Evidence has been mounting that in the rotational cycle of F1-ATPase there is a concerted ATP binding and ADP release that yields a million-fold acceleration in the rate of the product ADP release. We developed a theory of reaction kinetics to investigate the relationship between the concerted nucleotide exchange and previous single-molecule forced rotation data from Adachi, K. Nat. Commun. 2012, 3, 1022. We extracted from these data angle-dependent rate constants for nucleotide binding and release. The rate constants were then used in a unified kinetic scheme, also consistent with other single-molecule and ensemble experiments, to obtain analytical equations for nucleotide occupancy change events from nano- to millimolar ATP concentrations. A theory-experiment comparison revealed novel evidence about the concerted mechanism: it is determined by correlated conformational changes in the F1-ATPase ring, and its kinetic signature is a unified angle-dependent function of the nucleotide binding and release rate constants, which is independent of ATP concentration.

Earlier sum frequency generation (SFG) experiments involve one infrared and one visible laser, and a measurement of the intensity of the response, yielding data on the surface sensitive properties of the sample. Recently, both the real and imaginary components of the susceptibility were measured in two different sets of experiments. In one set, a broadband infrared laser was used, permitting observations at very short times, while in another set the infrared laser was narrowband, permitting higher spectral resolution. The differences in the spectrum obtained by the two will be most evident in studying narrow absorption bands, e.g., the band due to dangling OH bonds at a water interface. The direct comparisons in the integrated amplitude (sum rule) of the imaginary part of the dangling OH bond region differ by a factor of 3. Due to variations in experimental setup and data processing, corrections were made for the quartz reference, Fresnel factors, and the incident visible laser wavelength. After the corrections, the agreement differs now by the factors of 1.1 within broadband and narrowband groups and the two groups now differ by a factor of 1.5. The 1.5 factor may arise from the extra heating of the more powerful broadband laser system on the water surface. The convolution from the narrowband SFG spectrum to the broadband SFG spectrum is also investigated and it does not affect the sum rule. Theory and narrowband experiments are compared using the sum rule and agree to a factor of 1.3 with no adjustable parameters.

Recently, we proposed a theory-basaed method to analyze fast (10-100 µs timestep) single-molecule imaging trajectories in F1-ATPase. A key quantity in this method is the angular velocity vs. rotation angle extracted from both experimental data and computer simulations. When applying the method on Thermophilic Bacillus F1-ATPase rotation data, we detected a short-lived substep previously not detectable in the angular histograms. The comparison between the experimental and theory reveals that an 80O substep of the “concerted” ATP binding and ADP release involves an intermediate state reminiscent of a 3-occupancy structure. Its lifetime (∼10 µs) is about six orders of magnitude smaller than the lifetime for “spontaneous” ADP release from a singly occupied state. The theory-based method was also applied to single-molecule imaging data from Paracoccus Denitrificans F1-ATPase and it yielded a similar hidden state in the transitions between subsequent long dwells. The ∼20 µs lifetime is several times shorter then the experimental imaging frame time (100 µs), so by detecting this short-lived state the method was used to achieve temporal "super-resolution". Our recent findings indicate a common mechanism for the acceleration of ADP release in the F1-ATPase motor of the two species.

F1-ATPase enzyme is a biological motor which hydrolyzes ATP. The enzyme has been observed via single-molecule imaging experiments wherein the enzyme is allowed to rotate freely while being recorded using a gold nanoparticle. During hydrolysis, the enzyme causes Brownian noise when rotating to new chemical states because of a size difference between the 4 nm enzyme and 40 nm probe. The unconvoluted rotary movement has been revealed using techniques including rotational correction, correlation functions, and comparison of average rotational jumps which contribute to developing a multistate model.

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.

In this chapter we discuss the interaction between theory and single molecule experiments on a biological motor, F1-ATPase. In particular, we consider an interplay between the experiment, analytical theory, and computer simulations. The complementarity of the millisecond and microsecond experiments is noted. For example, the limited experiments on the latter indicate that the ATP binding in one β subunit precedes an ADP release in another β subunit, whereas the millisecond experiments do not time-resolve the two steps.

The Drude-Smith equation is widely used for treating the frequency-dependent electrical conductivity of materials in the terahertz region. An attractive feature is its sparsity of adjustable parameters. A significant improvement over Drude theory for these materials, the theory includes backscattering of the charge carriers. It has nevertheless been criticized, including by Smith himself, because of the arbitrariness of a step in the derivation. We recall a somewhat similar behavior of back scattering in fluids observed in molecular dynamics computations and discussed in terms of memory functions. We show how theories such as Drude-Smith and Cocker et al. are examples of a broader class of theories by showing how they also arise as particular cases of a memory function formalism that divides the interactions into short and long range.

Experimental studies on single-molecule junctions are typically in need of a simple theoretical approach that can reproduce or be fitted to experimentally measured transport data. In this context, the single-level variant of the Landauer approach is most commonly used, but methods based on Marcus theory are also gaining popularity. Recently, a generalized theory unifying these two approaches has also been developed. In the present work, we extend this theory so that it includes entropic effects (which can be important when polar solvents are involved but are likely minor for solid-state systems). We investigate the temperature-dependence of the electric current and compare it to the behavior predicted by the Landauer and the conventional Marcus theory. We argue that this generalized theory provides a simple yet effective framework for understanding charge transport through molecular junctions. Furthermore, we explore the role of the entropic effects in different transport regimes and suggest experimental criteria for detecting them in solvated molecular junctions. Finally, in order to account for nuclear tunneling effects, we also demonstrate how lifetime broadening can be introduced into the Marcus-Levich-Dogonadze-Jortner-type description of electron transport.

The Front Cover depicts an electron undergoing backscattering in a disordered solid, a phenomenon commonly described using the Drude-Smith equation. The equation, graphically represented by the damping curve, is shown to emerge as the “critically damped” case when treated with a more general formulation using a memory function. More information can be found in the Article by Rudolph A. Marcus and Wei-Chen Chen.

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.

: An objective of the research performed on this grant is the understanding of the detailed behavior of a variety of electron transfer processes. Theories were developed for (a) the rate of electron transfer between a reagent in one liquid phase and another in a second (immiscible) liquid or polymer, (b) the rate of long distance electron transfer in proteins, (c) charge transfers spectra in frozen media, (d) scanning tunneling microscopy (stm) of molecular adsorbates, and (e) analysis of models of solvents used in computer simulations of electron transfer, particularly examining the error incurred by their common neglect of the electronic and vibrational contributions of the solvent's dielectric response.

Significance The water surface structures, where a dangling OH bond plays a role in catalysis, can be studied by the sum frequency generation (SFG) experimentally and theoretically. The dangling OH bond region of the SFG is investigated using the classical molecular dynamics simulation for the ssp, sps, and ppp combinations of the different electric field polarizations. Reasonable agreement with no adjustable parameters within a factor of 1.3 is obtained with the experimental single-term SFG polarization combinations ssp and sps. The comparison of the calculated absolute SFG intensities with the experimental values provides a test of the various approximations in the literature, such as the values used in some cases for the two polarizability derivatives of a surface OH.

Single-molecule imaging experiments provide information that is not available from ensemble experiments. We are interested in the interpretation of dynamical studies imaging and manipulation in F₁-ATPase single-molecules. One key question that has arisen in single molecule stalling experiments is the erratic behavior a rotor angle of 55° between the binding and hydrolysis dwell angles of 0 and 80°, respectively. In our previous theoretical work, we used the elastic property of the rotor-stator structure to treat the experiments on controlled rotation. Our modeling suggests that there has to be a change in the bonding network, for example, of hydrogen bonds, as the system transitions between the two dwell points, perhaps at 55°, as indicated by an unusual stalling behavior around that angle. Therefore, in order to get further insights on these events, we have performed full-atomistic molecular dynamics (MD) simulations on the F₁-ATPase to explore the relationship between the conformational changes and the phosphate displacement. The pK_a values for phosphoric acid are 2.2, 7.2, and 12.3; although the effective pK_a for H₂PO₄⁻ within the binding site could be different. Hence, both protonation states (i.e., H₂PO₄⁻ and HPO₄²⁻) were assessed, performing molecular dynamics for 650 ns at 298.15K (Nose-Hoover thermostat) using the CHARMM36 force field. After the analysis of five MD trajectories, we found that the displacement of the phosphate was, to some extent, correlated with the dynamics of the protein. Moreover, one of the MD trajectories led to the complete release of the diprotic phosphate.

The idea of a concerted mechanism of molecular group transfer and large conformational change in power stroke generation serves as the basis for an elasto-chemical theory of the rate constants that dictate the stepping kinetics in single F1-ATPase motors. The theory was employed to calculate the rate and equilibrium constant dependence on the rotor angle in experiments with magnetic tweezers. Using independent biochemical and single-molecule imaging data, the model was used to correctly predict the rates of binding and release of fluorescently labeled ATP without any adjustable parameters. This initial success of the model prompted its further development to other systems and experiments. In particular, the idea of group transfer elastically coupled to conformational change monitored by the probe is applied and extended to (1) calculate power stroke rate constants in two different force-spectroscopy experiments on single and double-headed myosin V; to (2) extract rate constants for hydrolysis and synthesis from controlled rotation data on F1-ATPase; and to (3) account for "dynamical effects" due to the internal and probe-related friction in the single-molecule imaging experiments. An important aspect of the analysis is the statistical modeling of biasing effects of finite time resolution in the photon-counting trajectories and the finite response time of the imaging probes.

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 β subunit and the adenosine 5'-diphosphate (ADP) release from another β 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.

Lead halide perovskite semiconductors have low-frequency phonon modes within the lead halide sublattice and thus are considered to be soft. The soft lattice is considered to be important in defining their interesting optoelectronic properties. Electron-phonon coupling governs hot-carrier relaxation, carrier mobilities, carrier lifetimes, among other important electronic characteristics. Directly observing the interplay between free charge carriers and phonons can provide details on how phonons impact these properties (e.g., exciton populations and other collective modes). Here, we observe a delicate interplay among carriers, phonons, and excitons in mixed-cation and mixed-halide perovskite films by simultaneously resolving the contribution of charge carriers and phonons in time-resolved terahertz photoconductivity spectra. We are able to observe directly the increase in phonon population during carrier cooling and discuss how thermal equilibrium populations of carriers and phonons modulate the carrier transport properties, as well as reduce the population of carriers within band tails. We are also able to observe directly the formation of free charge carriers when excitons interact with phonons and dissociate and to describe how free carriers and exciton populations exchange through phonon interactions. Finally, we also time-resolve how the carriers are screened via the Coulomb interaction at low and room temperatures. Our studies shed light on how charge carriers interact with the low-energy phonons and discuss implications.

Oxygen Production and Reduction in Artificial and Natural Systems, pp. 31-51 (2019) No AccessChapter 2: Theory of Rate Constants of Substeps in Single Molecule Experiments on F1-ATPaseSándor Volkán-Kacsó and Rudolph A. MarcusSándor Volkán-KacsóSegerstrom Science Center, Azusa Pacific University, 901 E. Alosta Ave., Azusa, CA 91702, USANoyes Laboratory of Chemical Physics, California Institute of Technology, 1200 E. California Blvd., Pasadena, CA 91125, USA and Rudolph A. MarcusNoyes Laboratory of Chemical Physics, California Institute of Technology, 1200 E. California Blvd., Pasadena, CA 91125, USAhttps://doi.org/10.1142/9789813276925_0002Cited by:1 PreviousNext AboutSectionsPDF/EPUB ToolsAdd to favoritesDownload CitationsTrack CitationsRecommend to Library ShareShare onFacebookTwitterLinked InRedditEmail Abstract: We review a recently proposed elastic group transfer theory and its application to free, stalling, and controlled rotation single molecule experiments on F1-ATPase. In these experiments ATP is hydrolyzed to form ADP and inorganic phosphate, Pi. The main task is how to combine the mechanical and the chemical processes so as to form a theory, largely analytic, of the overall process and its substeps. Using the theory predictions can be made and tested within this class of experiments for the angle dependent rate constants in this motor enzyme. Data from some experiments are used to predict and compare with data from different experiments. The theory also serves as a basis for the statistical analysis of single molecule imaging and force spectroscopy trajectories, in which the enzyme kinetic processes are a convolution of processes. The theory is also used to correct for time-resolution limitations and "shot noise" often inherent in single-molecule experiments. Some new features involve (1) the unidirectionality of the rotation, (2) a postulated approximate conservation of open cleft space during the rotation of the γ-shaft in the motor, and (3) the relevance of single-molecule experiments, in which the ATPase is fixed on a microscope slide, to the physiological system. FiguresReferencesRelatedDetailsCited By 1Single Molecule Studies of a Biological Motor F1-ATPase: Interplay of Experiment, Analytic Theory and ComputationSándor Volkán-Kacsó and Rudolph A. Marcus26 January 2021 Oxygen Production and Reduction in Artificial and Natural SystemsMetrics History PDF download

I present an elastic chemo-mechanical theory to treat single molecule imaging and "stalling" experiments in the F1-ATPase enzyme. Using a molecular group transfer approach the theory couples chemical reactions in the stator and the physics of torsional elasticity in the rotor. In the theory we predicted and compared with experiment the rate and equilibrium constant dependence of steps such as ATP binding as a function of the rotor angle.[PNAS, 112, 14230 (2015)] Using independent experimental data from biochemical ensemble and single-molecule imaging experiments, the model correctly predicts the controlled rotation data on fluorescent ATP without any adjustable parameters. We took into account the biasing effect of finite experimental time resolution in the single fluorescence trajectories and treated these data by developing computational statistical methods.[PNAS, 113 (48), 12029 (2016)] A theory-based method for the extraction of rate constants for hydrolysis and synthesis from controlled rotation data was also provided for angular range where no such data is currently available [PNAS, 114, 7272 (2016)] The framework is generic and we plan to apply it to other biomolecular motors.

We show how an elastic group transfer theory can be used to interpret and treat free, stalling, and controlled rotation single molecule experiments on F-1-ATPase. It is shown how predictions can be made and tested within this class of experiments using our recent theoretical treatment of rotor angle dependent rate constants for this biological motor. The theory is also used to suggest an additional type of analysis of single molecule trajectories involving dwell angles.