Along the road not taken: how many myosin heads act on a single actin filament at any instant in working muscle?

We reconsider the use of stiffness measurements to estimate N, the number of myosin heads acting (working at any instant to produce tension) on a single actin filament in vertebrate striated muscle, and give reasons for our rejection of numbers produced from such measurements. We go on to present a different approach to the problem, citing and extending a model bearing on the value of N which is derived from other physiological and biochemical data and which offers insight into the fundamental actin-myosin contractile event as an impulsive force. New experimental data accumulating over the past decade support this model, in which the myosin heads act sequentially along the actin filament (this is an example of Conformational Spread). In this model only a single myosin head acts on a single actin filament to produce an impulse at any given instant in normally-contracting muscle, either in the isometric or the isotonic mode, so N = 1. However, extra impulses occur within the same time frame after quick release of length or tension. The predictions of this sequential model are in striking agreement with a large body of recent detailed biophysical and biochemical evidence. We suggest that this warrants further in-depth experimental work, specifically to explore and test the sequential model and its implications.

Muscle contraction: a new interpretation of the transient behaviour of muscle.

Piazzesi et al. [G. Piazzesi, L. Lucii, V. Lombardi, J. Physiol. 545 (2002) 145-151] made a study on the muscle transients due to step changes in force using improved time resolution and recorded filament movement and shortening velocities in the four phases. They point to Phase 2 and to Phase 4 (working muscle) and claim that their results do not contradict the swinging-cross-bridge (SCB) model which has a much-quoted constant power stroke of about 150 A (their value of 70 A was smaller). Siding with the SCB model, they nevertheless record that the power stroke decreases with load. We are pleased with this experimental result as it conforms to our theory, published in 1996, of an impulsive model with a much smaller step-size distance z (approximately 20 A). Using their data we obtain precise interval times and estimates of filament movement in Phase 2 and in working muscle. Our first result is that the time frames (interval times) for Phase 2 are the same as in working muscle. Moreover, we demonstrate that the authors' data verify the correctness of our calculated z values. There are eight active ATP events in Phase 2 in time frame t compared to one in working muscle in the same time frame t. This gives, for the first time, precise numbers for contractile events. We show that the SCB model is incorrect and our analysis supports the impulsive model with a much smaller filament (zero-load) motion, approximately 20 A per ATP split.

Muscle contraction: energy rate equations in relation to efficiency and step-size distance.

We derive the energy rate equation for muscle contraction. Our equation has only two parameters m, the maintenance heat rate and 1/S, the shortening heat coefficient. The impulsive model (previously described in earlier papers) provides a physical basis for parameter 1/S as well as for constants a and b in Hill's force-velocity equation. We develop new theory and relate the efficiency and the step-size distance to our energy rate equation. Correlation between the efficiency and the step-size distance is established. The various numbers are listed in Table 1: we use data from five different muscles in the literature. In summary, our analysis strongly supports the impulsive model as the correct model of contraction.

Muscle contraction: viscous-like frictional forces and the impulsive model.

Apart from a few experimental studies muscle viscosity has not received much recent analytical attention as a determinant of the contractile process. This is surprising, since any muscle cell is 80% water, and may undergo large shape changes during its working cycle. Intuitively, one might expect the viscosity of the solvent to be an important determinant of the physiological activity of muscle tissue. This was apparent to pioneers of the study of muscle contraction such as Hill and his contemporaries, whose putative theoretical formulations contained terms related to muscle viscosity. More recently, though, a hydrodynamic calculation by Huxley, using a solvent viscosity close to that of water, has been held to demonstrate that viscous forces are negligible in muscle contraction. We have re-examined the role of viscosity in contraction, postulating impulsive acto-myosin forces that are opposed by a viscous resistance between the filaments. The viscous force required, 10(4) times the hydrodynamic estimate, is close to recent experimental measurements, themselves 10(2)-10(3) times the hydrodynamic estimate. This also agrees with contemporary measurements of cytoplasmic viscosity in other biological cells using magnetic bead micro-rheometry. These are several orders of magnitude greater than the viscosity of water. In the course of the analysis, we have derived the force-velocity equation for an isolated half-sarcomere containing a single actin filament for the first time, and from first principles. We conclude that muscle viscosity is indeed important for the contractile process, and that it has been too readily discounted.

Muscle contraction: viscous-like frictional forces and the impulsive model.

Apart from a few experimental studies muscle viscosity has not received much recent analytical attention as a determinant of the contractile process. This is surprising, since any muscle cell is 80% water, and may undergo large shape changes during its working cycle. Intuitively one might expect the viscosity of the solvent to be an important determinant of the physiological activity of muscle tissue. This was apparent to pioneers of the study of muscle contraction such as Hill and his contemporaries, whose putative theoretical formulations contained terms related to muscle viscosity. More recently, though, a hydrodynamic calculation by Huxley, using a solvent viscosity close to that of water, has been held to demonstrate that viscous forces are negligible in muscle contraction. We have re-examined the role of viscosity in contraction, postulating impulsive acto-myosin forces that are opposed by a viscous resistance between the filaments. The viscous force required, 10(4) times the hydrodynamic estimate, is close to recent experimental measurements, themselves 10(2)-10(3) times the hydrodynamic estimate. This also agrees with contemporary measurements of cytoplasmic viscosity in other biological cells using magnetic bead micro-rheometry. These are several orders of magnitude greater than the viscosity of water. In the course of the analysis we have derived the force-velocity equation for an isolated half-sarcomere containing a single actin filament for the first time, and from first principles. We conclude that muscle viscosity is indeed important for the contractile process, and that it has been too readily discounted.

The muscle motor: 'simultaneous' levers or sequential impulses?

We use the step-size distance equation z = u/n developed in our two previous papers (z is the step-size distance, u is the actin filament relative velocity and n is the rate of ATP splitting on a given actin filament), and introduce one additional concept: that the impulsive contractile forces developed on an actin filament should proceed sequentially along a given actin-myosin train. This enables us to elucidate some unexplained and puzzling data in the literature, and to predict the surprisingly high values of ATPase in intact muscle that have recently been found experimentally. It seems that a sequential impulsive model of the actin myosin interaction may give a better explanation of many phenomena in muscle physiology than does the current model of the action of simultaneous levers.

The step-size distance in muscle contraction: properties and estimates.

The step-size distance in muscle contraction is obtained using the step-size distance equation z = u/n, where z is the step-size distance, u is the actin filament velocity and n is the ATPase rate of splitting. In a previous study a step-size distance of about 17 A at no load was determined for intact frog muscle. Some properties of the step-size equation are described. We have now made estimates of the step-size distance z for a variety of muscles using existing physiological and biochemical data in the literature. The estimates are listed in Tables 1 and 2. We find that the step-size distances are clustered in the range 13-17 A for nearly all muscles.

Muscle contraction: the step-size distance and the impulse-time per ATP.

We derive the step-size distance, and the impulse time per ATP split, from a consideration of Hill's energy rate equation coupled with the enthalpy available per ATP split. This definition of step-size distance is model-independent, and is calculated to have a maximum of 17 A at no load and to reduce to zero at isometric tension, since it will depend on the velocity of shortening. We revisit a derivation of Hill's force-velocity equation depending on impulsive forces working against frictional forces and show that this gives a physical meaning to Hill's constants a and b. This is particularly elegant for Hill's constant b, which is directly related to the impulse time; the value of this impulse time is 1/2 ms. The question that muscle contraction may involve overlapping interactions is considered. However, we find that the step-size distance is not dependent on the possibility of overlapping interactions.

How muscle may contract.

A new molecular model is proposed for muscle contraction, that involves the electrical charging of the long (C-terminal) alpha-helical part of the head of the myosin molecule (S1) while the head is attached to actin; as it charges the alpha-helical part moves in the radial electric field between the filaments. The alpha-helical part snaps back when the myosin molecule is discharged electrically, at the moment that ATP binds to the active enzymatic site. This snap-back model explains several puzzling phenomena in contractility, as well as providing a physical explanation for the origin of an impulsive force that drives muscle contraction.

Measurement of chain tilt angle in fully hydrated bilayers of gel phase lecithins.

The tilt angle theta tilt of the hydrocarbon chains has been determined for fully hydrated gel phase of a series of saturated lecithins. Oriented samples were prepared on glass substrates and hydrated with supersaturated water vapor. Evidence for full hydration was the same intensity pattern of the low angle lamellar peaks and the same lamellar repeat D as unoriented multilamellar vesicles. Tilting the sample permitted observation of all the wide angle arcs necessary to verify the theoretical diffraction pattern corresponding to tilting of the chains towards nearest neighbors. The length of the scattering unit corresponds to two hydrocarbon chains, requiring each bilayer to scatter coherently rather than each monolayer. For DPPC, theta tilt was determined to be 32.0 +/- 0.5 degrees at 19 degrees C, slightly larger than previous direct determinations and considerably smaller than the value required by recent gravimetric measurements. This new value allows more accurate determinations of a variety of structural parameters, such as area per lipid molecule, A = 47.2 +/- 0.5 A2, and number of water molecules of hydration, nw = 11.8 +/- 0.7. As the chain length n of the lipids was increased from 16 to 20 carbons, the parameters A and nw remained constant, suggesting that the headgroup packing is at its excluded volume limit for this range. However, theta tilt increased by 3 degrees and the chain area Ac decreased by 0.5 A2. This behavior is explained in terms of a competition between a bulk free energy term and a finite or end effect term.

Synchrotron x-ray diffraction studies of the cornea, with implications for stromal hydration.

The intermolecular and interfibrillar spacings of collagen in bovine corneal stroma have been measured as a function of tissue hydration. Data were recorded from low- and high-angle x-ray diffraction patterns obtained using a high intensity synchrotron source. The most frequently occurring interfibrillar spacing varied from 34 nm in dry corneas to 76 nm at H = 5 (the hydration, H, is defined as the ratio of the weight of water to the dry weight). The most frequently occurring intermolecular Bragg spacing increased from 1.15 nm (dry) to approximately 1.60 nm at normal hydration (H approximately 3.2) and continued to increase only slowly above normal hydration. Most of the increase in the intermolecular spacing occurred between H = O and H = 1. Over this hydration range the interfibrillar and intermolecular spacings moved in tandem, which suggests that the initial water goes equally within and between the fibrils. Above H = 1 water goes preferentially between the fibrils. The results suggest that, even at normal hydration, water does not fill the interfibrillar space uniformly, and a proportion is located in another space or compartment. In dried-then-rehydrated corneas, a larger proportion of the water goes into this other compartment. In both cases, it is possible to postulate a second set or population of fibrils that are more widely and irregularly separated and therefore do not contribute significantly to the diffraction pattern.

X-ray diffraction evidence for the presence of discrete water layers on the surface of membranes.

X-ray diffraction spacings in multilayered membranes obtained from frog sciatic nerves were found to increase in discrete steps of approx. 5 A during swelling. These observed jumps in the repeat period suggest that the lipid bilayers exist in distinct states of hydration, and perhaps the swelling occurs by step-wise addition of water layers between the polar head groups. Our analysis and statistical tests of this hypothesis are presented.

Helical diffraction. I. The paracrystalline helix and disorder analysis.

In a new approach to helical diffraction a helix generating function is defined, and thence an expression for the autocorrelation function (a.c.f.) for a helix is obtained. The Fourier transform of this a.c.f. gives a new expression for the diffracted intensity, which is shown to be equivalent formally to the classical expression of Cochran, Crick & Vand [Acta Cryst. (1952), 5, 581-586] and A. R. Stokes (unpublished). The new expression allows straightforward examination of the effects of helical disorders on the diffracted intensity. The thermal and paracrystalline effects of disorders with cylindrical symmetry are shown, and examples are given from the diffraction of a model of the actin helix. The general case, disorder with no symmetry, is derived and the effects of axial and radial disorder, separately and together, are computed, again for the model actin helix. Translational disorder is also included, and its effects are explained. The new results are compared with existing accounts of the effects of helical disorders on fibre diffraction.

X-ray diffraction analysis of dehydrated myelin.

Improved X-ray diffraction data from dry nerve myelin are presented. In addition to the spacing of approx. 150 A, 44 A and 34.6 A, which have been previously reported, we identify a 14 A series. The data suggests that the hydrocarbon chains in the single bilayer (approximately equal to 60 A) is ordered, whereas in the double bilayer (approximately equal to 150 A) and in the fluid phase (approximately equal to 44 A) it is disordered. It is shown that cholesterol (approximately equal to 34.6 A) exists as a bilayer, and the 14 A series is probably another cholesterol phase.

The structure of oriented sphingomyelin bilayers.

X-ray diffraction from oriented bilayers of sphingomyelin gave up to 14 orders of diffraction of a lamellar repeat of 68.5 A on the merididan and up to eight reflections, including a strong reflection at 4.2 A, on the equator. The diffraction spacings did not change when the sphingomyelin bilayers were exposed to different humidities. A direct analysis of the low resolution X-ray data, using deconvolution is presented. A comparison of the Patterson functions of sphingomyelin with those of phosphatidylcholine and phosphatidylethanolamine suggests that the molecular structure of sphingomyelin in oriented bilayers resembles the structure of both phosphatidylcholine and phosphatidylethanolamine. Molecular model calculations for sphingomyelin bilayers have also been performed. Electron density profiles of sphingomyelin bilayers at resolution of about 6 A and about 2.5 A are presented. Our results indicate that the phosphorylcholine head group of sphingomyelin is in the plane of the membrane and at right angles to the hydrocarbon chains, the hydrocarbon chains are nearly parallel to each other, and there is only a limited, if any, interdigitation of the hydrocarbon chains of the adjacent sphingomyelin molecules in the bilayer.

Structure determination of lipid bilayers.

A method of determining the phases of X-ray reflections from oriented model membrane systems at low resolution is described. The method involves deconvolution and requires that d less than or equal to 2v where v is the width of the head group region within the bilayer and d is the thickness of the bilayer. The method can be used with a single set of X-ray data and applies to lipid bilayers which have a relatively constant density in the hydrocarbon region. Phases for the first five or six orders of phosphatidylethanolamine and lecithin are derived. A refined analysis based upon deconvolution but using information inherent in the Fourier profile is also described.