Cross-bridge Distribution Research Articles

The smooth muscle myosin seven amino acid heavy chain insert's kinetic role in the crossbridge cycle for mouse bladder.

The seven amino acid insert in the smooth muscle myosin heavy chain is thought to regulate the kinetics of contraction, contributing to the differences between fast and slow smooth muscle. The effects of this insert on force and stiffness were determined in bladder tissue of a transgenic mouse line expressing the insert SMB at one of three levels: an SMB wild type (+/+), an SMA homozygous type (-/-) and a heterozygous type (+/-). For skinned muscle, an increase in MgADP or inorganic phosphate (Pi) should shift the distribution of crossbridges in the actomyosin ATPase (AMATPase) to increase the relative population of the crossbridge state prior to ADP release and Pi release, respectively. Exogenous ADP increased force and stiffness in a manner consistent with increasing the Ca2+ concentration in both the +/+ and +/- mouse types. However, the -/- type showed a significantly greater increase in force than in stiffness suggesting that immediately prior to ADP release, the AMATPase either has an additional force producing isomerization state or a slower ADP dissociation rate for the -/- type compared to the +/+ or +/- types. Exogenous Pi led to a significantly greater decrease in stiffness than in force for all three mouse types suggesting that there is a force producing state prior to Pi release. In addition, the increase in Pi showed similar changes in the +/+ and -/- types whereas in the +/- type the decreases in both force and stiffness were greater than the other two mouse types indicating that the insert can affect the cooperativity between myosin heads. In conclusion, the seven amino acid insert modulates the kinetics and/or states of the AMATPase, which could lead to differences in the kinetics of contraction between fast and slow smooth muscle.

Perturbed Equilibria of Myosin Binding in Airway Smooth Muscle: Bond-Length Distributions, Mechanics, and ATP Metabolism

We carried out a detailed mathematical analysis of the effects of length fluctuations on the dynamically evolving cross-bridge distributions, simulating those that occur in airway smooth muscle during breathing. We used the latch regulation scheme of Hai and Murphy ( Am. J. Physiol. Cell Physiol. 255:C86–C94, 1988) integrated with Huxley’s sliding filament theory of muscle contraction. This analysis showed that imposed length fluctuations decrease the mean number of attached bridges, depress muscle force and stiffness, and increase force-length hysteresis. At frequencies >0.1 Hz, the bond-length distribution of slowly cycling latch bridges changed little over the stretch cycle and contributed almost elastically to muscle force, but the rapidly cycling cross-bridge distribution changed substantially and dominated the hysteresis. By contrast, at frequencies <0.033 Hz this behavior was reversed: the rapid cycling cross-bridge distribution changed little, effectively functioning as a constant force generator, while the latch bridge bond distribution changed substantially and dominated the stiffness and hysteresis. The analysis showed the dissociation of force/length hysteresis and cross-bridge cycling rates when strain amplitude exceeds 3%; that is, there is only a weak coupling between net external mechanical work and the ATP consumption required for cycling cross-bridges during the oscillatory steady state. Although these results are specific to airway smooth muscle, the approach generalizes to other smooth muscles subjected to cyclic length fluctuations.

The Two-state Cross-bridge Model of Muscle is an Asymptotic Limit of Multi-state Models

The relationship between the two-state model of muscle contraction and multi-state models is examined from the perspective of matched asymptotic expansions, under the assumption that transition rates between attached states are fast compared to those between detached and attached states. A detailed formal analysis of a three-state model reveals that the classic Huxley (1957. Prog. Biophys. Biophys. Chem.7, 225–318) rate equation, as modified for thermodynamic self-consistency by Hill et al. (1975. Biophys. J.15, 335–372), governs the “outer” solution of the three-state equations. Thus, the two-state model remains a valid description of muscle dynamics on physiologically relevant time scales, which are slow compared to millisecond-scale transitions between attached states. But the asymptotic analysis reveals also that the cross-bridge force must be considered to be a nonlinear function of the cross-bridge strain, in contrast to the usual assumption of two-state models. This apparent, or effective, force is determined by both the intrinsic stiffness of the cross-bridge and the equilibrium distribution of cross-bridges among attached states. Further, the asymptotic analysis yields an expression for the energy liberation rate that implies a reduced rate in stretch vs. shortening. Some behaviors of multi-state models that are suggested by the three-state analysis are discussed in qualitative terms.

Force generation and phosphate release steps in skinned rabbit soleus slow-twitch muscle fibers

The force-generation and phosphate-release steps of the cross-bridge cycle in rabbit soleus slow-twitch muscle fibers (STF) were investigated using sinusoidal analysis, and the results were compared with those of rabbit psoas fast-twitch fibers (FTF). Single fiber preparations were activated at pCa 4.40 and ionic strength 180 mM at 20 degrees C. The effects of inorganic phosphate (Pi) concentrations on three exponential processes, B, C, and D, were studied. Results are consistent with the following cross-bridge scheme: [formula: see text] where A is actin, M is myosin, D is MgADP, and P is inorganic phosphate. The values determined are k4 = 5.7 +/- 0.5 s-1 (rate constant of isomerization step, N = 9, mean +/- SE), k-4 = 4.5 +/- 0.5 s-1 (rate constant of reverse isomerization), K4 = 1.37 +/- 0.13 (equilibrium constant of the isomerization), and K5 = 0.18 +/- 0.01 mM-1 (Pi association constant). The isomerization step (k4) in soleus STF is 20 times slower, and its reversal (k-4) is 20 times slower than psoas fibers. Consequently, the equilibrium constant of the isomerization step (K4) is the same in these two types of fibers. The Pi association constant (K5) is slightly higher in STF than in FTF, indicating that Pi binds to cross-bridges slightly more tightly in STF than FTF. By correlating the cross-bridge distribution with isometric tension, it was confirmed that force is generated during the isomerization (step 4) of the AMDP state and before Pi release in soleus STF.

Theoretical treatment of striated muscle: Dynamic extension of four-state model

We constructed a muscle model, based on the model first proposed by Gray and Gonda [6,7), that simulates the twitch contraction of striated muscle. Their original model postulated four basic states in the contraction cycle and predicted the properties of steady state contraction in striated muscle. Using the relationship between steady state tension and calcium concentration, we described several rate constants as functions of calcium concentration and calculated the number of attached crossbridges at various calcium concentration values. The results for both skeletal and cardiac muscle were approximately consistent with those of X-ray studies. Assuming that rate constants change immediately with the phasic alteration of intracellular calcium concentration, we estimated the time course of crossbridge distribution during twitch contraction; these findings were also consistent with those of X-ray studies. We also simulated the effects of calcium concentration and sarcomere length on the magnitude of twitch tension. These simulations suggest that the major determinants of crossbridge distribution during twitch contraction are the time courses of calcium transients and the rate constants of crossbridge kinetics. Our findings suggest that the model used in this study provides a theoretical basis for interpreting the characteristics of cardiac muscle encountered in the clinical setting.

A real-time observation of X-ray diffraction from frog skeletal muscle during and after slow length changes.

X-ray diffraction patterns from a frog skeletal muscle were recorded in real time using an X-ray image intensifier and a CCD video camera with a time resolution of 1/60s. The muscle was stimulated and then released or stretched by 6% at a constant speed. The equatorial (1,1) reflection decreased in intensity during stretches, suggesting a conformational change of cross-bridges. The integrated intensity of the 3rd meridional reflection from the thick filament decreased during both releases and stretches. The decrease was larger in stretches than in releases and at higher velocities. The intensity change correlated well with the change in tension, showing that the axial mass distribution of cross-bridges becomes wider as the tension increases. The axial Bragg spacing of the 3rd meridional reflection tended to increase in stretches and decrease in releases by less than 0.1%, possibly reflecting the elasticity in the thick filament backbone. The only significant effect of a length change remaining at the end of the stimulation was the broadening of the 3rd meridional reflection, suggesting non-uniformity of sarcomere lengths or axial misalignment of thick filaments as causes of enhancement and deficit of tension.

Effects of pH on myofibrillar ATPase activity in fast and slow skeletal muscle fibers of the rabbit

In permeabilized single fibers of fast (psoas) and slow (soleus) muscle from the rabbit, the effect of pH on isometric myofibrillar ATPase activity and force was studied at 15 degrees C, in the pH range 6.4-7.9. ATPase activity was measured photometrically by enzymatic coupling of the regeneration of ATP to the oxidation of NADH, present in the bathing solution. NADH absorbance at 340 nm was determined inside a measuring chamber. To measure ATP turnover in single soleus fibers accurately, a new measuring chamber (volume 4 microliters) was developed that produced a sensitivity approximately 8 times higher than the system previously used. Under control conditions (pH 7.3), the isometric force was 136 and 115 kN/m2 and the ATP turnover was 0.43 and 0.056 mmol per liter fiber volume per second in psoas and soleus fibers, respectively. Over the pH range studied, isometric force increased monotonically by a factor 1.7 for psoas and 1.2 for soleus fibers. In psoas the isometric ATPase activity remained constant, whereas in soleus it slightly decreased with increasing pH. The pH dependency of relative tension cost (isometric ATPase activity divided by force) was practically identical for psoas and soleus fibers. In both cases it decreased by about a factor 0.57 as pH increased from 6.4 to 7.9. The implications of these findings are discussed in terms of cross-bridge kinetics. For both fiber types, estimates of the reaction rates and the distribution of cross-bridges and of their pH dependencies were obtained. A remarkable similarity was found between fast- and slow-twitch fibers in the effects of pH on the reaction rate constants.

A fluorescence depolarization study of the orientational distribution of crossbridges in muscle fibres.

A fluorescence depolarization study of the orientational distribution of crossbridges in dye-labelled muscle fibres is presented. The characterization of this distribution is important since the rotation of crossbridges is a key element in the theory of muscle contraction. In this study we exploited the advantages of angle-resolved experiments to characterize the principal features of the orientational distribution of the crossbridges in the muscle fibre. The directions of the transition dipole moments in the frame of the dye and the orientation and motion of the dye relative to the crossbridge determined previously were explicitly incorporated into the analysis of the experimental data. This afforded the unequivocal determination of all the second and fourth rank order parameters. Moreover, this additional information provided discrimination between different models for the orientational behaviour of the crossbridges. Our results indicate that no change of orientation takes place upon a transition from rigor to relaxation. The experiments, however, do no rule out a conformational change of the myosin S1 during the transition.

Application of angle-resolved fluorescence depolarization in muscle research

Angle-resolved fluorescence depolarization (AFD) experiments have been used for over a decade in studies of fluorescent molecules in macroscopically aligned systems such as lipid bilayers and stretched polymer films. The importance of this technique lies in the fact that it affords the determination of both the second- and the fourth-rank order parameters of the orientational distribution of the probe molecules in the sample. Here we apply the technique to the study of the orientational distribution of crossbridges in muscle fibers. This orientational distribution is particularly relevant in muscle research, as crossbridge rotation is commonly regarded to be the driving mechanism in force development. An unfortunate consequence of the fact that the crossbridges have an average orientation of approximately 45(o) relative to the fiber axis is that the values of the second-rank order parameter [Symbol: see text]P 2[Symbol: see text] of the crossbridge distribution are close to 0. Therefore, knowledge of [Symbol: see text]P 4[Symbol: see text] is essential for a reliable reconstruction of the form of the distribution function. AFD of dyelabeled muscle was measured under rigor and relaxation conditions. The results indicate that no significant changes in depolarization take place upon a transition from the rigor to the relaxed state in the muscle and seem not to support the rotating crossbridge model, which postulates a clear change of orientation of the crossbridges.

Following the rotational trajectory of the principal hydrodynamic frame of a protein using multiple probes.

A generalized set of fluorescence polarization intensities and electron paramagnetic resonance (EPR) spectra from multiple fluorescent and EPR probes of a specific site on an oriented and immobilized protein element in a biological assembly, combined with anisotropic rotational relaxation studies of the purified and labeled protein element freely rotationally diffusing in solution, are used to determine the angular distribution of the principal hydrodynamic frame of the protein element in the biological assembly. This multiprobe analysis method removes all of the basic ambiguities in the measured principal frame angular distribution introduced by limitations or symmetries intrinsic to standard fluorescence polarization intensity ratios and EPR spectra. The angular distribution of the principal frame is also more highly resolved than that previously reported for multiprobe determinations of probe angular distributions and is more useful for determining biological mechanisms because it indicates the order and orientation of the protein element rather than that of the extrinsic probe. Application of this method to the determination of the principal hydrodynamic frame distribution of myosin cross-bridges in muscle fibers in four physiological states, including the active isometric state, demonstrates the method's practicality by indicating the path of cross-bridge orientation changes during the active cycle [Ajtai, Toft, & Burghardt (1994) Biochemistry (following paper in this issue)].

Path and Extent of Cross-Bridge Rotation during Muscle Contraction

The angular distribution of myosin cross-bridges in muscle fibers was investigated in four physiological states using a multiple probe analysis of varied extrinsic probes of the cross-bridge [Burghardt & Ajtai (1994) Biochemistry (preceding paper in this issue)]. The analysis combines data of complementary techniques from different probes giving the highest possible angular resolution. Four extrinsic probes of the fast reactive sulfhydryl (SH1) on myosin subfragment 1 (S1) were employed. Electron paramagnetic resonance (EPR) spectra from paramagnetic probes, deuterium- and 15N-substituted for greater sensitivity to orientation, on S1 were measured when the protein was freely tumbling in solution and when it was decorating muscle fibers. The EPR spectra from labeled S1 tumbling in solution were measured at X- and Q-band microwave frequencies to uniquely specify the orientation of the probe relative to the S1 principal hydrodynamic frame. The EPR spectra from labeled S1 decorating muscle fibers in rigor and in the presence of MgADP were measured at X-band and used in the multiple probe analysis of cross-bridge orientation. The time-resolved fluorescence anisotropy decay (TRFAD) of fluorescent probes on S1 was measured when the protein was freely tumbling in solution, and fluorescence polarization (FP) intensities from fluorescent probes modifying SH1 in intact muscle fibers were measured for fibers in rigor, in the presence of MgADP, in isometric contraction, and in relaxation at low ionic strength. The TRFAD measurements limit the range of possible orientations of the probe relative to the S1 principal hydrodynamic frame. The FP intensity measurements were used in the multiple probe analysis of cross-bridge orientation. The combination of the EPR and FP data determined a highly resolved cross-bridge angular distribution in rigor, in the presence of MgADP, in isometric contraction, and in relaxation at low ionic strength. These findings confirm earlier observations of a rigid body rotation of the SH1 region in the myosin head group upon physiological state changes and indicate the path and extent of cross-bridge rotation during contraction. The rotation of the cross-bridge is visualized with computer-generated space-filling models of actomysin in six states of the contraction cycle.

On the Solutions of Huxley-type Models in Cardiac Muscle Fiber Contractions

Huxley's sliding filament crossbridge muscle model coupled with parallel and series elastic components was simulated to examine the effect of various solution techniques in cardiac contractions. Solutions of both isometric and isotonic contraction cases showed that the force versus time curves were not significantly altered by solving the three-element Hill model with Huxley's Equation written as either an ordinary or partial differential equation (ODE or PDE), but this makes a difference in the solution time required. Various theoretical studies have used either the ODE or PDE representation. The crossbridge cycles at the end of a contraction showed approximately 25% and 15% difference in the isometric and isotonic cases when Huxley's Equation was written as either an ODE or PDE. Examination of the crossbridge distribution (distribution among states of reach) showed that assuming that the crossbridge distribution is a Gaussian function is a poor approximation since the shape changes considerably between the cardiac contracting and expanding phases, and using a technique such as a distribution moment approximation is questionable. Recent experimental studies have demonstrated that solving Huxley-type relations as ordinary differential equations gives good agreement with cardiac data, implying that as a first approximation, this can be successfully used.

Freeze-Fracture Studies on the Cross-Bridge Angle Distribution at Various States and the Thin Filament Stiffness in Single Skinned Frog Muscle Fibers

To give information about the changes of cross-bridge (myosin head) orientation during muscle contraction, mechanically skinned frog muscle fibers were rapidly frozen at various states, and the cross-bridges were observed on freeze-etch replicas. The number of cross-bridges per unit length of thick filament in relaxed state was less than one-third of that in contracting and rigor states. The interval between adjacent cross-bridges was maximum around 35 nm, a value close to the crossover repeat of actin helix. The cross-bridge angle distribution, as measured with a digital image processor, showed a peak around 90 degrees in all the states examined. The proportion of cross-bridges with angles around 90 degrees decreased either after stretch or after release of fibers in rigor state. The axial spacing of actin monomers in the thin filament was found to increase with increasing rigor force, to give the thin filament stiffness (times unit length) of about 1.8 x 10(4) pN. These results are discussed in connection with the mechanical properties of cross-bridges.

Effects of Ca2+ on the kinetics of phosphate release in skeletal muscle.

The process of phosphate dissociation during the muscle cross-bridge cycle has been investigated by photoliberation of inorganic phosphate (Pi) within skinned fibers of rabbit psoas muscle. This permitted a test of the idea that Ca2+ controls muscle contraction by regulating the Pi release step of the cycle. Photoliberation of Pi from structurally distinct "caged" Pi precursors initiated a rapid tension decline of up to 12% of active tension, and this was followed by a slower tension decline. The apparent rate constant of the fast phase, kPi, depended on both [Pi] and [Ca2+], whereas the slow phase generally occurred at 2-4 s-1. At maximal Ca2+, kPi increased in a nonlinear manner from 43 +/- 2 s-1 to 118 +/- 7 s-1, as Pi was raised from 0.9 to 12 mM. This was analyzed in terms of a three-state kinetic model in which a force-generating transition is coupled to Pi dissociation from the cross-bridge. As Ca(2+)-activated tension was reduced from maximal (Pmax) to 0.1 Pmax, (i) kPi decreased by up to 2.5-fold, (ii) the relative amplitude of the rapid phase increased 2-fold, and (iii) the relative amplitude of the slow phase increased about 6-fold. Changes in the rapid phase are compatible with Ca2+ influencing an apparent equilibrium constant for the force-generating transition. By comparison, kPi was faster than the rate constant of tension redevelopment, ktr, and was influenced less by Ca2+. Ca2+ effects on the caged Pi transient cannot account for the large effects of Ca2+ on actomyosin ATPase rates or cross-bridge cycling kinetics but may be a manifestation of reciprocal interactions between the thin filament and force-generating cross-bridges, and may represent Ca2+ regulation of the distribution of cross-bridges between non-force-and force-generating states.

Distribution of cross-bridges in isometrically contracting muscle: its deduction from the Huxley-Simmons theory and the effect of filament elasticity.

The author attempted to deduce a possible cross-bridge (XB) distribution as a function of x, the amount of stretch of the XB, by applying the Huxley-Simmons theory with their recent experimental data to a steady isometric contraction. I found no XB distribution which satisfied at once the Huxley-Simmons theory, their data and observations in muscle energetics and biochemistry. Further, a computation with a simple model suggested that the XB distribution could be broadened by the elasticity of the thick and/or the thin filament if it existed.

A model of flagellar movement based on cooperative dynamics of dynein-tubulin cross-bridges

A theoretical model based on molecular mechanisms of both dynein cross-bridges and radial spokes is used to study bend propagation by eukaryotic flagella. Though nine outer doublets are arranged within an axoneme, a simplified model with four doublets is constructed on the assumption that cross-bridges between two of the four doublets are opposed to those between the other two, corresponding to the geometric array of cross-bridges on the 6-9 and the 1-4 doublets in the axoneme. We also assume that external viscosity is zero, whereas internal viscosity is non-zero in order to reduce numerical complexity. For demonstrating flagellar movement, computer simulations are available by dividing a long flagellum into many straight segments. Considering the fact that dynein cross-bridge spacing is almost equal to attachment site spacing, we may use a localized cross-bridge distribution along attachment sites in each straight segment. Dynamics of cross-bridges are determined by a three-state model, and effects of radial spokes are represented by a periodic mechanical potential whose periodicity is considered to be a stroke distance of the radial spoke. First of all, we examine the model of a short segment to know basic properties of the system. Changing parameters relating to "activation" of cross-bridges, our model demonstrates various phenomena; for example "excitable properties with threshold phenomena" and "limit cycle oscillation". Here, "activation" and "inactivation" (i.e. switching mechanisms) between a pair of oppositely-directed cross-bridges are essential for generation of excitable or oscillatory properties. Next, the model for a flagellar segment is incorporated into a flagellum with a whole length to show bending movement. When excitable properties of cross-bridges, not oscillatory properties, are provided along the length of the flagellum and elastic links between filaments are presented at the base, then our model can demonstrate self-organization of bending waves as well as wave propagation without special feedback control by the curvature of the flagellum. Here, "cooperative interaction" between adjacent short segments, based on "cooperative dynamics" of cross-bridges, is important for wave propagation.

Fine structure of the A-band in cryo-sections: III. Crossbridge distribution and the axial structure of the human C-zone

Optical analysis of the C-zones of negatively stained cryo-sectioned A-bands from human m. tibialis anterior muscle, combined with computer modelling of the C-zone staining pattern, has led to the following conclusions. 1. (1) At a 99.9% confidence level there are two slightly different axial repeats in the C-zone. One repeat of 434 (± 1.2) Å probably corresponds to the C-protein repeat, whereas the other of 429 Å (used for calibration) is probably due to myosin crossbridges. 2. (2) The myosin crossbridge array in (1) does not consist of crossbridge rows evenly spaced at 143 Å. The crossbridge positions are periodically shifted from such positions to give a true 429 Å crossbridge repeat. 3. (3) The myosin crossbridges are oriented almost perpendicular to the long axis of the muscle; the axial extent of the crossbridge profiles is less than 80 Å. It is shown that computed model structures which explain satisfactorily the appearances of cryo-sectioned C-zones can also explain several anomalous features of published X-ray diffraction patterns from vertebrate muscle.

Muscle structure, cryo-methods and image analysis.

Negatively stained cryo-sections from glutaraldehyde fixed, anti-freeze treated muscle, quench-frozen in Freon cooled by liquid nitrogen, show improved preservation of axial structure of the myofibrils compared with conventional plastic sections. Such sections are being used both to characterize the structural differences inthe M-bands of different vertebrate muscles and fibre types and also to define the axial distribution of myosin crossbridges and non-myosin proteins in the crossbridge region of the A-band. Combined with analysis of the transverse A-band structure from plastic sections, the cryo-sections are helping to reconstruct a three-dimensional picture of the molecular architecture of the A-band. This, in turn, is providing the necessary structural background with which to interpret the wealth of published X-ray diffraction data on muscle. Such data should reveal the nature of the contractile event itself. Since good X-ray diffraction patterns can be obtained from living muscles, these can be compared with optical diffraction patterns from muscle cryo-sections as a means of testing the degree of preservation in the sections. Muscle is therefore an excellent tissue with which to evaluate new cryo-techniques.

Three-dimensional structure of the vertebrate muscle A-band: II. The myosin filament superlattice

The three-dimensional arrangement of the myosin filaments in the A-band of frog sartorius muscle was studied using electron micrographs of very thin and accurately cut transverse sections through the bare region (on each side of the M-band) where the thick filament shafts are roughly triangular in shape. It was found that the orientations of these triangular profiles are arranged to give a superlattice of the same size and shape as that proposed by Huxley & Brown (1967) on the basis of X-ray diffraction evidence, but the contents of the superlattice may not be as they suggested. The results from detailed image analysis strongly suggest that myosin filaments (which have been shown to have 3-fold rotational symmetry, Luther, 1978; Luther, Munro and Squire, unpublished results) are arranged with one of two orientations which are 60 ° (or 180 °) apart. This arrangement of filaments with 3-fold symmetry is not that predicted for a superlattice with the symmetry suggested by Huxley & Brown. Two rules define the way in which the orientations of neighbouring filaments are defined. Rule (1): no three mutually adjacent filaments in the hexagonal array of filaments in the A-band can all have identical orientations; and rule (2): no three successive filaments along a 10 1 ̄ row in the filament array can have identical orientations. These two no-three-alike rules are sufficient to describe the observed arrangement of filament profiles in the frog bare region (except for some minor violations discussed in the text), and they lead automatically to the generation of the required superlattice. The A-band structure in fish muscle is different; there is no superlattice and the triangular bare region profiles have only one orientation. The frog superlattice and fish simple lattice are explained directly in terms of different interactions between the M-bridges in the M-bands of these muscles. The observed structures require that the myosin filament symmetry at the centre of the M-band is that of the dihedral point group 32. The two possible forms of interaction between filaments with this symmetry (apart from a completely random structure) give rise to the observed A-band lattices in frog and fish muscles. The 3-fold rotational symmetry of the myosin filaments required to explain the observed micrographs also requires that the myosin crossbridge arrangements around the actin filaments in frog and fish muscles will be different. It is suggested that the structure in the frog A-band (and in the A-bands of other higher vertebrates) has evolved from that in fish to improve the distribution of crossbridges around the aotin filaments. The X-ray diffraction evidence of Huxley & Brown (1967) will be accounted for in terms of the proposed A-band structure in a further paper in this series.

Geometrical factors influencing muscle force development. I. The effect of filament spacing upon axial forces

The influence of geometry on the force and stiffness measured during muscle contraction at different sarcomere lengths is examined by using three specific models of muscle cross-bridge geometry which are based upon the double-hinge model of H. E. Huxley (Science [Wash. D.C.]. 1969, 164:1356-1366) extended to three dimensions. The force generated during muscle contraction depends upon the orientation of the individual cross-bridge force vectors and the distribution of the cross-bridges between various states. For the simplest models, in which filament separation has no effect upon cross-bridge distribution, it is shown that changes in force vectors accompanying changes in myofilament separation between sarcomere lengths 2.0 and 3.65 microgram in an intact frog skeletal muscle fiber have only a small effect upon axial force. The simplest models, therefore, produce a total axial force proportional to the overlap between the actin and myosin filaments and independent of filament separation. However, the analysis shows that it is possible to find assumptions that produce a cross-bridge model in which the axial force is not independent of filament spacing. It is also shown that for some modes of attachment of subfragment-1 (S1) to actin the azimuthal location of the actin site is important in determining the axial force. A mode of S1 attachment to actin similar to that deduced by Moore et al. (J. Mol. Biol., 1970, 50:279-294), however, exhibits rather constant cross-bridge behavior over a wide range of actin site location.