishable and thus the fit was made with two identical rates k: constant·t·exp(−kt). The estimated rates and associated s.e. are: k^{4 mM}=0.36±0.01 ms⁻¹, k_a^{200 μM}=0.17±0.02 ms⁻¹, k_b^{200 μM}=0.28±0.03 ms⁻¹, k_a^{40 μM}=31±1 s⁻¹, k_b^{40 μM}=0.40±0.03 ms⁻¹, and k_a^{4 μM}=6.1±0.1 s⁻¹, k_b^{4 μM}=0.26±0.02 ms⁻¹. If we assume that k_a represents the rate of ATP binding (k_a=k_on[ATP]), k_on is given as 0.85×10⁶ M⁻¹ s⁻¹ at 200 μM ATP, 0.78×10⁶ M⁻¹ s⁻¹ at 40 μM and 1.5×10⁶ M⁻¹ s⁻¹ at 4 μM. At 4 μM, k_on should dominate the histogram, and the green fit with constant·exp(−k_on[ATP]t) gave k_on of 1.5×10⁶ M⁻¹ s⁻¹. Blue curves show a global fit to f–h (equal weight for each count), with sequential reactions starting with ATP binding at the rate k_on[ATP] and two ATP-independent reactions with rates k₁ and k₂: constant·{(k₂−k₁)·exp(−k_on[ATP]t) + (k_on[ATP]−k₂)·exp(−k₁t) + (k₁−k_on[ATP])·exp(−k₂t)} with k_on=(1.2±0.1)×10⁶ M⁻¹ s⁻¹, k₁=0.49±0.05 ms⁻¹, k₂=0.34±0.04 ms⁻¹.