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.

Calcium-dependence of Donnan potentials in glycerinated rabbit psoas muscle in rigor, at and beyond filament overlap; a role for titin in the contractile process.

In glycerinated rabbit psoas muscle, Donnan potential measurements demonstrated that the net electric charge on the actin-myosin matrix undergoes a sharp switch-like transition at pCa(50) = 6.8. The potentials are 2 mV less negative at the lower pCa(2+) (P < 0.001). If ATP is present, the muscle contracts and breaks the microelectrode. Therefore the rigor state was studied. There is no reason to suppose a priori that a similar voltage switch does not occur during contraction, however. Calcium dependence is still apparent in muscles stretched beyond overlap (sarcomere length>3.8 µm) and is also seen in the gap filaments between the A- and I-band ends; further stretching abolishes the dependence. These experiments strongly suggest that calcium dependence is controlled initially by the titin component, and that this control is lost when titin filaments break. We suppose that that effect is mediated by the titin kinase in the M-line region and may involve the extensible PEVK region of titin. There is great interest in the electric charge on proteins in muscle within the structural system. We suggest how changes in these charges may control the calcium activation process. We also suggest some simple experimental approaches that could clarify these effects.

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.

Structural changes in alpha-crystallin and whole eye lens during heating, observed by low-angle X-ray diffraction.

Whole eye lens and alpha-crystallin gels and solutions were investigated using X-ray scattering techniques at temperatures ranging from 20 degrees C to 70 degrees C. In whole lens isolated in phosphate-buffered saline, the spacing of the dominant X-ray reflection seen with low-angle scattering was constant from 20 degrees C to 45 degrees C but increased at 50 degrees C from 15.2 nm to 16.5 nm. At room temperature, the small-angle X-ray diffraction pattern of the intact lens was very similar to the pattern of alpha-crystallin gels at near-physiological concentration (approximately 300 mg/ml), so it is reasonable to assume that the alpha-crystallin pattern dominates the pattern of the intact lens. Our results therefore indicate that in whole lens alpha-crystallin is capable of maintaining its structural properties over a wide range of temperature. This property would be useful in providing protection for other lens proteins super-aggregating. In the alpha-crystallin gels, a moderate increase in both the spacing and intensity of the reflection was observed from 20 degrees C to 45 degrees C, followed by an accelerated increase from 45 degrees C to 70 degrees C. Upon cooling, this effect was found to be irreversible over 11 hours. Qualitatively similar results were observed for alpha-crystallin solutions at a variety of lower concentrations.

The ordering of corneal collagen fibrils with increasing ionic strength.

The fixed stromal charge of bovine corneas, osmotically clamped at physiological hydration, was altered by regulating the amount of chloride ions bound to the matrix. We measured the local fibrillar collagen order using X-ray diffraction methods. As the bound anions increased up to physiological values, the local fibrillar order increased to an optimal value. The coherence distance (t) approximately doubles to a maximum value (409 nm) from 10 mM NaCl to 154 mM NaCl. This then slowly decreased as the bathing solution increased to 1000 mM. In contrast the diameter of the collagen fibrils were minimal at physiological NaCl.

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.

Neutron and X-ray scattering by ox corneal stroma differentially loaded with bound anions.

Ox corneas at near physiological hydration were subjected to two variables: the amount of chloride ions bound to them and exposure of various mixtures of H(2)O/D(2)O as solvent. The preparations were then exposed to a neutron beam and the contrast match points, at which the collagen fibrils of the corneal stroma most nearly matched the scattering density of the various H(2)O/D(2)O mixtures, were measured. In both cases of high and low bound chloride, the contrast match points of the collagen fibril were equal, indicating that there were no significant changes in the water of electrostriction at the fibril surface when chloride ions bind to the stroma. The data suggest that the ligands which bind anions to corneal stroma are not located at the collagen fibril surface. When the chloride binding ligands were extracted from the corneal stroma there were significant changes in the structure of the fibrils. We suggest that the chloride binding ligands may be located within the collagen fibril.

Ion-dependence of Z-line and M-line response to calcium in striated muscle fibres in rigor.

The calcium-dependent contraction of vertebrate skeletal muscle is thought to be primarily controlled through the interaction of the thick and thin filaments. Through measurement of the Donnan potential, we have shown that an electrical switching mechanism (sensitive to both anions and cations) is present in both A- and I-bands [1]. Here we show that this mechanism is not confined to the contractile apparatus and report for the first time the presence of M-line potentials. The Z-line responds to Ca2+ ions in a similar manner to the A-band under the same solution conditions (phosphate-chloride and imidazole buffers), even though it has no reported Ca2+ binding sites. Z-line potentials were not observed in tris-acetate buffer. The M-line has a markedly different response to any of the other subsarcomeric regions, however, and can only be detected in the phosphate-chloride buffer. Preliminary observations of the M-line potential in creatine kinase-deficient mouse muscle (phosphate-chloride buffer) reveal significant differences in the calcium-induced transitions between two of the genotypes and demonstrate definitively that it is the M-line potential that is being recorded. From these results, it seems likely that the charge response of the Z-line and M-line is being mediated by titin in an anion-dependent manner. Our evidence comes from several observations. First, the similarity between the response of the Z-line potentials to the A-band potentials, where titin is the only link between these structures and second, the differential observation of M-line and Z-line potentials in a range of buffers containing different anion(s). Both Z-line and M-line potentials were seen in phosphate-chloride buffer, but only the Z-line potentials could be detected in chloride-only (imidazole) buffer and neither was observed in the acetate buffer. The latter observations can be attributed to two sources. The first is the effect of acetate buffer on the conformation of myosin [2]; the second is the absence of binding of the M-line protein, myomesin, to titin in the absence of phosphate ions [3].

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 effect of temperature on the Donnan potentials in biological polyelectrolyte gels: cornea and striated muscle.

The effect of temperature on the fixed electric charge in biological polyelectrolyte gels was studied between 10 and 35 degrees C using the Donnan microelectrode technique. Two tissues; cornea and striated muscle were used. In cornea, there is a gentle and uniform decrease in fixed charge over the temperature range. In rigor muscle, there is a dramatic step-function decrease in charge at around 28 degrees C. There is a charge decrease in relaxed muscle at around the same temperature, but the step function is less distinct. The significance of these different experimental relationships is discussed in relation to the Saroff model for ion binding to proteins, linked to the possible disordering effects of excess electric charge. The diverse effects in these systems are important for the physiological functions of the different tissues.

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.

Calcium dependence of Donnan potentials in rigor: the effects of [Mg2+] and anions in isolated rabbit psoas muscle fibres.

Contraction in vertebrate striated muscle is known to be dependent upon the binding of calcium ions to the regulatory protein troponin C (TnC). Our electrical (Donnan potential) studies of the subsarcomeric regions have revealed an electrical switching mechanism, which is sensitive to both cation concentration and to particular anions. In a buffer containing phosphate and chloride ions and at 2.7 mM Mg2+ we observe a single charge transition at pCa50 6.8 in both A- and I-bands. At zero Mg2+ the pCa50 of the A-band transition is shifted to 8.0 and the I-band shows two transitions (pCa50 approximately 6.8 and approximately 8.2). Increasing [Mg2+] to 4.5 mM produces a complex effect between pCas 7 and 9 in both bands. All effects are abolished at 9 mM Mg2+. In a chloride-only buffer (imidazole) at zero Mg2+ the direction of the charge transitions is reversed. In addition, two transitions (pCa50 approximately 8.5 and approximately 7.0) are evident in the A-band and three in the I-band (pCa50 approximately 8.5, approximately 7.4, approximately 6.7). In the presence of Mg2+, again the effects of pCa upon the Donnan potential are complex. In the A-band at 2.7 mM Mg2+ two transitions of opposite sign predominate (pCa approximately 7 and approximately 8), whilst in the I band a single transition (pCa approximately 8.3) occurs in the same direction as that observed in phosphate buffer. At 4.5 mM Mg2+ the 'W' shape observed in the corresponding phosphate buffer is preserved in both bands with similar pCa50s. This shape is also apparent in the 9 mM Mg2+ solution. In these two buffer systems, the magnitude of the charge change in terms of electron binding is far larger than expected from simple Ca2+/Mg2+ binding to troponin. In an acetate-only buffer, however, the Donnan potentials of the A-band and I-band were very similar in magnitude and the charge change across the full pCa curve is close to the expected value for Ca2+/Mg2+ binding to troponin. We speculate that titin has a role in the calcium activation of striated muscle in vertebrates for four reasons. First, the effects of long-term storage of the glycerinated muscle; second, the action of [Mg2+]ions; third the effect of anions; and fourth, our published and unpublished observations of sarcomere-length dependence. We also demonstrate the validity of our methodology, relating the charge transitions that we observe to cation-binding studies of a more traditional nature.

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.

The myosin molecule--charge response to nucleotide binding.

A decrease in the net fixed electric charge in the A-bands of cross-striated muscle was observed by Bartels and Elliott [2,10] when the muscle went from the rigor to the relaxed condition. The current work localises the source of the charge decrease by following the net charge on myosin (in the form of concentrated gels) and also myosin rod and light meromyosin gels when the gels are exposed to different concentrations of ATP. The work includes a study of muscle A-bands when the muscle is exposed to the same variations in ATP concentrations as the protein gels. The work shows that (i) Only 100-200 microns ATP is needed to initiate the charge decrease between the rigor and relaxed conditions; (ii) the effect of ATP is seen in the muscle A-band and the myosin and myosin rod gels, but not in LMM gels; (iii) pyrophosphate (PPi) shows a similar charge effect to ATP. ADP does not affect the charge on myosin gels, on the other hand. The results suggest that the charge decrease caused by ATP or PPi is due to ligand interaction with one or more sites on the myosin molecule. This interaction causes a disseminated effect in the protein, and a consequent loss in net negative charge either by a decrease in the absorption, of anions to Saroff sites on the protein, or, less probably, by an increase in the absorption of cations at those sites.

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.

Macular corneal dystrophy: the macromolecular structure of the stroma observed using electron microscopy and synchrotron X-ray diffraction.

The distribution of sulphated proteoglycans within the stromas of three patients (A,B,C) suffering from macular corneal dystrophy was studied using the specific dye Cuprolinic Blue in a 'critical electrolyte concentration' method. The corneas were examined using transmission electron microscopy and A and C were further studied by low-angle synchroton X-ray diffraction. Sera from all three patients were analyzed for the presence of keratan sulphate using a monoclonal antibody in an enzyme-linked immunosorbent assay. The serum from Patient A contained keratan sulphate, but the chains were thought to be shorter or less sulphate in their sera. Electron microscopy showed many electron-transparent lacunae randomly distributed throughout the specimens. The average collagen fibril diameter was normal but there were differences in packing between the specimens. Specimen A was closely-packed with most collagen fibrils in contact with their neighbours. Specimens B and C showed fewer regions of close packing; in most of the tissue the interfibrillar spacing appeared normal. Staining with Cuprolinic Blue revealed an unusual distribution of proteoglycans in some parts of the interfibrillar matrix, particularly in A, with 'small' proteoglycans running exclusively parallel to the collagen fibrils. Furthermore in A, and to a lesser extent in B and C, some lacunae were filled with clusters of abnormal sulphated proteoglycan filaments (of various sizes) which were chondroitinase ABC susceptible. Clearly defined regions, both within the lacunae and elsewhere, failed to stain with Cuprolinic Blue; this suggests an absence of sulphated proteoglycans within these areas. Equatorial X-ray diffraction of the wet tissues (A and C) gave values for the mean interfibrillar centre-to-centre separation of 43 +/- 2 nm in Specimen A and 52 +/- 3 nm in Specimen C. The differences observed in the serum keratan sulphate levels, the packing of the collagen fibrils and the distribution of chondroitin/dermatan sulphate proteoglycans confirm the heterogeneity that exists within the macular corneal dystrophies.

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.