Making ATP

Unlocking the potential of information flow: Maximizing free-energy transduction in a model of an autonomous rotary molecular motor

Molecular motors fulfill critical functions within all living beings. Understanding their underlying working principles is therefore of great interest. Here we develop a simple model inspired by the two-component biomolecular motor F_{o}-F_{1} ATP synthase. We analyze its energetics and characterize information flows between the machine's components. At maximum output power we find that information transduction plays a minor role for free-energy transduction. However, when the two components are coupled to different environments (e.g., when in contact with heat baths at different temperatures), we show that information flow becomes a resource worth exploiting to maximize free-energy transduction. Our findings suggest that real-world powerful and efficient information engines could be found in machines whose components are subjected to fluctuations of different strength, since in this situation the benefit gained from using information for work extraction can outweigh the costs of information generation.

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.

ATP synthase: Evolution, energetics, and membrane interactions

The synthesis of ATP, life’s “universal energy currency,” is the most prevalent chemical reaction in biological systems and is responsible for fueling nearly all cellular processes, from nerve impulse propagation to DNA synthesis. ATP synthases, the family of enzymes that carry out this endless task, are nearly as ubiquitous as the energy-laden molecule they are responsible for making. The F-type ATP synthase (F-ATPase) is found in every domain of life and has facilitated the survival of organisms in a wide range of habitats, ranging from the deep-sea thermal vents to the human intestine. Accordingly, there has been a large amount of work dedicated toward understanding the structural and functional details of ATP synthases in a wide range of species. Less attention, however, has been paid toward integrating these advances in ATP synthase molecular biology within the context of its evolutionary history. In this review, we present an overview of several structural and functional features of the F-type ATPases that vary across taxa and are purported to be adaptive or otherwise evolutionarily significant: ion channel selectivity, rotor ring size and stoichiometry, ATPase dimeric structure and localization in the mitochondrial inner membrane, and interactions with membrane lipids. We emphasize the importance of studying these features within the context of the enzyme’s particular lipid environment. Just as the interactions between an organism and its physical environment shape its evolutionary trajectory, ATPases are impacted by the membranes within which they reside. We argue that a comprehensive understanding of the structure, function, and evolution of membrane proteins—including ATP synthase—requires such an integrative approach.

Modeling work-speed-accuracy trade-offs in a stochastic rotary machine

Molecular machines are stochastic systems that catalyze the energetic processes keeping living cells alive and structured. Inspired by the examples of F_{1}-ATP synthase and the bacterial flagellum, we present a minimal model of an externally driven stochastic rotary machine. We explore the trade-offs of work, driving speed, and driving accuracy when changing driving strength, speed, and the underlying system dynamics. We find an upper bound on accuracy and work for a particular speed. Our results favor slow driving when tasked with minimizing the work-accuracy ratio and maximizing the rate of successful cycles. Finally, in the parameter regime mapping to the dynamics of F_{1}-ATP synthase, we find a significant decay of driving accuracy at physiological rotation rates, raising questions about how ATP synthase achieves reasonable or even remarkable efficiency in vivo.

Nonequilibrium Energy Transduction in Stochastic Strongly Coupled Rotary Motors

Living systems at the molecular scale are composed of many constituents with strong and heterogeneous interactions, operating far from equilibrium, and subject to strong fluctuations. These conditions pose significant challenges to efficient, precise, and rapid free energy transduction, yet nature has evolved numerous molecular machines that do just this. Using a simple model of the ingenious rotary machine F_oF₁-ATP synthase, we investigate the interplay between nonequilibrium driving forces, thermal fluctuations, and interactions between strongly coupled subsystems. This model reveals design principles for effective free energy transduction. Most notably, while tight coupling is intuitively appealing, we find that output power is maximized at intermediate-strength coupling, which permits lubrication by stochastic fluctuations with only minimal slippage.

Theory of Nonequilibrium Free Energy Transduction by Molecular Machines

Biomolecular machines are protein complexes that convert between different forms of free energy. They are utilized in nature to accomplish many cellular tasks. As isothermal nonequilibrium stochastic objects at low Reynolds number, they face a distinct set of challenges compared with more familiar human-engineered macroscopic machines. Here we review central questions in their performance as free energy transducers, outline theoretical and modeling approaches to understand these questions, identify both physical limits on their operational characteristics and design principles for improving performance, and discuss emerging areas of research.

Tight-binding derivation of a discrete-continuous description of mechanochemical coupling in a molecular motor

We present a tight-binding derivation of a discrete-continuous description of mechanochemical coupling in a molecular motor. Our derivation is based on the continuous diffusion equation for overdamped Brownian motion on a time-independent tilted periodic potential in two dimensions. The free-energy potential is nonseparable to allow coupling between the chemical and mechanical degrees of freedom. We formally discretize the chemical coordinate by expanding in Wannier states that are localized along the chemical coordinate and parametrized along the mechanical coordinate. A discrete-continuous equation is derived that is valid for anisotropic systems with weak mechanochemical coupling and deep potential wells along the chemical coordinate. The discrete-continuous description is consistent with established theoretical models of molecular motors with discrete chemical states but is constrained by the underlying continuous two-dimensional potential. In particular, we derive analytic expressions for the effective potential along the mechanical coordinate and for the rate of thermal hopping between chemical states. We determine the thermodynamic efficiency of energy conversion and find that, for a molecular motor with one chemical state per cycle, the derived discrete-continuous equation can accurately describe mechanochemical coupling but cannot describe energy conversion.

Phylogenetic Profiling of Mitochondrial Proteins and Integration Analysis of Bacterial Transcription Units Suggest Evolution of F1Fo ATP Synthase from Multiple Modules

ATP synthase is a complex universal enzyme responsible for ATP synthesis across all kingdoms of life. The F-type ATP synthase has been suggested to have evolved from two functionally independent, catalytic (F1) and membrane bound (Fo), ancestral modules. While the modular evolution of the synthase is supported by studies indicating independent assembly of the two subunits, the presence of intermediate assembly products suggests a more complex evolutionary process. We analyzed the phylogenetic profiles of the human mitochondrial proteins and bacterial transcription units to gain additional insight into the evolution of the F-type ATP synthase complex. In this study, we report the presence of intermediary modules based on the phylogenetic profiles of the human mitochondrial proteins. The two main intermediary modules comprise the α₃β₃ hexamer in the F1 and the c-subunit ring in the Fo. A comprehensive analysis of bacterial transcription units of F1Fo ATP synthase revealed that while a long and constant order of F1Fo ATP synthase genes exists in a majority of bacterial genomes, highly conserved combinations of separate transcription units are present among certain bacterial classes and phyla. Based on our findings, we propose a model that includes the involvement of multiple modules in the evolution of F1Fo ATP synthase. The central and peripheral stalk subunits provide a link for the integration of the F1/Fo modules.

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.

Biophysical comparison of ATP synthesis mechanisms shows a kinetic advantage for the rotary process

The ATP synthase (F-ATPase) is a highly complex rotary machine that synthesizes ATP, powered by a proton electrochemical gradient. Why did evolution select such an elaborate mechanism over arguably simpler alternating-access processes that can be reversed to perform ATP synthesis? We studied a systematic enumeration of alternative mechanisms, using numerical and theoretical means. When the alternative models are optimized subject to fundamental thermodynamic constraints, they fail to match the kinetic ability of the rotary mechanism over a wide range of conditions, particularly under low-energy conditions. We used a physically interpretable, closed-form solution for the steady-state rate for an arbitrary chemical cycle, which clarifies kinetic effects of complex free-energy landscapes. Our analysis also yields insights into the debated "kinetic equivalence" of ATP synthesis driven by transmembrane pH and potential difference. Overall, our study suggests that the complexity of the F-ATPase may have resulted from positive selection for its kinetic advantage.

ATP Synthesis by Oxidative Phosphorylation

The F1F0-ATP synthase (EC 3.6.1.34) is a remarkable enzyme that functions as a rotary motor. It is found in the inner membranes of Escherichia coli and is responsible for the synthesis of ATP in response to an electrochemical proton gradient. Under some conditions, the enzyme functions reversibly and uses the energy of ATP hydrolysis to generate the gradient. The ATP synthase is composed of eight different polypeptide subunits in a stoichiometry of α3β3γδεab2c10. Traditionally they were divided into two physically separable units: an F1 that catalyzes ATP hydrolysis (α3β3γδε) and a membrane-bound F0 sector that transports protons (ab2c10). In terms of rotary function, the subunits can be divided into rotor subunits (γεc10) and stator subunits (α3β3δab2). The stator subunits include six nucleotide binding sites, three catalytic and three noncatalytic, formed primarily by the β and α subunits, respectively. The stator also includes a peripheral stalk composed of δ and b subunits, and part of the proton channel in subunit a. Among the rotor subunits, the c subunits form a ring in the membrane, and interact with subunit a to form the proton channel. Subunits γ and ε bind to the c-ring subunits, and also communicate with the catalytic sites through interactions with α and β subunits. The eight subunits are expressed from a single operon, and posttranscriptional processing and translational regulation ensure that the polypeptides are made at the proper stoichiometry. Recent studies, including those of other species, have elucidated many structural and rotary properties of this enzyme.

Rotation triggers nucleotide-independent conformational transition of the empty β subunit of F₁-ATPase

F1-ATPase (F1) is the catalytic portion of ATP synthase, a rotary motor protein that couples proton gradients to ATP synthesis. Driven by a proton flux, the F1 asymmetric γ subunit undergoes a stepwise rotation inside the α3β3 headpiece and causes the β subunits' binding sites to cycle between states of different affinity for nucleotides. These concerted transitions drive the synthesis of ATP from ADP and phosphate. Here, we study the coupling between the mechanical progression of γ and the conformations of α3β3. Using molecular dynamics simulations, we show that the nucleotide-free β subunit, initially in the open, low-affinity state, undergoes a spontaneous closing transition to the half-open state in response to the γ rotation in the synthesis direction. We estimate the kinetics of this spontaneous conformational change and analyze its mechanism and driving forces. By computing free energy profiles, we find that the isolated empty β subunit preferentially adopts the half-open conformation and that the transition to this conformation from the fully open state is accompanied by well-defined changes in the structure and interactions of the active site region. These results suggest that ADP binding to F1 occurs via conformational selection and is preceded by the transition of the active site to the half-open conformation, driven by the intrinsic elasticity of β. Our results also indicate that opening of the nucleotide-free β during hydrolysis is not spontaneous, as previously assumed. Rather, the fully open conformation observed in the F1 X-ray structure is enforced sterically by the γ subunit whose orientation is stabilized by interactions with the two other β subunits in the completely closed state. This finding supports the notion that γ acts by coupling the extreme conformational states of β subunits within the α3β3 hexamer and therefore is responsible for high efficiency of the coordinated catalysis.

Motor proteins and molecular motors: how to operate machines at the nanoscale

Several classes of biological molecules that transform chemical energy into mechanical work are known as motor proteins or molecular motors. These nanometer-sized machines operate in noisy stochastic isothermal environments, strongly supporting fundamental cellular processes such as the transfer of genetic information, transport, organization and functioning. In the past two decades motor proteins have become a subject of intense research efforts, aimed at uncovering the fundamental principles and mechanisms of molecular motor dynamics. In this review, we critically discuss recent progress in experimental and theoretical studies on motor proteins. Our focus is on analyzing fundamental concepts and ideas that have been utilized to explain the non-equilibrium nature and mechanisms of molecular motors.

Invadolysin, a conserved lipid-droplet-associated metalloproteinase, is required for mitochondrial function in Drosophila

Mitochondria are the main producers of ATP, the principal energy source of the cell, and reactive oxygen species (ROS), important signaling molecules. Mitochondrial morphogenesis and function depend on a hierarchical network of mechanisms in which proteases appear to be center stage. The invadolysin gene encodes an essential conserved metalloproteinase of the M8 family that is necessary for mitosis and cell migration during Drosophila development. We previously demonstrated that invadolysin is found associated with lipid droplets in cells. Here, we present data demonstrating that invadolysin interacts physically with three mitochondrial ATP synthase subunits. Our studies have focused on the genetic phenotypes of invadolysin and bellwether, the Drosophila homolog of ATP synthase α, mutants. The invadolysin mutation presents defects in mitochondrial physiology similar to those observed in bellwether mutants. The invadolysin and bellwether mutants have parallel phenotypes that affect lipid storage and mitochondrial electron transport chain activity, which result in a reduction in ATP production and an accumulation of ROS. As a consequence, invadolysin mutant larvae show lower energetic status and higher oxidative stress. Our data demonstrate an essential role for invadolysin in mitochondrial function that is crucial for normal development and survival.

Mechanical transduction mechanisms of RecA-like molecular motors

A majority of ATP-dependent molecular motors are RecA-like proteins, performing diverse functions in biology. These RecA-like molecular motors consist of a highly conserved core containing the ATP-binding site. Here I examined how ATP binding within this core is coupled to the conformational changes of different RecA-like molecular motors. Conserved hydrogen bond networks and conformational changes revealed two major mechanical transduction mechanisms: (1) intra-domain conformational changes and (2) inter-domain conformational changes. The intra-domain mechanism has a significant hydrogen bond rearrangement within the domain containing the P-loop, causing relative motion between two parts of the protein. The inter-domain mechanism exhibits little conformational change in the P-loop domain. Instead, the major conformational change is observed between the P-loop domain and an adjacent domain or subunit containing the arginine finger. These differences in the mechanical transduction mechanisms may link to the underlying energy surface governing a Brownian ratchet or a power stroke.

Physics of bacterial morphogenesis

Bacterial cells utilize three-dimensional (3D) protein assemblies to perform important cellular functions such as growth, division, chemoreception, and motility. These assemblies are composed of mechanoproteins that can mechanically deform and exert force. Sometimes, small-nucleotide hydrolysis is coupled to mechanical deformations. In this review, we describe the general principle for an understanding of the coupling of mechanics with chemistry in mechanochemical systems. We apply this principle to understand bacterial cell shape and morphogenesis and how mechanical forces can influence peptidoglycan cell wall growth. We review a model that can potentially reconcile the growth dynamics of the cell wall with the role of cytoskeletal proteins such as MreB and crescentin. We also review the application of mechanochemical principles to understand the assembly and constriction of the FtsZ ring. A number of potential mechanisms are proposed, and important questions are discussed.

Monte Carlo simulation from proton slip to "coupled" proton flow in ATP synthase based on the bi-site mechanism

ATP synthase couples proton flow to ATP synthesis, but is leaky to protons at very low nucleotide concentration. Based on the bi-site mechanism, we simulated the proton conduction from proton slip to "coupled" proton flow in ATP synthase using the Monte Carlo method. Good agreement is obtained between the simulated and available experimental results. Our model provides deeper insight into the nucleotide dependence of ATP catalysis, and the kinetic cooperativity in three catalysis subunits. The results of simulation support the bi-site mechanism in ATP synthesis.

Torsional elasticity and energetics of F1-ATPase

F(o)F(1)-ATPase is a rotary motor protein synthesizing ATP from ADP driven by a cross-membrane proton gradient. The proton flow through the membrane-embedded F(o) generates the rotary torque that drives the rotation of the asymmetric shaft of F(1). Mechanical energy of the rotating shaft is used by the F(1) catalytic subunit to synthesize ATP. It was suggested that elastic power transmission with transient storage of energy in some compliant part of the shaft is required for the observed high turnover rate. We used atomistic simulations to study the spatial distribution and structural determinants of the F(1) torsional elasticity at the molecular level and to comprehensively characterize the elastic properties of F(1)-ATPase. Our fluctuation analysis revealed an unexpected heterogeneity of the F(1) shaft elasticity. Further, we found that the measured overall torsional moduli of the shaft arise from two distinct contributions, the intrinsic elasticity and the effective potential imposed on the shaft by the catalytic subunit. Separation of these two contributions provided a quantitative description of the coupling between the rotor and the catalytic subunit. This description enabled us to propose a minimal quantitative model of the F(1) energetics along the rotary degrees of freedom near the resting state observed in the crystal structures. As opposed to the usually employed models where the motor mechanical progression is described by a single angular variable, our multidimensional treatment incorporates the spatially inhomogeneous nature of the shaft and its interactions with the stator and offers new insight into the mechanoenzymatics of F(1)-ATPase.

FoF1-ATPase, rotary motor and biosensor

F(o)F(1)-ATPase is an amazing molecular rotary motor at the nanoscale. Single molecule technologies have contributed much to the understanding of the motor. For example, fluorescence imaging and spectroscopy revealed the physical rotation of isolated F(1) and F(o), or F(o)F(1) holoenzyme. Magnetic tweezers were employed to manipulate the ATP synthesis/hydrolysis in F(1), and proton translation in F(o). Here, we briefly review our recent works including a systematic kinetics study of the holoenzyme, the mechanochemical coupling mechanism, reconstituting the delta-free F(o)F(1)-ATPase, direct observation of F(o) rotation at single molecule level and activity regulation through external links on the stator.

Models of protein linear molecular motors for dynamic nanodevices

Protein molecular motors are natural nano-machines that convert the chemical energy from the hydrolysis of adenosine triphosphate into mechanical work. These efficient machines are central to many biological processes, including cellular motion, muscle contraction and cell division. The remarkable energetic efficiency of the protein molecular motors coupled with their nano-scale has prompted an increasing number of studies focusing on their integration in hybrid micro- and nanodevices, in particular using linear molecular motors. The translation of these tentative devices into technologically and economically feasible ones requires an engineering, design-orientated approach based on a structured formalism, preferably mathematical. This contribution reviews the present state of the art in the modelling of protein linear molecular motors, as relevant to the future design-orientated development of hybrid dynamic nanodevices.

Acidic lipids, H(+)-ATPases, and mechanism of oxidative phosphorylation. Physico-chemical ideas 30 years after P. Mitchell's Nobel Prize award

Peter D. Mitchell, who was awarded the Nobel Prize in Chemistry 30 years ago, in 1978, formulated the chemiosmotic theory of oxidative phosphorylation. This review initially analyzes the major aspects of this theory, its unresolved problems, and its modifications. A new physico-chemical mechanism of energy transformation and coupling of oxidation and phosphorylation is then suggested based on recent concepts regarding proteins, including ATPases that work as molecular motors, and acidic lipids that act as hydrogen ion (H(+)) carriers. According to this proposed mechanism, the chemical energy of a redox substrate is transformed into nonequilibrium states of electron-transporting chain (ETC) coupling proteins. This leads to nonequilibrium pumping of H(+) into the membrane. An acidic lipid, cardiolipin, binds with this H(+) and carries it to the ATP-synthase along the membrane surface. This transport generates gradients of surface tension or electric field along the membrane surface. Hydrodynamic effects on a nanolevel lead to rotation of ATP-synthase and finally to the release of ATP into aqueous solution. This model also explains the generation of a transmembrane protonmotive force that is used for regulation of transmembrane transport, but is not necessary for the coupling of electron transport and ATP synthesis.

ATP hydrolysis-driven H(+) translocation is stimulated by sulfate, a strong inhibitor of mitochondrial ATP synthesis

Sulfate is a partial inhibitor at low and a non-essential activator at high [ATP] of the ATPase activity of F(1). Therefore, a catalytically-competent ternary F(1) x ATP x sulfate complex can be formed. In addition, the ANS fluorescence enhancement driven by ATP hydrolysis in submitochondrial particles is also stimulated by sulfate, clearly showing that the ATP hydrolysis in its presence is coupled to H(+) translocation. However, sulfate is a strong linear inhibitor of the mitochondrial ATP synthesis. The inhibition was competitive (K (i) = 0.46 mM) with respect to Pi and mixed (K (i) = 0.60 and K'(i) = 5.6 mM) towards ADP. Since it is likely that sulfate exerts its effects by binding at the Pi binding subdomain of the catalytic site, we suggest that the catalytic site involved in the H(+) translocation driven by ATP hydrolysis has a more open conformation than the half-closed one (beta(HC)), which is an intermediate in ATP synthesis. Accordingly, ATP hydrolysis is not necessarily the exact reversal of ATP synthesis.

The concerted nature between three catalytic subunits driving the F1 rotary motor

F(1), a rotational molecular motor, shows strong cooperativity during ATP catalysis when driving the rotation of the central gamma subunit surrounded by the alpha(3)beta(3) subunits. To understand how the three catalytic beta subunits cooperate to drive rotation, we made a hybrid F(1) containing one or two mutant beta subunits with altered catalytic kinetics and observed its rotations. Analysis of the asymmetric stepwise rotations elucidated a concerted nature inside the F(1) complex where all three beta subunits participate to rotate the gamma subunit with a 120 degrees phase. In addition, observing hybrid F(1) rotations at various solution conditions, such as ADP, P(i) and the ATPase inhibitor 2,3-butanedione 2-monoxime (BDM) provides additional information for each elementary event. This novel experimental system, which combines single molecule observations and biochemical methods, enables us to dynamically visualize the catalytic coordination inside active enzymes and shed light on how biological machines provide unidirectional functions and rectify information from stochastic reactions.

The structural basis for unidirectional rotation of thermoalkaliphilic F1-ATPase

The ATP synthase of the thermoalkaliphilic Bacillus sp. TA2.A1 operates exclusively in ATP synthesis direction. In the crystal structure of the nucleotide-free alpha(3)beta(3)gamma epsilon subcomplex (TA2F(1)) at 3.1 A resolution, all three beta subunits adopt the open beta(E) conformation. The structure shows salt bridges between the helix-turn-helix motif of the C-terminal domain of the beta(E) subunit (residues Asp372 and Asp375) and the N-terminal helix of the gamma subunit (residues Arg9 and Arg10). These electrostatic forces pull the gamma shaft out of the rotational center and impede rotation through steric interference with the beta(E) subunit. Replacement of Arg9 and Arg10 with glutamines eliminates the salt bridges and results in an activation of ATP hydrolysis activity, suggesting that these salt bridges prevent the native enzyme from rotating in ATP hydrolysis direction. A similar bending of the gamma shaft as in the TA2F(1) structure was observed by single-particle analysis of the TA2F(1)F(o) holoenzyme.

Inventing the dynamo machine: the evolution of the F-type and V-type ATPases

The rotary proton- and sodium-translocating ATPases are reversible molecular machines present in all cellular life forms that couple ion movement across membranes with ATP hydrolysis or synthesis. Sequence and structural comparisons of F- and V-type ATPases have revealed homology between their catalytic and membrane subunits, but not between the subunits of the central stalk that connects the catalytic and membrane components. Based on this pattern of homology, we propose that these ATPases originated from membrane protein translocases, which, themselves, evolved from RNA translocases. We suggest that in these ancestral translocases, the position of the central stalk was occupied by the translocated polymer.

F1-ATPase rotates by an asymmetric, sequential mechanism using all three catalytic subunits

F1-ATPase, the catalytic part of FoF1-ATP synthase, rotates the central gamma subunit within the alpha3beta3 cylinder in 120 degrees steps, each step consuming a single ATP molecule. However, how the catalytic activity of each beta subunit is coordinated with the other two beta subunits to drive rotation remains unknown. Here we show that hybrid F1 containing one or two mutant beta subunits with altered catalytic kinetics rotates in an asymmetric stepwise fashion. Analysis of the rotations reveals that for any given beta subunit, the subunit binds ATP at 0 degrees, cleaves ATP at approximately 200 degrees and carries out a third catalytic event at approximately 320 degrees. This demonstrates the concerted nature of the F1 complex activity, where all three beta subunits participate to drive each 120 degrees rotation of the gamma subunit with a 120 degrees phase difference, a process we describe as a 'sequential three-site mechanism'.

Molecular Motors: A Theorist's Perspective

Individual molecular motors, or motor proteins, are enzymatic molecules that convert chemical energy, typically obtained from the hydrolysis of ATP (adenosine triphosphate), into mechanical work and motion. Processive motor proteins, such as kinesin, dynein, and certain myosins, step unidirectionally along linear tracks, specifically microtubules and actin filaments, and play a crucial role in cellular transport processes, organization, and function. In this review some theoretical aspects of motor-protein dynamics are presented in the light of current experimental methods that enable the measurement of the biochemical and biomechanical properties on a single-molecule basis. After a brief discussion of continuum ratchet concepts, we focus on discrete kinetic and stochastic models that yield predictions for the mean velocity, V(F, [ATP], ...), and other observables as a function of an imposed load force F, the ATP concentration, and other variables. The combination of appropriate theory with single-molecule observations should help uncover the mechanisms underlying motor-protein function.

Dwell time symmetry in random walks and molecular motors

The statistics of steps and dwell times in reversible molecular motors differ from those of cycle completion in enzyme kinetics. The reason is that a step is only one of several transitions in the mechanochemical cycle. As a result, theoretical results for cycle completion in enzyme kinetics do not apply to stepping data. To allow correct parameter estimation, and to guide data analysis and experiment design, a theoretical treatment is needed that takes this observation into account. In this article, we model the distribution of dwell times and number of forward and backward steps using first passage processes, based on the assumption that forward and backward steps correspond to different directions of the same transition. We extend recent results for systems with a single cycle and consider the full dwell time distributions as well as models with multiple pathways, detectable substeps, and detachments. Our main results are a symmetry relation for the dwell time distributions in reversible motors, and a relation between certain relative step frequencies and the free energy per cycle. We demonstrate our results by analyzing recent stepping data for a bacterial flagellar motor, and discuss the implications for the efficiency and reversibility of the force-generating subunits.

Extending the absorbing boundary method to fit dwell-time distributions of molecular motors with complex kinetic pathways

Dwell-time distributions, waiting-time distributions, and distributions of pause durations are widely reported for molecular motors based on single-molecule biophysical experiments. These distributions provide important information concerning the functional mechanisms of enzymes and their underlying kinetic and mechanical processes. We have extended the absorbing boundary method to simulate dwell-time distributions of complex kinetic schemes, which include cyclic, branching, and reverse transitions typically observed in molecular motors. This extended absorbing boundary method allows global fitting of dwell-time distributions for enzymes subject to different experimental conditions. We applied the extended absorbing boundary method to experimental dwell-time distributions of single-headed myosin V, and were able to use a single kinetic scheme to fit dwell-time distributions observed under different ligand concentrations and different directions of optical trap forces. The ability to use a single kinetic scheme to fit dwell-time distributions arising from a variety of experimental conditions is important for identifying a mechanochemical model of a molecular motor. This efficient method can be used to study dwell-time distributions for a broad class of molecular motors, including kinesin, RNA polymerase, helicase, F(1) ATPase, and to examine conformational dynamics of other enzymes such as ion channels.

Novel features of the rotary catalytic mechanism revealed in the structure of yeast F1 ATPase

The crystal structure of yeast mitochondrial F(1) ATPase contains three independent copies of the complex, two of which have similar conformations while the third differs in the position of the central stalk relative to the alpha(3)beta(3) sub-assembly. All three copies display very similar asymmetric features to those observed for the bovine enzyme, but the yeast F(1) ATPase structures provide novel information. In particular, the active site that binds ADP in bovine F(1) ATPase has an ATP analog bound and therefore this structure does not represent the ADP-inhibited form. In addition, one of the complexes binds phosphate in the nucleotide-free catalytic site, and comparison with other structures provides a picture of the movement of the phosphate group during initial binding and subsequent catalysis. The shifts in position of the central stalk between two of the three copies of yeast F(1) ATPase and when these structures are compared to those of the bovine enzyme give new insight into the conformational changes that take place during rotational catalysis.

Exploration of the conformational space of myosin recovery stroke via molecular dynamics

Muscle contractions are driven by cyclic conformational changes of myosin, whose molecular mechanisms of operation are being elucidated by recent advances in crystallographic studies and single molecule experiments. To complement such structural studies and consider the energetics of the conformational changes of myosin head, umbrella sampling molecular dynamics (MD) simulations were performed with the all-atom model of the scallop myosin sub-fragment 1 (S1) with a bound ATP in solution in explicit water using the crystallographic near-rigor and transition state conformations as two references. The constraints on RMSD reaction coordinates used for the umbrella sampling were found to steer the conformational changes efficiently, and relatively close correlations have been observed between the set of characteristic structural changes including the lever arm rotation and the closing of the nucleotide binding pocket. The lever arm angle and key residue interaction distances in the nucleotide binding pocket and the relay helix show gradual changes along the recovery stroke reaction coordinate, consistent with previous crystallographic and computational minimum energy studies. Thermal fluctuations, however, appear to make the switch-2 coordination of ATP more flexible than suggested by crystal structures. The local solvation environment of the fluorescence probe, Trp 507 (scallop numbering), also appears highly mobile in the presence of thermal fluctuations.

Analytical theory of the nonequilibrium spatial distribution of RNA polymerase translocations

A continuum Fokker-Planck model is considered for the RNA polymerase in the elongation phase, where the topology of a single free energy profile as a function of the translocation variable distinguishes the Brownian ratchet and power stroke mechanisms. The model yields a simple analytical stationary solution for arbitrary functional forms of the free energy. With the translocation potential of mean force estimated by the time-series data of the recent high-resolution single-molecule experiment [Abbondanzieri et al., Nature (London) 438, 460 (2005)], predictions of the model for the mechanical properties agree with experiments quantitatively with reasonable values of parameters. The evolution of the spatial distribution of translocation variable away from equilibrium with increasing nucleoside triphosphate concentration shows qualitatively different behavior in the two alternative scenarios, which could serve as an additional measurable signature of the underlying mechanism.

Torque-speed relationship of the bacterial flagellar motor

Many swimming bacteria are propelled by flagellar filaments driven by a rotary motor. Each of these tiny motors can generate an impressive torque. The motor torque vs. speed relationship is considered one of the most important measurable characteristics of the motor and therefore is a major criterion for judging models proposed for the working mechanism. Here we give an explicit explanation for this torque-speed curve. The same physics also can explain certain puzzling properties of other motors.