Fluctuation theorem applied to F1-ATPase

General theory for localizing the where and when of entropy production meets single-molecule experiments

The laws of thermodynamics apply to biophysical systems on the nanoscale as described by the framework of stochastic thermodynamics. This theory provides universal, exact relations for quantities like work, which have been verified in experiments where a fully resolved description allows direct access to such quantities. Complementary studies consider partially hidden, coarse-grained descriptions, in which the mean entropy production typically is not directly accessible but can be bounded in terms of observable quantities. Going beyond the mean, we introduce a fluctuating entropy production that applies to individual trajectories in a coarse-grained description under time-dependent driving. Thus, this concept is applicable to the broad and experimentally significant class of driven systems in which not all relevant states can be resolved. We provide a paradigmatic example by studying an experimentally verified protein unfolding process. As a consequence, the entire distribution of the coarse-grained entropy production rather than merely its mean retains spatial and temporal information about the microscopic process. In particular, we obtain a bound on the distribution of the physical entropy production of individual unfolding events.

Harnessing Information Thermodynamics: Conversion of DNA Information into Mechanical Work in RNA Transcription and Nanopore Sequencing

Recent advancements in information thermodynamics have revealed that information can be directly converted into mechanical work. Specifically, RNA transcription and nanopore sequencing serve as prime examples of this conversion, by reading information from a DNA template. This paper introduces an information thermodynamic model in which these molecular motors can move along the DNA template by converting the information read from the template DNA into their own motion. This process is a stochastic one, characterized by significant fluctuations in forward movement and is described by the Fokker-Planck equation, based on drift velocity and diffusion coefficients. In the current study, it is hypothesized that by utilizing the sequence information of the template DNA as mutual information, the fluctuations can be reduced, thereby biasing the forward movement on DNA and, consequently, reducing reading errors. Further research into the conversion of biological information by molecular motors could unveil new applications, insights, and important findings regarding the characteristics of information processing in biology.

Work distribution of a colloid in an elongational flow field and under Ornstein-Uhlenbeck noise

The study of thermodynamic properties of microscopic systems, such as a colloid in a fluid, has been of great interest to researchers since the discovery of the fluctuation theorem and associated laws of stochastic thermodynamics. However, most of these studies confine themselves to systems where effective fluctuations acting on the colloid are in the form of delta-correlated Gaussian white noise (GWN). In this study, instead, we look into the work distribution function when a colloid trapped in a harmonic potential moves from one position to another in a fluid medium with an elongational flow field where the effective fluctuations are given by the Ornstein-Uhlenbeck noise, a type of colored noise. We use path integrals to calculate this distribution function and compare and contrast its properties to the case with GWN. We find that the work distribution function turns out to be non-Gaussian as a result of the elongational flow field but continues to obey the fluctuation theorem in both types of noise. Further, we also look into the effects of the various system parameters on the behavior of work fluctuations and find that although the distribution tends to broaden with increasing noise intensity, increased correlation in fluctuations acts to oppose this effect. Additionally, the system is found to consume heat from the surroundings at early times and dissipate it into the media at later times. This study, therefore, is a step towards gaining a better understanding of the thermodynamic properties of colloidal systems under nonlinear complex flows that also display correlated fluctuations.

Rotations of F-ATPase and V-ATPase analyzed by a torque approach

F-type ATP synthase (F-ATPase) and vacuolar ATP hydrolase (V-ATPase) are well-known biomolecular motors, which play significant catalytic roles in ATP synthesis and ATP hydrolysis reactions. Their rotational torques are important factors involved in their rotational behavior that can be measured experimentally but with considerable difficulty. To overcome this difficulty and thereby provide an in-depth understanding of their operation mechanism, we herein carry out simple and fast computer modelling to study the two proteins, using our torque approach that relies on interatomic forces and coordinates of unequilibrated configurations taken from brief molecular dynamics (MD) simulations. As predicted by the torque approach, F-ATPase is demonstrated to be a random rotor, but it prefers to rotate in clockwise direction (as seen from the membrane toward the protein) for ATP synthesis, owing to the predominantly negative angle-averaged torques. By contrast, V-ATPase tends to rotate only in counterclockwise direction for ATP hydrolysis, due to the almost uniform averaged positive torques generated by the unidirectional rotation near the three catalytic sites. The rotational behaviors of both proteins are also affected by the surrounding solvent which can promote or hinder the internal rotation. By combining the torque approach with classic force-field MD simulations, the torques of two biomolecular motors can be calculated economically, and are found to agree with previous experiments and theoretical calculations. This work demonstrates that our torque approach can be extended to the field of biology and can help gain a deeper insight into the mechanistic rotation of biomolecular motors with modest computation time.Communicated by Ramaswamy H. Sarma.

Waiting Time Distributions in Hybrid Models of Motor-Bead Assays: A Concept and Tool for Inference

In single-molecule experiments, the dynamics of molecular motors are often observed indirectly by measuring the trajectory of an attached bead in a motor-bead assay. In this work, we propose a method to extract the step size and stalling force for a molecular motor without relying on external control parameters. We discuss this method for a generic hybrid model that describes bead and motor via continuous and discrete degrees of freedom, respectively. Our deductions are solely based on the observation of waiting times and transition statistics of the observable bead trajectory. Thus, the method is non-invasive, operationally accessible in experiments and can, in principle, be applied to any model describing the dynamics of molecular motors. We briefly discuss the relation of our results to recent advances in stochastic thermodynamics on inference from observable transitions. Our results are confirmed by extensive numerical simulations for parameters values of an experimentally realized F1-ATPase assay.

Direct identification of the rotary angle of ATP cleavage in F1-ATPase from Bacillus PS3

F1-ATPase is the world's smallest biological rotary motor driven by ATP hydrolysis at three catalytic β subunits. The 120° rotational step of the central shaft γ consists of 80° substep driven by ATP binding and a subsequent 40° substep. In order to correlate timing of ATP cleavage at a specific catalytic site with a rotary angle, we designed a new F1-ATPase (F1) from thermophilic Bacillus PS3 carrying β(E190D/F414E/F420E) mutations, which cause extremely slow rates of both ATP cleavage and ATP binding. We produced an F1 molecule that consists of one mutant β and two wild-type βs (hybrid F1). As a result, the new hybrid F1 showed two pausing angles that are separated by 200°. They are attributable to two slowed reaction steps in the mutated β, thus providing the direct evidence that ATP cleavage occurs at 200° rather than 80° subsequent to ATP binding at 0°. This scenario resolves the long-standing unclarified issue in the chemomechanical coupling scheme and gives insights into the mechanism of driving unidirectional rotation.

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.

Internal energy and information flows mediate input and output power in bipartite molecular machines

Microscopic biological systems operate far from equilibrium, are subject to strong fluctuations, and are composed of many coupled components with interactions varying in nature and strength. Researchers are actively investigating the general design principles governing how biomolecular machines achieve effective free-energy transduction in light of these challenges. We use a model of two strongly coupled stochastic rotary motors to explore the effect of coupling strength between components of a molecular machine. We observe prominent thermodynamic characteristics at intermediate coupling strength, near that which maximizes output power: a maximum in power and information transduced from the upstream to the downstream system, and equal subsystem entropy production rates. These observations are unified through a bound on the machine's input and output power, which accounts for both the energy and information transduced between subsystems.

Cycle counts and affinities in stochastic models of nonequilibrium systems

For nonequilibrium systems described by finite Markov processes, we consider the number of times that a system traverses a cyclic sequence of states (a cycle). The joint distribution of the number of forward and backward instances of any given cycle is described by universal formulas which depend on the cycle affinity, but are otherwise independent of system details. We discuss the similarities and differences of this result to fluctuation theorems, and generalize the result to families of cycles, relevant under coarse graining. Finally, we describe the application of large deviation theory to this cycle-counting problem.

Work fluctuation relation of an active Brownian particle in a viscoelastic fluid

We experimentally investigate the work fluctuations of an active Brownian particle (ABP) during its self-propelled motion in a viscoelastic medium. Under such conditions, ABPs display a persistent circular motion which allows the determination of the orientational work fluctuations along its trajectory. Due to the nonlinear coupling to the non-Markovian bath, we find strong deviations from the work fluctuation theorem (WFT) due to observed increased rotational ABP dynamics. Taking this enhanced rotational diffusion into account, the orientational work distributions can be recasted to be in accordance with the WFT by considering an effective temperature of about two orders of magnitude larger than k_{B}T. This approach is confirmed by the good agreement of the torque exerted by the viscoelastic bath on the ABP obtained from the WFT with the value obtained from the mean angular velocity and the friction coefficient of the ABP.

The Measurement of Information and Free Energy in Mechanical-Force-Driven Coil-Globule Transitions

We study the role of information (the relative entropy) for polymers undergoing coil-globule transitions driven by a time-dependent force. Pulling experiments at various speeds are performed by Brownian dynamics simulations. We obtain the work distributions for the forward and time-reversed backward processes and information stored at the end of the nonequilibrium pulling processes. We present the systematic method to measure the information from the pulling experiments and extract the information by analyzing slowly relaxing modes. When the information is incorporated, the work distributions modified by the information allow access to the proper free energy via the formulation of the generalized fluctuation theorems even if the initial states of the forward and time-reversed backward processes are out of equilibrium. This demonstrates that the work-information conversion works well for a single-molecule system with many degrees of freedom.

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.

Faster Uphill Relaxation in Thermodynamically Equidistant Temperature Quenches

We uncover an unforeseen asymmetry in relaxation: for a pair of thermodynamically equidistant temperature quenches, one from a lower and the other from a higher temperature, the relaxation at the ambient temperature is faster in the case of the former. We demonstrate this finding on hand of two exactly solvable many-body systems relevant in the context of single-molecule and tracer-particle dynamics. We prove that near stable minima and for all quadratic energy landscapes it is a general phenomenon that also exists in a class of non-Markovian observables probed in single-molecule and particle-tracking experiments. The asymmetry is a general feature of reversible overdamped diffusive systems with smooth single-well potentials and occurs in multiwell landscapes when quenches disturb predominantly intrawell equilibria. Our findings may be relevant for the optimization of stochastic heat engines.

Tight Chemomechanical Coupling of the F1 Motor Relies on Structural Stability

The F₁ motor is a rotating molecular motor that ensures a tight chemomechanical coupling between ATP hydrolysis/synthesis reactions and rotation steps. However, the mechanism underlying this tight coupling remains to be elucidated. In this study, we used electrorotation in single-molecule experiments using an F₁βE190D mutant to demonstrate that the stall torque was significantly smaller than the wild-type F₁, indicating a loose coupling of this mutant, despite showing similar stepping torque as the wild-type. Experiments on the ATPase activity after heat treatment and gel filtration of the α₃β₃-subcomplex revealed the unstable structure of the βE190D mutant. Our results suggest that the tight chemomechanical coupling of the F₁ motor relies on the structural stability of F₁. We also discuss the difference between the stepping torque and the stall torque.

Entropy production estimation with optimal current

Entropy production characterizes the thermodynamic irreversibility and reflects the amount of heat dissipated into the environment and free energy lost in nonequilibrium systems. According to the thermodynamic uncertainty relation, we propose a deterministic method to estimate the entropy production from a single trajectory of system states. We explicitly and approximately compute an optimal current that yields the tightest lower bound using predetermined basis currents. Notably, the obtained tightest lower bound is intimately related to the multidimensional thermodynamic uncertainty relation. By proving the saturation of the thermodynamic uncertainty relation in the short-time limit, the exact estimate of the entropy production can be obtained for overdamped Langevin systems, irrespective of the underlying dynamics. For Markov jump processes, because the attainability of the thermodynamic uncertainty relation is not theoretically ensured, the proposed method provides the tightest lower bound for the entropy production. When entropy production is the optimal current, a more accurate estimate can be further obtained using the integral fluctuation theorem. We illustrate the proposed method using three systems: a four-state Markov chain, a periodically driven particle, and a multiple bead-spring model. The estimated results in all examples empirically verify the effectiveness and efficiency of the proposed method.

Nonequilibrium driven by an external torque in the presence of a magnetic field

We investigate a two-dimensional motion of a colloid in a harmonic trap driven out of equilibrium by an external nonconservative force producing a torque in the presence of a uniform magnetic field applied perpendicular to the plane of motion. We find a circulating steady-state current diagnostic to nonequilibrium. Unlikely in the overdamped limit, inertial motion requires a sufficient central force to reach steady state. The magnetic field can enhance or depress central force depending on its direction. We find that steady state exists only for a proper range of parameters such as mass, viscosity coefficient, stiffness of the harmonic potential, and the magnetic field. We rigorously derive the existence condition for the steady state. We examine the combined influence of nonconservative force and magnetic field on nonequilibrium characteristics. We find non-Boltzmann steady-state probability density function and circulating probability current. We show that nonnegative entropy production is composed of usual heat dissipation and unconventional contribution from velocity-dependence of the Lorentz force. We derive the full list of correlation functions, including position-velocity correlation function originated from nonequilibrium circulation. We finally give rigorous expression for the violation of fluctuation-dissipation relation. We verify our analytical results by using the Monte Carlo simulation.

Application of the fluctuation theorem for noninvasive force measurement in living neuronal axons

Although its importance is recently widely accepted, force measurement has been difficult in living biological systems, mainly due to the lack of the versatile noninvasive force measurement methods. The fluctuation theorem, which represents the thermodynamic properties of small fluctuating nonequilibrium systems, has been applied to the analysis of the thermodynamic properties of motor proteins in vitro. Here we extend it to the axonal transport (displacement) of endosomes. The distribution of the displacement fluctuation had three or four distinct peaks around multiples of a unit value, which the fluctuation theorem can convert into the drag force exerted on the endosomes. The results demonstrated that a single cargo vesicle is conveyed by one to three or four units of force production.

Application of the fluctuation theorem to motor proteins: from F1-ATPase to axonal cargo transport by kinesin and dynein

The fluctuation theorem is a representative theorem in non-equilibrium statistical physics actively studied in the 1990s. Relating to entropy production in non-equilibrium states, the theorem has been used to estimate the driving power of motor proteins from fluctuation in their motion. In this review, usage of the fluctuation theorem in experiments on motor proteins is illustrated for biologists, especially those who study mechanobiology, in which force measurement is a central issue. We first introduce the application of the fluctuation theorem in measuring the rotary torque of the rotary motor protein F1-ATPase. Next, as an extension of this application, a recent trial estimating the force generated during cargo transport in vivo by the microtubule motors kinesin and dynein is introduced. Elucidation of the physical mechanism of such transport is important, especially for neurons, in which deficits in cargo transport are deeply related to neuronal diseases. Finally, perspectives on the fluctuation theorem as a new technique in the field of neuroscience are discussed.

Broken detailed balance and non-equilibrium dynamics in living systems: a review

Living systems operate far from thermodynamic equilibrium. Enzymatic activity can induce broken detailed balance at the molecular scale. This molecular scale breaking of detailed balance is crucial to achieve biological functions such as high-fidelity transcription and translation, sensing, adaptation, biochemical patterning, and force generation. While biological systems such as motor enzymes violate detailed balance at the molecular scale, it remains unclear how non-equilibrium dynamics manifests at the mesoscale in systems that are driven through the collective activity of many motors. Indeed, in several cellular systems the presence of non-equilibrium dynamics is not always evident at large scales. For example, in the cytoskeleton or in chromosomes one can observe stationary stochastic processes that appear at first glance thermally driven. This raises the question how non-equilibrium fluctuations can be discerned from thermal noise. We discuss approaches that have recently been developed to address this question, including methods based on measuring the extent to which the system violates the fluctuation-dissipation theorem. We also review applications of this approach to reconstituted cytoskeletal networks, the cytoplasm of living cells, and cell membranes. Furthermore, we discuss a more recent approach to detect actively driven dynamics, which is based on inferring broken detailed balance. This constitutes a non-invasive method that uses time-lapse microscopy data, and can be applied to a broad range of systems in cells and tissue. We discuss the ideas underlying this method and its application to several examples including flagella, primary cilia, and cytoskeletal networks. Finally, we briefly discuss recent developments in stochastic thermodynamics and non-equilibrium statistical mechanics, which offer new perspectives to understand the physics of living systems.

Non-invasive force measurement reveals the number of active kinesins on a synaptic vesicle precursor in axonal transport regulated by ARL-8

Kinesin superfamily protein UNC-104, a member of the kinesin-3 family, transports synaptic vesicle precursors (SVPs). In this study, the number of active UNC-104 molecules hauling a single SVP in axons in the worm Caenorhabditis elegans was counted by applying a newly developed non-invasive force measurement technique. The distribution of the force acting on a SVP transported by UNC-104 was spread out over several clusters, implying the presence of several force-producing units (FPUs). We then compared the number of FPUs in the wild-type worms with that in arl-8 gene-deletion mutant worms. ARL-8 is a SVP-bound arf-like small guanosine triphosphatase, and is known to promote unlocking of the autoinhibition of the motor, which is critical for avoiding unnecessary consumption of adenosine triphosphate when the motor does not bind to a SVP. There were fewer FPUs in the arl-8 mutant worms. This finding indicates that a lack of ARL-8 decreased the number of active UNC-104 motors, which then led to a decrease in the number of motors responsible for SVP transport.

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.

Rotation Measurements of Tethered Cells

Direct observation of the rotation of tethered cells using optical microscopy is a simple method to examine dynamics of the bacterial flagellar motor. The rotational speed indicates not only the existence of the rotary motor, but also approximate number of stators that are torque-generating units in a motor. Since "run" and "tumble" of the peritrichous cells as Escherichia coli are regulated by the counterclockwise rotation and the clockwise rotation, respectively, rotational direction of the tethered cell is an important clue to understand the chemotactic system of the cells.

Iterative control strategies for nonlinear systems

In this paper, we focus on the control of the mean-field equilibrium of nonlinear networks of the Langevin type in the limit of small noise. Using iterative linear approximations, we derive a formula that prescribes a control strategy in order to displace the equilibrium state of a given system and remarkably find that the control function has a "universal" form under certain physical conditions. This result can be employed to define universal protocols useful, for example, in the optimal work extraction from a given reservoir. Generalizations and limits of application of the method are 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.

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.

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.

Entropy production by active particles: Coupling of odd and even functions of velocity

Nonequilibrium stochastic dynamics of several active Brownian systems are modeled in terms of nonlinear velocity dependent force. In general, this force may consist of both even and odd functions of velocity. We derive the expression for total entropy production in such systems using the Fokker-Planck equation. The result is consistent with the expression for stochastic entropy production in the reservoir that we obtain from probabilities of time-forward and time-reversed trajectories, leading to fluctuation theorems. Numerical simulation is used to find probability distribution of entropy production, which shows good agreement with the detailed fluctuation theorem.

F1 rotary motor of ATP synthase is driven by the torsionally-asymmetric drive shaft

F₁F₀ ATP synthase (ATPase) either facilitates the synthesis of ATP in a process driven by the proton moving force (pmf), or uses the energy from ATP hydrolysis to pump protons against the concentration gradient across the membrane. ATPase is composed of two rotary motors, F₀ and F₁, which compete for control of their shared γ -shaft. We present a self-consistent physical model of F₁ motor as a simplified two-state Brownian ratchet using the asymmetry of torsional elastic energy of the coiled-coil γ -shaft. This stochastic model unifies the physical concepts of linear and rotary motors, and explains the stepped unidirectional rotary motion. Substituting the model parameters, all independently known from recent experiments, our model quantitatively reproduces the ATPase operation, e.g. the ‘no-load’ angular velocity is ca. 400 rad/s anticlockwise at 4 mM ATP. Increasing the pmf torque exerted by F₀ can slow, stop and overcome the torque generated by F₁, switching from ATP hydrolysis to synthesis at a very low value of ‘stall torque’. We discuss the motor efficiency, which is very low if calculated from the useful mechanical work it produces - but is quite high when the ‘useful outcome’ is measured in the number of H⁺ pushed against the chemical gradient.

Universal bounds on current fluctuations

For current fluctuations in nonequilibrium steady states of Markovian processes, we derive four different universal bounds valid beyond the Gaussian regime. Different variants of these bounds apply to either the entropy change or any individual current, e.g., the rate of substrate consumption in a chemical reaction or the electron current in an electronic device. The bounds vary with respect to their degree of universality and tightness. A universal parabolic bound on the generating function of an arbitrary current depends solely on the average entropy production. A second, stronger bound requires knowledge both of the thermodynamic forces that drive the system and of the topology of the network of states. These two bounds are conjectures based on extensive numerics. An exponential bound that depends only on the average entropy production and the average number of transitions per time is rigorously proved. This bound has no obvious relation to the parabolic bound but it is typically tighter further away from equilibrium. An asymptotic bound that depends on the specific transition rates and becomes tight for large fluctuations is also derived. This bound allows for the prediction of the asymptotic growth of the generating function. Even though our results are restricted to networks with a finite number of states, we show that the parabolic bound is also valid for three paradigmatic examples of driven diffusive systems for which the generating function can be calculated using the additivity principle. Our bounds provide a general class of constraints for nonequilibrium systems.

Torque generation through the random movement of an asymmetric rotor: A potential rotational mechanism of the γ subunit of F(1)-ATPase

The rotation of the γ subunit of F(1)-ATPase is stochastic, processive, unidirectional, reversible through an external torque, and stepwise with a slow rotation. We propose a mechanism that can explain these properties of the rotary molecular motor, and that can determine the direction of rotation. The asymmetric structures of the γ subunit, both at the tip of the shaft (C and N termini) and at the part (ε subunit) protruding from the α(3)β(3) subunits, are critical. The torque required for stochastic rotation is generated from the impulsive reactive force due to the random collisions between the γ subunit and the quasihexagonal α(3)β(3) subunits. The rotation is the result of the random motion of the confined asymmetric γ subunit. The steps originate from the chemical reactions of the γ subunit and physical interaction between the γ subunit and the flexible protrusions of the α(3)β(3) subunits. An external torque as well as a configurational modification in the γ subunit (the central rotor) can reverse the rotational direction. We demonstrate the applicability of the mechanism to a macroscopic simulation system, which has the essential ingredients of the F(1)-ATPase structure, by reproducing the dynamic properties of the rotation.

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.

Anomalous free energy changes induced by topology

We report that nontrivial topology of a driven Brownian particle restricted on a ring leads to anomalous behaviors on free energy change. Starting from steady states with identical distribution and current on the ring, free energy changes are distinct and nonperiodic after the system is driven by the same periodic force protocol. We demonstrate our observation in examples through both exact solutions and numerical simulations. The free energy calculated here can be measured in recent experimental systems.

Stochastic thermodynamics of macrospins with fluctuating amplitude and direction

We consider stochastic energy balance and entropy production (EP) in a generalized Langevin dynamics of macrospins, allowing for both amplitude and direction fluctuations, under external magnetic field. EP is calculated using a Fokker-Planck equation, distinguishing between reversible and irreversible parts of probability currents. The system entropy increases due to irreversible non-equilibrium processes, and reduces as heat dissipates to the surrounding environment. Using path probability distributions of time-forward trajectories and conjugate trajectories under time reversal, we obtain fluctuation theorems (FT) for total stochastic EP. We show that the choice of conjugate trajectories is crucial in obtaining entropy-like quantities that obey FTs.

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.

Maxwell's demon in biochemical signal transduction with feedback loop

Signal transduction in living cells is vital to maintain life itself, where information transfer in noisy environment plays a significant role. In a rather different context, the recent intensive research on ‘Maxwell's demon'—a feedback controller that utilizes information of individual molecules—have led to a unified theory of information and thermodynamics. Here we combine these two streams of research, and show that the second law of thermodynamics with information reveals the fundamental limit of the robustness of signal transduction against environmental fluctuations. Especially, we find that the degree of robustness is quantitatively characterized by an informational quantity called transfer entropy. Our information-thermodynamic approach is applicable to biological communication inside cells, in which there is no explicit channel coding in contrast to artificial communication. Our result could open up a novel biophysical approach to understand information processing in living systems on the basis of the fundamental information–thermodynamics link.

The connection between information and thermodynamics is embodied in the figure of Maxwell's demon, a feedback controller. Here, the authors apply thermodynamics of information to signal transduction in chemotaxis of E. coli, predicting that its robustness is quantified by transfer entropy.

Rotational mechanism of Enterococcus hirae V1-ATPase by crystal-structure and single-molecule analyses

In ion-transporting rotary ATPases, the mechanical rotation of inner rotor subunits against other stator subunits in the complex mediates conversion of chemical free energy from ATP hydrolysis into electrochemical potential by pumping ions across the cell membrane. To fully understand the rotational mechanism of energy conversion, it is essential to analyze a target sample by multiple advanced methods that differ in spatiotemporal resolutions and sample environments. Here, we describe such a strategy applied to the water-soluble V1 moiety of Enterococcus hirae V-ATPase; this strategy involves integration of crystal structure studies and single-molecule analysis of rotary dynamics and torque generation. In addition, we describe our current model of the chemo-mechanical coupling scheme obtained by this approach, as well as future prospects.

Minimal model of stochastic athermal systems: origin of non-Gaussian noise

For a wide class of stochastic athermal systems, we derive Langevin-like equations driven by non-Gaussian noise, starting from master equations and developing a new asymptotic expansion. We found an explicit condition whereby the non-Gaussian properties of the athermal noise become dominant for tracer particles associated with both thermal and athermal environments. Furthermore, we derive an inverse formula to infer microscopic properties of the athermal bath from the statistics of the tracer particle. We apply our formulation to a granular motor under viscous friction and analytically obtain the angular velocity distribution function. Our theory demonstrates that the non-Gaussian Langevin equation is the minimal model of athermal systems.

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 fluctuations for a single-particle analog of gas in a soft wall

We investigate the motion of a colloidal particle driven out of equilibrium by a time-varying stiffness of the optical trap that produces persistent nonequilibrium work. Measurements of work production for repeated cycles composed of the compression and expansion processes for the optical potential show huge fluctuations due to thermal motion. Using a precise technique to modulate the stiffness in time, we accurately estimate the probability distributions of work produced for the compression and expansion processes. We confirm the fluctuation theorem from the ratio of the two distributions. We also show that the average values of work for the two processes comply with the Jarzynski equality. This system has an analogy with a gas in a breathing soft wall. We discuss about its applicability to a heat engine and an information engine operated by feedback control.

Effects of an ATP analogue, adenosine 5'-[α-thio]-triphosphate, on F1-ATPase rotary catalysis, torque generation, and inhibited intermediated formation

F1-ATPase (F1), an important rotary motor protein, converts the chemical energy of ATP hydrolysis into mechanical energy using rotary motion with extremely high efficiency. The energy-conversion mechanism for this molecular motor has been extensively clarified by previous studies, which indicate that the interactions between the catalytic residues and the β- and γ-phosphates of ATP are indispensable for efficient catalysis and torque generation. However, the role of α-phosphate is largely unknown. In this study, we observed the rotation of F1 fuelled with an ATP analogue, adenosine 5'-[α-thio]-triphosphate (ATPαS), in which the oxygen has been substituted with a sulfur ion to perturb the α-phosphate/F1 interactions. In doing so, we have revealed that ATPαS does not appear to have any impact on the kinetic properties of the motor or on torque generation compared to ATP. On the other hand, F1 was observed to lapse into the ADP-inhibited intermediate states when in the presence of ATPαS more severely than in the presence of ATP, suggesting that the α-phosphate group of ATP contributes to the avoidance of ADP-inhibited intermediate formation.

High-speed angle-resolved imaging of a single gold nanorod with microsecond temporal resolution and one-degree angle precision

We developed two types of high-speed angle-resolved imaging methods for single gold nanorods (SAuNRs) using objective-type vertical illumination dark-field microscopy and a high-speed CMOS camera to achieve microsecond temporal and one-degree angle resolution. These methods are based on: (i) an intensity analysis of focused images of SAuNR split into two orthogonally polarized components and (ii) the analysis of defocused SAuNR images. We determined the angle precision (statistical error) and accuracy (systematic error) of the resultant SAuNR (80 nm × 40 nm) images projected onto a substrate surface (azimuthal angle) in both methods. Although both methods showed a similar precision of ∼1° for the azimuthal angle at a 10 μs temporal resolution, the defocused image analysis showed a superior angle accuracy of ∼5°. In addition, the polar angle was also determined from the defocused SAuNR images with a precision of ∼1°, by fitting with simulated images. By taking advantage of the defocused image method's full revolution measurement range in the azimuthal angle, the rotation of the rotary molecular motor, F1-ATPase, was measured with 3.3 μs temporal resolution. The time constants of the pauses waiting for the elementary steps of the ATP hydrolysis reaction and the torque generated in the mechanical steps have been successfully estimated. The high-speed angle-resolved SAuNR imaging methods will be applicable to the monitoring of the fast conformational changes of many biological molecular machines.

Effective rates from thermodynamically consistent coarse-graining of models for molecular motors with probe particles

Many single-molecule experiments for molecular motors comprise not only the motor but also large probe particles coupled to it. The theoretical analysis of these assays, however, often takes into account only the degrees of freedom representing the motor. We present a coarse-graining method that maps a model comprising two coupled degrees of freedom which represent motor and probe particle to such an effective one-particle model by eliminating the dynamics of the probe particle in a thermodynamically and dynamically consistent way. The coarse-grained rates obey a local detailed balance condition and reproduce the net currents. Moreover, the average entropy production as well as the thermodynamic efficiency is invariant under this coarse-graining procedure. Our analysis reveals that only by assuming unrealistically fast probe particles, the coarse-grained transition rates coincide with the transition rates of the traditionally used one-particle motor models. Additionally, we find that for multicyclic motors the stall force can depend on the probe size. We apply this coarse-graining method to specific case studies of the F(1)-ATPase and the kinesin motor.

Fluctuation theorem for partially masked nonequilibrium dynamics

We establish a generalization of the fluctuation theorem for partially masked nonequilibrium dynamics. We introduce a partial entropy production with a subset of all possible transitions, and show that the partial entropy production satisfies the integral fluctuation theorem. Our result reveals the fundamental properties of a broad class of autonomous as well as nonautonomous nanomachines. In particular, our result gives a unified fluctuation theorem for both autonomous and nonautonomous Maxwell's demons, where mutual information plays a crucial role. Furthermore, we derive a fluctuation-dissipation theorem that relates nonequilibrium stationary current to two kinds of equilibrium fluctuations.

Key chemical factors of arginine finger catalysis of F1-ATPase clarified by an unnatural amino acid mutation

A catalytically important arginine, called Arg finger, is employed in many enzymes to regulate their functions through enzymatic hydrolysis of nucleotide triphosphates. F1-ATPase (F1), a rotary motor protein, possesses Arg fingers which catalyze hydrolysis of adenosine triphosphate (ATP) for efficient chemomechanical energy conversion. In this study, we examined the Arg finger catalysis by single-molecule measurements for a mutant of F1 in which the Arg finger is substituted with an unnatural amino acid of a lysine analogue, 2,7-diaminoheptanoic acid (Lyk). The use of Lyk, of which the side chain is elongated by one CH2 unit so that its chain length to the terminal nitrogen of amine is set to be equal to that of arginine, allowed us to resolve key chemical factors in the Arg finger catalysis, i.e., chain length matching and chemical properties of the terminal groups. Rate measurements by single-molecule observations showed that the chain length matching of the side-chain length is not a sole requirement for the Arg finger to catalyze the ATP hydrolysis reaction step, indicating the crucial importance of chemical properties of the terminal guanidinium group in the Arg finger catalysis. On the other hand, the Lyk mutation prevented severe formation of an ADP inhibited state observed for a lysine mutant and even improved the avoidance of inhibition compared with the wild-type F1. The present study demonstrated that incorporation of unnatural amino acids can widely extend with its high "chemical" resolution biochemical approaches for elucidation of the molecular mechanism of protein functions and furnishing novel characteristics.

Concepts, challenges, and successes in modeling thermodynamics of metabolism

The modeling of the chemical reactions involved in metabolism is a daunting task. Ideally, the modeling of metabolism would use kinetic simulations, but these simulations require knowledge of the thousands of rate constants involved in the reactions. The measurement of rate constants is very labor intensive, and hence rate constants for most enzymatic reactions are not available. Consequently, constraint-based flux modeling has been the method of choice because it does not require the use of the rate constants of the law of mass action. However, this convenience also limits the predictive power of constraint-based approaches in that the law of mass action is used only as a constraint, making it difficult to predict metabolite levels or energy requirements of pathways. An alternative to both of these approaches is to model metabolism using simulations of states rather than simulations of reactions, in which the state is defined as the set of all metabolite counts or concentrations. While kinetic simulations model reactions based on the likelihood of the reaction derived from the law of mass action, states are modeled based on likelihood ratios of mass action. Both approaches provide information on the energy requirements of metabolic reactions and pathways. However, modeling states rather than reactions has the advantage that the parameters needed to model states (chemical potentials) are much easier to determine than the parameters needed to model reactions (rate constants). Herein, we discuss recent results, assumptions, and issues in using simulations of state to model metabolism.

Torque generation of Enterococcus hirae V-ATPase

V-ATPase (V(o)V1) converts the chemical free energy of ATP into an ion-motive force across the cell membrane via mechanical rotation. This energy conversion requires proper interactions between the rotor and stator in V(o)V1 for tight coupling among chemical reaction, torque generation, and ion transport. We developed an Escherichia coli expression system for Enterococcus hirae V(o)V1 (EhV(o)V1) and established a single-molecule rotation assay to measure the torque generated. Recombinant and native EhV(o)V1 exhibited almost identical dependence of ATP hydrolysis activity on sodium ion and ATP concentrations, indicating their functional equivalence. In a single-molecule rotation assay with a low load probe at high ATP concentration, EhV(o)V1 only showed the "clear" state without apparent backward steps, whereas EhV1 showed two states, "clear" and "unclear." Furthermore, EhV(o)V1 showed slower rotation than EhV1 without the three distinct pauses separated by 120° that were observed in EhV1. When using a large probe, EhV(o)V1 showed faster rotation than EhV1, and the torque of EhV(o)V1 estimated from the continuous rotation was nearly double that of EhV1. On the other hand, stepping torque of EhV1 in the clear state was comparable with that of EhV(o)V1. These results indicate that rotor-stator interactions of the V(o) moiety and/or sodium ion transport limit the rotation driven by the V1 moiety, and the rotor-stator interactions in EhV(o)V1 are stabilized by two peripheral stalks to generate a larger torque than that of isolated EhV1. However, the torque value was substantially lower than that of other rotary ATPases, implying the low energy conversion efficiency of EhV(o)V1.

Active Brownian particles: entropy production and fluctuation response

Within the Rayleigh-Helmholtz model of active Brownian particles, activity is due to a nonlinear velocity-dependent force. In the presence of external trapping potential or constant force, the steady state of the system breaks detailed balance producing a net entropy. Using molecular dynamics simulations, we obtain the probability distributions of entropy production in these steady states. The distribution functions obey fluctuation theorems for entropy production. Using the simulation, we further show that the steady-state response function obeys a modified fluctuation-dissipation relation.