Hopping and stalling of processive molecular motors

When a two-headed molecular motor such as kinesin is attached to its track by just a single head in the presence of an applied load, thermally activated head detachment followed by rapid re-attachment at another binding site can cause the motor to ‘hop’ backwards. Such hopping, on its own, would produce a linear force-velocity relation. However, for kinesin, we must incorporate hopping into the motor's alternating-head scheme, where we expect it to be most important for the state prior to neck-linker docking. We show that hopping can account for the backward steps, run length and stalling of conventional kinesin. In particular, although hopping does not hydrolyse ATP, we find that the hopping rate obeys the same Michaelis-Menten relation as the ATP hydrolysis rate. Hopping can also account for the reduced processivity observed in kinesins with mutations in their tubulin-binding loop. Indeed, it may provide a general mechanism for the breakdown of perfect processivity in two-headed molecular motors.     

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.     

Theoretical formalism for kinesin motility I. Bead movement powered by single one-headed kinesins

The directional movement on a microtubule of a plastic bead connected elastically to a single one-headed kinesin motor is studied theoretically. The kinesin motor can bind and unbind to periodic binding sites on the microtubule and undergo conformational changes while catalyzing the hydrolysis of ATP. An analytic formalism relating the dynamics of the bead and the ATP hydrolysis cycle of the motor is derived so that the calculation of the average velocity of the bead can be easily carried out. The formalism was applied to a simple three-state biochemical model to investigate how the velocity of the bead movement is affected by the external load, the diffusion coefficient of the bead, and the stiffness of the elastic element connecting the bead and the motor. The bead velocity was found to be critically dependent on the diffusion coefficient of the bead and the stiffness of the elastic element. A linear force-velocity relation was found for the model no matter whether the bead velocity was modulated by the diffusion coefficient of the bead or by the externally applied load. The formalism should be useful in modeling the mechanisms of chemimechanical coupling in kinesin motors based on in vitro motility data.     

Thermodynamics and Kinetics of Molecular Motors

Molecular motors are first and foremost molecules, governed by the laws of chemistry rather than of mechanics. The dynamical behavior of motors based on chemical principles can be described as a random walk on a network of states. A key insight is that any molecular motor in solution explores all possible motions and configurations at thermodynamic equilibrium. By using input energy and chemical design to prevent motion that is not wanted, what is left behind is the motion that is desired. This review is focused on two-headed motors such as kinesin and Myosin V that move on a polymeric track. By use of microscopic reversibility, it is shown that the ratio between the number of forward steps and the number of backward steps in any sufficiently long time period does not directly depend on the mechanical properties of the linker between the two heads. Instead, this ratio is governed by the relative chemical specificity of the heads in the front-versus-rear position for the fuel, adenosine triphosphate and its products, adenosine diphosphate and inorganic phosphate. These insights have been key factors in the design of biologically inspired synthetic molecular walkers constructed out of DNA or out of small organic molecules.     

Properties of the Kinesin-1 motor DdKif3 from Dictyostelium discoideum

The amoeba Dictyostelium discoideum possesses genes for 13 different kinesins. Here we characterize DdKif3, a member of the Kinesin-1 family. Kinesin-1 motors form homodimers that can move micrometer-long distances on microtubules using the energy derived from ATP hydrolysis. We expressed recombinant motors in Escherichia coli and tested them in different in vitro assays. Full-length and truncated Kif3 motors were active in gliding and ATPase assays. They showed a strong dependence on ionic strength. Like the full-length motor, the truncated DdKif3-592 motor (aa 1-592; comprising motor domain, neck, and partial stalk) reached its maximum speed of around 2.0micrcom s(-1) at a potassium acetate concentration of 200mM. The shortened DdKif3-342 motor (aa 1-342; comprising motor domain, partial neck) showed a high ATP turnover, comparable to that of the fungal Kinesin-1, Nkin. Results from the duty cycle calculations and gliding assays indicate that DdKif3 is a processive motor. A GFP-fusion protein revealed a mainly cytoplasmic localization of DdKif3. Immunofluorescence staining makes an association with the endoplasmic reticulum or mitochondria unlikely. Despite a similar phylogenetic distance to both metazoa and fungi, in terms of its biochemical properties DdKif3 revealed a closer similarity to fungal than animal kinesins.     

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.     

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.     

Single molecule measurements and molecular motors

Single molecule imaging and manipulation are powerful tools in describing the operations of molecular machines like molecular motors. The single molecule measurements allow a dynamic behaviour of individual biomolecules to be measured. In this paper, we describe how we have developed single molecule measurements to understand the mechanism of molecular motors. The step movement of molecular motors associated with a single cycle of ATP hydrolysis has been identified. The single molecule measurements that have sensitivity to monitor thermal fluctuation have revealed that thermal Brownian motion is involved in the step movement of molecular motors. Several mechanisms have been suggested in different motors to bias random thermal motion to directional movement.     

The kinesin-13 MCAK has an unconventional ATPase cycle adapted for microtubule depolymerization

Unlike other kinesins, members of the kinesin-13 subfamily do not move directionally along microtubules but, instead, depolymerize them. To understand how kinesins with structurally similar motor domains can have such dissimilar functions, we elucidated the ATP turnover cycle of the kinesin-13, MCAK. In contrast to translocating kinesins, ATP cleavage, rather than product release, is the rate-limiting step for ATP turnover by MCAK; unpolymerized tubulin and microtubules accelerate this step. Further, microtubule ends fully activate the ATPase by accelerating the exchange of ADP for ATP. This tuning of the cycle adapts MCAK for its depolymerization activity: lattice-stimulated ATP cleavage drives MCAK into a weakly bound nucleotide state that reaches microtubule ends by diffusion, and end-specific acceleration of nucleotide exchange drives MCAK into a strongly bound state that promotes depolymerization. This altered cycle accounts well for the different mechanical behaviour of this kinesin, which depolymerizes microtubules from their ends, compared to translocating kinesins that walk along microtubules. Thus, the kinesin motor domain is a nucleotide-dependent engine that can be differentially tuned for transport or depolymerization functions.     

A numerical algorithm for investigating the role of the motor-cargo linkage in molecular motor-driven transport

We extend the numerical algorithm developed by Wang et al. (2003. J. Theor. Biol. 221, 491-511) for studying biomolecular transport processes to include the linkage that connects molecular motors to their cargo. The new algorithm is used to investigate how the stiffness of the linkage affects the average velocity, effective diffusion coefficient, and randomness parameter. Three different models for molecular motors are considered: (1) a discrete stepping motor (2) a motor moving in a tilted-periodic potential and (3) a motor driven by a flashing potential. We demonstrate that a flexible motor-cargo linkage can make inferences on motor behavior based on measurements of the cargo's position difficult. We also show that even for the case of a tilted-periodic potential there exists an optimal stiffness of the linkage at which transport is maximized. The MATLAB code used in this paper is available at: http://www.unc.edu/approximatelytelston/code/.     

Intramolecular strain coordinates kinesin stepping behavior along microtubules

Kinesin advances 8 nm along a microtubule per ATP hydrolyzed, but the mechanism responsible for coordinating the enzymatic cycles of kinesin's two identical motor domains remains unresolved. Here, we have tested whether such coordination is mediated by intramolecular tension generated by the "neck linkers," mechanical elements that span between the motor domains. When tension is reduced by extending the neck linkers with artificial peptides, the coupling between ATP hydrolysis and forward stepping is impaired and motor's velocity decreases as a consequence. However, speed recovers to nearly normal levels when external tension is applied by an optical trap. Remarkably, external load also induces bidirectional stepping of an immotile kinesin that lacks its mechanical element (neck linker) and fuel (ATP). Our results indicate that the kinesin motor domain senses and responds to strain in a manner that facilitates its plus-end-directed stepping and communication between its two motor domains.     

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.     

Torque generation by the Fo motor of the sodium ATPase

Based on recent structural and functional findings, we have constructed a mathematical model for the sodium-driven Fo motor of the F1Fo-ATPase from the anaerobic bacterium Propionigenium modestum. The model reveals the mechanochemical principles underlying the Fo motor's operation, and explains all of the existing experimental data on wild-type and mutant Fo motors. In particular, the model predicts a nonmonotonic dependence of the ATP hydrolysis activity on the sodium concentration, a prediction confirmed by new experiments. To explain experimental observations, the positively charged stator residue (R227) must assume different positions in the ATP synthesis and hydrolysis directions. This work also illustrates how to extract a motor mechanism from dynamical experimental observations in the absence of complete structural information.     

ATP synthase: two motors, two fuels

FoF1 ATPase is the universal protein responsible for ATP synthesis. The enzyme comprises two reversible rotary motors: Fo is either an ion 'turbine' or an ion pump, and F1 is either a hydrolysis motor or an ATP synthesizer. Recent biophysical and biochemical studies have helped to elucidate the operating principles for both motors.     

Stepping, strain gating, and an unexpected force-velocity curve for multiple-motor-based transport

BACKGROUND: Intracellular transport via processive kinesin, dynein, and myosin molecular motors plays an important role in maintaining cell structure and function. In many cases, cargoes move distances longer than expected for single motors; there is significant evidence that this increased travel is in part due to multiple motors working together to move the cargoes. Although we understand single motors experimentally and theoretically, our understanding of multiple motors working together is less developed. RESULTS: We theoretically investigate how multiple kinesin motors function. Our model includes stochastic fluctuations of each motor as it proceeds through its enzymatic cycle. Motors dynamically influence each other and function in the presence of thermal noise and viscosity. We test the theory via comparison with the experimentally observed distribution of step sizes for two motors moving a cargo, and by predicting slightly subadditive stalling force for two motors relative to one. In the presence of load, our predictions for travel distances and mean velocities are different from the steady-state model: with high motor-motor coupling, we predict a form of strain-gating, where-because of the underlying motor's dynamics-the motors share load unevenly, leading to increased mean travel distance of the multiple-motor system under load. Surprisingly, we predict that in the presence of small load, two-motor cargoes move slightly slower than do single-motor cargoes. Unpublished data from G.T. Shubeita, B.C. Carter, and S.P.G. confirm this prediction in vivo. CONCLUSIONS: When only a few motors are active, fluctuations and unequal load sharing between motors can result in significant alterations of ensemble function.     

Effectiveness, active energy produced by molecular motors, and nonlinear capacitance of the cochlear outer hair cell

Cochlear outer hair cells are crucial for active hearing. These cells have a unique form of motility, named electromotility, whose main features are the cell's length changes, active force production, and nonlinear capacitance. The molecular motor, prestin, that drives outer hair cell electromotility has recently been identified. We reveal relationships between the active energy produced by the outer hair cell molecular motors, motor effectiveness, and the capacitive properties of the cell membrane. We quantitatively characterize these relationships by introducing three characteristics: effective capacitance, zero-strain capacitance, and zero-resultant capacitance. We show that zero-strain capacitance is smaller than zero-resultant capacitance, and that the effective capacitance is between the two. It was also found that the differences between the introduced capacitive characteristics can be expressed in terms of the active energy produced by the cell's molecular motors. The effectiveness of the cell and its molecular motors is introduced as the ratio of the motors'active energy to the energy of the externally applied electric field. It is shown that the effectiveness is proportional to the difference between zero-strain and zero-resultant capacitance. We analyze the cell and motor's effectiveness within a broad range of cellular parameters and estimate it to be within a range of 12%-30%.     

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.     

Three phase model of the processive motor protein kinesin

Kinesin is a stepping motor that successively produces forward and backward 8-nm steps along microtubules. Under physiological conditions, the steps powering kinesin's motility are biased in one direction and drive various biological motile processes. So far, the physical mechanism underlying the unidirectional bias of the kinesin is not fully understood. Recently, Martin Bier have provided a stepper model [Martin Bier, 2003, Processive motor protein as an overdamped Brownian stepper, Phys. Rev. Lett. 91, 148104], in which the stepping cycle of kinesin includes two distinguished phases: (i) a power stroke phase and (ii) a ratcheted diffusion phase which is characterized as a "random diffusional search". At saturating ATP level, this model can fit the experimental results accurately. In this paper, we'll provide a modified Brownian stepper model, in which the dependence of ATP concentration is considered. In our model, the stepping cycle of kinesin is distinguished into three phases: an ATP-binding phase, a power stroke phase and a ratcheted diffusion phase. This modified model can reconstruct most of the experimental results accurately.