Entropy rectifies the Brownian steps of kinesin

Effect of small molecular crowders on dynamics of kinesin molecular motors

Kinesin is a motor protein that can convert chemical energy of ATP hydrolysis into mechanical energy of moving processively on microtubules. Apart from the load and ATP concentration affecting the dynamics of the motor such as velocity, run length, dissociation rate, etc., the increase of solution viscosity by supplementing crowding agents of low molecular weight into the buffer can also affect the dynamics. Here, based on our proposed model for the chemomechanical coupling of the kinesin motor, a systematically theoretical study of the motor dynamics under the variation of the viscosity and load is presented. Both the load on the motor's stalk and that on one of the two heads are considered. The theoretical results provide a consistent explanation of the available contradictory experimental results, with some showing that increasing viscosity decreases sensitively the velocity whereas others showing that increasing viscosity has little effect on the velocity. The theoretical results reproduce quantitatively the puzzling experimental data showing that under different directions of the load on the stalk, increasing viscosity has very different effects on the change of run length or dissociation rate. The theoretical results predict that in both the pure and crowded buffers the dependence of the run length on the load acting one of the two heads has very different feature from that on the load acting on the stalk.

Fast backward steps and detachment of single kinesin molecules measured under a wide range of loads

To understand force generation under a wide range of loads, the stepping of single kinesin molecules was measured at loads from -20 to 42 pN by optical tweezers with high temporal resolution. The optical trap has been improved to halve positional noise and increase bandwidth by using 200-nm beads. The step size of the forward and backward steps was 8.2 nm even over a wide range of loads. Histograms of the dwell times of backward steps and detachment fit well to two independent exponential equations with fast (~0.4 ms) and slow (>3 ms) time constants, indicating the existence of a fast step in addition to the conventional slow step. The dwell times of the fast steps were almost independent of the load and ATP concentration, while those of the slow backward steps and detachment depended on those. We constructed the kinetic model to explain the fast and slow steps under a wide range of loads.

Effect of the Neck Linker on Processive Stepping of Kinesin Motor

Kinesin motor protein, which is composed of two catalytic domains connected together by a long coiled-coil stalk via two flexible neck linkers (NLs), can step processively on a microtubule towards the plus end by hydrolyzing adenosine triphosphate (ATP) molecules. To understand what the role is that the NL plays in the processive stepping, the dynamics of the kinesin motor are studied theoretically here by considering the mutation or deletion of an N-terminal cover strand that contributes to the docking of the NL in kinesin-1, the extension of the NL in kinesin-1, the mutation of the NL in kinesin-1, the swapping of the NL of kinesin-2 with that of kinesin-1, the joining of the stalk and neck of Ncd that moves towards the minus end of MT to the catalytic domain of kinesin-1, the replacement of catalytic domain of kinesin-1 with that of Ncd, and so on. The theoretical results give a consistent and quantitative explanation of various available experimental results about the effects of these mutations on motor dynamics and, moreover, provide predicted results. Additionally, the processive motility of kinesin-6 MKLP2 without NL docking is also explained. The available experimental data about the effect of NL mutations on the dynamics of the bi-directional kinesin-5 Cin8 are also explained. The studies are critically implicative to the mechanism of the stepping of the kinesin motor.

Reaction cycle of operating pump protein studied with single-molecule spectroscopy

Biological machinery relies on nonequilibrium dynamics to maintain stable directional fluxes through complex reaction cycles. For such reaction cycles, the presence of microscopically irreversible conformational transitions of the protein, and the accompanying entropy production, is of central interest. In this work, we use multidimensional single-molecule fluorescence lifetime correlation spectroscopy to measure the forward and reverse conformational transitions of bacteriorhodopsin during trans-membrane H + pumping. We quantify the flux, affinity, enthalpy and entropy production through portions of the reaction cycle as a function of temperature. We find that affinity of irreversible conformational transitions decreases with increasing temperature, resulting in diminishing flux and entropy production. We show that the temperature dependence of the transition affinity is well fit by the Gibbs-Helmholtz relation, allowing the DH trans to be experimentally extracted.

Transport and Diffusion Enhancement in Experimentally Realized Non-Gaussian Correlated Ratchets

Living cells are known to generate non-Gaussian active fluctuations significantly larger than thermal fluctuations owing to various active processes. Understanding the effect of these active fluctuations on various physicochemical processes, such as the transport of molecular motors, is a fundamental problem in nonequilibrium physics. Therefore, we experimentally and numerically studied an active Brownian ratchet comprising a colloidal particle in an optically generated asymmetric periodic potential driven by non-Gaussian noise having finite-amplitude active bursts, each arriving at random and decaying exponentially. We find that the particle velocity is maximum for relatively sparse bursts with finite correlation time and non-Gaussian distribution. These occasional kicks, which produce Brownian yet non-Gaussian diffusion, are more efficient for transport and diffusion enhancement of the particle than the incessant kicks of active Ornstein-Uhlenbeck noise.

Noise-Induced Acceleration of Single Molecule Kinesin-1

The movement of single kinesin molecules was observed while applying noisy external forces that mimic intracellular active fluctuations. We found kinesin accelerates under noise, especially when a large hindering load is added. The behavior quantitatively conformed to a theoretical model that describes the kinesin movement with simple two-state reactions. The universality of the kinetic theory suggests that intracellular enzymes share a similar noise-induced acceleration mechanism, i.e., active fluctuations in cells are not just noise but are utilized to promote various physiological processes.

Molecular Mechanism of Processive Stepping of Kinesin Motors

Kinesin-1 is a motor protein that can step processively on microtubule by hydrolyzing ATP molecules, playing an essential role in intracellular transports. To better understand the mechanochemical coupling of the motor stepping cycle, numerous structural, biochemical, single molecule, theoretical modeling and numerical simulation studies have been undertaken for the kinesin-1 motor. Recently, a novel ultraresolution optical trapping method was employed to study the mechanics of the kinesin-1 motor and new results were supplemented to its stepping dynamics. In this commentary, the new single molecule results are explained well theoretically with one of the models presented in the literature for the mechanochemical coupling of the kinesin-1 motor. With the model, various prior experimental results for dynamics of different families of N-terminal kinesin motors have also been explained quantitatively.

Diffusion Biased by a Soft Neck Linker Regulates Kinesin Stepping

Conventional kinesin is a high-performance motor that moves primarily toward the plus end of microtubules and occasionally toward the opposite direction. The physical mechanism of this directional stepping remains unclear. Here we develop a kinetic two-cycle model incorporating kinesin forward and backward stepping, in which the neck linker zippering and ATP catalysis process are conserved in backward steps. This model is quantitatively validated by a variety of experimental data, including load dependence of velocity, stepping ratio, and dwell time. The physical mechanism of kinesin stepping regulated by a biased diffusion process is identified by analyzing the load dependence and relevant thermodynamic properties of the model. Furthermore, the model suggests the kinesin directionality is optimized resulting from fulfilling a thermodynamic constraint. Our modeling provides a chemomechanical coupling mechanism that connects the flexibility of the neck linker zippering effect for direction rectification and the measured performance into a consistent frame.

Backstepping Mechanism of Kinesin-1

Kinesin-1 is an ATP-driven molecular motor that transports cellular cargo along microtubules. At low loads, kinesin-1 almost always steps forward, toward microtubule plus ends, but at higher loads, it can also step backward. Backsteps are usually 8 nm but can be larger. These larger backward events of 16 nm, 24 nm, or more are thought to be slips rather than steps because they are too fast to consist of multiple, tightly coupled 8-nm steps. Here, we propose that not only these larger backsteps, but all kinesin-1 backsteps, are slips. We show first that kinesin waits before forward steps for less time than before backsteps and detachments; second, we show that kinesin waits for the same amount of time before backsteps and detachments; and third, we show that by varying the microtubule type, we can change the ratio of backsteps to detachments without affecting forward stepping. Our findings indicate that backsteps and detachments originate from the same state and that this state arises later in the mechanochemical cycle than the state that gives rise to forward steps. To explain our data, we propose that, in each cycle of ATP turnover, forward kinesin steps can only occur before Pi release, whereas backslips and detachments can only occur after Pi release. In the scheme we propose, Pi release gates access to a weak binding K⋅ADP-K⋅ADP state that can slip back along the microtubule, re-engage, release ADP, and try again to take an ATP-driven forward step. We predict that this rescued detachment pathway is key to maintaining kinesin processivity under load.

Experimental and theoretical energetics of walking molecular motors under fluctuating environments

Molecular motors are nonequilibrium open systems that convert chemical energy to mechanical work. Their energetics are essential for various dynamic processes in cells, but largely remain unknown because fluctuations typically arising in small systems prevent investigation of the nonequilibrium behavior of the motors in terms of thermodynamics. Recently, Harada and Sasa proposed a novel equality to measure the dissipation of nonequilibrium small systems. By utilizing this equality, we have investigated the nonequilibrium energetics of the single-molecule walking motor kinesin-1. The dissipation from kinesin movement was measured through the motion of an attached probe particle and its response to external forces, indicating that large hidden dissipation exists. In this short review, aiming to readers who are not familiar with nonequilibrium physics, we briefly introduce the theoretical basis of the dissipation measurement as well as our recent experimental results and mathematical model analysis and discuss the physiological implications of the hidden dissipation in kinesin. In addition, further perspectives on the efficiency of motors are added by considering their actual working environment: living cells.

A Generalized Kinetic Model for Coupling between Stepping and ATP Hydrolysis of Kinesin Molecular Motors

A general kinetic model is presented for the chemomechanical coupling of dimeric kinesin molecular motors with and without extension of their neck linkers (NLs). A peculiar feature of the model is that the rate constants of ATPase activity of a kinesin head are independent of the strain on its NL, implying that the heads of the wild-type kinesin dimer and the mutant with extension of its NLs have the same force-independent rate constants of the ATPase activity. Based on the model, an analytical theory is presented on the force dependence of the dynamics of kinesin dimers with and without extension of their NLs at saturating ATP. With only a few adjustable parameters, diverse available single molecule data on the dynamics of various kinesin dimers, such as wild-type kinesin-1, kinesin-1 with mutated residues in the NLs, kinesin-1 with extension of the NLs and wild-type kinesin-2, under varying force and ATP concentration, can be reproduced very well. Additionally, we compare the power production among different kinesin dimers, showing that the mutation in the NLs reduces the power production and the extension of the NLs further reduces the power production.

Generic maps of optimality reveal two chemomechanical coupling regimes for motor proteins: from F1-ATPase and kinesin to myosin and cytoplasmic dynein

Many motor proteins achieve high efficiency for chemomechanical conversion, and single-molecule force-resisting experiments are a major tool to detect the chemomechanical coupling of efficient motors. Here, we introduce several quantitative relations that involve only parameters extracted from force-resisting experiments and offer new benchmarks beyond mere efficiency to judge the chemomechanical optimality or deficit of evolutionary remote motors on the same footing. The relations are verified by the experimental data from F₁-ATPase, kinesin-1, myosin V and cytoplasmic dynein, which are representative members of four motor protein families. A double-fitting procedure yields the chemomechanical parameters that can be cross-checked for consistency. Using the extracted parameters, two generic maps of chemomechanical optimality are constructed on which motors across families can be quantitatively compared. The maps reveal two chemomechanical coupling regimes, one conducive to high efficiency and high directionality, and the other advantageous to force generation. Surprisingly, an F₁ rotor and a kinesin-1 walker belong to the first regime despite their obvious evolutionary gap, while myosin V and cytoplasmic dynein follow the second regime. This analysis also predicts the symmetries of directional biases and heat productions for the motors, which impose constraints on their chemomechanical coupling and are open to future experimental tests. The verified relations, six in total, present a unified fitting framework to analyze force-resisting experiments. The generic maps of optimality, to which many more motors can be added in future, provide a rigorous method for a systematic cross-family comparison of motors to expose their evolutionary connections and mechanistic similarities.

Nonequilibrium Energetics of Molecular Motor Kinesin

Nonequilibrium energetics of single molecule translational motor kinesin was investigated by measuring heat dissipation from the violation of the fluctuation-response relation of a probe attached to the motor using optical tweezers. The sum of the dissipation and work did not amount to the input free energy change, indicating large hidden dissipation exists. Possible sources of the hidden dissipation were explored by analyzing the Langevin dynamics of the probe, which incorporates the two-state Markov stepper as a kinesin model. We conclude that internal dissipation is dominant.

A Unified Walking Model for Dimeric Motor Proteins

Dimeric motor proteins, kinesin-1, cytoplasmic dynein-1, and myosin-V, move stepwise along microtubules and actin filaments with a regular step size. The motors take backward as well as forward steps. The step ratio r and dwell time τ, which are the ratio of the number of backward steps to the number of forward steps and the time between consecutive steps, respectively, were observed to change with the load. To understand the movement of motor proteins, we constructed a unified and simple mathematical model to explain the load dependencies of r and of τ measured for the above three types of motors quantitatively. Our model consists of three states, and the forward and backward steps are represented by the cycles of transitions visiting different pairs of states among the three, implying that a backward step is not the reversal of a forward step. Each of r and τ is given by a simple expression containing two exponential functions. The experimental data for r and τ for dynein available in the literature are not sufficient for a quantitative analysis, which is in contrast to those for kinesin and myosin-V. We reanalyze the data to obtain r and τ of native dynein to make up the insufficient data to fit them to the model. Our model successfully describes the behavior of r and τ for all of the motors in a wide range of loads from large assisting loads to superstall loads.

Processivity of dimeric kinesin-1 molecular motors

Kinesin‐1 is a homodimeric motor protein that can move along microtubule filaments by hydrolyzing ATP with a high processivity. How the two motor domains are coordinated to achieve such high processivity is not clear. To address this issue, we computationally studied the run length of the dimer with our proposed model. The computational data quantitatively reproduced the puzzling experimental data, including the dramatically asymmetric character of the run length with respect to the direction of external load acting on the coiled‐coil stalk, the enhancement of the run length by addition of phosphate, and the contrary features of the run length for different types of kinesin‐1 with extensions of their neck linkers compared with those without extension of the neck linker. The computational data on other aspects of the movement dynamics such as velocity and durations of one‐head‐bound and two‐head‐bound states in a mechanochemical coupling cycle were also in quantitative agreement with the available experimental data. Moreover, predicted results are provided on dependence of the run length upon external load acting on one head of the dimer, which can be easily tested in the future using single‐molecule optical trapping assays.

Recent progress in structural biology: lessons from our research history

The recent 'resolution revolution' in structural analyses of cryo-electron microscopy (cryo-EM) has drastically changed the research strategy for structural biology. In addition to X-ray crystallography and nuclear magnetic resonance spectroscopy, cryo-EM has achieved the structural analysis of biological molecules at near-atomic resolution, resulting in the Nobel Prize in Chemistry 2017. The effect of this revolution has spread within the biology and medical science fields affecting everything from basic research to pharmaceutical development by visualizing atomic structure. As we have used cryo-EM as well as X-ray crystallography since 2000 to elucidate the molecular mechanisms of the fundamental phenomena in the cell, here we review our research history and summarize our findings. In the first half of the review, we describe the structural mechanisms of microtubule-based motility of molecular motor kinesin by using a joint cryo-EM and X-ray crystallography method. In the latter half, we summarize our structural studies on transcriptional regulation by X-ray crystallography of in vitro reconstitution of a multi-protein complex.

Inferring energy dissipation from violation of the fluctuation-dissipation theorem

The Harada-Sasa equality elegantly connects the energy dissipation rate of a moving object with its measurable violation of the Fluctuation-Dissipation Theorem (FDT). Although proven for Langevin processes, its validity remains unclear for discrete Markov systems whose forward and backward transition rates respond asymmetrically to external perturbation. A typical example is a motor protein called kinesin. Here we show generally that the FDT violation persists surprisingly in the high-frequency limit due to the asymmetry, resulting in a divergent FDT violation integral and thus a complete breakdown of the Harada-Sasa equality. A renormalized FDT violation integral still well predicts the dissipation rate when each discrete transition produces a small entropy in the environment. Our study also suggests a way to infer this perturbation asymmetry based on the measurable high-frequency-limit FDT violation.

Mechanical transduction via a single soft polymer

Molecular machines from biology and nanotechnology often depend on soft structures to perform mechanical functions, but the underlying mechanisms and advantages or disadvantages over rigid structures are not fully understood. We report here a rigorous study of mechanical transduction along a single soft polymer based on exact solutions to the realistic three-dimensional wormlike-chain model and augmented with analytical relations derived from simpler polymer models. The results reveal surprisingly that a soft polymer with vanishingly small persistence length below a single chemical bond still transduces biased displacement and mechanical work up to practically significant amounts. This "soft" approach possesses unique advantages over the conventional wisdom of rigidity-based transduction, and potentially leads to a unified mechanism for effective allosterylike transduction and relay of mechanical actions, information, control, and molecules from one position to another in molecular devices and motors. This study also identifies an entropy limit unique to the soft transduction, and thereby suggests a possibility of detecting higher efficiency for kinesin motor and mutants in future experiments.

Unbinding of Kinesin from Microtubule in the Strongly Bound States Enhances under Assisting Forces

The ability to predict the cellular dynamics of intracellular transport has enormous potential to impact human health. A key transporter is kinesin-1, an ATP-driven molecular motor that shuttles cellular cargos along microtubules (MTs). The dynamics of kinesins depends critically on their unbinding rate from MT, which varies depending on the force direction applied on the motor, i.e. the force-unbinding rate relation is asymmetric. However, it remains unclear how changing the force direction from resisting (applied against the motion direction) to assisting (applied in the motion direction) alters the kinesin's unbinding and stepping. Here, we propose a theoretical model for the influence of the force direction on the stepping dynamics of a single kinesin. The model shows that the asymmetry of the force-unbinding rate relation is independent of ATP concentration. It also reveals that the synthesis of ATP from backward stepping under assisting forces is less likely than under resisting forces. It then finds that the unbinding of kinesin in the strongly MT-bound kinetic states enhances under assisting forces.

Parsing the roles of neck-linker docking and tethered head diffusion in the stepping dynamics of kinesin

Kinesin walks processively on microtubules (MTs) in an asymmetric hand-over-hand manner consuming one ATP molecule per 16-nm step. The individual contributions due to docking of the approximately 13-residue neck linker to the leading head (deemed to be the power stroke) and diffusion of the trailing head (TH) that contributes in propelling the motor by 16 nm have not been quantified. We use molecular simulations by creating a coarse-grained model of the MT-kinesin complex, which reproduces the measured stall force as well as the force required to dislodge the motor head from the MT, to show that nearly three-quarters of the step occurs by bidirectional stochastic motion of the TH. However, docking of the neck linker to the leading head constrains the extent of diffusion and minimizes the probability that kinesin takes side steps, implying that both the events are necessary in the motility of kinesin and for the maintenance of processivity. Surprisingly, we find that during a single step, the TH stochastically hops multiple times between the geometrically accessible neighboring sites on the MT before forming a stable interaction with the target binding site with correct orientation between the motor head and the [Formula: see text] tubulin dimer.

Dynamics of dimeric kinesins: Limping, effect of longitudinal force, effects of neck linker extension and mutation, and comparison between kinesin-1 and kinesin-2

Conventional kinesin (kinesin-1) can walk on microtubule filaments in an asymmetric hand-over-hand manner, exhibiting a marked alternation in the mean dwell time in successive steps. Here, we study computationally the asymmetric stepping dynamics of the kinesin-1 homodimer, revealing its origin and providing quantitative explanations of the available experimental data. The alternation in the mean dwell time in successive steps arises from the alternation in the mechanochemical coupling ratio, which is in turn caused by the alternation in the slight variation of the stretched neck linker length. Both the vertical and backward longitudinal forces can enhance the asymmetric ratio. Additionally, other aspects of the stepping dynamics of the dimer such as the velocity versus longitudinal force, extended neck linker, etc., are also studied. In particular, the conflicting experimental data, with some showing that the velocity does not change with the forward longitudinal load while others showing that the velocity increases largely with the forward longitudinal load, are explained quantitatively and consistently. The intriguing experimental data showing that cysteine-light Drosophila and human kinesin-1 mutants have different load-dependent velocity from the wild-type cases as well as that kinesin-2 dimers have different load-dependent velocity from the kinesin-1 are also explained consistently and quantitatively.

The Kinesin-1 Chemomechanical Cycle: Stepping Toward a Consensus

Kinesin-1 serves as a model for understanding fundamentals of motor protein mechanochemistry and for interpreting functional diversity across the kinesin superfamily. Despite sustained work over the last three decades, disagreements remain regarding the events that trigger the two key transitions in the stepping cycle: detachment of the trailing head from the microtubule and binding of the tethered head to the next tubulin binding site. This review describes the conflicting views of these events and highlights recent work that sheds light on these long-standing controversies. It concludes by presenting a consensus kinesin-1 chemomechanical that incorporates recent work, resolves discrepancies, and highlights key questions for future experimental work. It is hoped that this model provides a framework for understanding how diverse kinesins are tuned for their specific cellular roles.

Number Density Distribution of Small Particles around a Large Particle: Structural Analysis of a Colloidal Suspension

Some colloidal suspensions contain two types of particles-small and large particles-to improve the lubricating ability, light absorptivity, and so forth. Structural and chemical analyses of such colloidal suspensions are often performed to understand their properties. In a structural analysis study, the observation of the number density distribution of small particles around a large particle (g_LS) is difficult because these particles are randomly moving within the colloidal suspension by Brownian motion. We obtain g_LS using the data from a line optical tweezer (LOT) that can measure the potential of mean force between two large colloidal particles (Φ_LL). We propose a theory that transforms Φ_LL into g_LS. The transform theory is explained in detail and tested. We demonstrate for the first time that LOT can be used for the structural analysis of a colloidal suspension. LOT combined with the transform theory will facilitate structural analyses of the colloidal suspensions, which is important for both understanding colloidal properties and developing colloidal products.

Discrete Step Sizes of Molecular Motors Lead to Bimodal Non-Gaussian Velocity Distributions under Force

Fluctuations in the physical properties of biological machines are inextricably linked to their functions. Distributions of run lengths and velocities of processive molecular motors, like kinesin-1, are accessible through single-molecule techniques, but rigorous theoretical models for these probabilities are lacking. Here, we derive exact analytic results for a kinetic model to predict the resistive force (F)-dependent velocity [P(v)] and run length [P(n)] distribution functions of generic finitely processive molecular motors. Our theory quantitatively explains the zero force kinesin-1 data for both P(n) and P(v) using the detachment rate as the only parameter. In addition, we predict the F dependence of these quantities. At nonzero F, P(v) is non-Gaussian and is bimodal with peaks at positive and negative values of v, which is due to the discrete step size of kinesin-1. Although the predictions are based on analyses of kinesin-1 data, our results are general and should hold for any processive motor, which walks on a track by taking discrete steps.

Retrograde NGF axonal transport--motor coordination in the unidirectional motility regime

We present a detailed motion analysis of retrograde nerve growth factor (NGF) endosomes in axons to show that mechanical tugs-of-war and intracellular motor regulation are complimentary features of the near-unidirectional endosome directionality. We used quantum dots to fluorescently label NGF and acquired trajectories of retrograde quantum-dot-NGF-endosomes with <20-nm accuracy at 32 Hz in microfluidic neuron cultures. Using a combination of transient motion analysis and Bayesian parsing, we partitioned the trajectories into sustained periods of retrograde (dynein-driven) motion, constrained pauses, and brief anterograde (kinesin-driven) reversals. The data shows many aspects of mechanical tugs-of-war and multiple-motor mechanics in NGF-endosome transport. However, we found that stochastic mechanical models based on in vitro parameters cannot simulate the experimental data, unless the microtubule-binding affinity of kinesins on the endosome is tuned down by 10 times. Specifically, the simulations suggest that the NGF-endosomes are driven on average by 5-6 active dyneins and 1-2 downregulated kinesins. This is also supported by the dynamics of endosomes detaching under load in axons, showcasing the cooperativity of multiple dyneins and the subdued activity of kinesins. We discuss the possible motor coordination mechanism consistent with motor regulation and tugs-of-war for future investigations.

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.

Unwinding forward and sliding back: an intermittent unwinding mode of the BLM helicase

There are lines of evidence that the Bloom syndrome helicase, BLM, catalyzes regression of stalled replication forks and disrupts displacement loops (D-loops) formed during homologous recombination (HR). Here we constructed a forked DNA with a 3′ single-stranded gap and a 5′ double-stranded handle to partly mimic a stalled DNA fork and used magnetic tweezers to study BLM-catalyzed unwinding of the forked DNA. We have directly observed that the BLM helicase may slide on the opposite strand for some distance after duplex unwinding at different forces. For DNA construct with a long hairpin, progressive unwinding of the hairpin is frequently interrupted by strand switching and backward sliding of the enzyme. Quantitative study of the uninterrupted unwinding length (time) has revealed a two-state-transition mechanism for strand-switching during the unwinding process. Mutational studies revealed that the RQC domain plays an important role in stabilizing the helicase/DNA interaction during both DNA unwinding and backward sliding of BLM. Especially, Lys1125 in the RQC domain, a highly conserved amino acid among RecQ helicases, may be involved in the backward sliding activity. We have also directly observed the in vitro pathway that BLM disrupts the mimic stalled replication fork. These results may shed new light on the mechanisms for BLM in DNA repair and homologous recombination.

High-Pressure Microscopy for Studying Molecular Motors

Movement is a fundamental characteristic of all living things. This biogenic function is carried out by various nanometer-sized molecular machines. Molecular motor is a typical molecular machinery in which the characteristic features of proteins are integrated; these include enzymatic activity, energy conversion, molecular recognition and self-assembly. These biologically important reactions occur with the association of water molecules that surround the motors. Applied pressures can alter the intermolecular interactions between the motors and water. In this chapter we describe the development of a high-pressure microscope and a new motility assay that enables the visualization of the motility of molecular motors under conditions of high-pressure. Our results demonstrate that applied pressure dynamically changes the motility of molecular motors such as kinesin, F1-ATPase and bacterial flagellar motors.

Computational modeling of kinesin stepping

Kinesin is a walking motor protein that shuttles cellular cargoes along microtubules (MTs). This protein is considered as an information processor capable of sensing cellular inputs and transforming them into mechanical steps. Here, we propose a computational model to describe the mechanochemical kinetics underlying forward and backward stepping behavior of kinesin motor as a digital circuit designed based on an adenosine triphosphate (ATP)-driven finite state machine. Kinetic analysis suggests that the backward stepping of kinesin is mainly driven by ATP hydrolysis, whereas ATP synthesis rises the duration of this stepping. It is shown that kinesin pausing due to waiting for ATP binding at limiting ATP concentration ([ATP]) and low backward loads could be longer than that caused by low rate of ATP synthesis under high backward loads. These findings indicate that the pausing duration of kinesin in MT-bound (M·K) kinetic state is affected by [ATP], which in turn affects its velocity at fixed loads. We show that the proposed computational model accurately simulates the forward and backward stepping behavior of kinesin motor under different [ATP] and loads.

A stochastic automaton model for simulating kinesin processivity

MOTIVATION

Cellular interactions of kinesin-1, an adenosine triphosphate (ATP)-driven motor protein capable of undergoing multiple steps on a microtubule (MT), affect its mechanical processivity, the number of steps taken per encounter with MT. Even though the processivity of kinesin has been widely studied, a detailed study of the factors that affect the stepping of the motor along MT is still lacking.

RESULTS

We model the cellular interactions of kinesin as a probabilistic timed automaton and use the model to simulate the mechanical processivity of the motor. Theoretical analysis suggests: (i) backward stepping tends to be powered by ATP hydrolysis, rather than ATP synthesis, (ii) backward stepping powered by ATP synthesis is more likely to happen with limiting ATP concentration ([ATP]) at high loads and (iii) with increasing load the frequency of backward stepping powered by ATP hydrolysis at high [ATP] is greater than that powered by ATP synthesis at limiting [ATP]. Together, the higher frequency of backward stepping powered by ATP hydrolysis than by ATP synthesis is found to be a reason for the more dramatic falling of kinesin processivity with rising load at high [ATP] compared with that at low [ATP]. Simulation results further show that the processivity of kinesin can be determined by the number of ATP hydrolysis and synthesis kinetic cycles taken by the motor before becoming inactive. It is also found that the duration of a backward stepping cycle at high loads is more likely to be less than that of a forward stepping cycle.

CONTACT

h.r.khataee@griffithuni.edu.au or a.liew@griffith.edu.au.

A mathematical model describing the mechanical kinetics of kinesin stepping

MOTIVATION

Kinesin is a smart motor protein that steps processively forward and backward along microtubules (MTs). The mechanical kinetics of kinesin affecting its stepping behavior is not fully understood. Here, we propose a mathematical model to study the mechanical kinetics of forward and backward stepping of kinesin motor based on the four-state discrete stochastic model of the motor.

RESULTS

Results show that the probabilities of forward and backward stepping can be modeled using the mean probabilities of forward and backward kinetic transitions, respectively. We show that the backward stepping of kinesin motor starts when the probability of adenosine diphosphate (ADP) binding to the motor is much higher than that of adenosine triphosphate (ATP) binding. Furthermore, our results indicate that the backward stepping is related to both ATP hydrolysis and synthesis with rate limiting factor being ATP synthesis. Low rate of ATP synthesis under high backward loads above 10 pN is also suggested as a reason for the detachment of kinesin motor from MT in the kinetic state MTċKinesinċADPċPi.

AVAILABILITY AND IMPLEMENTATION

The code for this work is written in Visual C# and is available by request from the authors.

The load dependence of the physical properties of a molecular motor

The physical properties of a molecular motor with load changing in a wide range will be discussed in this study, in particular the mean velocity, output power and energy efficiency. The main difficulty of this study is that both the states of the molecular motor and the energy barriers between them change with the loading force. Moreover, with the change of load, the number of motor states may also change, so different models should be used to calculate the corresponding physical quantities in different ranges of load. The results show that, in contrast to the usual intuition, the mean velocity and output power of the molecular motor do not change continuously with load.

Role of directional fidelity in multiple aspects of extreme performance of the F(1)-ATPase motor

Quantitative understanding of the best possible performance of nanomotors allowed by physical laws pertains to the study of nanomotors from biology as well as nanotechnology. The biological nanomotor F(1) ATPase is the best available model system as it is the only nanomotor known for extreme energy conversion near the limit of energy conservation. Using a unified theoretical framework centered on a concept called directional fidelity, we analyze recent experiments in which the F(1) motor's performance was measured for controlled chemical potentials and expose from the experiments quantitative evidence for the motor's multiple extreme performances in directional fidelity, speed, and catalytic capability close to physical limits. Specifically, the motor nearly exhausts the available energy from the fuel to retain the highest possible directional fidelity for an arbitrary load, encompassing the motor's extreme energy conversion and beyond. The theory-experiment comparison implies a tight chemomechanical coupling up to stalemate as futile steps occur, but unlikely involve fuel consumption. The F(1)-motor data also help clarify the relation between directional fidelity and experimentally measured stepping ratio.

Metrics for characterizing collective transport by multiple dimeric kinesins

Kinesin is a processive molecular motor which transports various cellular cargos by converting chemical energy into mechanical movements. Although the motion of a single molecule has been characterized in several studies, the dynamics of collective transport remains controversial. Since the chemical reactions fueling molecular motors are stochastic processes, the movements of coupled motors are not perfectly synchronized. The goal of this study is to develop metrics to analyze the level of synchronization of coupled (stochastic) motors. The correlation among movements of coupled motors, the slackness, the cooperativity, and the power loss of kinesins are explored using the developed metrics. These metrics can be extended to characterize collective work done by other molecular motors also.

Dissecting the kinematics of the kinesin step

Kinesin walks processively on microtubules in an asymmetric hand-over-hand manner with each step spanning 16 nm. We used molecular simulations to determine the fraction of a single step due to conformational changes in the neck linker, and that due to diffusion of the tethered head. Stepping is determined largely by two energy scales, one favoring neck-linker docking and the other, ε(h)(MT-TH), between the trailing head (TH) and the microtubule. Neck-linker docking and an optimal value of ε(h)(MT-TH) are needed to minimize the probability that the TH takes side steps. There are three major stages in the kinematics of a step. In the first, the neck linker docks, resulting in ∼(5-6) nm movements of the trailing head. The TH moves an additional (6-8) nm in stage II by anisotropic translational diffusion. In the third stage, spanning ∼(3-4) nm, the step is complete with the TH binding to the αβ-tubulin binding site.

Electrostatically biased binding of kinesin to microtubules

An electrostatic field rotates, slides, and guides the kinesin head to bind the microtubule at a site a short distance ahead, thus determining the direction of movement of the motor.

The minimum motor domain of kinesin-1 is a single head. Recent evidence suggests that such minimal motor domains generate force by a biased binding mechanism, in which they preferentially select binding sites on the microtubule that lie ahead in the progress direction of the motor. A specific molecular mechanism for biased binding has, however, so far been lacking. Here we use atomistic Brownian dynamics simulations combined with experimental mutagenesis to show that incoming kinesin heads undergo electrostatically guided diffusion-to-capture by microtubules, and that this produces directionally biased binding. Kinesin-1 heads are initially rotated by the electrostatic field so that their tubulin-binding sites face inwards, and then steered towards a plus-endwards binding site. In tethered kinesin dimers, this bias is amplified. A 3-residue sequence (RAK) in kinesin helix alpha-6 is predicted to be important for electrostatic guidance. Real-world mutagenesis of this sequence powerfully influences kinesin-driven microtubule sliding, with one mutant producing a 5-fold acceleration over wild type. We conclude that electrostatic interactions play an important role in the kinesin stepping mechanism, by biasing the diffusional association of kinesin with microtubules.

Animal and plant cells contain a molecular-scale “railway” network, in which the tracks, called microtubules, radiate out from the cell centre and locomotive proteins, called kinesins, haul their molecular cargoes along the microtubule tracks. This railway system transports many different cargoes to where they are needed, so it is crucial for the cell's organization and function. Breakdowns in this transport system can cause diseases like Alzheimer's, and drugs that temporarily halt transport make powerful anti-cancer agents. Precisely how kinesin motor proteins move along their microtubule tracks is an important question in biology. We know that some kinesins have twin “heads” that alternately bind to and step along microtubules in a coordinated walking action. But more usually, kinesins have only one head. How single-headed kinesins produce force and movement is poorly understood. In this study, we address this question and show that electrical attraction between single kinesin heads and microtubules is a critical factor deciding the direction of movement: each time the head approaches a microtubule, it slides forwards by the electrical attraction between the engine and the track.

Single macromolecule diffusion in confined environments

We consider the behaviour of single molecules on surfaces and, more generally, in confined environments. These are loosely split into three sections: single molecules in biology, the physics of single molecules on surfaces and controlled (directed) diffusion. With recent advances in single molecule detection techniques, the importance and mechanisms of single molecule processes such as localised enzyme production and intracellular diffusion across membranes has been highlighted, emphasising the extra information that cannot be obtained with techniques that present average behaviour. Progress has also been made in producing artificial systems that can control the rate and direction of diffusion, and because these are still in their infancy (especially in comparison to complex biological systems), we discuss the new physics revealed by these phenomena.

Mechanism of processive movement of monomeric and dimeric kinesin molecules

Kinesin molecules are motor proteins capable of moving along microtubule by hydrolyzing ATP. They generally have several forms of construct. This review focuses on two of the most studied forms: monomers such as KIF1A (kinesin-3 family) and dimers such as conventional kinesin (kinesin-1 family), both of which can move processively towards the microtubule plus end. There now exist numerous models that try to explain how the kinesin molecules convert the chemical energy of ATP hydrolysis into the mechanical energy to "power" their processive movement along microtubule. Here, we attempt to present a comprehensive review of these models. We further propose a new hybrid model for the dimeric kinesin by combining the existing models and provide a framework for future studies in this subject.

Switch between large hand-over-hand and small inchworm-like steps in myosin VI

Many biological motor molecules move within cells using stepsizes predictable from their structures. Myosin VI, however, has much larger and more broadly distributed stepsizes than those predicted from its short lever arms. We explain the discrepancy by monitoring Qdots and gold nanoparticles attached to the myosin-VI motor domains using high-sensitivity nanoimaging. The large stepsizes were attributed to an extended and relatively rigid lever arm; their variability to two stepsizes, one large (72 nm) and one small (44 nm). These results suggest that there exist two tilt angles during myosin-VI stepping, which correspond to the pre- and postpowerstroke states and regulate the leading head. The large steps are consistent with the previously reported hand-over-hand mechanism, while the small steps follow an inchworm-like mechanism and increase in frequency with ADP. Switching between these two mechanisms in a strain-sensitive, ADP-dependent manner allows myosin VI to fulfill its multiple cellular tasks including vesicle transport and membrane anchoring.

Mechanical characterization of one-headed myosin-V using optical tweezers

Class V myosin (myosin-V) is a cargo transporter that moves along an actin filament with large (approximately 36-nm) successive steps. It consists of two heads that each includes a motor domain and a long (23 nm) neck domain. One of the more popular models describing these steps, the hand-over-hand model, assumes the two-headed structure is imperative. However, we previously succeeded in observing successive large steps by one-headed myosin-V upon optimizing the angle of the acto-myosin interaction. In addition, it was reported that wild type myosin-VI and myosin-IX, both one-headed myosins, can also generate successive large steps. Here, we describe the mechanical properties (stepsize and stepping kinetics) of successive large steps by one-headed and two-headed myosin-Vs. This study shows that the stepsize and stepping kinetics of one-headed myosin-V are very similar to those of the two-headed one. However, there was a difference with regards to stability against load and the number of multisteps. One-headed myosin-V also showed unidirectional movement that like two-headed myosin-V required 3.5 k(B)T from ATP hydrolysis. This value is also similar to that of smooth muscle myosin-II, a non-processive motor, suggesting the myosin family uses a common mechanism for stepping regardless of the steps being processive or non-processive. In this present paper, we conclude that one-headed myosin-V can produce successive large steps without following the hand-over-hand mechanism.

One-dimensional Brownian motion of charged nanoparticles along microtubules: a model system for weak binding interactions

Various proteins are known to exhibit one-dimensional Brownian motion along charged rodlike polymers, such as microtubules (MTs), actin, and DNA. The electrostatic interaction between the proteins and the rodlike polymers appears to be crucial for one-dimensional Brownian motion, although the underlying mechanism has not been fully clarified. We examined the interactions of positively-charged nanoparticles composed of polyacrylamide gels with MTs. These hydrophilic nanoparticles bound to MTs and displayed one-dimensional Brownian motion in a charge-dependent manner, which indicates that nonspecific electrostatic interaction is sufficient for one-dimensional Brownian motion. The diffusion coefficient decreased exponentially with an increasing particle charge (with the exponent being 0.10 kBT per charge), whereas the duration of the interaction increased exponentially (exponent of 0.22 kBT per charge). These results can be explained semiquantitatively if one assumes that a particle repeats a cycle of binding to and movement along an MT until it finally dissociates from the MT. During the movement, a particle is still electrostatically constrained in the potential valley surrounding the MT. This entire process can be described by a three-state model analogous to the Michaelis-Menten scheme, in which the two parameters of the equilibrium constant between binding and movement, and the rate of dissociation from the MT, are derived as a function of the particle charge density. This study highlights the possibility that the weak binding interactions between proteins and rodlike polymers, e.g., MTs, are mediated by a similar, nonspecific charge-dependent mechanism.

Direct observation of the binding state of the kinesin head to the microtubule

The dimeric motor protein kinesin-1 converts chemical energy from ATP hydrolysis into mechanical work used to transport cargo along microtubules. Cargo attached to the kinesin stalk moves processively in 8-nm increments as its twin motor domains (heads) carry out an asymmetric, 'hand-over-hand' walk. The extent of individual head interactions with the microtubule during stepping, however, remains controversial. A major experimental limitation has been the lack of a means to monitor the attachment of an individual head to the microtubule during movement, necessitating indirect approaches. Here we report the development of a single-molecule assay that can directly report head binding in a walking kinesin molecule, and show that only a single head is bound to the microtubule between steps at low ATP concentrations. A bead was linked to one of the two kinesin heads by means of a short DNA tether and used to apply rapidly alternating hindering and assisting loads with an optical trap. The time-dependent difference between forwards and backwards displacements of the bead alternated between two discrete values during stepping, corresponding to those intervals when the linked head adopted a bound or an unbound state. The linked head could only rebind the microtubule once ATP had become bound to its partner head.

Properties of tug-of-war model for cargo transport by molecular motors

Molecular motors are essential components for the biophysical functions of the cell. Current quantitative understanding of how multiple motors move along a single track is not complete, even though models and theories for a single motor mechanochemistry abound. Recently, Müller et al. have developed a tug-of-war model to describe the bidirectional movement of the cargo [Proc. Natl. Acad. Sci. U.S.A. 105, 4609 (2008)]. They found that the tug-of-war model exhibits several qualitative different motility regimes, which depend on the precise value of single motor parameters, and they suggested that the sensitivity can be used by a cell to regulate its cargo traffic. In the present paper, we will carry out a detailed theoretical analysis of a special case of tug-of-war model: in which the numbers of the two different motor species which bound to the cargo tend to infinite. Through the analysis, all the stable, i.e., biophysically observable, steady states and their stability domains can be obtained. Depending on values of the several parameters, the tug-of-war model exhibits uni-, bi-, or tristability. The steady-state movement of the cargo, which is transported by two different molecular motor species, is determined by the initial numbers of the motors which bound to the track.

A simulation model of the conventional kinesin based on the Driven-by-Detachment mechanism

Kinesins are molecular motors that unidirectionally move along microtubules using the chemical energy of ATP. Although the core structure of kinesins is similar to that of myosins, the lever-arm hypothesis, which is widely accepted as a plausible mechanism to explain the behaviors of myosins, cannot be directly applied to kinesins. Masuda has proposed a mechanochemical process called the 'Driven-by-Detachment (DbD)' mechanism to explain the characteristic behaviors of myosins, including the backward movement of myosin VI and the loose coupling phenomenon of myosin II. The DbD mechanism assumes that the energy of ATP is mainly used to detach a myosin head from an actin filament by temporarily reducing the affinity of the myosin against the actin. After the affinity is recovered, the detached head has potential energy originating from the attractive force between the myosin and the actin. During the docking process, the potential energy is converted into elastic energy within the myosin molecule, and the intramolecular elastic energy is finally used to produce the power strokes. In the present paper, the DbD mechanism was used to explain the hand-over-hand motion of the conventional kinesin. The neck linker of the kinesin is known to determine the directionality of the motility but, in this paper, it was assumed that the neck linker was not directly engaged in the power strokes, which were driven by the attractive force between the kinesin head and the microtubule. Based on this assumption, simple mechanical simulations showed that the model of a kinesin dimer processively moved along a microtubule protofilament, if the affinity of the kinesin against the microtubule is appropriately controlled. Moreover, if an external force was applied to the center of the kinesin dimer, the dimer moved backward along a microtubule, as observed in experimental motility assays.

Mechanical design of translocating motor proteins

Translocating motors generate force and move along a biofilament track to achieve diverse functions including gene transcription, translation, intracellular cargo transport, protein degradation, and muscle contraction. Advances in single molecule manipulation experiments, structural biology, and computational analysis are making it possible to consider common mechanical design principles of these diverse families of motors. Here, we propose a mechanical parts list that include track, energy conversion machinery, and moving parts. Energy is supplied not just by burning of a fuel molecule, but there are other sources or sinks of free energy, by binding and release of a fuel or products, or similarly between the motor and the track. Dynamic conformational changes of the motor domain can be regarded as controlling the flow of free energy to and from the surrounding heat reservoir. Multiple motor domains are organized in distinct ways to achieve motility under imposed physical constraints. Transcending amino acid sequence and structure, physically and functionally similar mechanical parts may have evolved as nature's design strategy for these molecular engines.

Kinesin's backsteps under mechanical load

Kinesins move processively toward the plus end of microtubules by hydrolyzing ATP for each step. From an enzymatic perspective, the mechanism of mechanical motion coupled to the nucleotide chemistry is often well explained using a single-loop cyclic reaction. However, several difficulties arise in interpreting kinesin's backstepping within this framework, especially when external forces oppose the motion of kinesin. We review evidence, such as an ATP-independent stall force and a slower cycle time for backsteps, that has emerged to challenge the idea that kinesin backstepping is due to ATP synthesis, i.e., the reverse cycle of kinesin's forward-stepping chemomechanics. Supplementing the conventional single-loop chemomechanics with routes for ATP-hydrolyzing backward steps and nucleotide-free steps, especially under load, gives a better physical interpretation of the experimental data on backsteps.

The intrinsic load-resisting capacity of kinesin

Conventional kinesin is a homodimeric motor protein that is capable of walking unidirectionally along a cytoskeletal filament. While previous experiments indicated unyielding unidirectionality against an opposing load up to the so-called stall force, recent experiments also observed limited processive backwalking under superstall loads. This theoretical study seeks to elucidate the molecular mechanical basis for kinesin's steps over the full range of external loads that can possibly be applied to the dimer. We found that kinesin's load-resisting capacity is largely determined by a synergic ratchet-and-pawl mechanism inherent in the dimer. Load susceptibility of this inner molecular mechanical mechanism underlies kinesin's response to various levels of external loads. Computational implementation of the mechanism enabled us to rationalize major trends observed experimentally in kinesin's stalemate and consecutive back steps. The study also predicts several distinct features of kinesin's load-affected motility, which are seemingly counterintuitive but readily verifiable by future experiment.

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 approximately 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.