Single-molecule spectroscopies revealed the stepping rotation of an F1-ATPase enzyme in which a substep can have microsecond transition dynamics. Here we describe a method for analyzing fast single molecule rotation trajectories in F1-ATPase monitored by a 40 nanometer probe. This method focuses on the rotation jumps that occur in the transitions during the steps between dwells in single molecule trajectories. These jumps are related to the "instantaneous" rotation velocity and they exhibit a bimodal distribution at certain angles, indicating that the system produces both forward and a backward torques at the same angle, due to being in either of two states. Two states at the same angle is a key assumption used to extracts rate constants in stalling experiments, so by observing the bimodal distributions we provide support for this assumption. To calculate the distribution of jumps, we use a multi-state theory to describe the visco-elastic fluctuation of the imaging probe. The predicted jump distribution in the transitions yields a relaxation time which agrees with its value of 14 microseconds in the dwell fluctuations. Using a sequence of three states, the theoretical profile of angular jumps agrees with experiment for most of the angular range, but full agreement between theory and experiment is reached if a fourth, 10 microsecond lifetime state is assumed with an effective dwell half way through the 80 degree substep. It suggests that the ATP binding in one subunit and the ADP release from another subunit occurs via this transient. The ability to detect a state that is comparable or shorter than the instrumental relaxation time indicates that this jump distribution based method can be used to effectively increase the time resolution of the imaging apparatus.
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