Angle-dependent rotation velocity consistent with ADP release in bacterial F1-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 120ø 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, 40o 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 ∼10µs), suggesting thereby a common mechanism for removing a nucleotide release bottleneck in the rotary mechanism.

F1-ATPase Rotary Mechanism: Interpreting Results of Diverse Experimental Modes With an Elastic Coupling Theory.

In this chapter, we review single-molecule observations of rotary motors, focusing on the general theme that their mechanical motion proceeds in substeps with each substep described by an angle-dependent rate constant. In the molecular machine F1-ATPase, the stepping rotation is described for individual steps by forward and back reaction rate constants, some of which depend strongly on the rotation angle. The rotation of a central shaft is typically monitored by an optical probe. We review our recent work on the theory for the angle-dependent rate constants built to treat a variety of single-molecule and ensemble experiments on the F1-ATPase, and relating the free energy of activation of a step to the standard free energy of reaction for that step. This theory, an elastic molecular transfer theory, provides a framework for a multistate model and includes the probe used in single-molecule imaging and magnetic manipulation experiments. Several examples of its application are the following: (a) treatment of the angle-dependent rate constants in stalling experiments, (b) use of the model to enhance the time resolution of the single-molecule imaging apparatus and to detect short-lived states with a microsecond lifetime, states hidden by the fluctuations of the imaging probe, (c) treatment of out-of-equilibrium "controlled rotation" experiments, (d) use of the model to predict, without adjustable parameters, the angle-dependent rate constants of nucleotide binding and release, using data from other experiments, and (e) insights obtained from correlation of kinetic and cryo-EM structural data. It is also noted that in the case where the release of ADP would be a bottleneck process, the binding of ATP to another site acts to accelerate the release by 5-6 orders of magnitude. The relation of the present set of studies to previous and current theoretical work in the field is described. An overall goal is to gain mechanistic insight into the biological function in relation to structure.

Method to extract multiple states in F1-ATPase rotation experiments from jump distributions.

A method is proposed for analyzing fast (10 µs) single-molecule rotation trajectories in F1 adenosinetriphosphatase ([Formula: see text]-ATPase). This method is based on the distribution of jumps in the rotation angle that occur in the transitions during the steps between subsequent catalytic dwells. The method is complementary to the "stalling" technique devised by H. Noji et al. [Biophys. Rev. 9, 103-118, 2017], and can reveal multiple states not directly detectable as steps. A bimodal distribution of jumps is observed at certain angles, due to the system being in either of 2 states at the same rotation angle. In this method, a multistate theory is used that takes into account a viscoelastic fluctuation of the imaging probe. Using an established sequence of 3 specific states, a theoretical profile of angular jumps is predicted, without adjustable parameters, that agrees with experiment for most of the angular range. Agreement can be achieved at all angles by assuming a fourth state with an ∼10 µs lifetime and a dwell angle about 40° after the adenosine 5'-triphosphate (ATP) binding dwell. The latter result suggests that the ATP binding in one ß subunit and the adenosine 5'-diphosphate (ADP) release from another ß subunit occur via a transient whose lifetime is ∼10 µs and is about 6 orders of magnitude smaller than the lifetime for ADP release from a singly occupied [Formula: see text]-ATPase. An internal consistency test is given by comparing 2 independent ways of obtaining the relaxation time of the probe. They agree and are ∼15 µs.

What can be learned about the enzyme ATPase from single-molecule studies of its subunit F1?

We summarize the different types of single molecule experiments on the F1 component of FOF1-ATP Synthase and what has been learned from them. We also describe results from our recent studies on interpreting the experiments using a chemical-mechanical theory for these biological motors.

Theory of long binding events in single-molecule-controlled rotation experiments on F1-ATPase.

The theory of elastic group transfer for the binding and release rate constants for nucleotides in F1-ATPase as a function of the rotor angle is further extended in several respects. (i) A method is described for predicting the experimentally observed lifetime distribution of long binding events in the controlled rotation experiments by taking into account the hydrolysis and synthesis reactions occurring during these events. (ii) A method is also given for treating the long binding events in the experiments and obtaining the rate constants for the hydrolysis and synthesis reactions occurring during these events. (iii) The theory in the previous paper is given in a symmetric form, an extension that simplifies the application of the theory to experiments. It also includes a theory-based correction of the reported "on" and "off" rates by calculating the missed events. A near symmetry of the data about the angle of -40° and a "turnover" in the binding rate data vs. rotor angle for angles greater than [Formula: see text]40° is also discussed.

Theory of single-molecule controlled rotation experiments, predictions, tests, and comparison with stalling experiments in F1-ATPase.

A recently proposed chemomechanical group transfer theory of rotary biomolecular motors is applied to treat single-molecule controlled rotation experiments. In these experiments, single-molecule fluorescence is used to measure the binding and release rate constants of nucleotides by monitoring the occupancy of binding sites. It is shown how missed events of nucleotide binding and release in these experiments can be corrected using theory, with F1-ATP synthase as an example. The missed events are significant when the reverse rate is very fast. Using the theory the actual rate constants in the controlled rotation experiments and the corrections are predicted from independent data, including other single-molecule rotation and ensemble biochemical experiments. The effective torsional elastic constant is found to depend on the binding/releasing nucleotide, and it is smaller for ADP than for ATP. There is a good agreement, with no adjustable parameters, between the theoretical and experimental results of controlled rotation experiments and stalling experiments, for the range of angles where the data overlap. This agreement is perhaps all the more surprising because it occurs even though the binding and release of fluorescent nucleotides is monitored at single-site occupancy concentrations, whereas the stalling and free rotation experiments have multiple-site occupancy.

Theory for rates, equilibrium constants, and Brønsted slopes in F1-ATPase single molecule imaging experiments.

A theoretical model of elastically coupled reactions is proposed for single molecule imaging and rotor manipulation experiments on F1-ATPase. Stalling experiments are considered in which rates of individual ligand binding, ligand release, and chemical reaction steps have an exponential dependence on rotor angle. These data are treated in terms of the effect of thermodynamic driving forces on reaction rates, and lead to equations relating rate constants and free energies to the stalling angle. These relations, in turn, are modeled using a formalism originally developed to treat electron and other transfer reactions. During stalling the free energy profile of the enzymatic steps is altered by a work term due to elastic structural twisting. Using biochemical and single molecule data, the dependence of the rate constant and equilibrium constant on the stall angle, as well as the Børnsted slope are predicted and compared with experiment. Reasonable agreement is found with stalling experiments for ATP and GTP binding. The model can be applied to other torque-generating steps of reversible ligand binding, such as ADP and Pi release, when sufficient data become available.

Heterogeneous Fluorescence Intermittency in Single Layer Reduced Graphene Oxide.

We provide, for the first time, direct experimental evidence for heterogeneous blinking in reduced graphene oxide (rGO) during photolysis. The spatially resolved intermittency originates from regions within individual rGO sheets and shows 1/f-like power spectral density. We describe the evolution of rGO blinking using the multiple recombination center (MRC) model that captures common features of nanoscale blinking. Our results illustrate the universal nature of blinking and suggest a common microscopic origin for the effect.

Two-state theory of binned photon statistics for a large class of waiting time distributions and its application to quantum dot blinking.

A theoretical method is proposed for the calculation of the photon counting probability distribution during a bin time. Two-state fluorescence and steady excitation are assumed. A key feature is a kinetic scheme that allows for an extensive class of stochastic waiting time distribution functions, including power laws, expanded as a sum of weighted decaying exponentials. The solution is analytic in certain conditions, and an exact and simple expression is found for the integral contribution of "bright" and "dark" states. As an application for power law kinetics, theoretical results are compared with experimental intensity histograms from a number of blinking CdSe/ZnS quantum dots. The histograms are consistent with distributions of intensity states around a "bright" and a "dark" maximum. A gap of states is also revealed in the more-or-less flat inter-peak region. The slope and to some extent the flatness of the inter-peak feature are found to be sensitive to the power-law exponents. Possible models consistent with these findings are discussed, such as the combination of multiple charging and fluctuating non-radiative channels or the multiple recombination center model. A fitting of the latter to experiment provides constraints on the interaction parameter between the recombination centers. Further extensions and applications of the photon counting theory are also discussed.

Universality of the fluorescence intermittency in nanoscale systems: experiment and theory.

A variety of optically active nanoscale objects show extremely long correlations in the fluctuations of fluorescence intensity (blinking). Here we performed a systematic study to quantitatively estimate the power spectral density (PSD) of the fluorescence trajectories of colloidal and self-assembled quantum dots (QDs), nanorods (NRs), nanowires (NWs), and organic molecules. We report for the first time a statistically correct method of PSD estimation suitable for these systems. Our method includes a detailed analysis of the confidence intervals. The striking similarity in the spectra of these nanoscale systems, including even a "nonblinking" quantum dot investigated by Wang and collaborators (Nature2009, 459, 685-689), is powerful evidence for the existence of a universal physical mechanism underlying the blinking phenomenon in all of these fluorophores (Frantsuzov et al. Nat. Phys.2008, 4, 519-522). In this paper we show that the features of this universal mechanism can be captured phenomenologically by the multiple recombination center model (MRC) we suggested recently for explaining single colloidal QD intermittency. Within the framework of the MRCs we qualitatively explain all of the important features of fluorescence intensity fluctuations for a broad spectrum of nanoscale emitters.

Correlations between subsequent blinking events in single quantum dots.

We explain the long-range correlations found by Stefani and his co-workers between blinking times of single colloidal quantum dot emission. Our explanation is based on the multiple recombination center model we recently suggested. The model produces positive correlations between subsequent on--on and off--off times and negative on--off correlations, as observed in the experiment. We also reproduce qualitatively the dependence of correlations between subsequent on--on, on-off, and off--off times on the number of switching events separating them.

Model of fluorescence intermittency of single colloidal semiconductor quantum dots using multiple recombination centers.

We present a new physical model resolving a long-standing mystery of the power-law distributions of the blinking times in single colloidal quantum dot fluorescence. The model considers the nonradiative relaxation of the exciton through multiple recombination centers. Each center is allowed to switch between two quasistationary states. We point out that the conventional threshold analysis method used to extract the exponents of the distributions for the on times and off times has a serious flaw: the qualitative properties of the distributions strongly depend on the threshold value chosen for separating the on and off states. Our new model explains naturally this threshold dependence, as well as other key experimental features of the single quantum dot fluorescence trajectories, such as the power-law power spectrum (1/f noise).