A rotary molecular motor that can work at near 100% efficiency

The cluster model of energy transduction in biological systems

The central problem in transduction is to explain how the energy caught from sunlight by chloroplasts becomes biological work. Or to express it in different terms: how does the energy remain trapped in the biological network and not get lost through thermalization into the environment? The pathway consists of an immensely large number of steps crossing hierarchical levels - some upwards, to larger assemblies, others downwards into energy rich molecules - before fuelling an action potential or a contracting cell. Accepting the assumption that steps are executed by protein domains, we expect that transduction mechanisms are the result of conformational changes, which in turn involve rearrangements of the bonds responsible for the protein fold. But why are these essential changes so difficult to detect? In this presentation, the metabolic pathway is viewed as equivalent to an energy conduit composed of equally sized units - the protein domains - rather than a row of catalysts. The flow of energy through them occurs by the same mechanism as through the cytoplasmic medium (water). This mechanism is based on the cluster-wave model of water structure, which successfully explains the transfer of energy through the liquid medium responsible for the build up of osmotic pressure. The analogy to the line of balls called "Newton's cradle" provides a useful comparison, since there the transfer is also invisible to us because the intermediate balls are motionless. It is further proposed that the spatial arrangements of the H-bonds of the α and β secondary structures support wave motion, with the linear and lateral forms of the groups of bonds belonging to the helices and sheets executing the longitudinal and transverse modes, respectively.

The computational power of the human brain

At the end of the 20th century, analog systems in computer science have been widely replaced by digital systems due to their higher computing power. Nevertheless, the question keeps being intriguing until now: is the brain analog or digital? Initially, the latter has been favored, considering it as a Turing machine that works like a digital computer. However, more recently, digital and analog processes have been combined to implant human behavior in robots, endowing them with artificial intelligence (AI). Therefore, we think it is timely to compare mathematical models with the biology of computation in the brain. To this end, digital and analog processes clearly identified in cellular and molecular interactions in the Central Nervous System are highlighted. But above that, we try to pinpoint reasons distinguishing in silico computation from salient features of biological computation. First, genuinely analog information processing has been observed in electrical synapses and through gap junctions, the latter both in neurons and astrocytes. Apparently opposed to that, neuronal action potentials (APs) or spikes represent clearly digital events, like the yes/no or 1/0 of a Turing machine. However, spikes are rarely uniform, but can vary in amplitude and widths, which has significant, differential effects on transmitter release at the presynaptic terminal, where notwithstanding the quantal (vesicular) release itself is digital. Conversely, at the dendritic site of the postsynaptic neuron, there are numerous analog events of computation. Moreover, synaptic transmission of information is not only neuronal, but heavily influenced by astrocytes tightly ensheathing the majority of synapses in brain (tripartite synapse). At least at this point, LTP and LTD modifying synaptic plasticity and believed to induce short and long-term memory processes including consolidation (equivalent to RAM and ROM in electronic devices) have to be discussed. The present knowledge of how the brain stores and retrieves memories includes a variety of options (e.g., neuronal network oscillations, engram cells, astrocytic syncytium). Also epigenetic features play crucial roles in memory formation and its consolidation, which necessarily guides to molecular events like gene transcription and translation. In conclusion, brain computation is not only digital or analog, or a combination of both, but encompasses features in parallel, and of higher orders of complexity.

Six states of Enterococcus hirae V-type ATPase reveals non-uniform rotor rotation during turnover

The vacuolar-type ATPase from Enterococcus hirae (EhV-ATPase) is a thus-far unique adaptation of V-ATPases, as it performs Na⁺ transport and demonstrates an off-axis rotor assembly. Recent single molecule studies of the isolated V₁ domain have indicated that there are subpauses within the three major states of the pseudo three-fold symmetric rotary enzyme. However, there was no structural evidence for these. Herein we activate the EhV-ATPase complex with ATP and identified multiple structures consisting of a total of six states of this complex by using cryo-electron microscopy. The orientations of the rotor complex during turnover, especially in the intermediates, are not as perfectly uniform as expected. The densities in the nucleotide binding pockets in the V₁ domain indicate the different catalytic conditions for the six conformations. The off-axis rotor and its' interactions with the stator a-subunit during rotation suggests that this non-uniform rotor rotation is performed through the entire complex.

Theoretical efficiency limits and speed-efficiency trade-off in myosin motors

Muscle myosin is a non-processive molecular motor generates mechanical work when cooperating in large ensembles. During its cyle, each individual motor keeps attaching and detaching from the actin filament. The random nature of attachment and detachment inevitably leads to losses and imposes theoretical limits on the energetic efficiency. Here, we numerically determine the theoretical efficiency limit of a classical myosin model with a given number of mechano-chemical states. All parameters that are not bounded by physical limits (like rate limiting steps) are determined by numerical efficiency optimization. We show that the efficiency is limited by the number of states, the stiffness and the rate-limiting kinetic steps. There is a trade-off between speed and efficiency. Slow motors are optimal when most of the available free energy is allocated to the working stroke and the stiffness of their elastic element is high. Fast motors, on the other hand, work better with a lower and asymmetric stiffness and allocate a larger fraction of free energy to the release of ADP. Overall, many features found in myosins coincide with the findings from the model optimization: there are at least 3 bound states, the largest part of the working stroke takes place during the first transition, the ADP affinity is adapted differently in slow and fast myosins and there is an asymmetry in elastic elements.

A physically realizable molecular motor driven by the Landauer blowtorch effect

We propose a model for a molecular motor in a molecular electronic junction driven by a natural manifestation of Landauer's blowtorch effect. The effect emerges via the interplay of electronic friction and diffusion coefficients, each calculated quantum mechanically using nonequilibrium Green's functions, within a semiclassical Langevin description of the rotational dynamics. The motor functionality is analyzed through numerical simulations where the rotations exhibit a directional preference according to the intrinsic geometry of the molecular configuration. The proposed mechanism for motor function is expected to be ubiquitous for a range of molecular geometries beyond the one examined here.

Torque generation mechanism in Fo motor of ATP synthase elucidated by free-energy and Coulomb-energy landscapes along the c-ring rotation

F_o portion of ATP synthase is a proton-motive rotary motor. The Coulombic attraction between the conserved acidic residues in the c-ring and the arginine in the a-subunit (aR) was early proposed to drive the c-ring rotation relative to the a-subunit, and has been actually observed in our previous molecular dynamics simulation with full atomistic description of F_o embedded in the membrane. In this study, to quantify the driving force, we conducted the umbrella sampling (US) and obtained the free-energy landscape for the c-ring rotation. We first show that the free-energy gradient toward the ATP-synthesis direction appears in the deprotonated state of cE. Using the sampled snapshots that cover a wide range of the rotational angle, we further analyzed the rotational-angle dependence of the hydration and the protonation states and obtained the Coulomb-energy landscapes with a focus on the cE-aR interaction. The results indicate that both the Coulombic solvation energy of cE and the interaction energy between cE and aR contribute to the torque generation for the c-ring rotation.

Revolving ATPase motors as asymmetrical hexamers in translocating lengthy dsDNA via conformational changes and electrostatic interactions in phi29, T7, herpesvirus, mimivirus, E. coli, and Streptomyces

Investigations of the parallel architectures of biomotors in both prokaryotic and eukaryotic systems suggest a similar revolving mechanism in the use of ATP to drive translocation of the lengthy double-stranded (ds)DNA genomes. This mechanism is exemplified by the dsDNA packaging motor of bacteriophage phi29 that operates through revolving but not rotating dsDNA to "Push through a one-way valve". This unique and novel revolving mechanism discovered in phi29 DNA packaging motor was recently reported in other systems including the dsDNA packaging motor of herpesvirus, the dsDNA ejecting motor of bacteriophage T7, the plasmid conjugation machine TraB in Streptomyces, the dsDNA translocase FtsK of gram-negative bacteria, and the genome-packaging motor in mimivirus. These motors exhibit an asymmetrical hexameric structure for transporting the genome via an inch-worm sequential action. This review intends to delineate the revolving mechanism from a perspective of conformational changes and electrostatic interactions. In phi29, the positively charged residues Arg-Lys-Arg in the N-terminus of the connector bind the negatively charged interlocking domain of pRNA. ATP binding to an ATPase subunit induces the closed conformation of the ATPase. The ATPase associates with an adjacent subunit to form a dimer facilitated by the positively charged arginine finger. The ATP-binding induces a positive charging on its DNA binding surface via an allostery mechanism and thus the higher affinity for the negatively charged dsDNA. ATP hydrolysis induces an expanded conformation of the ATPase with a lower affinity for dsDNA due to the change of the surface charge, but the (ADP+Pi)-bound subunit in the dimer undergoes a conformational change that repels dsDNA. The positively charged lysine rings of the connector attract dsDNA stepwise and periodically to keep its revolving motion along the channel wall, thus maintaining the one-way translocation of dsDNA without reversal and sliding out. The finding of the presence of the asymmetrical hexameric architectures of many ATPases that use the revolving mechanism may provide insights into the understanding of translocation of the gigantic genomes including chromosomes in complicated systems without coiling and tangling to speed up dsDNA translocation and save energy.

Changes within the central stalk of E. coli F1Fo ATP synthase observed after addition of ATP

F₁F_o ATP synthase functions as a biological generator and makes a major contribution to cellular energy production. Proton flow generates rotation in the F_o motor that is transferred to the F₁ motor to catalyze ATP production, with flexible F₁/F_o coupling required for efficient catalysis. F₁F_o ATP synthase can also operate in reverse, hydrolyzing ATP and pumping protons, and in bacteria this function can be regulated by an inhibitory ε subunit. Here we present cryo-EM data showing E. coli F₁F_o ATP synthase in different rotational and inhibited sub-states, observed following incubation with 10 mM MgATP. Our structures demonstrate how structural transitions within the inhibitory ε subunit induce torsional movement in the central stalk, thereby enabling its rotation within the F_ο motor. This highlights the importance of the central rotor for flexible coupling of the F₁ and F_o motors and provides further insight into the regulatory mechanism mediated by subunit ε.

Roles of local nonequilibrium free energy in the description of biomolecules

When a system is in equilibrium, external perturbations yield a time series of nonequilibrium distributions, and recent experimental techniques give access to the nonequilibrium data that may contain critical information. Jinwoo and Tanaka [Sci. Rep. 5, 7832 (2015)2045-232210.1038/srep07832] have provided mathematical proof that such a process's nonequilibrium free energy profile over a system's substates has Jarzynski's work as content, which spontaneously dissipates while molecules perform their tasks. Here we numerically verify this fact and give a practical example where we analyze a computer simulation of RNA translocation by a ring-shaped ATPase motor. By interpreting the cyclic process of substrate translocation as a series of quenching, relaxation, and second quenching, the theory gives how the ATPase motor allocates the hydrolysis energy between individual substates until the end of the process. It turns out that the efficiency of RNA translocation is 48%-60% for most molecules, but 12% of molecules achieve 80%-100% efficiency, which is consistent with the literature. This theory would be a valuable tool for extracting quantitative information about molecular nonequilibrium behavior from experimental observations.

Optimal Control of the F1-ATPase Molecular Motor

F₁-ATPase is a rotary molecular motor that in vivo is subject to strong nonequilibrium driving forces. There is great interest in understanding the operational principles governing its high efficiency of free-energy transduction. Here we use a near-equilibrium framework to design a nontrivial control protocol to minimize dissipation in rotating F₁ to synthesize adenosine triphosphate. We find that the designed protocol requires much less work than a naive (constant-velocity) protocol across a wide range of protocol durations. Our analysis points to a possible mechanism for energetically efficient driving of F₁ in vivo and provides insight into free-energy transduction for a broader class of biomolecular and synthetic machines.

A dynamical model for Brownian molecular motors driven by inelastic electron tunneling

In recent years, several artificial molecular motors driven and controlled by electric currents have been proposed. Similar to Brownian machines, these systems work by turning random inelastic tunneling events into a directional rotation of the molecule. Despite their importance as the ultimate component of future molecular machines, their modeling has not been sufficiently studied. Here, we develop a dynamical model to describe these systems. We illustrate the validity and usefulness of our model by applying it to a well-known molecular motor, showing that the obtained results are consistent with the available experimental data. Moreover, we demonstrate how to use our model to extract some difficult-to-access microscopic parameters. Finally, we include an analysis of the expected effects of current-induced forces (CIFs). Our analysis suggests that, although nonconservative contributions of the CIFs can be important in some scenarios, they do not seem important in the analyzed case. Despite this, the conservative contributions of CIFs could be strong enough to significantly alter the system’s dynamics.

Modulation of the H+/ATP coupling ratio by ADP and ATP as a possible regulatory feature in the F-type ATP synthases

F-type ATP synthases are transmembrane enzymes, which play a central role in the metabolism of all aerobic and photosynthetic cells and organisms, being the major source of their ATP synthesis. Catalysis occurs via a rotary mechanism, in which the free energy of a transmembrane electrochemical ion gradient is converted into the free energy of ATP phosphorylation from ADP and Pi, and vice versa. An ADP, tightly bound to one of the three catalytic sites on the stator head, is associated with catalysis inhibition, which is relieved by the transmembrane proton gradient and by ATP. By preventing wasteful ATP hydrolysis in times of low osmotic energy and low ATP/ADP ratio, such inhibition constitutes a classical regulatory feedback effect, likely to be an integral component of in vivo regulation. The present miniview focuses on an additional putative regulatory phenomenon, which has drawn so far little attention, consisting in a substrate-induced tuning of the H+/ATP coupling ratio during catalysis, which might represent an additional key to energy homeostasis in the cell. Experimental pieces of evidence in support of such a phenomenon are reviewed.

F1FO ATP synthase molecular motor mechanisms

The F-ATP synthase, consisting of F1 and FO motors connected by a central rotor and the stators, is the enzyme responsible for synthesizing the majority of ATP in all organisms. The F1 (αβ)3 ring stator contains three catalytic sites. Single-molecule F1 rotation studies revealed that ATP hydrolysis at each catalytic site (0°) precedes a power-stroke that rotates subunit-γ 120° with angular velocities that vary with rotational position. Catalytic site conformations vary relative to subunit-γ position (βE, empty; βD, ADP bound; βT, ATP-bound). During a power stroke, βE binds ATP (0°–60°) and βD releases ADP (60°–120°). Årrhenius analysis of the power stroke revealed that elastic energy powers rotation via unwinding the γ-subunit coiled-coil. Energy from ATP binding at 34° closes βE upon subunit-γ to drive rotation to 120° and forcing the subunit-γ to exchange its tether from βE to βD, which changes catalytic site conformations. In F1FO, the membrane-bound FO complex contains a ring of c-subunits that is attached to subunit-γ. This c-ring rotates relative to the subunit-a stator in response to transmembrane proton flow driven by a pH gradient, which drives subunit-γ rotation in the opposite direction to force ATP synthesis in F1. Single-molecule studies of F1FO embedded in lipid bilayer nanodisks showed that the c-ring transiently stopped F1-ATPase-driven rotation every 36° (at each c-subunit in the c10-ring of E. coli F1FO) and was able to rotate 11° in the direction of ATP synthesis. Protonation and deprotonation of the conserved carboxyl group on each c-subunit is facilitated by separate groups of subunit-a residues, which were determined to have different pKa’s. Mutations of any of any residue from either group changed both pKa values, which changed the occurrence of the 11° rotation proportionately. This supports a Grotthuss mechanism for proton translocation and indicates that proton translocation occurs during the 11° steps. This is consistent with a mechanism in which each 36° of rotation the c-ring during ATP synthesis involves a proton translocation-dependent 11° rotation of the c-ring, followed by a 25° rotation driven by electrostatic interaction of the negatively charged unprotonated carboxyl group to the positively charged essential arginine in subunit-a.

Optimal control with a strong harmonic trap

Quadratic trapping potentials are widely used to experimentally probe biopolymers and molecular machines and drive transitions in steered molecular-dynamics simulations. Approximating energy landscapes as locally quadratic, we design multidimensional trapping protocols that minimize dissipation. The designed protocols are easily solvable and applicable to a wide range of systems. The approximation does not rely on either fast or slow limits and is valid for any duration provided the trapping potential is sufficiently strong. We demonstrate the utility of the designed protocols with a simple model of a periodically driven rotary motor. Our results elucidate principles of effective single-molecule manipulation and efficient nonequilibrium free-energy estimation.

Light-Driven Molecular Whirligig

A unidirectional light-driven rotary motor was looped in a figure-of-eight molecule by linking two polymer chains between its stator and rotor parts. By properly tuning the size of these linkers, clockwise rotation of the motor under UV light was shown to create conformationally strained twists between the polymer chains, and in this tensed conformation, the energy stored in the molecular object was sufficient to trigger the reverse rotation of the motor back to its fully relaxed state. The functioning principle of this motorized molecular device appears very similar to that of macroscopic whirligig crafts used by children for fun. In addition, we found that in its out-of-equilibrium tensed state, the fluorescence emission of the molecular motor increased by 500% due to the mechanical constraints imposed by the polymer chains on its conjugated core. Finally, by calculating the apparent thermal energies of activation for the backward rotations at different levels of twisting, we quantitatively determined a lower estimate of the work generated by this rotary motor, from which a torque and a force were extracted, thus answering a long-term open question in this field of research.

Computational Design of Inhibitors Targeting the Catalytic β Subunit of Escherichia coli FOF1-ATP Synthase

With the uncontrolled growth of multidrug-resistant bacteria, there is an urgent need to search for new therapeutic targets, to develop drugs with novel modes of bactericidal action. FoF1-ATP synthase plays a crucial role in bacterial bioenergetic processes, and it has emerged as an attractive antimicrobial target, validated by the pharmaceutical approval of an inhibitor to treat multidrug-resistant tuberculosis. In this work, we aimed to design, through two types of in silico strategies, new allosteric inhibitors of the ATP synthase, by targeting the catalytic β subunit, a centerpiece in communication between rotor subunits and catalytic sites, to drive the rotary mechanism. As a model system, we used the F1 sector of Escherichia coli, a bacterium included in the priority list of multidrug-resistant pathogens. Drug-like molecules and an IF1-derived peptide, designed through molecular dynamics simulations and sequence mining approaches, respectively, exhibited in vitro micromolar inhibitor potency against F1. An analysis of bacterial and Mammalia sequences of the key structural helix-turn-turn motif of the C-terminal domain of the β subunit revealed highly and moderately conserved positions that could be exploited for the development of new species-specific allosteric inhibitors. To our knowledge, these inhibitors are the first binders computationally designed against the catalytic subunit of FOF1-ATP synthase.

Insights from an information thermodynamics analysis of a synthetic molecular motor

Information is physical, a realization that has transformed the physics of measurement and communication. However, the flow between information, energy and mechanics in chemical systems remains largely unexplored. Here we analyse a minimalist autonomous chemically driven molecular motor in terms of information thermodynamics, a framework that quantitatively relates information to other thermodynamic parameters. The treatment reveals how directional motion is generated by free energy transfer from chemical to mechanical (conformational and/or co-conformational) processes by 'energy flow' and 'information flow'. It provides a thermodynamic level of understanding of molecular motors that is general, complements previous analyses based on kinetics and has practical implications for machine design. In line with kinetic analysis, we find that power strokes do not affect the directionality of chemically driven machines. However, we find that power strokes can modulate motor velocity, the efficiency of free energy transfer and the number of fuel molecules consumed per cycle. This may help explain the role of such (co-)conformational changes in biomachines and illustrates the interplay between energy and information in chemical systems.

Current fluctuations in nonequilibrium discontinuous phase transitions

Discontinuous phase transitions out of equilibrium can be characterized by the behavior of macroscopic stochastic currents. But while much is known about the average current, the situation is much less understood for higher statistics. In this paper, we address the consequences of the diverging metastability lifetime-a hallmark of discontinuous transitions-in the fluctuations of arbitrary thermodynamic currents, including the entropy production. In particular, we center our discussion on the conditional statistics, given which phase the system is in. We highlight the interplay between integration window and metastability lifetime, which is not manifested in the average current, but strongly influences the fluctuations. We introduce conditional currents and find, among other predictions, their connection to average and scaled variance through a finite-time version of large deviation theory and a minimal model. Our results are then further verified in two paradigmatic models of discontinuous transitions: Schlögl's model of chemical reactions, and a 12-state Potts model subject to two baths at different temperatures.

Highly Efficient Active Colloids Driven by Galvanic Exchange Reactions

Micromotors are propelled by a variety of chemical reactions, with most of them being of catalytic nature. There are, however, systems based on redox reactions, which show clear benefits for efficiency. Here, we broaden the spectrum of suitable reactions to galvanic exchange processes, or an electrochemical replacement of a solid metal layer with dissolved ionic species of a more noble metal. We study the details of motility and the influence of different reaction parameters to conclude that these galvanophoretic processes circumvent several steps that lose efficiency in catalytic micromotors. Furthermore, we investigate the chemical process, the charge, and flow conditions that lead to this highly efficient new type of active motility. Toward a better understanding of the underlying processes, we propose an electrokinetic model that we numerically solve via finite elements.

A Drive towards Thermodynamic Efficiency for Dissipative Structures in Chemical Reaction Networks

Dissipative accounts of structure formation show that the self-organisation of complex structures is thermodynamically favoured, whenever these structures dissipate free energy that could not be accessed otherwise. These structures therefore open transition channels for the state of the universe to move from a frustrated, metastable state to another metastable state of higher entropy. However, these accounts apply as well to relatively simple, dissipative systems, such as convection cells, hurricanes, candle flames, lightning strikes, or mechanical cracks, as they do to complex biological systems. Conversely, interesting computational properties-that characterize complex biological systems, such as efficient, predictive representations of environmental dynamics-can be linked to the thermodynamic efficiency of underlying physical processes. However, the potential mechanisms that underwrite the selection of dissipative structures with thermodynamically efficient subprocesses is not completely understood. We address these mechanisms by explaining how bifurcation-based, work-harvesting processes-required to sustain complex dissipative structures-might be driven towards thermodynamic efficiency. We first demonstrate a simple mechanism that leads to self-selection of efficient dissipative structures in a stochastic chemical reaction network, when the dissipated driving chemical potential difference is decreased. We then discuss how such a drive can emerge naturally in a hierarchy of self-similar dissipative structures, each feeding on the dissipative structures of a previous level, when moving away from the initial, driving disequilibrium.

Fast ATP-Dependent Subunit Rotation in Reconstituted FoF1-ATP Synthase Trapped in Solution

F_oF₁-ATP synthases are ubiquitous membrane-bound, rotary motor enzymes that can catalyze ATP synthesis and hydrolysis. Their enzyme kinetics are controlled by internal subunit rotation, by substrate and product concentrations, and by mechanical inhibitory mechanisms but also by the electrochemical potential of protons across the membrane. Single-molecule Förster resonance energy transfer (smFRET) has been used to detect subunit rotation within F_oF₁-ATP synthases embedded in freely diffusing liposomes. We now report that kinetic monitoring of functional rotation can be prolonged from milliseconds to seconds by utilizing an anti-Brownian electrokinetic trap (ABEL trap). These extended observation times allowed us to observe fluctuating rates of functional rotation for individual F_oF₁-liposomes in solution. Broad distributions of ATP-dependent catalytic rates were revealed. The buildup of an electrochemical potential of protons was confirmed to limit the maximum rate of ATP hydrolysis. In the presence of ionophores or uncouplers, the fastest subunit rotation speeds measured in single reconstituted F_oF₁-ATP synthases were 180 full rounds per second. This was much faster than measured by biochemical ensemble averaging, but not as fast as the maximum rotational speed reported previously for isolated single F₁ complexes uncoupled from the membrane-embedded F_o complex. Further application of ABEL trap measurements should help resolve the mechanistic causes of such fluctuating rates of subunit rotation.

Chemically-powered swimming and diffusion in the microscopic world

The past decade has seen intriguing reports and heated debates concerning the chemically-driven enhanced motion of objects ranging from small molecules to millimetre-size synthetic robots. These objects, in solutions in which chemical reactions were occurring, were observed to diffuse (spread non-directionally) or swim (move directionally) at rates exceeding those expected from Brownian motion alone. The debates have focused on whether observed enhancement is an experimental artefact or a real phenomenon. If the latter were true, then we would also need to explain how the chemical energy is converted into mechanical work. In this Perspective, we summarize and discuss recent observations and theories of active diffusion and swimming. Notably, the chemomechanical coupling and magnitude of diffusion enhancement are strongly size-dependent and should vanish as the size of the swimmers approaches the molecular scale. We evaluate the reliability of common techniques to measure diffusion coefficients and finish by considering the potential applications and chemical to mechanical energy conversion efficiencies of typical nanoswimmers and microswimmers.

Nonequilibrium thermodynamics of non-ideal chemical reaction networks

All current formulations of nonequilibrium thermodynamics of open chemical reaction networks rely on the assumption of non-interacting species. We develop a general theory that accounts for interactions between chemical species within a mean-field approach using activity coefficients. Thermodynamic consistency requires that rate equations do not obey standard mass-action kinetics but account for the interactions with concentration dependent kinetic constants. Many features of the ideal formulations are recovered. Crucially, the thermodynamic potential and the forces driving non-ideal chemical systems out of equilibrium are identified. Our theory is general and holds for any mean-field expression of the interactions leading to lower bounded free energies.

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.

Radio Waveguide-Double Ratchet Rotors Work in Unison on a Surface to Convert Heat into Power

Synchronizing thousands of 100% efficient rotors in a macrodevice for harvesting noise is unapprehended. Thermodynamically, realizing a thermal gradient at the atomic scale is critical. Harvesting free thermal energy or noise by resonance has a hidden clause; either externally activating a directed self-powered motion or constructing a nanoscale power supply. To accomplish this, we combined two rotor concepts, Brownian rotor, BR, and power stroke, PS, rotors available in living systems in two planes of a single molecule. Quantum tunneling images reveal how a radio-wave guided synchronization of PS-BR combination tweaks rotational dynamics of a rotor to bypass the necessity of temperature gradient (ΔT). Live imaging of thermal noise movement as electron density between a pair of molecular planes helped in optimizing the rotor design. The rotor's monolayer harvests heat from the liquid's Brownian noise and electromagnetic noise, together delivering a finite, usable power. The chip supplies the power if we wet the surface or shine electric noise.

FliI6-FliJ molecular motor assists with unfolding in the type III secretion export apparatus

The role of rotational molecular motors of the ATP synthase class is integral to the metabolism of cells. Yet the function of FliI6-FliJ complex, a homolog of the F1 ATPase motor, within the flagellar export apparatus remains unclear. We use a simple two-state model adapted from studies of linear molecular motors to identify key features of this motor. The two states are the 'locked' ground state where the FliJ coiled coil filament experiences angular fluctuations in an asymmetric torsional potential, and a 'free' excited state in which FliJ undergoes rotational diffusion. Michaelis-Menten kinetics was used to treat transitions between these two states, and obtain the average angular velocity of the unloaded FliJ filament within the FliI6 stator: ωmax ≈ 9.0 rps. The motor was then studied under external counter torque conditions in order to ascertain its maximal power output: Pmax ≈ 42 kBT/s (or 102 kW/mol), and the stall torque: Gstall ≈ 3 kBT/rad (or 0.01 nN·nm/rad). Two modes of action within the flagellar export apparatus are proposed, in which the motor performs useful work either by continuously 'grinding' through the resistive environment of the export gate, or by exerting equal and opposite stall force on it. In both cases, the resistance is provided by flagellin subunits entering the flagellar export channel prior to their unfolding. We therefore propose that the function of the FliI6-FliJ complex is to lower the energy barrier, and therefore assist in unfolding of the flagellar proteins before feeding them into the transport channel.

Optical Control of Cellular ATP Levels with a Photocaged Adenylate Kinase

We have developed a new tool for the optical control of cellular ATP concentrations with a photocaged adenylate kinase (Adk). The photocaged Adk is generated by substituting a catalytically essential lysine with a hydroxycoumarin-protected lysine through site-specific unnatural amino acid mutagenesis in both E. coli and mammalian cells. Caging of the critical lysine residue offers complete suppression of Adk's phosphotransferase activity and rapid restoration of its function both in vitro and in vivo upon optical stimulation. Light-activated Adk renders faster rescue of cell growth than chemically inducible expression of wild-type Adk in E. coli as well as rapid ATP depletion in mammalian cells. Thus, caging Adk provides a new tool for direct conditional perturbation of cellular ATP concentrations thereby enabling the investigation of ATP-coupled physiological events in temporally dynamic contexts.

Long-term observation of Magnetospirillum gryphiswaldense in a microfluidic channel

We controlled and observed individual magneto-tactic bacteria (Magnetospirillum gryphiswaldense) inside a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$5\, \upmu \hbox {m}$$\end{document}5μm-high microfluidic channel for over 4 h. After a period of constant velocity, the duration of which varied between bacteria, all observed bacteria showed a gradual decrease in their velocity of about \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$25\, \hbox {nm}/\hbox {s}^2$$\end{document}25nm/s2. After coming to a full stop, different behaviour was observed, ranging from rotation around the centre of mass synchronous with the direction of the external magnetic field, to being completely immobile. Our results suggest that the influence of the high-intensity illumination and the presence of the channel walls are important parameters to consider when performing observations of such long duration.

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

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

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.

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

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

Structure and Mechanisms of F-Type ATP Synthases

F₁F_o ATP synthases produce most of the ATP in the cell. F-type ATP synthases have been investigated for more than 50 years, but a full understanding of their molecular mechanisms has become possible only with the recent structures of complete, functionally competent complexes determined by electron cryo-microscopy (cryo-EM). High-resolution cryo-EM structures offer a wealth of unexpected new insights. The catalytic F₁ head rotates with the central γ-subunit for the first part of each ATP-generating power stroke. Joint rotation is enabled by subunit δ/OSCP acting as a flexible hinge between F₁ and the peripheral stalk. Subunit a conducts protons to and from the c-ring rotor through two conserved aqueous channels. The channels are separated by ∼6 Å in the hydrophobic core of F_o, resulting in a strong local field that generates torque to drive rotary catalysis in F₁. The structure of the chloroplast F₁F_o complex explains how ATPase activity is turned off at night by a redox switch. Structures of mitochondrial ATP synthase dimers indicate how they shape the inner membrane cristae. The new cryo-EM structures complete our picture of the ATP synthases and reveal the unique mechanism by which they transform an electrochemical membrane potential into biologically useful chemical energy.

Using a system's equilibrium behavior to reduce its energy dissipation in nonequilibrium processes

Cells must operate far from equilibrium, utilizing and dissipating energy continuously to maintain their organization and to avoid stasis and death. However, they must also avoid unnecessary waste of energy. Recent studies have revealed that molecular machines are extremely efficient thermodynamically compared with their macroscopic counterparts. However, the principles governing the efficient out-of-equilibrium operation of molecular machines remain a mystery. A theoretical framework has been recently formulated in which a generalized friction coefficient quantifies the energetic efficiency in nonequilibrium processes. Moreover, it posits that, to minimize energy dissipation, external control should drive the system along the reaction coordinate with a speed inversely proportional to the square root of that friction coefficient. Here, we demonstrate the utility of this theory for designing and understanding energetically efficient nonequilibrium processes through the unfolding and folding of single DNA hairpins.

A dynamic preferred direction model for the self-organization dynamics of bacterial microfluidic pumping

It is known that some flagellated bacteria like Serratia marcescens, when deposited and affixed onto a surface to form a "bacterial carpet", self-organize in a collective motion of the flagella that is capable of pumping fluid through microfluidic channels. We set up a continuum model comprising two macroscopic variables that is capable of describing this self-organization mechanism as well as quantifying it to the extent that an agreement with the experimentally observed channel width dependence of the pumping is reached. The activity is introduced through a collective angular velocity of the helical flagella rotation, which is an example of a dynamic macroscopic preferred direction. Our model supports and quantifies the view that the self-coordination is due to a positive feedback loop between the bacterial flagella and the local flow generated by their rotation. Moreover, our results indicate that this biological active system is operating close to the self-organization threshold.

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

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

Engineering with Biomolecular Motors

Biomolecular motors, such as the motor protein kinesin, can be used as off-the-shelf components to power hybrid nanosystems. These hybrid systems combine elements from the biological and synthetic toolbox of the nanoengineer and can be used to explore the applications and design principles of active nanosystems. Efforts to advance nanoscale engineering benefit greatly from biological and biophysical research into the operating principles of motor proteins and their biological roles. In return, the process of creating in vitro systems outside of the context of biology can lead to an improved understanding of the physical constraints creating the fitness landscape explored by evolution. However, our main focus is a holistic understanding of the engineering principles applying to systems integrating molecular motors in general. To advance this goal, we and other researchers have designed biomolecular motor-powered nanodevices, which sense, compute, and actuate. In addition to demonstrating that biological solutions can be mimicked in vitro, these devices often demonstrate new paradigms without parallels in current technology. Long-term trends in technology toward the deployment of ever smaller and more numerous motors and computers give us confidence that our work will become increasingly relevant. Here, our discussion aims to step back and look at the big picture. From our perspective, energy efficiency is a key and underappreciated metric in the design of synthetic motors. On the basis of an analogy to ecological principles, we submit that practical molecular motors have to have energy conversion efficiencies of more than 10%, a threshold only exceeded by motor proteins. We also believe that motor and system lifetime is a critical metric and an important topic of investigation. Related questions are if future molecular motors, by necessity, will resemble biomolecular motors in their softness and fragility and have to conform to the "universal performance characteristics of motors", linking the maximum force and mass of any motor, identified by Marden and Allen. The utilization of molecular motors for computing devices emphasizes the interesting relationship among the conversion of energy, extraction of work, and production of information. Our recent work touches upon these topics and discusses molecular clocks as well as a Landauer limit for robotics. What is on the horizon? Just as photovoltaics took advantage of progress in semiconductor fabrication to become commercially viable over a century, one can envision that engineers working with biomolecular motors leverage progress in biotechnology and drug development to create the engines of the future. However, the future source of energy is going to be electricity rather than fossil or biological fuels, a fact that has to be accounted for in our future efforts. In summary, we are convinced that past, ongoing, and future efforts to engineer with biomolecular motors are providing exciting demonstrations and fundamental insights as well as opportunities to wander freely across the borders of engineering, biology, and chemistry.

3D Hybrid Small Scale Devices

Interfacing nano/microscale elements with biological components in 3D contexts opens new possibilities for mimicry, bionics, and augmentation of organismically and anatomically inspired materials. Abiotic nanoscale elements such as plasmonic nanostructures, piezoelectric ribbons, and thin film semiconductor devices interact with electromagnetic fields to facilitate advanced capabilities such as communication at a distance, digital feedback loops, logic, and memory. Biological components such as proteins, polynucleotides, cells, and organs feature complex chemical synthetic networks that can regulate growth, change shape, adapt, and regenerate. Abiotic and biotic components can be integrated in all three dimensions in a well-ordered and programmed manner with high tunability, versatility, and resolution to produce radically new materials and hybrid devices such as sensor fabrics, anatomically mimetic microfluidic modules, artificial tissues, smart prostheses, and bionic devices. In this critical Review, applications of small scale devices in 3D hybrid integration, biomicrofluidics, advanced prostheses, and bionic organs are discussed.

Predictive regulation and human design

Organisms evolving toward greater complexity were selected across aeons to use energy and resources efficiently. Efficiency depended on prediction at every stage: first a clock to predict the planet's statistical regularities; then a brain to predict bodily needs and compute commands that dynamically adjust the flows of energy and nutrients. Predictive regulation (allostasis) frugally matches resources to needs and thus forms a core principle of our design. Humans, reaching a pinnacle of cognitive complexity, eventually produced a device (the steam engine) that converted thermal energy to work and were suddenly awash in resources. Today boundless consumption in many nations challenges all our regulatory mechanisms, causing obesity, diabetes, drug addiction and their sequelae. So far we have sought technical solutions, such as drugs, to treat complex circuits for metabolism, appetites and mood. Here I argue for a different approach which starts by asking: why does our regulatory system, which evolution tuned for small satisfactions, now constantly demand 'more'?

Reconstructing free-energy landscapes for nonequilibrium periodic potentials

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

Elastic coupling power stroke mechanism of the F1-ATPase molecular motor

The angular velocity profile of the 120° F₁-ATPase power stroke was resolved as a function of temperature from 16.3 to 44.6 °C using a Δμ_ATP = -31.25 k_BT at a time resolution of 10 μs. Angular velocities during the first 60° of the power stroke (phase 1) varied inversely with temperature, resulting in negative activation energies with a parabolic dependence. This is direct evidence that phase 1 rotation derives from elastic energy (spring constant, κ = 50 k_BT·rad^-2). Phase 2 of the power stroke had an enthalpic component indicating that additional energy input occurred to enable the γ-subunit to overcome energy stored by the spring after rotating beyond its 34° equilibrium position. The correlation between the probability distribution of ATP binding to the empty catalytic site and the negative E_a values of the power stroke during phase 1 suggests that this additional energy is derived from the binding of ATP to the empty catalytic site. A second torsion spring (κ = 150 k_BT·rad^-2; equilibrium position, 90°) was also evident that mitigated the enthalpic cost of phase 2 rotation. The maximum ΔG^ǂ was 22.6 k_BT, and maximum efficiency was 72%. An elastic coupling mechanism is proposed that uses the coiled-coil domain of the γ-subunit rotor as a torsion spring during phase 1, and then as a crankshaft driven by ATP-binding-dependent conformational changes during phase 2 to drive the power stroke.

Structural DNA Nanotechnology: Artificial Nanostructures for Biomedical Research

Structural DNA nanotechnology utilizes synthetic or biologic DNA as designer molecules for the self-assembly of artificial nanostructures. The field is founded upon the specific interactions between DNA molecules, known as Watson-Crick base pairing. After decades of active pursuit, DNA has demonstrated unprecedented versatility in constructing artificial nanostructures with significant complexity and programmability. The nanostructures could be either static, with well-controlled physicochemical properties, or dynamic, with the ability to reconfigure upon external stimuli. Researchers have devoted considerable effort to exploring the usability of DNA nanostructures in biomedical research. We review the basic design methods for fabricating both static and dynamic DNA nanostructures, along with their biomedical applications in fields such as biosensing, bioimaging, and drug delivery.

Detailed Analyses of Stall Force Generation in Mycoplasma mobile Gliding

Mycoplasma mobile is a bacterium that uses a unique mechanism to glide on solid surfaces at a velocity of up to 4.5 μm/s. Its gliding machinery comprises hundreds of units that generate the force for gliding based on the energy derived from ATP; the units catch and pull sialylated oligosaccharides fixed to solid surfaces. In this study, we measured the stall force of wild-type and mutant strains of M. mobile carrying a bead manipulated using optical tweezers. The strains that had been enhanced for binding exhibited weaker stall forces than the wild-type strain, indicating that stall force is related to force generation rather than to binding. The stall force of the wild-type strain decreased linearly from 113 to 19 picoNewtons after the addition of 0-0.5 mM free sialyllactose (a sialylated oligosaccharide), with a decrease in the number of working units. After the addition of 0.5 mM sialyllactose, the cells carrying a bead loaded using optical tweezers exhibited stepwise movements with force increments. The force increments ranged from 1 to 2 picoNewtons. Considering the 70-nm step size, this small-unit force may be explained by the large gear ratio involved in the M. mobile gliding machinery.

Stochastic efficiency of an isothermal work-to-work converter engine

We investigate the efficiency of an isothermal Brownian work-to-work converter engine, composed of a Brownian particle coupled to a heat bath at a constant temperature. The system is maintained out of equilibrium by using two external time-dependent stochastic Gaussian forces, where one is called load force and the other is called drive force. Work done by these two forces are stochastic quantities. The efficiency of this small engine is defined as the ratio of stochastic work done against load force to stochastic work done by the drive force. The probability density function as well as large deviation function of the stochastic efficiency are studied analytically and verified by numerical simulations.

Structural basis of proton translocation and force generation in mitochondrial ATP synthase

ATP synthases produce ATP by rotary catalysis, powered by the electrochemical proton gradient across the membrane. Understanding this fundamental process requires an atomic model of the proton pathway. We determined the structure of an intact mitochondrial ATP synthase dimer by electron cryo-microscopy at near-atomic resolution. Charged and polar residues of the a-subunit stator define two aqueous channels, each spanning one half of the membrane. Passing through a conserved membrane-intrinsic helix hairpin, the lumenal channel protonates an acidic glutamate in the c-ring rotor. Upon ring rotation, the protonated glutamate encounters the matrix channel and deprotonates. An arginine between the two channels prevents proton leakage. The steep potential gradient over the sub-nm inter-channel distance exerts a force on the deprotonated glutamate, resulting in net directional rotation.

Methodology for the Simulation of Molecular Motors at Different Scales

Millisecond-scale conformational transitions represent a seminal challenge for traditional molecular dynamics simulations, even with the help of high-end supercomputer architectures. Such events are particularly relevant to the study of molecular motors-proteins or abiological constructs that convert chemical energy into mechanical work. Here, we present a hybrid-simulation scheme combining an array of methods including elastic network models, transition path sampling, and advanced free-energy methods, possibly in conjunction with generalized-ensemble schemes to deliver a viable option for capturing the millisecond-scale motor steps of biological motors. The methodology is already implemented in large measure in popular molecular dynamics programs, and it can leverage the massively parallel capabilities of petascale computers. The applicability of the hybrid method is demonstrated with two examples, namely cyclodextrin-based motors and V-type ATPases.

Cryo-EM structures of the autoinhibited E. coli ATP synthase in three rotational states

A molecular model that provides a framework for interpreting the wealth of functional information obtained on the E. coli F-ATP synthase has been generated using cryo-electron microscopy. Three different states that relate to rotation of the enzyme were observed, with the central stalk’s ε subunit in an extended autoinhibitory conformation in all three states. The F_o motor comprises of seven transmembrane helices and a decameric c-ring and invaginations on either side of the membrane indicate the entry and exit channels for protons. The proton translocating subunit contains near parallel helices inclined by ~30° to the membrane, a feature now synonymous with rotary ATPases. For the first time in this rotary ATPase subtype, the peripheral stalk is resolved over its entire length of the complex, revealing the F₁ attachment points and a coiled-coil that bifurcates toward the membrane with its helices separating to embrace subunit a from two sides.

DOI:http://dx.doi.org/10.7554/eLife.21598.001

ATP synthase is a biological motor that produces a molecule called adenosine tri-phosphate (ATP for short), which acts as the major store of chemical energy in cells. A single molecule of ATP contains three phosphate groups: the cell can remove one of these phosphates to make a molecule called adenosine di-phosphate (ADP) and release energy to drive a variety of biological processes.

ATP synthase sits in the membranes that separate cell compartments or form barriers around cells. When cells break down food they transport hydrogen ions across these membranes so that each side of the membrane has a different level (or “concentration”) of hydrogen ions. Movement of hydrogen ions from an area with a high concentration to a low concentration causes ATP synthase to rotate like a turbine. This rotation of the enzyme results in ATP synthase adding a phosphate group to ADP to make a new molecule of ATP. In certain conditions cells need to switch off the ATP synthase and this is done by changing the shape of the central shaft in a process called autoinhibition, which blocks the rotation.

The ATP synthase from a bacterium known as E. coli – which is commonly found in the human gut –has been used as a model to study how this biological motor works. However, since the precise details of the three-dimensional structure of ATP synthase have remained unclear it has been difficult to interpret the results of these studies.

Sobti et al. used a technique called Cryo-electron microscopy to investigate the structure of ATP synthase from E. coli. This made it possible to develop a three-dimensional model of the ATP synthase in its autoinhibited form. The structural data could also be split into three distinct shapes that relate to dwell points in the rotation of the motor where the rotation has been inhibited. These models further our understanding of ATP synthases and provide a template to understand the findings of previous studies.

Further work will be needed to understand this essential biological process at the atomic level in both its inhibited and uninhibited form. This will reveal the inner workings of a marvel of the natural world and may also lead to the discovery of new antibiotics against related bacteria that cause diseases in humans.

DOI:http://dx.doi.org/10.7554/eLife.21598.002

Modulation of coupling in the Escherichia coli ATP synthase by ADP and Pi: Role of the ε subunit C-terminal domain

The ε-subunit of ATP-synthase is an endogenous inhibitor of the hydrolysis activity of the complex and its α-helical C-terminal domain (εCTD) undergoes drastic changes among at least two different conformations. Even though this domain is not essential for ATP synthesis activity, there is evidence for its involvement in the coupling mechanism of the pump. Recently, it was proposed that coupling of the ATP synthase can vary as a function of ADP and P_i concentration. In the present work, we have explored the possible role of the εCTD in this ADP- and P_i-dependent coupling, by examining an εCTD-lacking mutant of Escherichia coli. We show that the loss of P_i-dependent coupling can be observed also in the εCTD-less mutant, but the effects of P_i on both proton pumping and ATP hydrolysis were much weaker in the mutant than in the wild-type. We also show that the εCTD strongly influences the binding of ADP to a very tight binding site (half-maximal effect≈1nM); binding at this site induces higher coupling in EF_OF₁ and increases responses to P_i. It is proposed that one physiological role of the εCTD is to regulate the kinetics and affinity of ADP/P_i binding, promoting ADP/P_i-dependent coupling.

The thermodynamic efficiency of ATP synthesis in oxidative phosphorylation

As the chief energy source of eukaryotic cells, it is important to determine the thermodynamic efficiency of ATP synthesis in oxidative phosphorylation (OX PHOS). Previous estimates of the thermodynamic efficiency of this vital process have ranged from Lehninger's original back-of-the-envelope calculation of 38% to the often quoted value of 55-60% in current textbooks of biochemistry, to high values of 90% from recent information theoretic considerations, and reports of realizations of close to ideal 100% efficiencies by single molecule experiments. Hence this problem has been reinvestigated from first principles. The overall thermodynamic efficiency of ATP synthesis in the mitochondrial energy transduction OX PHOS process has been found to lie between 40 and 41% from four different approaches based on a) estimation using structural and biochemical data, b) fundamental nonequilibrium thermodynamic analysis, c) novel insights arising from Nath's torsional mechanism of energy transduction and ATP synthesis, and d) the overall balance of cellular energetics. The torsional mechanism also offers an explanation for the observation of a thermodynamic efficiency approaching 100% in some experiments. Applications of the unique, molecular machine mode of functioning of F₁F_O-ATP synthase involving direct inter-conversion of chemical and mechanical energies in the design and fabrication of novel, man-made mechanochemical devices have been envisaged, and some new ways to exorcise Maxwell's demon have been proposed. It is hoped that analysis of the fundamental problem of energy transduction in OX PHOS from a fresh perspective will catalyze new avenues of research in this interdisciplinary field.