Extracting dwell time sequences from processive molecular motor data

Processive molecular motors, such as kinesin, myosin, or dynein, convert chemical energy into mechanical energy by hydrolyzing ATP. The mechanical energy is used for moving in discrete steps along the cytoskeleton and carrying a molecular load. Single-molecule recordings of motor position along a substrate polymer appear as a stochastic staircase. Recordings of other single molecules, such as F1-ATPase, RNA polymerase, or topoisomerase, have the same appearance. We present a maximum likelihood algorithm that extracts the dwell time sequence from noisy data, and estimates state transition probabilities and the distribution of the motor step size. The algorithm can handle models with uniform or alternating step sizes, and reversible or irreversible kinetics. A periodic Markov model describes the repetitive chemistry of the motor, and a Kalman filter allows one to include models with variable step size and to correct for baseline drift. The data are optimized recursively and globally over single or multiple data sets, making the results objective over the full scale of the data. Local binary algorithms, such as the t-test, do not represent the behavior of the whole data set. Our method is model-based, and allows rapid testing of different models by comparing the likelihood scores. From data obtained with current technology, steps as small as 8 nm can be resolved and analyzed with our method. The kinetic consequences of the extracted dwell sequence can be further analyzed in detail. We show results from analyzing simulated and experimental kinesin and myosin motor data. The algorithm is implemented in the free QuB software.     

Molecular motors: thermodynamics and the random walk

The biochemical cycle of a molecular motor provides the essential link between its thermodynamics and kinetics. The thermodynamics of the cycle determine the motor's ability to perform mechanical work, whilst the kinetics of the cycle govern its stochastic behaviour. We concentrate here on tightly coupled, processive molecular motors, such as kinesin and myosin V, which hydrolyse one molecule of ATP per forward step. Thermodynamics require that, when such a motor pulls against a constant load f, the ratio of the forward and backward products of the rate constants for its cycle is exp [-(DeltaG + u(0)f)/kT], where -DeltaG is the free energy available from ATP hydrolysis and u(0) is the motor's step size. A hypothetical one-state motor can therefore act as a chemically driven ratchet executing a biased random walk. Treating this random walk as a diffusion problem, we calculate the forward velocity v and the diffusion coefficient D and we find that its randomness parameter r is determined solely by thermodynamics. However, real molecular motors pass through several states at each attachment site. They satisfy a modified diffusion equation that follows directly from the rate equations for the biochemical cycle and their effective diffusion coefficient is reduced to D-v(2)tau, where tau is the time-constant for the motor to reach the steady state. Hence, the randomness of multistate motors is reduced compared with the one-state case and can be used for determining tau. Our analysis therefore demonstrates the intimate relationship between the biochemical cycle, the force-velocity relation and the random motion of molecular motors.     

Improved hidden Markov models for molecular motors, part 1: basic theory

Hidden Markov models (HMMs) provide an excellent analysis of recordings with very poor signal/noise ratio made from systems such as ion channels which switch among a few states. This method has also recently been used for modeling the kinetic rate constants of molecular motors, where the observable variable—the position—steadily accumulates as a result of the motor's reaction cycle. We present a new HMM implementation for obtaining the chemical-kinetic model of a molecular motor's reaction cycle called the variable-stepsize HMM in which the quantized position variable is represented by a large number of states of the Markov model. Unlike previous methods, the model allows for arbitrary distributions of step sizes, and allows these distributions to be estimated. The result is a robust algorithm that requires little or no user input for characterizing the stepping kinetics of molecular motors as recorded by optical techniques.     

Measuring kinesin's first step

A variety of models have recently emerged to explain how the molecular motor kinesin is able to maintain processive movement for over 100 steps. Although these models differ in significant features, they all predict that kinesin's catalytic domains intermittently separate from each other as the motor takes 8-nm steps along the microtubule. Furthermore, at some point in this process, one molecule of ATP is hydrolyzed per step. However, exactly when hydrolysis and product release occur in relation to this forward step have not been established. Furthermore, the rate at which this separation occurs as well as the speed of motor stepping onto and release from the microtubule have not been measured. In the absence of this information, it is difficult to critically evaluate competing models of kinesin function. We have addressed this issue by developing spectroscopic probes whose fluorescence is sensitive to motor-motor separation or microtubule binding. The kinetics of these fluorescence changes allow us to directly measure how fast kinesin steps onto and releases from the microtubule and provide insight into how processive movement is maintained by this motor.     

Use of Stopped-Flow Fluorescence and Labeled Nucleotides to Analyze the ATP Turnover Cycle of Kinesins

The kinesin superfamily of microtubule associated motor proteins share a characteristic motor domain which both hydrolyses ATP and binds microtubules. Kinesins display differences across the superfamily both in ATP turnover and in microtubule interaction. These differences tailor specific kinesins to various functions such as cargo transport, microtubule sliding, microtubule depolymerization and microtubule stabilization. To understand the mechanism of action of a kinesin it is important to understand how the chemical cycle of ATP turnover is coupled to the mechanical cycle of microtubule interaction. To dissect the ATP turnover cycle, one approach is to utilize fluorescently labeled nucleotides to visualize individual steps in the cycle. Determining the kinetics of each nucleotide transition in the ATP turnover cycle allows the rate-limiting step or steps for the complete cycle to be identified. For a kinesin, it is important to know the rate-limiting step, in the absence of microtubules, as this step is generally accelerated several thousand fold when the kinesin interacts with microtubules. The cycle in the absence of microtubules is then compared to that in the presence of microtubules to fully understand a kinesin’s ATP turnover cycle. The kinetics of individual nucleotide transitions are generally too fast to observe by manually mixing reactants, particularly in the presence of microtubules. A rapid mixing device, such as a stopped-flow fluorimeter, which allows kinetics to be observed on timescales of as little as a few milliseconds, can be used to monitor such transitions. Here, we describe protocols in which rapid mixing of reagents by stopped-flow is used in conjunction with fluorescently labeled nucleotides to dissect the ATP turnover cycle of a kinesin.     

Mechanical modulation of catalytic power on F1-ATPase

The conformational fluctuation of enzymes has a crucial role in reaction acceleration. However, the contribution to catalysis enhancement of individual substates with conformations far from the average conformation remains unclear. We studied the catalytic power of the rotary molecular motor F(1)-ATPase from thermophilic Bacillus PS3 as it was stalled in transient conformations far from a stable pausing angle. The rate constants of ATP binding and hydrolysis were determined as functions of the rotary angle. Both rates exponentially increase with rotation, revealing the molecular basis of positive cooperativity among three catalytic sites: elementary reaction steps are accelerated via the mechanical rotation driven by other reactions on neighboring catalytic sites. The rate enhancement induced by ATP binding upon rotation was greater than that brought about by hydrolysis, suggesting that the ATP binding step contributes more to torque generation than does the hydrolysis step. Additionally, 9% of the ATP-driven rotary step was supported by thermal diffusion, suggesting that acceleration of the ATP docking process occurs via thermally agitated conformational fluctuations.     

Myosin motor domain lever arm rotation is coupled to ATP hydrolysis

We have investigated coupling of lever arm rotation to the ATP binding and hydrolysis steps for the myosin motor domain. In several current hypotheses of the mechanism of force production by muscle, the primary mechanical feature is the rotation of a lever arm that is a subdomain of the myosin motor domain. In these models, the lever arm rotates while the myosin motor domain is free, and then reverses the rotation to produce force while it is bound to actin. These mechanical steps are coupled to steps in the ATP hydrolysis cycle. Our hypothesis is that ATP hydrolysis induces lever arm rotation to produce a more compact motor domain that has stored mechanical energy. Our approach is to use transient electric birefringence techniques to measure changes in hydrodynamic size that result from lever arm rotation when various ligands are bound to isolated skeletal muscle myosin motor domain in solution. Results for ATP and CTP, which do support force production by muscle fibers, are compared to those of ATPgammaS and GTP, which do not. Measurements are also made of conformational changes when the motor domain is bound to NDP's and PP(i) in the absence and presence of the phosphate analogue orthovanadate, to determine the roles the nucleoside moieties of the nucleotides have on lever arm rotation. The results indicate that for the substrates investigated, rotation does not occur upon substrate binding, but is coupled to the NTP hydrolysis step. The data are consistent with a model in which only substrates that produce a motor domain-NDP-P(i) complex as the steady-state intermediate make the motor domain more compact, and only those substrates support force production.     

A direct optimization approach to hidden Markov modeling for single channel kinetics

Hidden Markov modeling (HMM) provides an effective approach for modeling single channel kinetics. Standard HMM is based on Baum's reestimation. As applied to single channel currents, the algorithm has the inability to optimize the rate constants directly. We present here an alternative approach by considering the problem as a general optimization problem. The quasi-Newton method is used for searching the likelihood surface. The analytical derivatives of the likelihood function are derived, thereby maximizing the efficiency of the optimization. Because the rate constants are optimized directly, the approach has advantages such as the allowance for model constraints and the ability to simultaneously fit multiple data sets obtained at different experimental conditions. Numerical examples are presented to illustrate the performance of the algorithm. Comparisons with Baum's reestimation suggest that the approach has a superior convergence speed when the likelihood surface is poorly defined due to, for example, a low signal-to-noise ratio or the aggregation of multiple states having identical conductances.     

Model-based fitting of single-channel dwell-time distributions

Single-channel recordings provide unprecedented resolutions on kinetics of conformational changes of ion channels. Several approaches exist for analysis of the data, including the dwell-time histogram fittings and the full maximal-likelihood approaches that fit either the idealized dwell-time sequence or more ambitiously the noisy data directly using hidden Markov modeling. Although the full maximum likelihood approaches are statistically advantageous, they can be time-consuming especially for large datasets and/or complex models. We present here an alternative approach for model-based fitting of one-dimensional and two-dimensional dwell-time histograms. To improve performance, we derived analytical expressions for the derivatives of one-dimensional and two-dimensional dwell-time distribution functions and employed the gradient-based variable metric method for fast search of optimal rate constants in a model. The algorithm also has the ability to allow for a first-order correction for the effects of missed events, global fitting across different experimental conditions, and imposition of typical constraints on rate constants including microscopic reversibility. Numerical examples are presented to illustrate the performance of the algorithm, and comparisons with the full maximum likelihood fitting are discussed.     

A Brownian Dynamics Model of Kinesin in Three Dimensions Incorporating the Force-Extension Profile of the Coiled-Coil Cargo Tether

The kinesin family of motor proteins are involved in a variety of cellular processes that transport materials and generate force. With recent advances in experimental techniques, such as optical tweezers can probe individual molecules, there has been an increasing interest in understanding the mechanisms by which motor proteins convert chemical energy into mechanical work. Here we present a mathematical model for the chemistry and three dimensional mechanics of the kinesin motor protein which captures many of the force dependent features of the motor. For the elasticity of the tether that attaches cargo to the motor we develop a method for deriving the non-linear force-extension relationship from optical trap data. For the kinesin heads, cargo, and microscope stage we formulate a three dimensional Brownian Dynamics model that takes into account excluded volume interactions. To efficiently compute statistics from the model, an algorithm is proposed which uses a two step protocol that separates the simulation of the mechanical features of the model from the chemical kinetics of the model. Using this approach for a bead transported by the motor, the force dependent average velocity and randomness parameter are computed and compared with the experimental data.     

Improved hidden Markov models for molecular motors, part 2: extensions and application to experimental data

Unbiased interpretation of noisy single molecular motor recordings remains a challenging task. To address this issue, we have developed robust algorithms based on hidden Markov models (HMMs) of motor proteins. The basic algorithm, called variable-stepsize HMM (VS-HMM), was introduced in the previous article. It improves on currently available Markov-model based techniques by allowing for arbitrary distributions of step sizes, and shows excellent convergence properties for the characterization of staircase motor timecourses in the presence of large measurement noise. In this article, we extend the VS-HMM framework for better performance with experimental data. The extended algorithm, variable-stepsize integrating-detector HMM (VSI-HMM) better models the data-acquisition process, and accounts for random baseline drifts. Further, as an extension, maximum a posteriori estimation is provided. When used as a blind step detector, the VSI-HMM outperforms conventional step detectors. The fidelity of the VSI-HMM is tested with simulations and is applied to in vitro myosin V data where a small 10 nm population of steps is identified. It is also applied to an in vivo recording of melanosome motion, where strong evidence is found for repeated, bidirectional steps smaller than 8 nm in size, implying that multiple motors simultaneously carry the cargo.     

Hidden Markov modeling for single channel kinetics with filtering and correlated noise

Hidden Markov modeling (HMM) can be applied to extract single channel kinetics at signal-to-noise ratios that are too low for conventional analysis. There are two general HMM approaches: traditional Baum's reestimation and direct optimization. The optimization approach has the advantage that it optimizes the rate constants directly. This allows setting constraints on the rate constants, fitting multiple data sets across different experimental conditions, and handling nonstationary channels where the starting probability of the channel depends on the unknown kinetics. We present here an extension of this approach that addresses the additional issues of low-pass filtering and correlated noise. The filtering is modeled using a finite impulse response (FIR) filter applied to the underlying signal, and the noise correlation is accounted for using an autoregressive (AR) process. In addition to correlated background noise, the algorithm allows for excess open channel noise that can be white or correlated. To maximize the efficiency of the algorithm, we derive the analytical derivatives of the likelihood function with respect to all unknown model parameters. The search of the likelihood space is performed using a variable metric method. Extension of the algorithm to data containing multiple channels is described. Examples are presented that demonstrate the applicability and effectiveness of the algorithm. Practical issues such as the selection of appropriate noise AR orders are also discussed through examples.     

Mechanochemical coupling of the motion of molecular motors to ATP hydrolysis.

The typical biochemical paradigm for coupling between hydrolysis of ATP and the performance of chemical or mechanical work involves a well-defined sequence of events (a kinetic mechanism) with a fixed stoichiometry between the number of ATP molecules hydrolyzed and the turnover of the output reaction. Recent experiments show, however, that such a deterministic picture of coupling may not be adequate to explain observed behavior of molecular motor proteins in the presence of applied forces. Here we present a general model in which the binding of ATP and release of ADP serve to modulate the binding energy of a motor protein as it travels along a biopolymer backbone. The mechanism is loosely coupled--the average number of ATPs hydrolyzed to cause a single step from one binding site to the next depends strongly on the magnitude of an applied force and on the effective viscous drag force. The statistical mechanical perspective described here offers insight into how local anisotrophy along the "track" for a molecular motor, combined with an energy-releasing chemical reaction to provide a source of nonequilibrium fluctuations, can lead to macroscopic motion.     

Dynamics of myosin-V processivity

Myosin-V is an actin-associated processive molecular motor. Single molecule experiments revealed that myosin-V walks in a stepwise fashion with occasional backward steps. By combining the mechanical structure of the motor with the ATP hydrolysis kinetics, we construct a dynamical model that accounts for the stepwise processivity. The molecular properties of the protein chains connecting the myosin heads are important. A simple elastic model demonstrates that the stress transmitted from the leading head to the trailing head leads to net forward motion. The step-sizes are non-uniform. We also predict there are several substeps. The translational speed and step-size distributions are computed for several different conditions. The computed force-versus-velocity curve shows that under an external load, myosin-V slows down. However, the sizes of the steps remain the same.     

Cooperativity in the motor activities of the ATP-fueled molecular motors

Kinesin, myosin and F1-ATPase are multi-domain molecular motors with multiple catalytic subunits. The motor mechanochemics are achieved via the conversion of ATP hydrolysis energy into forces and motions. We find that the catalysis of these molecular motors do not follow the simple Michaelis-Menten mechanism. The motor activities, such as the hydrolysis or processive rates, of kinesin, myosin and F1-ATPase have a complex ATP-dependent cooperativity. To understand this complexity in kinetics and mechanochemics, we develop a conformation correlation theory of cooperativity for the ATP-fueled motor proteins. The quantitative analysis and simulations indicate that cooperativity is induced by the conformational coupling of binding states of different subunits and prevails in the motor activities.     

Dwell time symmetry in random walks and molecular motors

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

Alternating-Site Mechanism of Kinesin-1 Characterized by Single-Molecule FRET Using Fluorescent ATP Analogues

Kinesin-1 motor proteins move along microtubules in repetitive steps of 8 nm at the expense of ATP. To determine nucleotide dwell times during these processive runs, we used a Förster resonance energy transfer method at the single-molecule level that detects nucleotide binding to kinesin motor heads. We show that the fluorescent ATP analog used produces processive motility with kinetic parameters altered <2.5-fold compared with normal ATP. Using our confocal fluorescence kinesin motility assay, we obtained fluorescence intensity time traces that we then analyzed using autocorrelation techniques, yielding a time resolution of ∼1 ms for the intensity fluctuations due to fluorescent nucleotide binding and release. To compare these experimental autocorrelation curves with kinetic models, we used Monte-Carlo simulations. We find that the experimental data can only be described satisfactorily on the basis of models assuming an alternating-site mechanism, thus supporting the view that kinesin's two motor domains hydrolyze ATP and step in a sequential way.     

Walking motion of an overdamped active particle in a ratchet potential

An active particle can convert its internal energy into mechanical work. We study a generalized energy-depot model of an overdamped active particle in a ratchet potential. Using well-known biological parameters for kinesin-1 and modeling ATP influx as a pulsed energy supply, we apply our model to the molecular motor system. We find that our simple model can capture the essential properties of the kinesin motor such as forward stepping, stalling, backward stepping, dependence on ATP concentration, and stall force. Our model might be quite universal in the sense that it is able to describe dynamics of various types of motors as long as realistic parameters for each motor species are adopted.