A chemically reversible Brownian motor: application to kinesin and Ncd.

Kinesin and nonclaret disjunctional protein (ncd) are two microtubule-based molecular motors that use energy from ATP hydrolysis to drive motion in opposite directions. They are structurally very similar and bind with similar orientations on microtubule. What is the origin of the different directionality? Is it some subtle feature of the structure of the motor domains, not apparent in x-ray diffraction studies, or possibly some difference near the neck regions far from the microtubule binding site? Perhaps because the motors function as dimers, the explanation involves differences in the strength of the interaction between the two motor monomers themselves. Here we present another possibility, based on a Brownian ratchet, in which the direction of motion of the motor is controlled by the chemical mechanism of ATP hydrolysis and is an inherent property of a single head. In contrast to conventional power stroke models, dissociation of the individual heads is not obligatory in the chemomechanical cycle, and the steps during which motion and force generation occurs are best described as one-dimensional thermally activated transitions that take place while both heads are attached to the microtubule. We show that our model is consistent with experiments on kinesin in which the velocity is measured as a function of external force and with the observed stiochiometry of one ATP/8-nm step at low load. Further, the model provides a way of understanding recent experiments on the ATP dependence of the variance (randomness) of the distance moved in a given time.     

Kinesin: walking, crawling or sliding along?

Kinesins are microtubule-based motor proteins that are involved in cargo transport and mitosis. They are called "motors" because they convert chemical energy to mechanical energy (i.e. force and motion). They use the energy of ATP hydrolysis for their enzymatic processes by walking on microtubules. However, the mechanism underlying their motion has been unclear. Recently, conventional kinesin, which was the first-identified member of the family, has been shown to walk by swapping its two heads in a "hand-over-hand" mechanism. There is also experimental evidence supporting an asymmetric walking of kinesin in which two identical heads of the motor take alternate slow and fast steps. Other cargo-carrier and mitotic kinesins remain uninvestigated and are of great interest to biophysicists.     

The force-velocity relationship for the actin-based motility of Listeria monocytogenes

The intracellular movement of the bacterial pathogen Listeria monocytogenes has helped identify key molecular constituents of actin-based motility (recent reviews ). However, biophysical as well as biochemical data are required to understand how these molecules generate the forces that extrude eukaryotic membranes. For molecular motors and for muscle, force-velocity curves have provided key biophysical data to distinguish between mechanistic theories. Here we manipulate and measure the viscoelastic properties of tissue extracts to provide the first force-velocity curve for Listeria monocytogenes. We find that the force-velocity relationship is highly curved, almost biphasic, suggesting a high cooperativity between biochemical catalysis and force generation. Using high-resolution motion tracking in low-noise extracts, we find long trajectories composed exclusively of molecular-sized steps. Robust statistics from these trajectories show a correlation between the duration of steps and macroscopic Listeria speed, but not between average step size and speed. Collectively, our data indicate how the molecular properties of the Listeria polymerization engine regulate speed, and that regulation occurs during molecular-scale pauses.     

Pressure-Induced Changes in the Structure and Function of the Kinesin-Microtubule Complex

Kinesin-1 is an ATP-driven molecular motor that “walks” along a microtubule by working two heads in a “hand-over-hand” fashion. The stepping motion is well-coordinated by intermolecular interactions between the kinesin head and microtubule, and is sensitively changed by applied forces. We demonstrate that hydrostatic pressure works as an inhibitory action on kinesin motility. We developed a high-pressure microscope that enables the application of hydrostatic pressures of up to 200 MPa (2000 bar). Under high-pressure conditions, taxol-stabilized microtubules were shortened from both ends at the same speed. The sliding velocity of kinesin motors was reversibly changed by pressure, and reached half-maximal value at ∼100 MPa. The pressure-velocity relationship was very close to the force-velocity relationship of single kinesin molecules, suggesting a similar inhibitory mechanism on kinesin motility. Further analysis showed that the pressure mainly affects the stepping motion, but not the ATP binding reaction. The application of pressure is thought to enhance the structural fluctuation and/or association of water molecules with the exposed regions of the kinesin head and microtubule. These pressure-induced effects could prevent kinesin motors from completing the stepping motion.     

Mobility of Molecular Motors Regulates Contractile Behaviors of Actin Networks

Cells generate mechanical forces primarily from interactions between F-actin, cross-linking proteins, myosin motors, and other actin-binding proteins in the cytoskeleton. To understand how molecular interactions between the cytoskeletal elements generate forces, a number of in vitro experiments have been performed but are limited in their ability to accurately reproduce the diversity of motor mobility. In myosin motility assays, myosin heads are fixed on a surface and glide F-actin. By contrast, in reconstituted gels, the motion of both myosin and F-actin is unrestricted. Because only these two extreme conditions have been used, the importance of mobility of motors for network behaviors has remained unclear. In this study, to illuminate the impacts of motor mobility on the contractile behaviors of the actin cytoskeleton, we employed an agent-based computational model based on Brownian dynamics. We find that if motors can bind to only one F-actin like myosin I, networks are most contractile at intermediate mobility. In this case, less motor mobility helps motors stably pull F-actins to generate tensile forces, whereas higher motor mobility allows F-actins to aggregate into larger clustering structures. The optimal intermediate motor mobility depends on the stall force and affinity of motors that are regulated by mechanochemical rates. In addition, we find that the role of motor mobility can vary drastically if motors can bind to a pair of F-actins. A network can exhibit large contraction with high motor mobility because motors bound to antiparallel pairs of F-actins can exert similar forces regardless of their mobility. Results from this study imply that the mobility of molecular motors may critically regulate contractile behaviors of actin networks in cells.     

Single molecule processes on the stepwise movement of ATP-driven molecular motors

Movement is a fundamental characteristic of all living things. This biogenic function that is attributed to the molecular motors such as kinesin, dynein and myosin. Molecular motors generate forces by using chemical energy derived from the hydrolysis reaction of ATP molecules. Despite a large number of studies on this topic, the chemomechanical energy transduction mechanism is still unsolved. In this study, we have investigated the chemomechanical coupling of the ATPase cycle to the mechanical events of the molecular motor kinesin using single molecule detection (SMD) techniques. The SMD techniques allowed to detection of the movement of single kinesin molecules along a microtubule and showed that kinesin steps mainly in the forward direction, but occasionally in the backward. The stepping direction is determined by a certain load-dependent process, on which the stochastic behavior is well characterized by Feynman's thermal ratchet model. The driving force of the stepwise movement is essentially Brownian motion, but it is biased in the forward direction by using the free energy released from the hydrolysis of ATP.     

Two kinesins transport cargo primarily via the action of one motor: implications for intracellular transport

The number of microtubule motors attached to vesicles, organelles, and other subcellular commodities is widely believed to influence their motile properties. There is also evidence that cells regulate intracellular transport by tuning the number and/or ratio of motor types on cargos. Yet, the number of motors responsible for cargo motion is not easily characterized, and the extent to which motor copy number affects intracellular transport remains controversial. Here, we examined the load-dependent properties of structurally defined motor assemblies composed of two kinesin-1 molecules. We found that a group of kinesins can produce forces and move with velocities beyond the abilities of single kinesin molecules. However, such capabilities are not typically harnessed by the system. Instead, two-kinesin assemblies adopt a range of microtubule-bound configurations while transporting cargos against an applied load. The binding arrangement of motors on their filament dictates how loads are distributed within the two-motor system, which in turn influences motor-microtubule affinities. Most configurations promote microtubule detachment and prevent both kinesins from contributing to force production. These results imply that cargos will tend to be carried by only a fraction of the total number of kinesins that are available for transport at any given time, and provide an alternative explanation for observations that intracellular transport depends weakly on kinesin number in vivo.     

Motor potential profile and a robust method for extracting it from time series of motor positions

Molecular motors are small, and, as a result, motor operation is dominated by high-viscous friction and large thermal fluctuations from the surrounding fluid environment. The small size has hindered, in many ways, the studies of physical mechanisms of molecular motors. For a macroscopic motor, it is possible to observe/record experimentally the internal operation details of the motor. This is not yet possible for molecular motors. The chemical reaction in a molecular motor has many occupancy states, each having a different effect on the motor motion. The overall effect of the chemical reaction on the motor motion can be characterized by the motor potential profile. The potential profile reveals how the motor force changes with position in a motor step, which may lead to insights into how the chemical reaction is coupled to force generation. In this article, we propose a mathematical formulation and a robust method for constructing motor potential profiles from time series of motor positions measured in single molecule experiments. Numerical examples based on simulated data are shown to demonstrate the method. Interestingly, it is the small size of molecular motors (negligible inertia) that makes it possible to recover the potential profile from time series of motor positions. For a macroscopic motor, the variation of driving force within a cycle is smoothed out by the large inertia.     

Submillisecond rotational dynamics of spin-labeled myosin heads in myofibrils.

The rotational motion of crossbridges, formed when myosin heads bind to actin, is an essential element of most molecular models of muscle contraction. To obtain direct information about this molecular motion, we have performed saturation transfer EPR experiments in which spin labels were selectively and rigidly attached to myosin heads in purified myosin and in glycerinated myofibrils. In synthetic myosin filaments, in the absence of actin, the spectra indicated rapid rotational motion of heads characterized by an effective correlation time of 10 microseconds. By contrast, little or no submillisecond rotational motion was observed when isolated myosin heads (subfragment-1) were attached to glass beads or to F-actin, indicating that the bond between the myosin head and actin is quite rigid on this time scale. A similar immobilization of heads was observed in spin-labeled myofibrils in rigor. Therefore, we conclude that virtually all of the myosin heads in a rigor myofibril are immobilized, apparently owing to attachment of heads to actin. Addition of ATP to myofibrils, either in the presence or absence of 0.1 mM Ca2+, produced spectra similar to those observed for myosin filaments in the absence of actin, indicating rapid submillisecond rotational motion. These results indicate that either (a) most of the myosin heads are detached at any instant in relaxed or activated myofibrils or (b) attached heads bearing the products of ATP hydrolysis rotate as rapidly as detached heads.     

Step length measurement--theory and simulation for tethered bead constant-force single molecule assay

Linear molecular motors translocate along polymeric tracks using discrete steps. The step length is usually measured using constant-force single molecule experiments in which the polymer is tethered to a force-clamped microsphere. During the enzymatic cycle the motor shortens the tether contour length. Experimental conditions influence the achievable step length resolution, and ideally experiments should be conducted with high clamp-force using slow motors linked to small beads via stiff short tethers. We focus on the limitations that the polymer-track flexibility, the thermal motion of the microsphere, and the motor kinetics pose for step-length measurement in a typical optical tweezers experiment. An expression for the signal/noise ratio in a constant-force, worm-like chain tethered particle, single-molecule experiment is developed. The signal/noise ratio is related to the Fourier transform of the pairwise distance distribution, commonly used to determine step length from a time-series. Monte Carlo simulations verify the proposed theory for experimental parameter values typically encountered with molecular motors (polymerases and helicases) translocating along single- or double-stranded nucleic acids. The predictions are consistent with recent experimental results for double-stranded DNA tethers. Our results map favorable experimental conditions for observing single motor steps on various substrates but indicate that principal resolution limits are set by thermal fluctuations.     

Differential labeling of myosin V heads with quantum dots allows direct visualization of hand-over-hand processivity

The double-headed myosin V molecular motor carries intracellular cargo processively along actin tracks in a hand-over-hand manner. To test this hypothesis at the molecular level, we observed single myosin V molecules that were differentially labeled with quantum dots having different emission spectra so that the position of each head could be identified with approximately 6-nm resolution in a total internal reflectance microscope. With this approach, the individual heads of a single myosin V molecule were observed taking 72-nm steps as they alternated positions on the actin filament during processive movement. In addition, the heads were separated by 36 nm during pauses in motion, suggesting attachment to actin along its helical repeat. The 36-nm interhead spacing, the 72-nm step size, and the observation that heads alternate between leading and trailing positions on actin are obvious predictions of the hand-over-hand model, thus confirming myosin V's mode of walking along an actin filament.     

Effects of Obstacles on the Dynamics of Kinesins,Including Velocity and Run Length,Predicted by a Model of Two Dimensional Motion

Kinesins are molecular motors which walk along microtubules by moving their heads to different binding sites. The motion of kinesin is realized by a conformational change in the structure of the kinesin molecule and by a diffusion of one of its two heads. In this study, a novel model is developed to account for the 2D diffusion of kinesin heads to several neighboring binding sites (near the surface of microtubules). To determine the direction of the next step of a kinesin molecule, this model considers the extension in the neck linkers of kinesin and the dynamic behavior of the coiled-coil structure of the kinesin neck. Also, the mechanical interference between kinesins and obstacles anchored on the microtubules is characterized. The model predicts that both the kinesin velocity and run length (i.e., the walking distance before detaching from the microtubule) are reduced by static obstacles. The run length is decreased more significantly by static obstacles than the velocity. Moreover, our model is able to predict the motion of kinesin when other (several) motors also move along the same microtubule. Furthermore, it suggests that the effect of mechanical interaction/interference between motors is much weaker than the effect of static obstacles. Our newly developed model can be used to address unanswered questions regarding degraded transport caused by the presence of excessive tau proteins on microtubules.     

Biased Brownian motion mechanism for processivity and directionality of single-headed myosin-VI

Conventional form to function as a vesicle transporter is not a 'single molecule' but a coordinated 'two molecules'. The coordinated two molecules make it complicated to reveal its mechanism. To overcome the difficulty, we adopted a single-headed myosin-VI as a model protein. Myosin-VI is an intracellular vesicle and organelle transporter that moves along actin filaments in a direction opposite to most other known myosin classes. The myosin-VI was expected to form a dimer to move processively along actin filaments with a hand-over-hand mechanism like other myosin organelle transporters. However, wild-type myosin-VI was demonstrated to be monomer and single-headed, casting doubt on its processivity. Using single molecule techniques, we show that green fluorescent protein (GFP)-fused single-headed myosin-VI does not move processively. However, when coupled to a 200 nm polystyrene bead (comparable to an intracellular vesicle in size) at a ratio of one head per bead, single-headed myosin-VI moves processively with large (40 nm) steps. Furthermore, we found that a single-headed myosin-VI-bead complex moved more processively in a high-viscous solution (40-fold higher than water) similar to cellular environment. Because diffusion of the bead is 60-fold slower than myosin-VI heads alone in water, we propose a model in which the bead acts as a diffusional anchor for the myosin-VI, enhancing the head's rebinding following detachment and supporting processive movement of the bead-monomer complex. This investigation will help us understand how molecular motors utilize Brownian motion in cells.     

Motor-like Properties of Nonmotor Enzymes

Molecular motors are thought to generate force and directional motion via nonequilibrium switching between energy surfaces. Because all enzymes can undergo such switching, we hypothesized that the ability to generate rotary motion and torque is not unique to highly adapted biological motor proteins but is instead a common feature of enzymes. We used molecular dynamics simulations to compute energy surfaces for hundreds of torsions in three enzymes—adenosine kinase, protein kinase A, and HIV-1 protease—and used these energy surfaces within a kinetic model that accounts for intersurface switching and intrasurface probability flows. When substrate is out of equilibrium with product, we find computed torsion rotation rates up ～140 cycles s^?1, with stall torques up to ～2 kcal mol^?1 cycle^?1, and power outputs up to ～50 kcal mol^?1 s^?1. We argue that these enzymes are instances of a general phenomenon of directional probability flows on asymmetric energy surfaces for systems out of equilibrium. Thus, we conjecture that cyclic probability fluxes, corresponding to rotations of torsions and higher-order collective variables, exist in any chiral molecule driven between states in a nonequilibrium manner; we call this the “Asymmetry-Directionality” conjecture. This is expected to apply as well to synthetic chiral molecules switched in a nonequilibrium manner between energy surfaces by light, redox chemistry, or catalysis.     

The influence of direct motor-motor interaction in models for cargo transport by a single team of motors

We analyze theoretically the effects of excluded-volume interactions between motors on the dynamics of a cargo driven by multiple motors. The model considered shares much in common with others recently proposed in the literature, with the addition of direct interaction between motors and motor back steps. The cargo is assumed to follow a continuum Langevin dynamics, while individual motors evolve following a Monte Carlo algorithm based on experimentally accessible probabilities for discrete forward and backward jumps, and attachment and detachment rates. The links between cargo and motors are considered as nonlinear springs. By means of numerical simulations we compute the relevant quantities characterizing the dynamical properties of the system, and we compare the results to those for noninteracting motors. We find that interactions lead to quite relevant changes in the force-velocity relation for cargo, with a considerable reduction of the stall force, and also cause a notable decrease of the run length. These effects are mainly due to traffic-like phenomena in the microtubule. The consideration of several parallel tracks for motors reduces such effects. However, we find that for realistic values of the number of motors and the number of tracks, the influence of interactions on the global parameters of transport of cargo are far from being negligible. Our studies also provide an analysis of the relevance of motor back steps on the modeling, and of the influence of different assumptions for the detachment rates. In particular, we discuss these two aspects in connection with the possibility of observing processive back motion of cargo at large load forces.     

Dynamics of myosin-driven skeletal muscle contraction: I. Steady-state force generation

Skeletal muscle contraction is a canonical example of motor-driven force generation. Despite the long history of research in this topic, a mechanistic explanation of the collective myosin force generation is lacking. We present a theoretical model of muscle contraction based on the conformational movements of individual myosins and experimentally measured chemical rate constants. Detailed mechanics of the myosin motor and the geometry of the sarcomere are taken into account. Two possible scenarios of force generation are examined. We find only one of the scenarios can give rise to a plausible contraction mechanism. We propose that the synchrony in muscle contraction is due to a force-dependent ADP release step. Computational results of a half sarcomere with 150 myosin heads can explain the experimentally measured force-velocity relationship and efficiency data. We predict that the number of working myosin motors increases as the load force is increased, thus showing synchrony among myosin motors during muscle contraction. We also find that titin molecules anchoring the thick filament are passive force generators in assisting muscle contraction.     

Low-frequency motions in protein molecules. Beta-sheet and beta-barrel.

Low-frequency internal motions in protein molecules play a key role in biological functions. Based on previous work with alpha-helical structure, the quasi-continuum model is extended to the beta-structure, another fundamental element in protein molecules. In terms of the equations derived here, one can easily calculate the low-frequency wave number of a beta-sheet in an accordionlike motion, and the low-frequency wave number of a beta-barrel in a breathing motion. The calculated results for immunoglobulin G and concanavalin A agree well with the observations. These findings further verify that the observed low-frequency motion (or the so-called dominant low-frequency mode) in a protein molecule is essentially governed by the collective fluctuations of its weak bonds, especially hydrogen bonds, and the internal displacement of the massive atoms therein, as described by the quasi-continuum model.     

Critical motor number for fractional steps of cytoskeletal filaments in gliding assays

In gliding assays, filaments are pulled by molecular motors that are immobilized on a solid surface. By varying the motor density on the surface, one can control the number [Formula: see text] of motors that pull simultaneously on a single filament. Here, such gliding assays are studied theoretically using Brownian (or Langevin) dynamics simulations and taking the local force balance between motors and filaments as well as the force-dependent velocity of the motors into account. We focus on the filament stepping dynamics and investigate how single motor properties such as stalk elasticity and step size determine the presence or absence of fractional steps of the filaments. We show that each gliding assay can be characterized by a critical motor number, [Formula: see text]. Because of thermal fluctuations, fractional filament steps are only detectable as long as [Formula: see text]. The corresponding fractional filament step size is [Formula: see text] where [Formula: see text] is the step size of a single motor. We first apply our computational approach to microtubules pulled by kinesin-1 motors. For elastic motor stalks that behave as linear springs with a zero rest length, the critical motor number is found to be [Formula: see text], and the corresponding distributions of the filament step sizes are in good agreement with the available experimental data. In general, the critical motor number [Formula: see text] depends on the elastic stalk properties and is reduced to [Formula: see text] for linear springs with a nonzero rest length. Furthermore, [Formula: see text] is shown to depend quadratically on the motor step size [Formula: see text]. Therefore, gliding assays consisting of actin filaments and myosin-V are predicted to exhibit fractional filament steps up to motor number [Formula: see text]. Finally, we show that fractional filament steps are also detectable for a fixed average motor number [Formula: see text] as determined by the surface density (or coverage) of the motors on the substrate surface.