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

On the Control of Directionality of Myosin

The origin of the unique directionality of myosin has been a problem of fundamental and practical importance. This work establishes in a conclusive way that the directionality is controlled by tuning the barrier for the rate-determining step, namely, the ADP release step. This conclusion is based on exploring the molecular origin behind the reverse directionality of myosins V and VI and the determination of the origin of the change in the barriers of the ADP release for the forward and backward motions. Our investigation is performed by combining different simulation methods such as steer molecular dynamics (SMD), umbrella sampling, renormalization method, and automated path searching method. It is found that in the case of myosin V, the ADP release from the postrigor (trailing head) state overcomes a lower barrier than the prepowerstroke (leading head) state, which is also evident from experimental observation. In the case of myosin VI, we noticed a different trend when compared to myosin V. Since the directionality of myosins V and VI follows a reverse trend, we conclude that such differences in the directionality are controlled by the free energy barrier for the ADP release. Overall, the proof that the directionality of myosin is determined by the activation barrier of the rate-determining step in the cycle, rather than by some unspecified dynamical effects, has general importance.

Catalytic mechanism study of ATP-citrate lyase during citryl-CoA synthesis process

ATP-citrate lyase (ACLY) is a critical metabolic enzyme and promising target for drug development. The structure determinations of ACLY have revealed its homotetramer states with various subunit symmetries, but catalytic mechanism of ACLY tetramer and the importance of subunit symmetry have not been clarified. Here, we constructed the free energy landscape of ACLY tetramer with arbitrary subunit symmetries and investigated energetic and conformational coupling of subunits during citryl-CoA synthesis process. The optimal conformational pathway indicates that ACLY tetramer encounters three critical conformational barriers and undergoes a loss of rigid-D2 symmetry to gain an energetic advantage. Energetic coupling of conformational changes and biochemical reactions suggests that these biological events are not independent but rather coupled with each other, showing a comparable energy barrier to the experimental data for the rate-limiting step. These findings could contribute to further research on catalytic mechanism, functional modulation, and inhibitor design of ACLY.

Mechanochemical active ratchet

Self-propelled nanoparticles moving through liquids offer the possibility of creating advanced applications where such nanoswimmers can operate as artificial molecular-sized motors. Achieving control over the motion of nanoswimmers is a crucial aspect for their reliable functioning. While the directionality of micron-sized swimmers can be controlled with great precision, steering nano-sized active particles poses a real challenge. One of the reasons is the existence of large fluctuations of active velocity at the nanoscale. Here, we describe a mechanism that, in the presence of a ratchet potential, transforms these fluctuations into a net current of active nanoparticles. We demonstrate the effect using a generic model of self-propulsion powered by chemical reactions. The net motion along the easy direction of the ratchet potential arises from the coupling of chemical and mechanical processes and is triggered by a constant, transverse to the ratchet, force. The current magnitude sensitively depends on the amplitude and the periodicity of the ratchet potential and the strength of the transverse force. Our results highlight the importance of thermodynamically consistent modeling of chemical reactions in active matter at the nanoscale and suggest new ways of controlling dynamics in such systems.

Long-Time Dynamics of Selected Molecular-Motor Components Using a Physics-Based Coarse-Grained Approach

Molecular motors are essential for the movement and transportation of macromolecules in living organisms. Among them, rotatory motors are particularly efficient. In this study, we investigated the long-term dynamics of the designed left-handed alpha/alpha toroid (PDB: 4YY2), the RBM2 flagellum protein ring from Salmonella (PDB: 6SD5), and the V-type Na+-ATPase rotor in Enterococcus hirae (PDB: 2BL2) using microcanonical and canonical molecular dynamics simulations with the coarse-grained UNRES force field, including a lipid-membrane model, on a millisecond laboratory time scale. Our results demonstrate that rotational motion can occur with zero total angular momentum in the microcanonical regime and that thermal motions can be converted into net rotation in the canonical regime, as previously observed in simulations of smaller cyclic molecules. For 6SD5 and 2BL2, net rotation (with a ratcheting pattern) occurring only about the pivot of the respective system was observed in canonical simulations. The extent and direction of the rotation depended on the initial conditions. This result suggests that rotatory molecular motors can convert thermal oscillations into net rotational motion. The energy from ATP hydrolysis is required probably to set the direction and extent of rotation. Our findings highlight the importance of molecular-motor structures in facilitating movement and transportation within living organisms.

Application of Coarse-Grained (CG) Models to Explore Conformational Pathway of Large-Scale Protein Machines

Protein machines are clusters of protein assemblies that function in order to control the transfer of matter and energy in cells. For a specific protein machine, its working mechanisms are not only determined by the static crystal structures, but also related to the conformational transition dynamics and the corresponding energy profiles. With the rapid development of crystallographic techniques, the spatial scale of resolved structures is reaching up to thousands of residues, and the concomitant conformational changes become more and more complicated, posing a great challenge for computational biology research. Previously, a coarse-grained (CG) model aiming at conformational free energy evaluation was developed and showed excellent ability to reproduce the energy profiles by accurate electrostatic interaction calculations. In this study, we extended the application of the CG model to a series of large-scale protein machine systems. The spike protein trimer of SARS-CoV-2, ATP citrate lyase (ACLY) tetramer, and P4-ATPases systems were carefully studied and discussed as examples. It is indicated that the CG model is effective to depict the energy profiles of the conformational pathway between two endpoint structures, especially for large-scale systems. Both the energy change and energy barrier between endpoint structures provide reasonable mechanism explanations for the associated biological processes, including the opening of receptor binding domain (RBD) of spike protein, the phospholipid transportation of P4-ATPase, and the loop translocation of ACLY. Taken together, the CG model provides a suitable alternative in mechanistic studies related to conformational change in large-scale protein machines.

Polyrotaxanes and the pump paradigm

The year 2022 marks the 30th anniversary of the first reports of polyrotaxanes in the scientific literature.

Kinetic analysis of the inhibition mechanism of bovine mitochondrial F1-ATPase inhibitory protein using biochemical assay

ATPase inhibitory factor 1 (IF1) is a mitochondrial regulatory protein that blocks ATP hydrolysis of F1-ATPase, by inserting its N-terminus into the rotor-stator interface of F1-ATPase. Although previous studies have proposed a two-step model for IF1-mediated inhibition, the underlying molecular mechanism remains unclear. Here, we analysed the kinetics of IF1-mediated inhibition under a wide range of [ATP]s and [IF1]s, using bovine mitochondrial IF1 and F1-ATPase. Typical hyperbolic curves of inhibition rates with [IF1]s were observed at all [ATP]s tested, suggesting a two-step mechanism: the initial association of IF1 to F1-ATPase and the locking process, where IF1 blocks rotation by inserting its N-terminus. The initial association was dependent on ATP. Considering two principal rotation dwells, binding dwell and catalytic dwell, in F1-ATPase, this result means that IF1 associates with F1-ATPase in the catalytic-waiting state. In contrast, the isomerization process to the locking state was almost independent of ATP, suggesting that it is also independent of the F1-ATPase state. Further, we investigated the role of Glu30 or Tyr33 of IF1 in the two-step mechanism. Kinetic analysis showed that Glu30 is involved in the isomerization, whereas Tyr33 contributes to the initial association. Based on these findings, we propose an IF1-mediated inhibition scheme.

Molecular Pumps and Motors

Pumps and motors are essential components of the world as we know it. From the complex proteins that sustain our cells, to the mechanical marvels that power industries, much we take for granted is only possible because of pumps and motors. Although molecular pumps and motors have supported life for eons, it is only recently that chemists have made progress toward designing and building artificial forms of the microscopic machinery present in nature. The advent of artificial molecular machines has granted scientists an unprecedented level of control over the relative motion of components of molecules through the development of kinetically controlled, away-from-thermodynamic equilibrium chemistry. We outline the history of pumps and motors, focusing specifically on the innovations that enable the design and synthesis of the artificial molecular machines central to this Perspective. A key insight connecting biomolecular and artificial molecular machines is that the physical motions by which these machines carry out their function are unambiguously in mechanical equilibrium at every instant. The operation of molecular motors and pumps can be described by trajectory thermodynamics, a theory based on the work of Onsager, which is grounded on the firm foundation of the principle of microscopic reversibility. Free energy derived from thermodynamically non-equilibrium reactions kinetically favors some reaction pathways over others. By designing molecules with kinetic asymmetry, one can engineer potential landscapes to harness external energy to drive the formation and maintenance of geometries of component parts of molecules away-from-equilibrium, that would be impossible to achieve by standard synthetic approaches.

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

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

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.

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

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

Evidence for a Partially Stalled γ Rotor in F1-ATPase from Hydrogen-Deuterium Exchange Experiments and Molecular Dynamics Simulations

F₁-ATPase uses ATP hydrolysis to drive rotation of the γ subunit. The γ C-terminal helix constitutes the rotor tip that is seated in an apical bearing formed by α₃β₃. It remains uncertain to what extent the γ conformation during rotation differs from that seen in rigid crystal structures. Existing models assume that the entire γ subunit participates in every rotation. Here we interrogated E. coli F₁-ATPase by hydrogen-deuterium exchange (HDX) mass spectrometry. Rotation of γ caused greatly enhanced deuteration in the γ C-terminal helix. The HDX kinetics implied that most F₁ complexes operate with an intact rotor at any given time, but that the rotor tip is prone to occasional unfolding. A molecular dynamics (MD) strategy was developed to model the off-axis forces acting on γ. MD runs showed stalling of the rotor tip and unfolding of the γ C-terminal helix. MD-predicted H-bond opening events coincided with experimental HDX patterns. Our data suggest that in vitro operation of F₁-ATPase is associated with significant rotational resistance in the apical bearing. These conditions cause the γ C-terminal helix to get "stuck" (and unfold) sporadically while the remainder of γ continues to rotate. This scenario contrasts the traditional "greasy bearing" model that envisions smooth rotation of the γ C-terminal helix. The fragility of the apical rotor tip in F₁-ATPase is attributed to the absence of a c₁₀ ring that stabilizes the rotation axis in intact F_oF₁. Overall, the MD/HDX strategy introduced here appears well suited for interrogating the inner workings of molecular motors.

Intramolecular Torque Study of a Molecular Rotation Stimulated by Electron Injection and Extraction

Rotation-inducing torque based on interatomic forces is a true indicator of internal molecular rotations. We use the induced intramolecular torque to study the underlying rotational mechanism stimulated by an electron injection or extraction for the rotor molecule 9-(2,4,7-trimethyl-2,3-dihydro-1 H-inden-1-ylidene)-9 H-fluorene, which consists of a "rotator" fragment and a "stator" fragment. The results show that the charged molecule in a quartet spin state can rotate internally, while that in the doublet state cannot. The torque on the rotator in the quartet state always maintains unidirectional rotation. In addition, the attachment/extraction of an electron leads to the reduction of the rotational energy barrier by about 18 kcal/mol, facilitating a more favorable molecular rotation than in the neutral singlet state. Our finding provides a molecular-level understanding of various transformation pathways for experimental designs and further demonstrates the effectiveness of the torque approach.

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.

The FOF1 ATP synthase: from atomistic three-dimensional structure to the rotary-chemical function

Molecular motors are multi-subunit complexes that are indispensable for accomplishing various tasks of the living cells. One such molecular motor is the F_OF₁ ATP synthase that synthesizes ATP at the expense of the membrane proton gradient. Elucidating the molecular origin of the motor function is challenging despite significant advances in various experimental fields. Currently atomic simulations of whole motor complexes cannot reach to functionally relevant time scales that extend beyond the millisecond regime. Moreover, to reveal the underlying molecular origin of the function, one must model the coupled chemical and conformational events using physically and chemically meaningful multiscaling techniques. In this review, we discuss our approach to model the action of the F₁ and F_O molecular motors, where emphasis is laid on elucidating the molecular origin of the driving force that leads to directional rotation at the expense of ATP hydrolysis or proton gradients. We have used atomic structures of the motors and used hierarchical multiscaling techniques to generate low dimensional functional free energy surfaces of the complete mechano-chemical process. These free energy surfaces were studied further to calculate important characteristics of the motors, such as, rotational torque, temporal dynamics, occurrence of intermittent dwell states, etc. We also studied the result of mutating various parts of the motor domains and our observations correspond very well with the experimental findings. Overall, our studies have generated a cumulative understanding of the motor action, and especially highlight the crucial role of electrostatics in establishing the mechano-chemical coupling.

Thermally induced micro-motion by inflection in optical potential

Recent technological progress in a precise control of optically trapped objects allows much broader ventures to unexplored territory of thermal motion in non-linear potentials. In this work, we exploit an experimental set-up of holographic optical tweezers to experimentally investigate Brownian motion of a micro-particle near the inflection point of the cubic optical potential. We present two complementary views on the non-linear Brownian motion. On an ensemble of stochastic trajectories, we simultaneously determine (i) the detailed short-time position statistics and (ii) the long-distance first-passage time statistics. We evaluate specific statistical moment ratios demonstrating strongly non-linear stochastic dynamics. This is a crucial step towards a possible massive exploitation of the broad class of complex non-linear stochastic effects with objects of more complex structure and shape including living ones.

Analysis of the Acting Forces in a Theory of Catalysis and Mechanochemistry

The theoretical description of a chemical process resulting from the application of mechanical or catalytical stress to a molecule is performed by the generation of an effective potential energy surface (PES). Changes for minima and saddle points by the stress are described by Newton trajectories (NTs) on the original PES. From the analysis of the acting forces we postulate the existence of pulling corridors built by families of NTs that connect the same stationary points. For different exit saddles of different height we discuss the corresponding pulling corridors; mainly by simple two-dimensional surface models. If there are different exit saddles then there can exist saddles of index two, at least, between. Then the case that a full pulling corridor crosses a saddle of index two is the normal case. It leads to an intrinsic hysteresis of such pullings for the forward or the backward reaction. Assuming such relations we can explain some results in the literature. A new finding is the existence of roundabout corridors that can switch between different saddle points by a reversion of the direction. The findings concern the mechanochemistry of molecular systems under a mechanical load as well as the electrostatic force and can be extended to catalytic and enzymatic accelerated reactions. The basic and ground ansatz includes both kinds of forces in a natural way without an extra modification.

Mechanochemical Energy Transduction during the Main Rotary Step in the Synthesis Cycle of F1-ATPase

F₁-ATPase is a highly efficient molecular motor that can synthesize ATP driven by a mechanical torque. Its ability to function reversibly in either direction requires tight mechanochemical coupling between the catalytic domain and the rotating central shaft, as well as temporal control of substrate binding and product release. Despite great efforts and significant progress, the molecular details of this synchronized and fine-tuned energy conversion mechanism are not fully understood. Here, we use extensive molecular dynamics simulations to reconcile recent single-molecule experiments with structural data and provide a consistent thermodynamic, kinetic and mechanistic description of the main rotary substep in the synthetic cycle of mammalian ATP synthase. The calculated free energy profiles capture a discrete pattern in the rotation of the central γ-shaft, with a metastable intermediate located-consistently with recent experimental findings-at 70° relative to the X-ray position. We identify this rotary step as the ATP-dependent substep, and find that the associated free energy input supports the mechanism involving concurrent nucleotide binding and release. During the main substep, our simulations show no significant opening of the ATP-bound β subunit; instead, we observe that mechanical energy is transmitted to its nucleotide binding site, thus lowering the affinity for ATP. Simultaneously, the empty subunit assumes a conformation that enables the enzyme to harness the free energy of ADP binding to drive ATP release. Finally, we show that ligand exchange is regulated by a checkpoint mechanism, an apparent prerequisite for high efficiency in protein nanomotors.

Simulating the dynamics of the mechanochemical cycle of myosin-V

The detailed dynamics of the cycle of myosin-V are explored by simulation approaches, examining the nature of the energy-driven motion. Our study started with Langevin dynamics (LD) simulations on a very coarse landscape with a single rate-limiting barrier and reproduced the stall force and the hand-over-hand dynamics. We then considered a more realistic landscape and used time-dependent Monte Carlo (MC) simulations that allowed trajectories long enough to reproduce the force/velocity characteristic sigmoidal correlation, while also reproducing the hand-over-hand motion. Overall, our study indicated that the notion of a downhill lever-up to lever-down process (popularly known as the powerstroke mechanism) is the result of the energetics of the complete myosin-V cycle and is not the source of directional motion or force generation on its own. The present work further emphasizes the need to use well-defined energy landscapes in studying molecular motors in general and myosin in particular.

The Physics and Physical Chemistry of Molecular Machines

The concept of a "power stroke"-a free-energy releasing conformational change-appears in almost every textbook that deals with the molecular details of muscle, the flagellar rotor, and many other biomolecular machines. Here, it is shown by using the constraints of microscopic reversibility that the power stroke model is incorrect as an explanation of how chemical energy is used by a molecular machine to do mechanical work. Instead, chemically driven molecular machines operating under thermodynamic constraints imposed by the reactant and product concentrations in the bulk function as information ratchets in which the directionality and stopping torque or stopping force are controlled entirely by the gating of the chemical reaction that provides the fuel for the machine. The gating of the chemical free energy occurs through chemical state dependent conformational changes of the molecular machine that, in turn, are capable of generating directional mechanical motions. In strong contrast to this general conclusion for molecular machines driven by catalysis of a chemical reaction, a power stroke may be (and often is) an essential component for a molecular machine driven by external modulation of pH or redox potential or by light. This difference between optical and chemical driving properties arises from the fundamental symmetry difference between the physics of optical processes, governed by the Bose-Einstein relations, and the constraints of microscopic reversibility for thermally activated processes.

Intramolecular torque, an indicator of the internal rotation direction of rotor molecules and similar systems

Rotation-inducing torque being ubiquitous in many molecular systems is the driving force of the molecular internal rotation and an indicator of the rotation direction.

Theory for rates, equilibrium constants, and Brønsted slopes in F1-ATPase single molecule imaging experiments

A theoretical model of elastically coupled reactions is proposed for single molecule imaging and rotor manipulation experiments on F1-ATPase. Stalling experiments are considered in which rates of individual ligand binding, ligand release, and chemical reaction steps have an exponential dependence on rotor angle. These data are treated in terms of the effect of thermodynamic driving forces on reaction rates, and lead to equations relating rate constants and free energies to the stalling angle. These relations, in turn, are modeled using a formalism originally developed to treat electron and other transfer reactions. During stalling the free energy profile of the enzymatic steps is altered by a work term due to elastic structural twisting. Using biochemical and single molecule data, the dependence of the rate constant and equilibrium constant on the stall angle, as well as the Børnsted slope are predicted and compared with experiment. Reasonable agreement is found with stalling experiments for ATP and GTP binding. The model can be applied to other torque-generating steps of reversible ligand binding, such as ADP and Pi release, when sufficient data become available.