How Fo-ATPase generates rotary torque

Group-theoretic analysis of symmetry-preserving deployable structures and metamaterials

Many deployable structures in nature, as well as human-made mechanisms, preserve symmetry as their configurations evolve. Examples in nature include blooming flowers, dilation of the iris within the human eye, viral capsid maturation and molecular and bacterial motors. Engineered examples include opening umbrellas, elongating scissor jacks, variable apertures in cameras, expanding Hoberman spheres and some kinds of morphing origami structures. In these cases, the structures either preserve a discrete symmetry group or are described as an evolution from one discrete symmetry group to another of the same type as the structure deploys. Likewise, elastic metamaterials built from lattice structures can also preserve symmetry type while passively deforming and changing lattice parameters. A mathematical formulation of such transitions/deployments is articulated here. It is shown that if [Formula: see text] is Euclidean space, [Formula: see text] is a continuous group of motions of Euclidean space and [Formula: see text] is the type of the discrete subgroup of [Formula: see text] describing the symmetries of the deploying structure, then the symmetry of the evolving structure can be described by time-dependent subgroups of [Formula: see text] of the form [Formula: see text], where [Formula: see text] is a time-dependent affine transformation. Then, instead of considering the whole structure in [Formula: see text], a 'sector' of it that lives in the orbit space [Formula: see text] can be considered at each instant in time, and instead of considering all motions in [Formula: see text], only representatives from right cosets in the space [Formula: see text] need to be considered. This article is part of the theme issue 'Current developments in elastic and acoustic metamaterials science (Part 1)'.

Ion-driven rotary membrane motors: From structure to function

Ion-driven membrane motors, essential across all domains of life, convert a gradient of ions across a membrane into rotational energy, facilitating diverse biological processes including ATP synthesis, substrate transport, and bacterial locomotion. Herein, we highlight recent structural advances in the understanding of two classes of ion-driven membrane motors: rotary ATPases and 5:2 motors. The recent structure of the human F-type ATP synthase is emphasised along with the gained structural insight into clinically relevant mutations. Furthermore, we highlight the diverse roles of 5:2 motors and recent mechanistic understanding gained through the resolution of ions in the structure of a sodium-driven motor, combining insights into potential unifying mechanisms of ion selectivity and rotational torque generation in the context of their function as part of complex biological systems.

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.

pH-dependent 11° F1FO ATP synthase sub-steps reveal insight into the FO torque generating mechanism

Most cellular ATP is made by rotary F₁F_O ATP synthases using proton translocation-generated clockwise torque on the F_O c-ring rotor, while F₁-ATP hydrolysis can force counterclockwise rotation and proton pumping. The F_O torque-generating mechanism remains elusive even though the F_O interface of stator subunit-a, which contains the transmembrane proton half-channels, and the c-ring is known from recent F₁F_O structures. Here, single-molecule F₁F_O rotation studies determined that the pKa values of the half-channels differ, show that mutations of residues in these channels change the pKa values of both half-channels, and reveal the ability of F_O to undergo single c-subunit rotational stepping. These experiments provide evidence to support the hypothesis that proton translocation through F_O operates via a Grotthuss mechanism involving a column of single water molecules in each half-channel linked by proton translocation-dependent c-ring rotation. We also observed pH-dependent 11° ATP synthase-direction sub-steps of the Escherichia coli c₁₀-ring of F₁F_O against the torque of F₁-ATPase-dependent rotation that result from H⁺ transfer events from F_O subunit-a groups with a low pKa to one c-subunit in the c-ring, and from an adjacent c-subunit to stator groups with a high pKa. These results support a mechanism in which alternating proton translocation-dependent 11° and 25° synthase-direction rotational sub-steps of the c₁₀-ring occur to sustain F₁F_O ATP synthesis.

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.

Molecular dynamics simulation of proton-transfer coupled rotations in ATP synthase FO motor

The F_O motor in F_OF₁ ATP synthase rotates its rotor driven by the proton motive force. While earlier studies elucidated basic mechanisms therein, recent advances in high-resolution cryo-electron microscopy enabled to investigate proton-transfer coupled F_O rotary dynamics at structural details. Here, taking a hybrid Monte Carlo/molecular dynamics simulation method, we studied reversible dynamics of a yeast mitochondrial F_O. We obtained the 36°-stepwise rotations of F_O per one proton transfer in the ATP synthesis mode and the proton pumping in the ATP hydrolysis mode. In both modes, the most prominent path alternatively sampled states with two and three deprotonated glutamates in c-ring, by which the c-ring rotates one step. The free energy transduction efficiency in the model F_O motor reached ~ 90% in optimal conditions. Moreover, mutations in key glutamate and a highly conserved arginine increased proton leakage and markedly decreased the coupling, in harmony with previous experiments. This study provides a simple framework of simulations for chemical-reaction coupled molecular dynamics calling for further studies in ATP synthase and others.

Catalytic robustness and torque generation of the F1-ATPase

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

The Phylogenetic Signature Underlying ATP Synthase c-Ring Compliance

The proton-driven ATP synthase (FOF1) is comprised of two rotary, stepping motors (FO and F1) coupled by an elastic power transmission. The elastic compliance resides in the rotor module that includes the membrane-embedded FO c-ring. Proton transport by FO is firmly coupled to the rotation of the c-ring relative to other FO subunits (ab2). It drives ATP synthesis. We used a computational method to investigate the contribution of the c-ring to the total elastic compliance. We performed principal component analysis of conformational ensembles built using distance constraints from the bovine mitochondrial c-ring x-ray structure. Angular rotary twist, the dominant ring motion, was estimated to show that the c-ring accounted in part for the measured compliance. Ring rotation was entrained to rotation of the external helix within each hairpin-shaped c-subunit in the ring. Ensembles of monomer and dimers extracted from complete c-rings showed that the coupling between collective ring and the individual subunit motions was independent of the size of the c-ring, which varies between organisms. Molecular determinants were identified by covariance analysis of residue coevolution and structural-alphabet-based local dynamics correlations. The residue coevolution gave a readout of subunit architecture. The dynamic couplings revealed that the hinge for both ring and subunit helix rotations was constructed from the proton-binding site and the adjacent glycine motif (IB-GGGG) in the midmembrane plane. IB-GGGG motifs were linked by long-range couplings across the ring, while intrasubunit couplings connected the motif to the conserved cytoplasmic loop and adjacent segments. The correlation with principal collective motions shows that the couplings underlie both ring rotary and bending motions. Noncontact couplings between IB-GGGG motifs matched the coevolution signal as well as contact couplings. The residue coevolution reflects the physiological importance of the dynamics that may link proton transfer to ring compliance.

Fo-driven Rotation in the ATP Synthase Direction against the Force of F1 ATPase in the FoF1 ATP Synthase

Living organisms rely on the FoF1 ATP synthase to maintain the non-equilibrium chemical gradient of ATP to ADP and phosphate that provides the primary energy source for cellular processes. How the Fo motor uses a transmembrane electrochemical ion gradient to create clockwise torque that overcomes F1 ATPase-driven counterclockwise torque at high ATP is a major unresolved question. Using single FoF1 molecules embedded in lipid bilayer nanodiscs, we now report the observation of Fo-dependent rotation of the c10 ring in the ATP synthase (clockwise) direction against the counterclockwise force of ATPase-driven rotation that occurs upon formation of a leash with Fo stator subunit a. Mutational studies indicate that the leash is important for ATP synthase activity and support a mechanism in which residues aGlu-196 and cArg-50 participate in the cytoplasmic proton half-channel to promote leash formation.

Electric field driven torque in ATP synthase

F_O-ATP synthase (F_O) is a rotary motor that converts potential energy from ions, usually protons, moving from high- to low-potential sides of a membrane into torque and rotary motion. Here we propose a mechanism whereby electric fields emanating from the proton entry and exit channels act on asymmetric charge distributions in the c-ring, due to protonated and deprotonated sites, and drive it to rotate. The model predicts a scaling between time-averaged torque and proton motive force, which can be hindered by mutations that adversely affect the channels. The torque created by the c-ring of F_O drives the γ-subunit to rotate within the ATP-producing complex (F₁) overcoming, with the aid of thermal fluctuations, an opposing torque that rises and falls with angular position. Using the analogy with thermal Brownian motion of a particle in a tilted washboard potential, we compute ATP production rates vs. proton motive force. The latter shows a minimum, needed to drive ATP production, which scales inversely with the number of proton binding sites on the c-ring.

Resonance phenomenon of the ATP motor as an ultrasensitive biosensor

We designed a rotary biosensor as a damping effector, with the rotation of the F(0)F(1)-ATPase driven by Adenosine Triphosphate (ATP) synthesis being indicated by the fluorescence intensity and a damping effect force being induced by the binding of an RNA molecule to its probe on the rotary biosensor. We found that the damping effect could contribute to the resonance phenomenon and energy transfer process of our rotary biosensor in the liquid phase. This result indicates that the ability of the rotary motor to operate in the vibration harmonic mode depends on the environmental conditions and mechanism in that a few molecules of the rotary biosensor could induce all of the sensor molecules to fluoresce together. These findings contribute to the theory study of the ATPase motor and future development of biosensors for ultrasensitive detection.

Modelling the influence of thermal effects induced by radio frequency electric field on the dynamics of the ATPase nano-biomolecular motors

We model the dynamics of the F(0) component of the F(0)F(1)-ATPase mitochondrion-based nano-motor operating in a stochastically-fluctuating medium that represents the intracellular environment. The stochastic dynamics are modeled via Langevin equation of motion wherein fluctuations are treated as white noise. We have investigated the influence of an applied alternating electric field on the rotary motion of the F(0) rotor in such an environment. The exposure to the field induces a temperature rise in the mitochondrion's membrane, within which the F(0) is embedded. The external field also induces an electric potential that promotes a change in the mitochondrion's transmembrane potential (TMP). Both the induced temperature and the change in TMP contribute to a change in the dynamics of the F(0). We have found that for external fields in the radio frequency (RF) range, normally present in the environment and encountered by biological systems, the contribution of the induced thermal effects, relative to that of the induced TMP, to the dynamics of the F(0) is more significant. The changes in the dynamics of the F(0) part affect the frequency of the rotary motion of the F(0)F(1)-ATPase protein motor which, in turn, affects the production rate of the ATP molecules.

Brain responses to chronic social defeat stress: effects on regional oxidative metabolism as a function of a hedonic trait, and gene expression in susceptible and resilient rats

Chronic social defeat stress, a depression model in rats, reduced struggling in the forced swimming test dependent on a hedonic trait-stressed rats with high sucrose intake struggled less. Social defeat reduced brain regional energy metabolism, and this effect was also more pronounced in rats with high sucrose intake. A number of changes in gene expression were identified after social defeat stress, most notably the down-regulation of Gsk3b and Map1b. The majority of differences were between stress-susceptible and resilient rats. Conclusively, correlates of inter-individual differences in stress resilience can be identified both at gene expression and oxidative metabolism levels.

Direct observation of stepped proteolipid ring rotation in E. coli F₀F₁-ATP synthase

Although single-molecule experiments have provided mechanistic insight for several molecular motors, these approaches have proved difficult for membrane bound molecular motors like the F₀F₁-ATP synthase, in which proton transport across a membrane is used to synthesize ATP. Resolution of smaller steps in F₀ has been particularly hampered by signal-to-noise and time resolution. Here, we show the presence of a transient dwell between F₀ subunits a and c by improving the time resolution to 10 μs at unprecedented S/N, and by using Escherichia coli F₀F₁ embedded in lipid bilayer nanodiscs. The transient dwell interaction requires 163 μs to form and 175 μs to dissociate, is independent of proton transport residues aR210 and cD61, and behaves as a leash that allows rotary motion of the c-ring to a limit of ∼36° while engaged. This leash behaviour satisfies a requirement of a Brownian ratchet mechanism for the F₀ motor where c-ring rotational diffusion is limited to 36°.

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.

Stoichiometry of energy coupling by proton-translocating ATPases: a history of variability

One of the central energy-coupling reactions in living systems is the intraconversion of ATP with a transmembrane proton gradient, carried out by proton-translocating F- and V-type ATPases/synthases. These reversible enzymes can hydrolyze ATP and pump protons, or can use the energy of a transmembrane proton gradient to synthesize ATP from ADP and inorganic phosphate. The stoichiometry of these processes (H(+)/ATP, or coupling ratio) has been studied in many systems for many years, with no universally agreed upon solution. Recent discoveries concerning the structure of the ATPases, their assembly and the stoichiometry of their numerous subunits, particularly the proton-carrying proteolipid (subunit c) of the F(O) and V(0) sectors, have shed new light on this question and raise the possibility of variable coupling ratios modulated by variable proteolipid stoichiometries.

Changes in regional long-term oxidative metabolism induced by partial serotonergic denervation and chronic variable stress in rat brain

Stressful experiences and genetic predisposition have both independent and interactive contributions to the development of depression. The serotonergic system is involved in the development of depression, and administration of neurotoxins that specifically compromise its function leads to symptoms of affective disorders. In order to find out which brain regions are most affected by stress, partial serotonergic denervation and their combination, chronic variable stress (CVS) was applied for 3 week. Serotonergic denervation was elicited by parachloroampetamine (PCA, 2mg/kg), and cytochrome oxidase histochemistry was used to characterize the long-term levels of neuronal oxidative energy metabolism. PCA pretreatment blocked the increase in oxidative activity in chronically stressed rats in medial preoptic area, cortical and medial amygdala. PCA raised oxidative activity compared to control animals in substantia nigra and ventrolateral division of laterodorsal thalamus. CVS reduced the oxidative activity induced by PCA in suprachiasmatic hypothalamus, anteroventral thalamus, hippocampal CA3 region and cortical amygdala. In the dorsal part of the anterior olfactory nucleus chronic stress blocked the decrease in oxidative activity evoked by PCA. Conclusively, partial serotonergic denervation with PCA and chronic variable stress both had independent effects on long-term energy metabolism in several rat brain structures, tending to increase it. However, partial serotonergic denervation by parachloroampetamine and chronic variable stress had in many brain regions an interactive effect on energy metabolism, each factor reducing the effect of the other, which could reflect the weakening of adaptive mechanisms.

Effects of oscillatory electric fields on internal membranes: an analytical model

We derive an analytical model of the potential differences induced across plasma and internal organelle membranes in suspended cells exposed to oscillatory electric fields. Multiple shells are modeled using iterative applications of the single-shell calculation with mobile charges. This work is motivated, in part, by recent results suggesting the ability to use alternating current (ac) fields to noninvasively monitor enzyme activity within internal membranes, particularly the mitochondrial electron transport chain. Previous work, on induced transmembrane voltages in cells subjected to ac fields, has mainly been limited to oscillatory potentials across the plasma membrane. Here we first develop a three-membrane model, consisting of a plasma membrane surrounding inner and outer membranes representing an internal organelle, such as a mitochondrion. Frequency-dependent transmembrane potentials are modeled for spherical, weakly conducting membrane shells enclosing a conductive cytoplasm surrounding an idealized internal organelle. We then use a two-shell model to simulate induced ac membrane potentials of a suspended isolated mitochondrion in which the outer membrane is usually much more permeable than the inner membrane.

An oscillating electric field with thermal noise increases the rotational diffusion and drives rotation in a dipole

Here we consider a dipole in a viscous medium under the influence of an oscillating electric field and thermal noise. Because of the very low Reynolds numbers involved in molecular processes, we considered overdamped Langevin dynamics. As a consequence the inertia term becomes negligible. We observed a great increase in the rotational diffusion and also net rotation for some values of the parameters.

Molecular machines

Molecular machines are tiny energy conversion devices on the molecular-size scale. Whether naturally occurring or synthetic, these machines are generally more efficient than their macroscale counterparts. They have their own mechanochemistry, dynamics, workspace, and usability and are composed of nature's building blocks: namely proteins, DNA, and other compounds, built atom by atom. With modern scientific capabilities it has become possible to create synthetic molecular devices and interface them with each other. Countless such machines exist in nature, and it is possible to build artificial ones by mimicking nature. Here we review some of the known molecular machines, their structures, features, and characteristics. We also look at certain devices in their early development stages, as well as their future applications and challenges.

Mechanical Processes in Biochemistry

Mechanical processes are involved in nearly every facet of the cell cycle. Mechanical forces are generated in the cell during processes as diverse as chromosomal segregation, replication, transcription, translation, translocation of proteins across membranes, cell locomotion, and catalyzed protein and nucleic acid folding and unfolding, among others. Because force is a product of all these reactions, biochemists are beginning to directly apply external forces to these processes to alter the extent or even the fate of these reactions hoping to reveal their underlying molecular mechanisms. This review provides the conceptual framework to understand the role of mechanical force in biochemistry.

Molecular mechanisms of energy transduction in cells: engineering applications and biological implications

The synthesis of ATP from ADP and inorganic phosphate by F1F0-ATP synthase, the universal enzyme in biological energy conversion, using the energy of a transmembrane gradient of ions, and the use of ATP by the myosin-actin system to cause muscular contraction are among the most fundamental processes in biology. Both the ATP synthase and the myosin-actin may be looked upon as molecular machines. A detailed analysis of the molecular mechanisms of energy transduction by these molecular machines has been carried out in order to understand the means by which living cells produce and consume energy. These mechanisms have been compared with each other and their biological implications have been discussed. The thermodynamics of energy coupling in the oxidative phosphorylation process has been developed and the consistency of the mechanisms with the thermodynamics has been explored. Novel engineering applications that can result have been discussed in detail and several directions for future work have been pointed out.

Developmental robustness

Developmental robustness, the capacity to stay "on track" despite the myriad vicissitudes that inevitably plague a developing organism, is, I argue, a prerequisite for natural selection and key to our understanding of the evolution of developmental processes. But how is such robustness achieved? And how can we reconcile this property with the delicate precision that seems to characterize so many developmental mechanisms, with what Michael Behe calls "irreducible complexity"? By looking at context, I argue. Developmental mechanisms must be robust with respect to the kinds of insults they are most likely to face, but with respect to less likely vicissitudes, they can be fragile. More specifically, I examine the relative absence of reaction-diffusion mechanisms in development and suggest that such mechanisms, theoretically attractive though they may be, have been judged by evolution to be ill suited for providing protection against the kinds of vicissitudes developing organisms are most likely to face, and have been supplanted by more intricate mechanisms that are protected from insult by structural design.

Arginine residue at position 573 in Enterococcus hirae vacuolar-type ATPase NtpI subunit plays a crucial role in Na+ translocation

The 76-kDa NtpI subunit constitutes the membrane-embedded V(0) moiety of Enterococcus hirae vacuolar type Na+-ATPase with a 16-kDa NtpK hexamer containing Na+ binding sites. In this study, we investigated the role of an arginine residue, which is highly conserved among the corresponding subunits of bacterial vacuolar-type ATPases, at position 573 of NtpI. Substitution of Glu, Leu, or Gln for Arg-573 abolished sodium transport and sodium-stimulated ATP hydrolysis of the enzyme. The conservative replacement of Arg by Lys lowered both activities about one-fifth of those of the wild type enzyme. We have reported previously on ATP-dependent negative cooperativity for Na+ coupling of this enzyme (Murata, T., Kakinuma, Y., and Yamato, I. (2001) J. Biol. Chem. 276, 48337-48340). The negative cooperativity for the Na+ dependence of ATPase activity was weakened by the mutation R573K; the Hill coefficients for the wild type and mutant enzymes at a saturated ATP concentration were 0.22 +/- 0.03 and 0.40 +/- 0.05, respectively. The Hill coefficients of both enzymes at limited ATP concentrations approached 1. These results indicate that NtpI Arg-573 is indispensable for sodium translocation and for the cooperative features of E. hirae vacuolar-type ATPase.