Tail currents in the myelinated axon of Xenopus laevis suggest a two-open-state Na channel.

Na tail currents in the myelinated axon of Xenopus laevis were measured at -70 mV after steps to -10 mV. The tail currents were biexponential, comprising a fast and a slow component. The time constant of the slow tail component, analyzed in the time window 0.35-0.50 ms, was independent of step duration, and had a value of 0.23 ms. The amplitude, extrapolated back to time 0, varied, however, with step duration. It reached a peak after 0.7 ms and inactivated relatively slowly (at 2.1 ms the absolute value was reduced by approximately 30%). The amplitude of the fast component, estimated by subtracting the amplitude of the slow component from the calculated total tail current amplitude, reached a peak (three times larger than that of the slow component) after 0.5 ms and inactivated relatively fast (at 2.1 ms it was reduced by approximately 65%). The results were explained by a novel Na channel model, comprising two open states bifurcating from a common closed state and with separate inactivating pathways. A voltage-regulated use of the two pathways explains a number of findings reported in the literature.     

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.     

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.     

Alternate fast and slow stepping of a heterodimeric kinesin molecule

A conventional kinesin molecule travels continuously along a microtubule in discrete 8-nm steps. This processive movement is generally explained by models in which the two identical heads of a kinesin move in a 'hand-over-hand' manner. Here, we show that a single heterodimeric kinesin molecule (in which one of the two heads is mutated in a nucleotide-binding site) exhibits fast and slow (with the dwell time at least 10 times longer than that of the fast step) 8-nm steps alternately, presumably corresponding to the displacement by the wild-type and mutant heads, respectively. Our results provide the first direct evidence for models in which the roles of the two heads alternate every 8-nm step.     

Mechanics of single kinesin molecules measured by optical trapping nanometry.

We have analyzed the mechanics of individual kinesin molecules by optical trapping nanometry. A kinesin molecule was adsorbed onto a latex bead, which was captured by an optical trap and brought into contact with an axoneme that was bound to a glass surface. The displacement of kinesin during force generation was determined by measuring the position of the beads with nanometer accuracy. As the displacement of kinesin was attenuated because of the compliance of the kinesin-to-bead and kinesin-to-microtubule linkages, the compliance was monitored during force generation and was used to correct the displacement of kinesin. Thus the velocity and the unitary steps could be obtained accurately over a wide force range. The force-velocity curves were linear from 0 to a maximum force at 10 microM and 1 mM ATP, and the maximum force was approximately 7 pN, which is larger by approximately 30% than values previously reported. Kinesin exhibited forward and occasionally backward stepwise displacements with a size of approximately 8 nm. The histograms of step dwell time show a monotonic decrease with time. Model calculations indicate that each kinesin head steps by 16-nm, whereas kinesin molecule steps by 8-nm.     

A kinetic mechanism for the fast movement of Chara myosin

Endoplasmic streaming of characean cells of Nitella or Chara is known to be in the range 30-100 microm/second. The Chara myosin extracted from the cells and fixed onto a glass surface was found to move muscle actin filaments at a velocity of 60 microm/second. This is ten times faster than that of skeletal muscle myosin (myosin II). In this study, the displacement caused by single Chara myosin molecules was measured using optical trapping nanometry. The step size of Chara myosin was approximately 19nm. This step size is longer than that of skeletal muscle myosin but shorter than that of myosin V. The dwell time of the steps was relatively long, and this most likely resulted from two rate-limiting steps, the dissociation of ADP and the binding of ATP. The rate of ADP release from Chara myosin after the completion of the force-generation step was similar to that of myosin V, but was considerably slower than that of skeletal muscle myosin. The 19nm step size and the dwell time obtained could not explain the fast movement. The fast movement could be explained by the load-dependent release of ADP. As the load imposed on the myosin decreased, the rate of ADP release increased. We propose that the interaction of Chara myosin with an actin filament resulted in a negative load being imposed on other myosin molecules interacting with the same actin filament. This resulted in an accelerated release of ADP and the fast sliding movement.     

Force-dependent stepping kinetics of myosin-V

Myosin-V is a processive two-headed actin-based motor protein involved in many intracellular transport processes. A key question for understanding myosin-V function and the communication between its two heads is its behavior under load. Since in vivo myosin-V colocalizes with other much stronger motors like kinesins, its behavior under superstall forces is especially relevant. We used optical tweezers with a long-range force feedback to study myosin-V motion under controlled external forward and backward loads over its full run length. We find the mean step size remains constant at approximately 36 nm over a wide range of forces from 5 pN forward to 1.5 pN backward load. We also find two force-dependent transitions in the chemomechanical cycle. The slower ADP-release is rate limiting at low loads and depends only weakly on force. The faster rate depends more strongly on force. The stronger force dependence suggests this rate represents the diffusive search of the leading head for its binding site. In contrast to kinesin motors, myosin-V's run length is essentially independent of force between 5 pN of forward to 1.5 pN of backward load. At superstall forces of 5 pN, we observe continuous backward stepping of myosin-V, indicating that a force-driven reversal of the power stroke is possible.     

The Kinesin-8 Kip3 Depolymerizes Microtubules with a Collective Force-Dependent Mechanism

Microtubules are highly dynamic filaments with dramatic structural rearrangements and length changes during the cell cycle. An accurate control of the microtubule length is essential for many cellular processes, in particular during cell division. Motor proteins from the kinesin-8 family depolymerize microtubules by interacting with their ends in a collective and length-dependent manner. However, it is still unclear how kinesin-8 depolymerizes microtubules. Here, we tracked the microtubule end-binding activity of yeast kinesin-8, Kip3, under varying loads and nucleotide conditions using high-precision optical tweezers. We found that single Kip3 motors spent up to 200 s at the microtubule end and were not stationary there but took several 8-nm forward and backward steps that were suppressed by loads. Interestingly, increased loads, similar to increased motor concentrations, also exponentially decreased the motors’ residence time at the microtubule end. On the microtubule lattice, loads also exponentially decreased the run length and time. However, for the same load, lattice run times were significantly longer compared to end residence times, suggesting the presence of a distinct force-dependent detachment mechanism at the microtubule end. The force dependence of the end residence time enabled us to estimate what force must act on a single motor to achieve the microtubule depolymerization speed of a motor ensemble. This force is higher than the stall force of a single Kip3 motor, supporting a collective force-dependent depolymerization mechanism that unifies the so-called “bump-off” and “switching” models. Understanding the mechanics of kinesin-8’s microtubule end activity will provide important insights into cell division with implications for cancer research.     

Stepping rotation of F(1)-ATPase with one, two, or three altered catalytic sites that bind ATP only slowly

F(1)-ATPase is an ATP hydrolysis-driven motor in which the gamma subunit rotates in the stator cylinder alpha(3)beta(3). To know the coordination of three catalytic beta subunits during catalysis, hybrid F(1)-ATPases, each containing one, two, or three "slow" mutant beta subunits that bind ATP very slowly, were prepared, and the rotations were observed with a single molecule level. Each hybrid made one, two, or three steps per 360 degrees revolution, respectively, at 5 microm ATP where the wild-type enzyme rotated continuously without step under the same observing conditions. The observed dwell times of the steps are explained by the slow binding rate of ATP. Except for the steps, properties of rotation, such as the torque forces exerted during rotary movement, were not significantly changed from those of the wild-type enzyme. Thus, it appears that the presence of the slow beta subunit(s) does not seriously affect other normal beta subunit(s) in the same F(1)-ATPase molecule and that the order of sequential catalytic events is faithfully maintained even when ATP binding to one or two of the catalytic sites is retarded.     

Mechanism of the binding of 2-(4'-hydroxyphenylazo)benzoic acid to bovine serum albumin

The mechanism of the binding of 2-(4'-hydroxyphenylazo)benzoic acid (HABA) to bovine serum albumin was studied by relaxation methods as well as the binding isotherm using gel chromatography. A single relaxation was observed over a wide range of HABA concentration except at the extremes of high concentration where another slow process was observed. The concentration dependence of the reciprocal relaxation time of the fast process decreased monotonically with increase in concentration of HABA at constant polymer concentration. The data were analyzed on the basis of Brown's domain structure model and were found to be consistent with a sequential binding mechanism. The azohydrazon tautomerism of HABA was identified with the intramolecular step of the complex. The activation parameters of the step, determined from the temperature dependence of the relaxation time of the fast process, showed that this step is rate limited by an enthalpy barrier in both forward and backward directions. Comparison of the activation parameters with those of other serum albumin-ligand systems suggests that there is an enthalpy-entropy compensation in the activation process of the intramolecular step with the compensation temperature at about 270 K; the enthalpy-entropy compensation is thought to be related to the hydrophobic nature of the ligand.     

Fast inactivation of Nav current in rat adrenal chromaffin cells involves two independent inactivation pathways

Voltage-dependent sodium (Nav) current in adrenal chromaffin cells (CCs) is rapidly inactivating and tetrodotoxin (TTX)–sensitive. The fractional availability of CC Nav current has been implicated in regulation of action potential (AP) frequency and the occurrence of slow-wave burst firing. Here, through recordings of Nav current in rat CCs, primarily in adrenal medullary slices, we describe unique inactivation properties of CC Nav inactivation that help define AP firing rates in CCs. The key feature of CC Nav current is that recovery from inactivation, even following brief (5 ms) inactivation steps, exhibits two exponential components of similar amplitude. Various paired pulse protocols show that entry into the fast and slower recovery processes result from largely independent competing inactivation pathways, each of which occurs with similar onset times at depolarizing potentials. Over voltages from −120 to −80 mV, faster recovery varies from ∼3 to 30 ms, while slower recovery varies from ∼50 to 400 ms. With strong depolarization (above −10 mV), the relative entry into slow or fast recovery pathways is similar and independent of voltage. Trains of short depolarizations favor recovery from fast recovery pathways and result in cumulative increases in the slow recovery fraction. Dual-pathway fast inactivation, by promoting use-dependent accumulation in slow recovery pathways, dynamically regulates Nav availability. Consistent with this finding, repetitive AP clamp waveforms at 1–10 Hz frequencies reduce Nav availability 80–90%, depending on holding potential. These results indicate that there are two distinct pathways of fast inactivation, one leading to conventional fast recovery and the other to slower recovery, which together are well-suited to mediate use-dependent changes in Nav availability.     

Linear impedance studies of voltage-dependent conductances in tissue cultured chick heart cells.

Plateau and pacemaker currents from tissue cultured clusters of embryonic chick heart cells were studied in the time domain, using voltage-clamp steps, and in the frequency domain, using a wide-band noise input superimposed on a steady holding voltage. In the presence of tetrodotoxin to block the sodium channel, a depolarizing voltage step into the plateau range elicited: (a) a rapid (approximately equal to 2 ms) activation of the slow inward current; (b) a subsequent slower (approximately equal to 25 ms) decline in the slow inward current; and (c) activation of a very slow (5 to 10 s) outward current. Impedance studies in this voltage range could clearly resolve two voltage-dependent processes, which appeared to correspond to points b and c above because of their voltage dependence, pharmacology, and time constants. A correlate of point a was also probably present but difficult to resolve owing to the fast time constant of activation for the slow inward channel. At voltages negative to -50 mV a new voltage-dependent process could be resolved, which, because of its voltage dependence and time constant, appeared to represent the pacemaker channel (also termed If or IK2). In the Appendix, linear models of voltage-dependent channels and ion accumulation/depletion are derived and these are compared with our data. Most of the above-mentioned processes could be attributed to voltage-dependent channels with kinetics similar to those observed in time domain, voltage-clamp studies. However, the frequency domain correlate of the decline of the slow inward current was incompatible with channel gating, rather, it appears accumulation/depletion of calcium may dominate the decline in this preparation.     

Extracting the Stepping Dynamics of Molecular Motors in Living Cells from Trajectories of Single Particles

Molecular motors are responsible of transporting a wide variety of cargos in the cytoplasm. Current efforts are oriented to characterize the biophysical properties of motors in cells with the aim of elucidating the mechanisms of these nanomachines in the complex cellular environment. In this study, we present an algorithm designed to extract motor step sizes and dwell times between steps from trajectories of motors or cargoes driven by motors in cells. The algorithm is based on finding patterns in the trajectory compatible with the behavior expected for a motor step, i.e., a region of confined motion followed by a jump in the position to another region of confined motion with similar characteristics to the previous one. We show that this algorithm allows the analysis of 2D trajectories even if they present complex motion patterns such as active transport interspersed with diffusion and does not require the assumption of a given step size or dwell period. The confidence on the step detection can be easily obtained and allows the evaluation of the confidence of the dwell and step size distributions. To illustrate the possible applications of this algorithm, we analyzed trajectories of myosin-V driven organelles in living cells.     

Spontaneous Detachment of the Leading Head Contributes to Myosin VI Backward Steps

Myosin VI is an ATP driven molecular motor that normally takes forward and processive steps on actin filaments, but also on occasion stochastic backward steps. While a number of models have attempted to explain the backwards steps, none offer an acceptable mechanism for their existence. We therefore performed single molecule imaging of myosin VI and calculated the stepping rates of forward and backward steps at the single molecule level. The forward stepping rate was proportional to the ATP concentration, whereas the backward stepping rate was independent. Using these data, we proposed that spontaneous detachment of the leading head is uncoupled from ATP binding and is responsible for the backward steps of myosin VI.     

Kinetics of the daunomycin--DNA interaction

The kinetics of the interaction of daunomycin with calf thymus DNA are described. Stopped-flow and temperature-jump relaxation methods, using absorption detection, were used to study the binding reaction. Three relaxation times were observed, all of which are concentration dependent, although the two slower relaxations approach constant values at high reactant concentrations. Relaxation times over a wide range of concentrations were gathered, and the data were fit by a minimal mechanism in which a rapid bimolecular association step is followed by two sequential isomerization steps. The six rate constants for this mechanism were extracted from our data by relaxation analysis. The values determined for the six rate constants may be combined to calculate an overall equilibrium constant that is in excellent agreement with that obtained by independent equilibrium measurements. Additional stopped-flow experiments, using first sodium dodecyl sulfate to dissociate bound drug and second pseudo-first-order conditions to study the fast bimolecular step, provide independent verification of three of the six rate constants. The temperature dependence of four of the six rate constants was measured, allowing estimates of the activation energy of some of the steps to be made. We speculate that the three steps in the proposed mechanism may correspond to a rapid "outside" binding of daunomycin to DNA, followed by intercalation of the drug, followed by either conformational adjustment of the drug or DNA binding site or redistribution of bound drug to preferred sites.     

The relevance of neck linker docking in the motility of kinesin

Conventional kinesin is a motor protein, which is able to walk along a microtubule processively. The exact mechanism of the stepping motion and force generation of kinesin is still far from clear. In this paper we argue that neck linker docking is a crucial element of this mechanism, without which the experimentally observed dwell times of the steps could not be explained under a wide range of loading forces. We also show that the experimental data impose very strict constraints on the lengths of both the neck linker and its docking section, which are compatible with the known structure of kinesin.     

A Refined Reaction-Diffusion Model of Tau-Microtubule Dynamics and Its Application in FDAP Analysis

Fluorescence decay after photoactivation (FDAP) and fluorescence recovery after photobleaching (FRAP) are well established approaches for studying the interaction of the microtubule (MT)-associated protein tau with MTs in neuronal cells. Previous interpretations of FDAP/FRAP data have revealed dwell times of tau on MTs in the range of several seconds. However, this is difficult to reconcile with a dwell time recently measured by single-molecule analysis in neuronal processes that was shorter by two orders of magnitude. Questioning the validity of previously used phenomenological interpretations of FDAP/FRAP data, we have generalized the standard two-state reaction-diffusion equations by 1), accounting for the parallel and discrete arrangement of MTs in cell processes (i.e., homogeneous versus heterogeneous distribution of tau-binding sites); and 2), explicitly considering both active (diffusion upon MTs) and passive (piggybacking upon MTs at rates of slow axonal transport) motion of bound tau. For some idealized cases, analytical solutions were derived. By comparing them with the full numerical solution and Monte Carlo simulations, the respective validity domains were mapped. Interpretation of our FDAP data (from processes of neuronally differentiated PC12 cells) in light of the heterogeneous formalism yielded independent estimates for the association (∼2 ms) and dwell (∼100 ms) times of tau to/on a single MT rather than in an MT array. The dwell time was shorter by orders of magnitude than that in a previous report where a homogeneous topology of MTs was assumed. We found that the diffusion of bound tau was negligible in vivo, in contrast to an earlier report that tau diffuses along the MT lattice in vitro. Methodologically, our results demonstrate that the heterogeneity of binding sites cannot be ignored when dealing with reaction-diffusion of cytoskeleton-associated proteins. Physiologically, the results reveal the behavior of tau in cellular processes, which is noticeably different from that in vitro.     

Binding of phytopolyphenol piceatannol disrupts β/γ subunit interactions and rate-limiting step of steady-state rotational catalysis in Escherichia coli F1-ATPase

In observations of single molecule behavior under V(max) conditions with minimal load, the F(1) sector of the ATP synthase (F-ATPase) rotates through continuous cycles of catalytic dwells (～0.2 ms) and 120° rotation steps (～0.6 ms). We previously established that the rate-limiting transition step occurs during the catalytic dwell at the initiation of the 120° rotation. Here, we use the phytopolyphenol, piceatannol, which binds to a pocket formed by contributions from α and β stator subunits and the carboxyl-terminal region of the rotor γ subunit. Piceatannol did not interfere with the movement through the 120° rotation step, but caused increased duration of the catalytic dwell. The duration time of the intrinsic inhibited state of F(1) also became significantly longer with piceatannol. All of the beads rotated at a lower rate in the presence of saturating piceatannol, indicating that the inhibitor stays bound throughout the rotational catalytic cycle. The Arrhenius plot of the temperature dependence of the reciprocal of the duration of the catalytic dwell (catalytic rate) indicated significantly increased activation energy of the rate-limiting step to trigger the 120° rotation. The activation energy was further increased by combination of piceatannol and substitution of γ subunit Met(23) with Lys, indicating that the inhibitor and the β/γ interface mutation affect the same transition step, even though they perturb physically separated rotor-stator interactions.     

Mechanochemical coupling of two substeps in a single myosin V motor

Myosin V is a double-headed processive molecular motor that moves along an actin filament by taking 36-nm steps. Using optical trapping nanometry with high spatiotemporal resolution, we discovered that there are two possible pathways for the 36-nm steps, one with 12- and 24-nm substeps, in this order, and the other without substeps. Based on the analyses of effects of ATP, ADP and 2,3-butanedione 2-monoxime (a reagent shown here to slow ADP release from actomyosin V) on the dwell time and the occurrence frequency of the main and the intermediate states, we propose that the 12-nm substep occurs after ATP binding to the bound trailing head and the 24-nm substep results from a mechanical step following the isomerization of an actomyosin-ADP state on the bound leading head. When the isomerization precedes the 12-nm substep, the 36-nm step occurs without substeps.     

Biosynthesis of pteridines. Stopped-flow kinetic analysis of GTP cyclohydrolase I.

GTP cyclohydrolase I catalyzes a mechanistically complex ring expansion affording dihydroneopterin triphosphate from GTP. The inherently slow enzyme reaction was studied under single turnover conditions monitored by multiwavelength ultraviolet spectroscopy. The spectroscopic data array was subjected to singular value decomposition and thereby shown to comprise six significant linearly independent optical processes. The data were fitted to a model of six consecutive unimolecular reaction steps where the first was considered to be reversible. The rate-limiting step was shown to occur rather late in the reaction sequence.