Mechanical modulation of catalytic power on F1-ATPase

Optical microscopic imaging, manipulation, and analysis methods for morphogenesis research

Morphogenesis is a developmental process of organisms being shaped through complex and cooperative cellular movements. To understand the interplay between genetic programs and the resulting multicellular morphogenesis, it is essential to characterize the morphologies and dynamics at the single-cell level and to understand how physical forces serve as both signaling components and driving forces of tissue deformations. In recent years, advances in microscopy techniques have led to improvements in imaging speed, resolution and depth. Concurrently, the development of various software packages has supported large-scale, analyses of challenging images at the single-cell resolution. While these tools have enhanced our ability to examine dynamics of cells and mechanical processes during morphogenesis, their effective integration requires specialized expertise. With this background, this review provides a practical overview of those techniques. First, we introduce microscopic techniques for multicellular imaging and image analysis software tools with a focus on cell segmentation and tracking. Second, we provide an overview of cutting-edge techniques for mechanical manipulation of cells and tissues. Finally, we introduce recent findings on morphogenetic mechanisms and mechanosensations that have been achieved by effectively combining microscopy, image analysis tools and mechanical manipulation techniques.

Towards the understanding of molecular motors and its relationship with local unfolding

Molecular motors are machines essential for life since they convert chemical energy into mechanical work. However, the precise mechanism by which nucleotide binding, catalysis, or release of products is coupled to the work performed by the molecular motor is still not entirely clear. This is due, in part, to a lack of understanding of the role of force in the mechanical-structural processes involved in enzyme catalysis. From a mechanical perspective, one promising hypothesis is the Haldane-Pauling hypothesis which considers the idea that part of the enzymatic catalysis is strain-induced. It suggests that enzymes cannot be efficient catalysts if they are fully complementary to the substrates. Instead, they must exert strain on the substrate upon binding, using enzyme-substrate energy interaction (binding energy) to accelerate the reaction rate. A novel idea suggests that during catalysis, significant strain energy is built up, which is then released by a local unfolding/refolding event known as 'cracking'. Recent evidence has also shown that in catalytic reactions involving conformational changes, part of the heat released results in a center-of-mass acceleration of the enzyme, raising the possibility that the heat released by the reaction itself could affect the enzyme's integrity. Thus, it has been suggested that this released heat could promote or be linked to the cracking seen in proteins such as adenylate kinase (AK). We propose that the energy released as a consequence of ligand binding/catalysis is associated with the local unfolding/refolding events (cracking), and that this energy is capable of driving the mechanical work.

Perspectives on Computational Enzyme Modeling: From Mechanisms to Design and Drug Development

Understanding enzyme mechanisms is essential for unraveling the complex molecular machinery of life. In this review, we survey the field of computational enzymology, highlighting key principles governing enzyme mechanisms and discussing ongoing challenges and promising advances. Over the years, computer simulations have become indispensable in the study of enzyme mechanisms, with the integration of experimental and computational exploration now established as a holistic approach to gain deep insights into enzymatic catalysis. Numerous studies have demonstrated the power of computer simulations in characterizing reaction pathways, transition states, substrate selectivity, product distribution, and dynamic conformational changes for various enzymes. Nevertheless, significant challenges remain in investigating the mechanisms of complex multistep reactions, large-scale conformational changes, and allosteric regulation. Beyond mechanistic studies, computational enzyme modeling has emerged as an essential tool for computer-aided enzyme design and the rational discovery of covalent drugs for targeted therapies. Overall, enzyme design/engineering and covalent drug development can greatly benefit from our understanding of the detailed mechanisms of enzymes, such as protein dynamics, entropy contributions, and allostery, as revealed by computational studies. Such a convergence of different research approaches is expected to continue, creating synergies in enzyme research. This review, by outlining the ever-expanding field of enzyme research, aims to provide guidance for future research directions and facilitate new developments in this important and evolving field.

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 ∼ 10 μ s), suggesting thereby a common mechanism for removing a nucleotide release bottleneck in the rotary mechanism.

Ordered-domain unfolding of thermophilic isolated β subunit ATP synthase

The flexibility of the ATP synthase's β subunit promotes its role in the ATP synthase rotational mechanism, but its domains stability remains unknown. A reversible thermal unfolding of the isolated β subunit (Tβ) of the ATP synthase from Bacillus thermophilus PS3, tracked through circular dichroism and molecular dynamics, indicated that Tβ shape transits from an ellipsoid to a molten globule through an ordered unfolding of its domains, preserving the β-sheet residual structure at high temperature. We determined that part of the stability origin of Tβ is due to a transversal hydrophobic array that crosses the β-barrel formed at the N-terminal domain and the Rossman fold of the nucleotide-binding domain (NBD), while the helix bundle of the C-terminal domain is the less stable due to the lack of hydrophobic residues, and thus the more flexible to trigger the rotational mechanism of the ATP synthase.

Molecular mechanism on forcible ejection of ATPase inhibitory factor 1 from mitochondrial ATP synthase

IF₁ is a natural inhibitor protein for mitochondrial F_oF₁ ATP synthase that blocks catalysis and rotation of the F₁ by deeply inserting its N-terminal helices into F₁. A unique feature of IF₁ is condition-dependent inhibition; although IF₁ inhibits ATP hydrolysis by F₁, IF₁ inhibition is relieved under ATP synthesis conditions. To elucidate this condition-dependent inhibition mechanism, we have performed single-molecule manipulation experiments on IF₁-inhibited bovine mitochondrial F₁ (bMF₁). The results show that IF₁-inhibited F₁ is efficiently activated only when F₁ is rotated in the clockwise (ATP synthesis) direction, but not in the counterclockwise direction. The observed rotational-direction-dependent activation explains the condition-dependent mechanism of IF₁ inhibition. Investigation of mutant IF₁ with N-terminal truncations shows that the interaction with the γ subunit at the N-terminal regions is crucial for rotational-direction-dependent ejection, and the middle long helix is responsible for the inhibition of F₁.

How Does F1-ATPase Generate Torque?: Analysis From Cryo-Electron Microscopy and Rotational Catalysis of Thermophilic F1

The F₁-ATPase is a rotary motor fueled by ATP hydrolysis. Its rotational dynamics have been well characterized using single-molecule rotation assays. While F₁-ATPases from various species have been studied using rotation assays, the standard model for single-molecule studies has been the F₁-ATPase from thermophilic Bacillus sp. PS3, named TF₁. Single-molecule studies of TF₁ have revealed fundamental features of the F₁-ATPase, such as the principal stoichiometry of chemo-mechanical coupling (hydrolysis of 3 ATP per turn), torque (approximately 40 pN·nm), and work per hydrolysis reaction (80 pN·nm = 48 kJ/mol), which is nearly equivalent to the free energy of ATP hydrolysis. Rotation assays have also revealed that TF₁ exhibits two stable conformational states during turn: a binding dwell state and a catalytic dwell state. Although many structures of F₁ have been reported, most of them represent the catalytic dwell state or its related states, and the structure of the binding dwell state remained unknown. A recent cryo-EM study on TF₁ revealed the structure of the binding dwell state, providing insights into how F₁ generates torque coupled to ATP hydrolysis. In this review, we discuss the torque generation mechanism of F₁ based on the structure of the binding dwell state and single-molecule studies.

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.

The nucleotide binding affinities of two critical conformations of Escherichia coli ATP synthase

ATP synthase is essential in aerobic energy metabolism, and the rotary catalytic mechanism is one of the core concepts to understand the energetic functions of ATP synthase. Disulfide bonds formed by oxidizing a pair of cysteine mutations halted the rotation of the γ subunit in two critical conformations, the ATP-waiting dwell (αE284C/γQ274C) and the catalytic dwell (αE284C/γL276C). Tryptophan fluorescence was used to measure the nucleotide binding affinities for MgATP, MgADP and MgADP-AlF4 (a transition state analog) to wild-type and mutant F1 under reducing and oxidizing conditions. In the reduced state, αE284C/γL276C F1 showed a wild-type-like nucleotide binding pattern; after oxidation to lock the enzyme in the catalytic dwell state, the nucleotide binding parameters remained unchanged. In contrast, αE284C/γQ274C F1 showed significant differences in the affinities of the oxidized versus the reduced state. Locking the enzyme in the ATP-waiting dwell reduced nucleotide binding affinities of all three catalytic sites. Most importantly, the affinity of the low affinity site was reduced to such an extent that it could no longer be detected in the binding assay (Kd > 5 mM). The results of the present study allow to present a model for the catalytic mechanism of ATP synthase under consideration of the nucleotide affinity changes during a 360° cycle of the rotor.

The six steps of the complete F1-ATPase rotary catalytic cycle

F₁F_o ATP synthase interchanges phosphate transfer energy and proton motive force via a rotary catalysis mechanism. Isolated F₁-ATPase catalytic cores can hydrolyze ATP, passing through six intermediate conformational states to generate rotation of their central γ-subunit. Although previous structural studies have contributed greatly to understanding rotary catalysis in the F₁-ATPase, the structure of an important conformational state (the binding-dwell) has remained elusive. Here, we exploit temperature and time-resolved cryo-electron microscopy to determine the structure of the binding- and catalytic-dwell states of Bacillus PS3 F₁-ATPase. Each state shows three catalytic β-subunits in different conformations, establishing the complete set of six states taken up during the catalytic cycle and providing molecular details for both the ATP binding and hydrolysis strokes. We also identify a potential phosphate-release tunnel that indicates how ADP and phosphate binding are coordinated during synthesis. Overall these findings provide a structural basis for the entire F₁-ATPase catalytic cycle.

Optimal Rectification without Forward-Current Suppression by Biological Molecular Motor

We experimentally show that biological molecular motor F_{1}-ATPase (F_{1}) implements an optimal rectification mechanism. The rectification mechanism hardly suppresses the synthesis of adenosine triphosphate by F_{1}, which is F_{1}'s physiological role, while inhibiting the unfavorable hydrolysis of adenosine triphosphate. This optimal rectification contrasts highly with that of a simple ratchet model, where the inhibition of the backward current is inevitably accompanied by the suppression of the forward current. Our detailed analysis of single-molecule trajectories demonstrates a novel but simple rectification mechanism of F_{1} with parallel landscapes and asymmetric transition rates.

The 3 × 120° rotary mechanism of Paracoccus denitrificans F1-ATPase is different from that of the bacterial and mitochondrial F1-ATPases

The rotation of Paracoccus denitrificans F₁-ATPase (PdF₁) was studied using single-molecule microscopy. At all concentrations of adenosine triphosphate (ATP) or a slowly hydrolyzable ATP analog (ATPγS), above or below K _m, PdF₁ showed three dwells per turn, each separated by 120°. Analysis of dwell time between steps showed that PdF₁ executes binding, hydrolysis, and probably product release at the same dwell. The comparison of ATP binding and catalytic pauses in single PdF₁ molecules suggested that PdF₁ executes both elementary events at the same rotary position. This point was confirmed in an inhibition experiment with a nonhydrolyzable ATP analog (AMP-PNP). Rotation assays in the presence of adenosine diphosphate (ADP) or inorganic phosphate at physiological concentrations did not reveal any obvious substeps. Although the possibility of the existence of substeps remains, all of the datasets show that PdF₁ is principally a three-stepping motor similar to bacterial vacuolar (V₁)-ATPase from Thermus thermophilus This contrasts with all other known F₁-ATPases that show six or nine dwells per turn, conducting ATP binding and hydrolysis at different dwells. Pauses by persistent Mg-ADP inhibition or the inhibitory ζ-subunit were also found at the same angular position of the rotation dwell, supporting the simplified chemomechanical scheme of PdF₁ Comprehensive analysis of rotary catalysis of F₁ from different species, including PdF₁, suggests a clear trend in the correlation between the numbers of rotary steps of F₁ and F_o domains of F-ATP synthase. F₁ motors with more distinctive steps are coupled with proton-conducting F_o rings with fewer proteolipid subunits, giving insight into the design principle the F₁F_o of ATP synthase.

The catalytic dwell in ATPases is not crucial for movement against applied torque

The ATPase-catalysed conversion of ATP to ADP is a fundamental process in biology. During the hydrolysis of ATP, the α₃β₃ domain undergoes conformational changes while the central stalk (γ/D) rotates unidirectionally. Experimental studies have suggested that different catalytic mechanisms operate depending on the type of ATPase, but the structural and energetic basis of these mechanisms remains unclear. In particular, it is not clear how the positions of the catalytic dwells influence the energy transduction. Here we show that the observed dwell positions, unidirectional rotation and movement against the applied torque are reflections of the free-energy surface of the systems. Instructively, we determine that the dwell positions do not substantially affect the stopping torque. Our results suggest that the three resting states and the pathways that connect them should not be treated equally. The current work demonstrates how the free-energy landscape determines the behaviour of different types of ATPases.

Opposing Pressures of Speed and Efficiency Guide the Evolution of Molecular Machines

Many biomolecular machines need to be both fast and efficient. How has evolution optimized these machines along the tradeoff between speed and efficiency? We explore this question using optimizable dynamical models along coordinates that are plausible evolutionary degrees of freedom. Data on 11 motors and ion pumps are consistent with the hypothesis that evolution seeks an optimal balance of speed and efficiency, where any further small increase in one of these quantities would come at great expense to the other. For F_oF₁-ATPases in different species, we also find apparent optimization of the number of subunits in the c-ring, which determines the number of protons pumped per ATP synthesized. Interestingly, these ATPases appear to more optimized for efficiency than for speed, which can be rationalized through their key role as energy transducers in biology. The present modeling shows how the dynamical performance properties of biomolecular motors and pumps may have evolved to suit their corresponding biological actions.

Rotary catalysis of bovine mitochondrial F1-ATPase studied by single-molecule experiments

The reaction scheme of rotary catalysis and the torque generation mechanism of bovine mitochondrial F₁ (bMF₁) were studied in single-molecule experiments. Under ATP-saturated concentrations, high-speed imaging of a single 40-nm gold bead attached to the γ subunit of bMF₁ showed 2 types of intervening pauses during the rotation that were discriminated by short dwell and long dwell. Using ATPγS as a slowly hydrolyzing ATP derivative as well as using a functional mutant βE188D with slowed ATP hydrolysis, the 2 pausing events were distinctively identified. Buffer-exchange experiments with a nonhydrolyzable analog (AMP-PNP) revealed that the long dwell corresponds to the catalytic dwell, that is, the waiting state for hydrolysis, while it remains elusive which catalytic state short pause represents. The angular position of catalytic dwell was determined to be at +80° from the ATP-binding angle, mostly consistent with other F₁s. The position of short dwell was found at 50 to 60° from catalytic dwell, that is, +10 to 20° from the ATP-binding angle. This is a distinct difference from human mitochondrial F₁, which also shows intervening dwell that probably corresponds to the short dwell of bMF₁, at +65° from the binding pause. Furthermore, we conducted "stall-and-release" experiments with magnetic tweezers to reveal how the binding affinity and hydrolysis equilibrium are modulated by the γ rotation. Similar to thermophilic F₁, bMF₁ showed a strong exponential increase in ATP affinity, while the hydrolysis equilibrium did not change significantly. This indicates that the ATP binding process generates larger torque than the hydrolysis process.

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 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.

Reconstructing free-energy landscapes for cyclic molecular motors using full multidimensional or partial one-dimensional dynamic information

Diffusion on a free-energy landscape is a fundamental framework for describing molecular motors. In the landscape framework, energy conversion between different forms of energy, e.g., chemical and mechanical, is explicitly described using multidimensional nonseparable potential landscapes. We present a k-space method for reconstructing multidimensional free-energy landscapes from stochastic single-molecule trajectories. For a variety of two-dimensional model potential landscapes, we demonstrate the robustness of the method by reconstructing the landscapes using full dynamic information, i.e., simulated two-dimensional stochastic trajectories. We then consider the case where the stochastic trajectory is known only along one dimension. With this partial dynamic information, the reconstruction of the full two-dimensional landscape is severely limited in the majority of cases. However, we reconstruct effective one-dimensional landscapes for the two-dimensional model potentials. We discuss the interpretation of the one-dimensional landscapes and identify signatures of energy conversion. Finally, we consider the implications of these results for biological molecular motors experiments.

Insights into the origin of the high energy-conversion efficiency of F1-ATPase

Our understanding of the rotary-coupling mechanism of F₁-ATPase has been greatly enhanced in the last decade by advances in X-ray crystallography, single-molecular imaging, and theoretical models. Recently, Volkán-Kacsó and Marcus [S. Volkán-Kacsó, R. A. Marcus, Proc. Natl. Acad. Sci. U.S.A. 112, 14230 (2015)] presented an insightful thermodynamic model based on the Marcus reaction theory coupled with an elastic structural deformation term to explain the observed γ-rotation angle dependence of the adenosine triphosphate (ATP)/adenosine diphosphate (ADP) exchange rates of F₁-ATPase. Although the model is successful in correlating single-molecule data, it is not in agreement with the available theoretical results. We describe a revision of the model, which leads to consistency with the simulation results and other experimental data on the F₁-ATPase rotor compliance. Although the free energy liberated on ATP hydrolysis by F₁-ATPase is rapidly dissipated as heat and so cannot contribute directly to the rotation, we show how, nevertheless, F₁-ATPase functions near the maximum possible efficiency. This surprising result is a consequence of the differential binding of ATP and its hydrolysis products ADP and P_i along a well-defined pathway.

Essential Role of the ε Subunit for Reversible Chemo-Mechanical Coupling in F1-ATPase

F₁-ATPase is a rotary motor protein driven by ATP hydrolysis. Among molecular motors, F₁ exhibits unique high reversibility in chemo-mechanical coupling, synthesizing ATP from ADP and inorganic phosphate upon forcible rotor reversal. The ε subunit enhances ATP synthesis coupling efficiency to > 70% upon rotation reversal. However, the detailed mechanism has remained elusive. In this study, we performed stall-and-release experiments to elucidate how the ε subunit modulates ATP association/dissociation and hydrolysis/synthesis process kinetics and thermodynamics, key reaction steps for efficient ATP synthesis. The ε subunit significantly accelerated the rates of ATP dissociation and synthesis by two- to fivefold, whereas those of ATP binding and hydrolysis were not enhanced. Numerical analysis based on the determined kinetic parameters quantitatively reproduced previous findings of two- to fivefold coupling efficiency improvement by the ε subunit at the condition exhibiting the maximum ATP synthesis activity, a physiological role of F₁-ATPase. Furthermore, fundamentally similar results were obtained upon ε subunit C-terminal domain truncation, suggesting that the N-terminal domain is responsible for the rate enhancement.

Is F1-ATPase a Rotary Motor with Nearly 100% Efficiency? Quantitative Analysis of Chemomechanical Coupling and Mechanical Slip

We present a chemomechanical network model of the rotary molecular motor F₁-ATPase which quantitatively describes not only the rotary motor dynamics driven by ATP hydrolysis but also the ATP synthesis caused by forced reverse rotations. We observe a high reversibility of F₁-ATPase, that is, the main cycle of ATP synthesis corresponds to the reversal of the main cycle in the hydrolysis-driven motor rotation. However, our quantitative analysis indicates that torque-induced mechanical slip without chemomechanical coupling occurs under high external torque and reduces the maximal efficiency of the free energy transduction to 40-80% below the optimal efficiency. Heat irreversibly dissipates not only through the viscous friction of the probe but also directly from the motor due to torque-induced mechanical slip. Such irreversible heat dissipation is a crucial limitation for achieving a 100% free-energy transduction efficiency with biological nanomachines because biomolecules are easily deformed by external torque.

Analysing single-molecule trajectories to reconstruct free-energy landscapes of cyclic motor proteins

Stochastic trajectories measured in single-molecule experiments have provided key insights into the microscopic behaviour of cyclic motor proteins. However, the fundamental free-energy landscapes of motor proteins are currently only able to be determined by computationally intensive numerical methods that do not take advantage of available single-trajectory data. In this paper we present a robust method for analysing single-molecule trajectories of cyclic motor proteins to reconstruct their free-energy landscapes. We use simulated trajectories on model potential landscapes to show the reliable reconstruction of the potentials. We determine the accuracy of the reconstruction method for common precision limitations and show that the method converges logarithmically. These results are then used to determine the experimental precision required to reconstruct a potential with a desired accuracy. The key advantages of the method are that it is simple to implement, is free of numerical difficulties that plague existing methods and is easily generalizable to higher dimensions.

Optimal control of rotary motors

Single-molecule experiments have found near-perfect thermodynamic efficiency in the rotary motor F_{1}-ATP synthase. To help elucidate the principles underlying nonequilibrium energetic efficiency in such stochastic machines, we investigate driving protocols that minimize dissipation near equilibrium in a simple model rotary mechanochemical motor, as determined by a generalized friction coefficient. Our simple model has a periodic friction coefficient that peaks near system energy barriers. This implies a minimum-dissipation protocol that proceeds rapidly when the system is overwhelmingly in a single macrostate but slows significantly near energy barriers, thereby harnessing thermal fluctuations to kick the system over energy barriers with minimal work input. This model also manifests a phenomenon not seen in otherwise similar nonperiodic systems: Sufficiently fast protocols can effectively lap the system. While this leads to a trade-off between accuracy of driving and energetic cost, we find that our designed protocols outperform naive protocols.

Reconstructing free-energy landscapes for nonequilibrium periodic potentials

We present a method for reconstructing the free-energy landscape of overdamped Brownian motion on a tilted periodic potential. Our approach exploits the periodicity of the system by using the k-space form of the Smoluchowski equation and we employ an iterative approach to determine the nonequilibrium tilt. We reconstruct landscapes for a number of example potentials to show the applicability of the method to both deep and shallow wells and near-to- and far-from-equilibrium regimes. The method converges logarithmically with the number of Fourier terms in the potential.

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.

Effects of non-equilibrium angle fluctuation on F1-ATPase kinetics induced by temperature increase

F₁-ATPase (F₁) is an efficient rotary protein motor, whose reactivity is modulated by the rotary angle to utilize thermal fluctuation. In order to elucidate how its kinetics are affected by the change in the fluctuation, we have extended the reaction-diffusion formalism [R. Watanabe et al., Biophys. J., 2013, 105, 2385] applicable to a wider range of temperatures based on experimental data analysis of F₁ derived from thermophilic Bacillus under high ATP concentration conditions. Our simulation shows that the rotary angle distribution manifests a stronger non-equilibrium feature as the temperature increases, because ATP hydrolysis and P_i release are more accelerated compared with the timescale of rotary angle relaxation. This effect causes the rate coefficient obtained from dwell time fitting to deviate from the Arrhenius relation in P_i release, which has been assumed in the previous activation thermodynamic quantities estimation using linear Arrhenius fitting. Larger negative correlation is also found between hydrolysis and P_i release waiting time in a catalytic dwell with the increase in temperature. This loss of independence between the two successive reactions at the catalytic dwell sheds doubt on the conventional dwell time fitting to obtain rate coefficients with a double exponential function at temperatures higher than 65 °C, which is close to the physiological temperature of the thermophilic Bacillus.

Steady diffusion in a drift field: A comparison of large-deviation techniques and multiple-scale analysis

A particle with internal unobserved states diffusing in a force field will generally display effective advection-diffusion. The drift velocity is proportional to the mobility averaged over the internal states, or effective mobility, while the effective diffusion has two terms. One is of the equilibrium type and satisfies an Einstein relation with the effective mobility while the other is quadratic in the applied force. In this contribution we present two new methods to obtain these results, on the one hand using large deviation techniques and on the other by a multiple-scale analysis, and compare the two. We consider both systems with discrete internal states and continuous internal states. We show that the auxiliary equations in the multiple-scale analysis can also be derived in second-order perturbation theory in a large deviation theory of a generating function (discrete internal states) or generating functional (continuous internal states). We discuss that measuring the two components of the effective diffusion give a way to determine kinetic rates from only first and second moments of the displacement in steady state.

Nanoscale imaging and force probing of biomolecular systems using atomic force microscopy: from single molecules to living cells

Due to the lack of adequate tools for observation, native molecular behaviors at the nanoscale have been poorly understood. The advent of atomic force microscopy (AFM) provides an exciting instrument for investigating physiological processes on individual living cells with molecular resolution, which attracts the attention of worldwide researchers. In the past few decades, AFM has been widely utilized to investigate molecular activities on diverse biological interfaces, and the performances and functions of AFM have also been continuously improved, greatly improving our understanding of the behaviors of single molecules in action and demonstrating the important role of AFM in addressing biological issues with unprecedented spatiotemporal resolution. In this article, we review the related techniques and recent progress about applying AFM to characterize biomolecular systems in situ from single molecules to living cells. The challenges and future directions are also discussed.

Single-molecule imaging and manipulation of biomolecular machines and systems

BACKGROUND

Biological molecular machines support various activities and behaviors of cells, such as energy production, signal transduction, growth, differentiation, and migration.

SCOPE OF REVIEW

We provide an overview of single-molecule imaging methods involving both small and large probes used to monitor the dynamic motions of molecular machines in vitro (purified proteins) and in living cells, and single-molecule manipulation methods used to measure the forces, mechanical properties and responses of biomolecules. We also introduce several examples of single-molecule analysis, focusing primarily on motor proteins and signal transduction systems.

MAJOR CONCLUSIONS

Single-molecule analysis is a powerful approach to unveil the operational mechanisms both of individual molecular machines and of systems consisting of many molecular machines.

GENERAL SIGNIFICANCE

Quantitative, high-resolution single-molecule analyses of biomolecular systems at the various hierarchies of life will help to answer our fundamental question: "What is life?" This article is part of a Special Issue entitled "Biophysical Exploration of Dynamical Ordering of Biomolecular Systems" edited by Dr. Koichi Kato.

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.

Torque-coupled thermodynamic model for F_{o}F_{1}-ATPase

F_{o}F_{1}-ATPase is a motor protein complex that utilizes transmembrane ion flow to drive the synthesis of adenosine triphosphate (ATP) from adenosine diphosphate (ADP) and phosphate (Pi). While many theoretical models have been proposed to account for its rotary activity, most of them focus on the F_{o} or F_{1} portions separately rather than the complex as a whole. Here, we propose a simple but new torque-coupled thermodynamic model of F_{o}F_{1}-ATPase. Solving this model at steady state, we find that the monotonic variation of each portion's efficiency becomes much more robust over a wide range of parameters when the F_{o} and F_{1} portions are coupled together, as compared to cases when they are considered separately. Furthermore, the coupled model predicts the dependence of each portion's kinetic behavior on the parameters of the other. Specifically, the power and efficiency of the F_{1} portion are quite sensitive to the proton gradient across the membrane, while those of the F_{o} portion as well as the related Michaelis constants for proton concentrations respond insensitively to concentration changes in the reactants of ATP synthesis. The physiological proton gradient across the membrane in the F_{o} portion is also shown to be optimal for the Michaelis constants of ADP and phosphate in the F_{1} portion during ATP synthesis. Together, our coupled model is able to predict key dynamic and thermodynamic features of the F_{o}F_{1}-ATPase in vivo semiquantitatively, and suggests that such coupling approach could be further applied to other biophysical systems.

Catalytic robustness and torque generation of the F1-ATPase

The F₁-ATPase is the catalytic portion of the F_oF₁ ATP synthase and acts as a rotary molecular motor when it hydrolyzes ATP. Two decades have passed since the single-molecule rotation assay of F₁-ATPase was established. Although several fundamental issues remain elusive, basic properties of F-type ATPases as motor proteins have been well characterized, and a large part of the reaction scheme has been revealed by the combination of extensive structural, biochemical, biophysical, and theoretical studies. This review is intended to provide a concise summary of the fundamental features of F₁-ATPases, by use of the well-described model F₁ from the thermophilic Bacillus PS3 (TF₁). In the last part of this review, we focus on the robustness of the rotary catalysis of F₁-ATPase to provide a perspective on the re-designing of novel molecular machines.

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.

Rotation of artificial rotor axles in rotary molecular motors

F₁- and V₁-ATPase are rotary molecular motors that convert chemical energy released upon ATP hydrolysis into torque to rotate a central rotor axle against the surrounding catalytic stator cylinder with high efficiency. How conformational change occurring in the stator is coupled to the rotary motion of the axle is the key unknown in the mechanism of rotary motors. Here, we generated chimeric motor proteins by inserting an exogenous rod protein, FliJ, into the stator ring of F₁ or of V₁ and tested the rotation properties of these chimeric motors. Both motors showed unidirectional and continuous rotation, despite no obvious homology in amino acid sequence between FliJ and the intrinsic rotor subunit of F₁ or V₁ These results showed that any residue-specific interactions between the stator and rotor are not a prerequisite for unidirectional rotation of both F₁ and V₁ The torque of chimeric motors estimated from viscous friction of the rotation probe against medium revealed that whereas the F₁-FliJ chimera generates only 10% of WT F₁, the V₁-FliJ chimera generates torque comparable to that of V₁ with the native axle protein that is structurally more similar to FliJ than the native rotor of F₁ This suggests that the gross structural mismatch hinders smooth rotation of FliJ accompanied with the stator ring of F₁.

Biophysical Characterization of a Thermoalkaliphilic Molecular Motor with a High Stepping Torque Gives Insight into Evolutionary ATP Synthase Adaptation

F₁F₀ ATP synthases are bidirectional molecular motors that translocate protons across the cell membrane by either synthesizing or hydrolyzing ATP. Alkaliphile ATP synthases are highly adapted, performing oxidative phosphorylation at high pH against an inverted pH gradient (acid_in/alkaline_out). Unlike mesophilic ATP synthases, alkaliphilic enzymes have tightly regulated ATP hydrolysis activity, which can be relieved in the presence of lauryldimethylamine oxide. Here, we characterized the rotary dynamics of the Caldalkalibacillus thermarum TA2.A1 F₁ ATPase (TA2F₁) with two forms of single molecule analysis, a magnetic bead duplex and a gold nanoparticle. TA2F₁ rotated in a counterclockwise direction in both systems, adhering to Michaelis-Menten kinetics with a maximum rotation rate (V_max) of 112.4 revolutions/s. TA2F₁ displayed 120° unitary steps coupled with ATP hydrolysis. Torque measurements revealed the highest torque (52.4 piconewtons) derived from an F₁ molecule using fluctuation theorem. The implications of high torque in terms of extreme environment adaptation are discussed.

Torque, chemistry and efficiency in molecular motors: a study of the rotary-chemical coupling in F1-ATPase

Detailed understanding of the action of biological molecular machines must overcome the challenge of gaining a clear knowledge of the corresponding free-energy landscape. An example for this is the elucidation of the nature of converting chemical energy to torque and work in the rotary molecular motor of F₁-ATPase. A major part of the challenge involves understanding the rotary–chemical coupling from a non-phenomenological structure/energy description. Here we focused on using a coarse-grained model of F₁-ATPase to generate a structure-based free-energy landscape of the rotary–chemical process of the whole system. In particular, we concentrated on exploring the possible impact of the position of the catalytic dwell on the efficiency and torque generation of the molecular machine. It was found that the experimentally observed torque can be reproduced with landscapes that have different positions for the catalytic dwell on the rotary–chemical surface. Thus, although the catalysis is undeniably required for torque generation, the experimentally observed position of the catalytic dwell at 80° might not have a clear advantage for the force generation by F₁-ATPase. This further implies that the rotary–chemical couplings in these biological motors are quite robust and their efficiencies do not depend explicitly on the position of the catalytic dwells. Rather, the specific positioning of the dwells with respect to the rotational angle is a characteristic arising due to the structural construct of the molecular machine and might not bear any clear connection to the thermodynamic efficiency for the system.

Operating principles of rotary molecular motors: differences between F1 and V1 motors

Among the many types of bioenergy-transducing machineries, F- and V-ATPases are unique bio- and nano-molecular rotary motors. The rotational catalysis of F₁-ATPase has been investigated in detail, and molecular mechanisms have been proposed based on the crystal structures of the complex and on extensive single-molecule rotational observations. Recently, we obtained crystal structures of bacterial V₁-ATPase (A₃B₃ and A₃B₃DF complexes) in the presence and absence of nucleotides. Based on these new structures, we present a novel model for the rotational catalysis mechanism of V₁-ATPase, which is different from that of F₁-ATPases.

Biological Nanomotors with a Revolution, Linear, or Rotation Motion Mechanism

The ubiquitous biological nanomotors were classified into two categories in the past: linear and rotation motors. In 2013, a third type of biomotor, revolution without rotation (http://rnanano.osu.edu/movie.html), was discovered and found to be widespread among bacteria, eukaryotic viruses, and double-stranded DNA (dsDNA) bacteriophages. This review focuses on recent findings about various aspects of motors, including chirality, stoichiometry, channel size, entropy, conformational change, and energy usage rate, in a variety of well-studied motors, including FoF1 ATPase, helicases, viral dsDNA-packaging motors, bacterial chromosome translocases, myosin, kinesin, and dynein. In particular, dsDNA translocases are used to illustrate how these features relate to the motion mechanism and how nature elegantly evolved a revolution mechanism to avoid coiling and tangling during lengthy dsDNA genome transportation in cell division. Motor chirality and channel size are two factors that distinguish rotation motors from revolution motors. Rotation motors use right-handed channels to drive the right-handed dsDNA, similar to the way a nut drives the bolt with threads in same orientation; revolution motors use left-handed motor channels to revolve the right-handed dsDNA. Rotation motors use small channels (<2 nm in diameter) for the close contact of the channel wall with single-stranded DNA (ssDNA) or the 2-nm dsDNA bolt; revolution motors use larger channels (>3 nm) with room for the bolt to revolve. Binding and hydrolysis of ATP are linked to different conformational entropy changes in the motor that lead to altered affinity for the substrate and allow work to be done, for example, helicase unwinding of DNA or translocase directional movement of DNA.

Torque transmission mechanism via DELSEED loop of F1-ATPase

F1-ATPase (F1) is an ATP-driven rotary motor in which the three catalytic β subunits in the stator ring sequentially induce the unidirectional rotation of the rotary γ subunit. Many lines of evidence have revealed open-to-closed conformational transitions in the β subunit that swing the C-terminal domain inward. This conformational transition causes a C-terminal protruding loop with conserved sequence DELSEED to push the γ subunit. Previous work, where all residues of DELSEED were substituted with glycine to disrupt the specific interaction with γ and introduce conformational flexibility, showed that F1 still rotated, but that the torque was halved, indicating a remarkable impact on torque transmission. In this study, we conducted a stall-and-release experiment on F1 with a glycine-substituted DELSEED loop to investigate the impact of the glycine substitution on torque transmission upon ATP binding and ATP hydrolysis. The mutant F1 showed a significantly reduced angle-dependent change in ATP affinity, whereas there was no change in the equilibrium for ATP hydrolysis. These findings indicate that the DELSEED loop is predominantly responsible for torque transmission upon ATP binding but not for that upon ATP hydrolysis.

ATP hydrolysis assists phosphate release and promotes reaction ordering in F1-ATPase

F1-ATPase (F1) is a rotary motor protein that can efficiently convert chemical energy to mechanical work of rotation via fine coordination of its conformational motions and reaction sequences. Compared with reactant binding and product release, the ATP hydrolysis has relatively little contributions to the torque and chemical energy generation. To scrutinize possible roles of ATP hydrolysis, we investigate the detailed statistics of the catalytic dwells from high-speed single wild-type F1 observations. Here we report a small rotation during the catalytic dwell triggered by the ATP hydrolysis that is indiscernible in previous studies. Moreover, we find in freely rotating F1 that ATP hydrolysis is followed by the release of inorganic phosphate with low synthesis rates. Finally, we propose functional roles of the ATP hydrolysis as a key to kinetically unlock the subsequent phosphate release and promote the correct reaction ordering.

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.

Torque generation mechanism of F1-ATPase upon NTP binding

Molecular machines fueled by NTP play pivotal roles in a wide range of cellular activities. One common feature among NTP-driven molecular machines is that NTP binding is a major force-generating step among the elementary reaction steps comprising NTP hydrolysis. To understand the mechanism in detail,in this study, we conducted a single-molecule rotation assay of the ATP-driven rotary motor protein F1-ATPase using uridine triphosphate (UTP) and a base-free nucleotide (ribose triphosphate) to investigate the impact of a pyrimidine base or base depletion on kinetics and force generation. Although the binding rates of UTP and ribose triphosphate were 10(3) and 10(6) times, respectively, slower than that of ATP, they supported rotation, generating torque comparable to that generated by ATP. Affinity change of F1 to UTP coupled with rotation was determined, and the results again were comparable to those for ATP, suggesting that F1 exerts torque upon the affinity change to UTP via rotation similar to ATP-driven rotation. Thus, the adenine-ring significantly enhances the binding rate, although it is not directly involved in force generation. Taking into account the findings from another study on F1 with mutated phosphate-binding residues, it was proposed that progressive bond formation between the phosphate region and catalytic residues is responsible for the rotation-coupled change in affinity.

Simple mechanism whereby the F1-ATPase motor rotates with near-perfect chemomechanical energy conversion

F1-ATPase is a motor enzyme in which a central shaft γ subunit rotates 120° per ATP in the cylinder made of α3β3 subunits. During rotation, the chemical energy of ATP hydrolysis (ΔGATP) is converted almost entirely into mechanical work by an elusive mechanism. We measured the force for rotation (torque) under various ΔGATP conditions as a function of rotation angles of the γ subunit with quasi-static, single-molecule manipulation and estimated mechanical work (torque × traveled angle) from the area of the function. The torque functions show three sawtooth-like repeats of a steep jump and linear descent in one catalytic turnover, indicating a simple physical model in which the motor is driven by three springs aligned along a 120° rotation angle. Although the second spring is unaffected by ΔGATP, activation of the first spring (timing of the torque jump) delays at low [ATP] (or high [ADP]) and activation of the third spring delays at high [Pi]. These shifts decrease the size and area of the sawtooth (magnitude of the work). Thus, F1-ATPase responds to the change of ΔGATP by shifting the torque jump timing and uses ΔGATP for the mechanical work with near-perfect efficiency.

Dissecting the role of the γ-subunit in the rotary-chemical coupling and torque generation of F1-ATPase

Unraveling the molecular nature of the conversion of chemical energy (ATP hydrolysis in the α/β-subunits) to mechanical energy and torque (rotation of the γ-subunit) in F1-ATPase is very challenging. A major part of the challenge involves understanding the rotary-chemical coupling by a nonphenomenological structure-energy description, while accounting for the observed torque generated on the γ-subunit and its change due to mutation of this unit. Here we extend our previous study that used a coarse-grained model of the F1-ATPase to generate a structure-based free energy landscape of the rotary-chemical process. Our quantitative analysis of the landscape reproduced the observed torque for the wild-type enzyme. In doing so, we found that there are several possibilities of torque generation from landscapes with various shapes and demonstrated that a downhill slope along the chemical coordinate could still result in negligible torque, due to ineffective coupling of the chemistry to the γ-subunit rotation. We then explored the relationship between the functionality and the underlying sequence through systematic examination of the effect of various parts of the γ-subunit on free energy surfaces of F1-ATPase. Furthermore, by constructing several types of γ-deletion systems and calculating the corresponding torque generation, we gained previously unknown insights into the molecular nature of the F1-ATPase rotary motor. Significantly, our results are in excellent agreement with recent experimental findings and indicate that the rotary-chemical coupling is primarily established through electrostatic effects, although specific contacts through γ-ionizable residue side chains are not essential for establishing the basic features of the coupling.

Nonequilibrium dissipation-free transport in F₁-ATPase and the thermodynamic role of asymmetric allosterism

F1-ATPase (or F1), the highly efficient and reversible biochemical engine, has motivated physicists as well as biologists to imagine the design principles governing machines in the fluctuating world. Recent experiments have clarified yet another interesting property of F1; the dissipative heat inside the motor is very small, irrespective of the velocity of rotation and energy transport. Conceptual interest is devoted to the fact that the amount of internal dissipation is not simply determined by the sequence of equilibrium pictures, but also relies on the rotational-angular dependence of nucleotide affinity, which is a truly nonequilibrium aspect. We propose that the totally asymmetric allosteric model (TASAM), where adenosine triphosphate (ATP) binding to F1 is assumed to have low dependence on the angle of the rotating shaft, produces results that are most consistent with the experiments. Theoretical analysis proves the crucial role of two time scales in the model, which explains the universal mechanism to produce the internal dissipation-free feature. The model reproduces the characteristic torque dependence of the rotational velocity of F1 and predicts that the internal dissipation upon the ATP synthesis direction rotation becomes large at the low nucleotide condition.

Trapping the ATP binding state leads to a detailed understanding of the F1-ATPase mechanism

The rotary motor enzyme FoF1-ATP synthase uses the proton-motive force across a membrane to synthesize ATP from ADP and Pi (H2PO4(-)) under cellular conditions that favor the hydrolysis reaction by a factor of 2 × 10(5). This remarkable ability to drive a reaction away from equilibrium by harnessing an external force differentiates it from an ordinary enzyme, which increases the rate of reaction without shifting the equilibrium. Hydrolysis takes place in the neighborhood of one conformation of the catalytic moiety F1-ATPase, whose structure is known from crystallography. By use of molecular dynamics simulations we trap a second structure, which is rotated by 40° from the catalytic dwell conformation and represents the state associated with ATP binding, in accord with single-molecule experiments. Using the two structures, we show why Pi is not released immediately after ATP hydrolysis, but only after a subsequent 120° rotation, in agreement with experiment. A concerted conformational change of the α3β3 crown is shown to induce the 40° rotation of the γ-subunit only when the βE subunit is empty, whereas with Pi bound, βE serves as a latch to prevent the rotation of γ. The present results provide a rationalization of how F1-ATPase achieves the coupling between the small changes in the active site of βDP and the 40° rotation of γ.