Katsuhisa Tawada

Protein friction exerted by motor enzymes through a weak-binding interaction

Recently Vale et al. (1989, Cell 59, 915-925.) reported an observation of the one-dimensional Brownian movement of microtubules bound to flagellar dynein through a weak-binding interaction. In this study, we propose a theoretical model of this phenomenon. Our model consists of a rigid microtubule associated with a number of elastic dynein heads through a weak-binding interaction at equilibrium. The model implies that (1) the Brownian motion of the microtubule is not directly driven by the atomic collision of the solvent particles, but is driven by the thermally-generated structural fluctuations of the dynein heads which interact with the microtubule; (2) dynein heads through a weak-binding interaction exert a frictional drag force on the sliding motion of the microtubule and the drag force is proportional to the sliding velocity the same as in hydrodynamic viscous friction. This protein friction, with such viscous-like characteristics, may well play a role as a velocity-limiting factor in the normal ATP-induced sliding movement of motile proteins.

Stiffness of carbodiimide-crosslinked glycerinated muscle fibres in rigor and relaxing solutions at high salt concentrations

SummaryIn this article, we have applied a crosslinking technique with a water-soluble carbodiimide to single glycerol-extracted muscle fibres from the rabbit. We have measured the stiffness of the fibres in a relaxing solution at high salt concentration. These fibres were crosslinked to varying extents in the rigor state. The relaxing solution caused uncrosslinked crossbridge heads (S1) to detach. High salt concentrations were used because the fibres were not activated by the crosslinked crossbridges under these conditions, although they were at physiological ionic strength. We found (1) a linear correlation between the extent of S1 crosslinking to thin filaments and the stiffness and (2) that the stiffness in the relaxing solution of muscle fibres with all the S1 heads crosslinked to thin filaments was the same as the rigor stiffness of the fibres before crosslinking. We conclude that the sarcomere compliance is mostly a property of the crossbridges (with more than 65% of the crossbridge compliance in the S1 portions and less than 35% in the S2 portion) and little of other sarcomere structures. In an earlier paper [Kimura & Tawada,Biophys. J.603–10 (1984)], we demonstrated that the S2 portion of the crossbridge was stiff. It then follows that the crossbridge compliance, and thus the sarcomere compliance, is a property of the S1 heads. Assuming that the S1 portion of the crossbridges in rigor strained muscle fibres is bent, we calculated the Youngs modulus of the S1 portion and found that it is about 102 MN m−2. Because this order of magnitude is reasonable in terms of globular protein elasticity, bending is likely to be the nature of the S1 compliance in rigor muscle fibres.

Covalent cross-linking of single fibers from rabbit psoas increases oscillatory power.

Single fibers from chemically skinned rabbit psoas muscle were treated with 1-ethyl-3-[3-dimethyl-amino)proyl]-carbodiimide (EDC) at 20 degrees C after rigor was induced. A 22-min treatment resulted in 18% covalent cross-linking between myosin heads and the thin filament as determined by stiffness measurements. This treatment also results in covalent cross-linking among rod portions of myosin molecules in the backbone of the thick filament. The fibers thus prepared are stable and do not dissolve in solutions at ionic strengths as high as 1,000 mM. The preparation was subjected to sinusoidal analysis, and the resulting complex modulus data were analyzed in terms of three exponential processes, (A), (B), and (C). Oscillatory work (process B) was much greater in the cross-linked fibers than in untreated ones in activating solutions of physiological ionic strength (200 mM); this difference was attributed to the decline of process (A) with EDC treatment. Consequently, the Nyquist plot of the EDC-treated preparation exhibited an insect-type response. We conclude that, under these conditions, both cross-linked and non-cross-linked myosin heads contribute to the production of oscillatory power. The cross-linked preparations also exhibited oscillatory work in high ionic strength (500-1,000 mM) solutions, indicating that cross-linked myosin heads are capable of utilizing ATP to produce work. We conclude that process (A) does not relate to an elementary step in a cross-bridge cycle, but it may relate to dynamics outside the cross-bridge such as filament sliding or sarcomere rearrangement.

FLUCTUATION IN THE MICROTUBULE SLIDING MOVEMENT DRIVEN BY KINESIN IN VITRO

We studied the fluctuation in the translational sliding movement of microtubules driven by kinesin in a motility assay in vitro. By calculating the mean-square displacement deviation from the average as a function of time, we obtained motional diffusion coefficients for microtubules and analyzed the dependence of the coefficients on microtubule length. Our analyses suggest that 1) the motional diffusion coefficient consists of the sum of two terms, one that is proportional to the inverse of the microtubule length (as the longitudinal diffusion coefficient of a filament in Brownian movement is) and another that is independent of the length, and 2) the length-dependent term decreases with increasing kinesin concentration. This latter term almost vanishes within the length range we studied at high kinesin concentrations. From the length-dependence relationship, we evaluated the friction coefficient for sliding microtubules. This value is much larger than the solvent friction and thus consistent with protein friction. The length independence of the motional diffusion coefficient observed at sufficiently high kinesin concentrations indicates the presence of correlation in the sliding movement fluctuation. This places significant constraint on the possible mechanisms of the sliding movement generation by kinesin motors in vitro.

Stiffness of glycerinated rabbit psoas fibers in the rigor state. Filament-overlap relation.

The stiffness of glycerinated rabbit psoas fibers in the rigor state was measured at various sarcomere lengths in order to determine the distribution of the sarcomere compliance between the cross-bridge and other structures. The stiffness was determined by measuring the tension increment at one end of a fiber segment while stretching the other end of the fiber. The contribution of the end compliance to the rigor segments was checked both by laser diffractometry of the sarcomere length change and by measuring the length dependence of the Youngs modulus; the contribution was found to be small. The stiffness in the rigor state was constant at sarcomere lengths of 2.4 microns or less; at greater sarcomere lengths the stiffness, when corrected for the contribution of resting stiffness, scaled with the amount of overlap between the thick and thin filaments. These results suggest that the source of the sarcomere compliance of the rigor fiber at the full overlapping of filaments is mostly the cross-bridge compliance.

Kinesin: a molecular motor with a spring in its step

A key step in the processive motion of two–headed kinesin along a microtubule is the ‘docking’ of the neck linker that joins each kinesin head to the motors dimerized coiled–coil neck. This process is similar to the folding of a protein β–hairpin, which starts in a highly mobile unfolded state that has significant entropic elasticity and finishes in a more rigid folded state. We therefore suggest that neck–linker docking is mechanically equivalent to the thermally activated shortening of a spring that has been stretched by an applied load. This critical tension–dependent step utilizes Brownian motion and it immediately follows the binding of ATP, the hydrolysis of which provides the free energy that drives the kinesin cycle. A simple three–state model incorporating neck–linker docking can account quantitatively for both the kinesin force–velocity relation and the unusual tension–dependence of its Michaelis constant. However, we find that the observed randomness of the kinesin motor requires a more detailed four–state model. Monte Carlo simulations of single–molecule stepping with this model illustrate the possibility of sub–8 nm steps, the size of which is predicted to vary linearly with the applied load.

Molecular motors: thermodynamics and the random walk.

The biochemical cycle of a molecular motor provides the essential link between its thermodynamics and kinetics. The thermodynamics of the cycle determine the motors ability to perform mechanical work, whilst the kinetics of the cycle govern its stochastic behaviour. We concentrate here on tightly coupled, processive molecular motors, such as kinesin and myosin V, which hydrolyse one molecule of ATP per forward step. Thermodynamics require that, when such a motor pulls against a constant load f, the ratio of the forward and backward products of the rate constants for its cycle is exp [−(ΔG + u0f)/kT], where −ΔG is the free energy available from ATP hydrolysis and u0 is the motors step size. A hypothetical one–state motor can therefore act as a chemically driven ratchet executing a biased random walk. Treating this random walk as a diffusion problem, we calculate the forward velocity v and the diffusion coefficient D and we find that its randomness parameter r is determined solely by thermodynamics. However, real molecular motors pass through several states at each attachment site. They satisfy a modified diffusion equation that follows directly from the rate equations for the biochemical cycle and their effective diffusion coefficient is reduced to D−v2τ, where τ is the time–constant for the motor to reach the steady state. Hence, the randomness of multistate motors is reduced compared with the one–state case and can be used for determining τ. Our analysis therefore demonstrates the intimate relationship between the biochemical cycle, the force–velocity relation and the random motion of molecular motors.

Co-operative regulation mechanism of muscle contraction: Inter-tropomyosin co-operation model

Abstract A new model is presented on the basis of our experimental data and the “tropomyosin-blocking theory” of muscle relaxation to explain the regulation of certain characteristics of muscle contraction, namely that the relation of contraction to pCa is co-operative while calcium-binding is essentially non-cooperative. Our experiments show that end-to-end interactions between adjacent tropomyosin molecules in the groove of the actin helix are essential for the co-operative regulation. The blocking theory says that the tropomyosin molecule in relaxed muscle sterically blocks the myosin attachment site on actin, whereas in contracting muscle it moves to a position away from the attachment site. In this model a concerted movement of tropomyosin molecules, brought about by their end-to-end interactions, is considered to be the essential mechanism of co-operative regulation, and it is assumed that the positional changes of tropomyosin occur primarily when the four calcium binding sites of troponin on the tropomyosin are saturated with calcium. Theoretical analysis of the model, based upon the two-state allosteric model, leads to a Michaelis-Menten equation for the Ca-binding function together with a co-operative equation for the state function, proportional to the contraction or ATPase activity. These two functions fit well the experimental data. With cardiac muscle the slope of the contraction versus pCa curve is slightly less steep than that obtained with skeletal muscle. This difference can be explained by the difference in the number of Ca-binding sites of troponins.

Fluctuations in sliding motion generated by independent and random actions of protein motors

We consider theoretical fluctuations in the in vitro sliding movement of individual cytoskeletal filaments generated by an ensemble of protein motors whose actions are assumed to be statistically independent and random. We show that the mean square deviation of the sliding distances of a filament for a given period of time around their average is proportional to the inverse of the filament length. This result provides a basis for an experimental test of the general assumption on the independent and random actions of protein motors.

Covalent crosslinking of myosin subfragment-1 and heavy meromyosin to actin at various molar ratios: different correlations between ATPase activity and crosslinking extent.

SummaryThis paper describes a systematic study of crosslinking of skeletal muscle myosin subfragment-1 (S1) and heavy meromyosin (HMM) to F-actin in the rigor state with 1-ethyl-3-[3-(dimethylamino)propyl]carbodiimide (EDC). We followed the time courses of S1 or HMM head crosslinking at various actin:S1 or actin:HMM head molar ratios and the resulting superactivation of ATPase activity. The ATPase activity of the covalent complexes was measured at 0.5 M KCl, where the covalent complexes retain superactivated ATPase activity but the activity of uncrosslinked myosin heads is not activated by actin. S1 crosslinking was slowest at the actin:S1 molar ratio of 1∶1, but faster at larger molar ratios, where more than 80% of added S1 could be crosslinked to actin. In spite of the dependence of crosslinking rate on actin∶S1 ratio, there were two linear correlations between ATPase activity and the extent of S1 crosslinking to actin: one for S1 crosslinked to actin at actin∶S1 molar ratios more than 2.7∶1 and the other for S1 crosslinked at a molar ratio of 1∶1. Extrapolation of the former correlation line to 100% crosslinked S1 gave an ATPase activity of 39 s−1 for actin-S1 covalent complex at 25 °C, whereas that of the other correlation line gave 21 s−1. The latter smaller activity suggests that the interface between actin and S1 in their rigor complexes at a molar ratio of 1∶1 is different from that at molar ratios of more than 2.7∶1. The acto-HMM crosslinking rate depended on the ratio of actin to HMM head, like that of S1 crosslinking to actin. The ATPase activity of crosslinked actin-HMM was, unlike that of actin-S1 covalent complexes, bell-shaped as a function of the crosslinked heads, but chymotryptic conversion of HMM to S1 in the covalent complexes made the bell-shaped characteristics disappear and increased the activity close to that of actin-Sl covalent complexes. These results indicate that some physical constraint imposed on myosin heads suppresses the actin-activated ATPase activity of HMM crosslinked to actin.