Angle-dependent rotation velocity consistent with ADP release in bacterial F 1 -ATPase

A model-based method is used to extract a short-lived state in the rotation kinetics of the F1-ATPase of a bacterial species, Paracoccus denitrificans (PdF1). Imaged as a single molecule, PdF1 takes large 120 ø steps during it rotation. The apparent lack of further substeps in the trajectories not only renders the rotation of PdF1 unlike that of other F-ATPases, but also hinders the establishment of its mechano-chemical kinetic scheme. We addressed these challenges using the angular velocity extracted from the single-molecule trajectories and compare it with its theoretically calculated counterpart. The theory-experiment comparison indicate the presence of a 20μs lifetime state, 40 o after ATP binding. We identify a kinetic cycle in which this state is a three-nucleotide occupancy state prior to ADP release from another site. A similar state was also reported in our earlier study of the Thermophilic bacillus F1-ATPase (lifetime ∼10μ s), suggesting thereby a common mechanism for removing a nucleotide release bottleneck in the rotary mechanism.

between subunits within some samples. The existing irregularities were believed to be caused by nanoscale imperfections on the substrate surface rather than abnormal operation of the enzyme itself, hence, a rotational correction was performed by tilting the z-axis of the PdF1. Rotational correction was performed using an algorithm that optimized similarity between the three subunits, with an example output shown in Fig. S2. The goodness-of-fit peak corresponded to optimal tilting and rotation values relative to the experimental origin.
The correction contributed to comparability between subunits, as shown in Fig. S3.
Despite some lingering differences after correction, subunits within respective samples yielded more consistent results overall, generally allowing for comparative analysis of subunits within samples. Torsional spring constant Corrected data was used to calculate an improved estimate of respective systems' torsional spring constants. Original estimates shown in Fig. S4 are asymmetric with significant variances in subunit height and width; trends that do not make sense on account of being recorded in the same system. The data in Fig. S5 provides significantly more similarity between calculated torsional spring constants.
The calculated uncorrected and corrected spring constants can be seen in Table S1 and  Table S1 and S2 were sequenced according to the sequential magnitudes of spring constant as opposed to grouping by biological structure.

Angular fluctuations in the dwell
Examining dwell behavior shows that fluctuations are centered around chemical equilibrium at 0, as seen in Fig. S6, a result consistent with Brownian noise. Within Fig. S6, any perceived change in dwell magnitude is due to a smaller quantity of data points rather than an actual behavior. The distribution of angular position within dwells can be seen in Fig.   S7, the samples mostly follow a Gaussian with a heavier right-tail and a recorded mean near zero.

Identifying definitive dwell states
Visually, dwell regions were majorly identifiable. In Fig. S8, the dwell areas are represented using a solid line. As dwells reflect when the enzyme is in chemical equilibrium, the net change in angle is zero. Practically there were some dwells that had a nonzero mean (± 2°), but this was believed to be caused by insufficiently long dwells. While data at dwell edges proved to be challenging to separate, the general location of moderately long dwell states (around n> 50) was readily apparent.
Within regions that behave indistinguishably from normal dwell states, there are jumps far beyond the calculated variance that could be the enzyme undergoing a failed rotation or a measurement artefact. Comparatively, within transitions, there are occasionally, sequences where the probe spends an uncharacteristic amount of time in a transitions ( 1-2 ms), which is shorter than a normal dwell, but longer than most transitions. Within the scope of this paper, the former events were considered as dwells (as they did not have a significant effect the net dwell average), and events where the probe appeared to fluctuate about for greater than 1 ms were considered dwells, as the focus was with transitions which appeared to have an average 0.5 ms.

Evaluating the dwell variance
After identifying parts of the trajectory that were definitively in a dwell state, the next step was to identify the variation within dwells to determine the magnitude of fluctuation.
The fluctuations can be extracted simply used the pdf as seen in Fig. S7.

Determining dwell state averages
Subsequently, individual dwells' energy equilibrium was obtained using the previously determined dwell points, verified by summing the normalised dwell points. Ultimately, the average appeared like the dashed line in Fig

Velocity methodology
After separating dwell and transition states, an additional analysis in the form of velocity versus position was used to elucidate enzyme behavior.
1. We separated dwells and transitions according to procedure described in the previous section.
2. We obtained and subtracted dwell averages to normalise data. 4. We binned the data according to angles and velocities thereby creating a 2-dimensional histogram. The data can be sorted using a hashmap or looping through predetermined angular position ranges averaging associated velocities at each range. 5. We plotted angular bin midpoints versus average rotation velocity (averaging is performed for each angular bin).

Robustness of velocity method compared to the angular position-based method in the transition range
Analysis of position versus time was found to be unreliable as the starting value majorly impacts the subsequent rate. Fig. S9 shows angular position versus time. As can be seen, the behavior varies significantly until approaching asymptotic behavior around 120°as it approaches the next dwell.
Rotational correction also proved valuable in enhancing the resolution of the average velocity as seen in Fig. S10. The comparability the subunits allowed greater resolution regarding transition behavior, as multiple subunits were able to be considered in tandem, effectively doubling or tripling the dataset.

Angle-dependent rate constant of the fast ADP release
For an arbitrary single-step transition, According to an elastic molecular group transfer model, 3  Theoretical calculations were performed both assuming an exponential angle dependence defined in Eqs. (1-1) using the value provided thereby. The results shown in Fig. S13 show that a best fit of the latter model cannot reproduce the dip and subsequent peak in the angular velocity vs. rotation angle in the experimental data. Instead, using an exponential angle dependence identical to that of ATP release or slow ADP release, the 2-state theory can be well fitted to the experimental angular velocity vs. angle data in the transition. Further assumptions in the multistate theory of rotation monitored by a nano-probe In the model 9 we assume that the processes occur on two markedly different timescales: (1) the transfer of nucleotides (ATP or ADP) and smaller ions (inorganic phosphate P i ) in the binding channel take microseconds or less; and (2) the waiting time between these transitions, the video frame time of the apparatus and the motion of the probe itself are slower process that relax in micro-or milliseconds. The monitored θ is quasi-static during any particular transition, 3,8 which is the basis for a kinetics of angular position-dependent rate constants.
Accordingly, a series of steps is described by a discrete 'chemical state' (or occupancy state) variable i. To each state i a dwell angle θ i corresponds, that may be 'observable' in the imaging trajectories or hidden, if the state is too short-lived. During the transitions from i to i + 1 transfer reactions occur with forward and back rate constants k f i (θ) and k bi (θ).
When the system is in the reactant minimum θ fluctuates around the dwell angle θ i ; after a fast transition the system is found in the product state and θ relaxes and fluctuates about the next (product) dwell angle θ i+1 .
The i dependence of D and κ r are assumed to be negligible since the former is determined by the size and shape of the probe, and the latter by linkage of the probe, the γ shaft and flexible elements of the stator ring. So ideally, none of these quantities change over the course of a trajectory. We note that the viscous drag of the probe is significantly affected by its interaction with the surface on which the F 1 -ATPase was deposited. The interaction is likely angle-dependent, due to nanoscale features of the surface.

Angular velocity distributions
To calculate the angular velocity distributions we define the angle-time conditional probability distribution ρ pos ii (θ + ∆θ, t + ∆t|θ, t) that the system survives in state i at time t + ∆t and is found at angular position θ + ∆θ, if at time t it was at θ. In this analysis ∆t is the time step (frame time) of the imaging apparatus. ρ ii (θ + ∆θ, t + ∆t|θ, t) satisfies Eq. ?? for a delta-function initial condition. 10 The solution is a Gaussian, 11 which has a θ-dependent peak, and is independent of t. A change of variable ∆θ → v from Eq. ??, yields the conditional probability for an angular velocity ρ vel ii (v, t + ∆t|θ, t) = ρ ii (v|θ) in a simplified time-suppressed notation, Then a summation over all states yields Extracting the spring constant from dwells in the single-

molecule trajectories
The fluctuations of the probe in single-molecule imaging can be detected accurately if the time step of the imaging apparatus ∆t is smaller then the relaxation time τ . 1 We calculate the measured standard deviation σ m of the angular position histogram compared to the true σ.
In single-molecule imaging, the position of the probe is detected during the imaging frame time ∆t. For an imaging frame, when the system is in a given chemical state i (in a dwell), the measured angle θ n = θ ∆t , is a time average over the frame time, Let's consider the auto-correlation function. The "true" time-time correlation function C(t− t ′ ) = ⟨θ(t ′ )θ(t)⟩ is defined for t ′ ≤ t. For simplicity, the angles θ 1 and θ m are shifted so that the dwell mean angle is 0. The true-time correlation is then: where σ 2 = k B T /κ. The discrete correlation function for zero lag is, The factor of 2 is due to interchangeable between t and t ′ when both have same range from 0 to ∆t. The variance is then calculated from equation 7, Since σ 2 ∝ κ, the variance allows relation between the true and measured value of the stiffness, κ and κ m , respectively, The measured angles will follow Gaussian distribution p m (θ m ) whose width is scaled by a proportionality factor, according to Eq. 9. For a TBF1 trajectory with ∆t = 10 µs analyzed The Gaussian angular histograms described in Eq. 9 and seen in experiment on Fig. S8 in the SI are used for the latter. Once these quantities are established, the 2-state model is used to calculate the velocities. Assuming as an initial condition that the system is in state 1 (the hidden state) and that the probability approaches zero at boundaries 2σ away from both θ 0 = 0 and θ 2 = 120, a two-state Eqs. 1-2 is solved numerically using a standard PDE solver in Matlab. The numerically calculated velocities then are compared with the angular velocity profile extracted from the points in the transitions (Fig. 4). The best fit then yields an estimate for the dwell angle θ 1 and lifetime 1/f f 0,1 of the intermediate state. The procedure is performed separately for each trajectory, due to the (usually small) differences of probe size, attachment conformation and local environment.

Mean discrete angular velocity and its relation to the true mean angular velocity
We note that the mean angular velocity from Eq. 4 tends to the true mean angular velocity,  the 'correction factor' can become so large that the peak angular position (and the average jump) will have little connection with the 'true' average angular velocity. However, the latter can be still extracted from experiment if τ is know accurately, e.g., from the angular autocorrelation function in the dwell.