A mathematical model for sarcomere mechanics in cross-striated muscles taking into account the stretch and twist of actin filaments

A model of sarcomere mechanics, which takes into account the elongation of the actin and myosin filaments and twisting of the actin filaments during muscle contraction is suggested. The model accounts for the experimentally observed phenomena of the stretch and twist of actin filaments due to strong binding of myosin heads and pulling force. Some model parameters were estimated from published experimental data. The results of modelling suggest that the twist of actin filaments may play an essential role in mechanical responses of contracting muscle fibres to stepwise changes in their length.     

The Three Filament Model of Skeletal Muscle Stability and Force Production

Ever since the 1950s, muscle force regulation has been associated with the cross-bridge interactions between the two contractile filaments, actin and myosin. This gave rise to what is referred to as the "two-filament sarcomere model". This model does not predict eccentric muscle contractions well, produces instability of myosin alignment and force production on the descending limb of the force-length relationship, and cannot account for the vastly decreased ATP requirements of actively stretched muscles. Over the past decade, we and others, identified that a third myofilament, titin, plays an important role in stabilizing the sarcomere and the myosin filament. Here, we demonstrate additionally how titin is an active participant in muscle force regulation by changing its stiffness in an activation/force dependent manner and by binding to actin, thereby adjusting its free spring length. Therefore, we propose that skeletal muscle force regulation is based on a three filament model that includes titin, rather than a two filament model consisting only of actin and myosin filaments.     

Architecture of the sarcomere matrix of skeletal muscle: immunoelectron microscopic evidence that suggests a set of parallel inextensible nebulin filaments anchored at the Z line

Nebulin, a giant myofibrillar protein (600-800 kD) that is abundant (3%) in the sarcomere of a wide range of skeletal muscles, has been proposed as a component of a cytoskeletal matrix that coexists with actin and myosin filaments within the sarcomere. Immunoblot analysis indicates that although polypeptides of similar size are present in cardiac and smooth muscles at low abundance, those proteins show no immunological cross-reactivity with skeletal muscle nebulin. Gel analysis reveals that nebulins in various skeletal muscles of rabbit belong to at least two classes of size variants. A monospecific antibody has been used to localize nebulin by immunoelectron microscopy in a mechanically split rabbit psoas muscle fiber preparation. Labeled split fibers exhibit six pairs of stripes of antibody-imparted transverse densities spaced at 0.1-1.0 micron from the Z line within each sarcomere. These epitopes maintain a fixed distance to the Z line irrespective of sarcomere length and do not exhibit the characteristic elastic stretch-response of titin epitopes within the I band domain. It is proposed that nebulin constitutes a set of inextensible filaments attached at one end to the Z line and that nebulin filaments are in parallel, and not in series, with titin filaments. Thus the skeletal muscle sarcomere may have two sets of nonactomyosin filaments: a set of I segment-linked nebulin filaments and a set of A segment-linked titin filaments. This four-filament sarcomere model raises the possibility that nebulin and titin might act as organizing templates and length- determining factors for actin and myosin respectively.     

Actin-myosin interaction: The role of myosin in determining the actin pattern in self-assembled ‘hybrid’ contractile units

Self-assembly of actin-myosin filamentous complexes was assayed by polymerizing rabbit G-ADP actin on formed filaments of lobster myosin. The resulting contractile units indicate a 12-member actin orbital rather than the six-member orbital obtained previously using rabbit myosin and actin. Furthermore, the pattern of actin distribution surrounding the myosin filament is similar to that of the lobster tonic muscle sarcomere rather than the trigonal actin position characteristic of vertebrate muscle. The results show that the pattern and mode of actin complexing is determined by the specific myosin and the arrangement of the cross-bridges on the organized filament.     

Titin organisation and the 3D architecture of the vertebrate-striated muscle I-band

The giant muscle protein titin (connectin) is known to serve as a cytoskeletal element in muscle sarcomeres. It elastically restrains lengthening sarcomeres, it aids the integrity and central positioning of the A-band in the sarcomere and it may act as a template upon which some sarcomeric components are laid down during myogenesis. A puzzle has been how titin molecules, arranged systematically within the hexagonal A-band lattice of myosin filaments, can redistribute through the I-band to their anchoring sites in the tetragonal Z-band lattice. Recent work by Liversage and colleagues has suggested that there are six titin molecules per half myosin filament. Since there are two actin filaments per half myosin filament in a half sarcomere, this means that there are three titin molecules interacting with each Z-band unit cell containing one actin filament in the same sarcomere and one of opposite polarity from the next sarcomere. Liversage et al. suggested that the three titins might be distributed with two on an actin filament of one polarity and one on the filament of opposite polarity. Here, we build on this suggestion and discuss the transition of titin from the A-band to the Z-band. We show that there are good structural and mechanical reasons why titin might be organised as Liversage et al., suggested and we discuss the possible relationships between A-band arrangements in successive sarcomeres along a myofibril.     

X-ray diffraction evidence for the extensibility of actin and myosin filaments during muscle contraction.

To clarify the extensibility of thin actin and thick myosin filaments in muscle, we examined the spacings of actin and myosin filament-based reflections in x-ray diffraction patterns at high resolution during isometric contraction of frog skeletal muscles and steady lengthening of the active muscles using synchrotron radiation as an intense x-ray source and a storage phosphor plate as a high sensitivity, high resolution area detector. Spacing of the actin meridional reflection at approximately 1/2.7 nm-1, which corresponds to the axial rise per actin subunit in the thin filament, increased about 0.25% during isometric contraction of muscles at full overlap length of thick and thin filaments. The changes in muscles stretched to approximately half overlap of the filaments, when they were scaled linearly up to the full isometric tension, gave an increase of approximately 0.3%. Conversely, the spacing decreased by approximately 0.1% upon activation of muscles at nonoverlap length. Slow stretching of a contracting muscle increased tension and increased this spacing over the isometric contraction value. Scaled up to a 100% tension increase, this corresponds to a approximately 0.26% additional change, consistent with that of the initial isometric contraction. Taken together, the extensibility of the actin filament amounts to 3-4 nm of elongation when a muscle switches from relaxation to maximum isometric contraction. Axial spacings of the layer-line reflections at approximately 1/5.1 nm-1 and approximately 1/5.9 nm-1 corresponding to the pitches of the right- and left-handed genetic helices of the actin filament, showed similar changes to that of the meridional reflection during isometric contraction of muscles at full overlap. The spacing changes of these reflections, which also depend on the mechanical load on the muscle, indicate that elongation is accompanied by slight changes of the actin helical structure possibly because of the axial force exerted by the actomyosin cross-bridges. Additional small spacing changes of the myosin meridional reflections during length changes applied to contracting muscles represented an increase of approximately 0.26% (scaled up to a 100% tension increase) in the myosin periodicity, suggesting that such spacing changes correspond to a tension-related extension of the myosin filaments. Elongation of the myosin filament backbone amounts to approximately 2.1 nm per half sarcomere. The results indicate that a large part (approximately 70%) of the sarcomere compliance of an active muscle is caused by the extensibility of the actin and myosin filaments; 42% of the compliance resides in the actin filaments, and 27% of it is in the myosin filaments.     

ANALYSIS OF MUSCLE CONTRACTION BY ULTRAVIOLET MICROBEAM DISRUPTION OF SARCOMERE STRUCTURE

In an effort to differentiate between the sliding filament theory for muscle contraction and alternative views which propose attachment between actin and myosin filaments at or across the H zone, rabbit psoas myofibrils were irradiated in various areas of the sarcomere with an ultraviolet microbeam. Irradiation of the I band appears to destroy the actin filaments; in vitro irradiation of F actin causes an irreversible depolymerization of the protein. Irradiation of the A band disorients the myosin but causes no apparent loss of dry mass. These effects are maximal at the wavelength of maximum absorption of the proteins involved. Actin filaments, released at the Z line of a sarcomere, are seen to slide into the A band on addition of ATP. Irradiation of a full A band prevents contraction, whereas irradiation of two-thirds of the A band, leaving a lateral edge intact, permits contraction at the non-irradiated edge. Thus contraction can occur in what is in essence only one-third of a sarcomere, eliminating any necessity for postulated H zone connections. These observations are in complete accord with the classical sliding filament theory but incompatible with either the contralateral filament hypothesis or the actin folding model for muscle contraction.     

Sarcomeric pattern formation by actin cluster coalescence

Contractile function of striated muscle cells depends crucially on the almost crystalline order of actin and myosin filaments in myofibrils, but the physical mechanisms that lead to myofibril assembly remains ill-defined. Passive diffusive sorting of actin filaments into sarcomeric order is kinetically impossible, suggesting a pivotal role of active processes in sarcomeric pattern formation. Using a one-dimensional computational model of an initially unstriated actin bundle, we show that actin filament treadmilling in the presence of processive plus-end crosslinking provides a simple and robust mechanism for the polarity sorting of actin filaments as well as for the correct localization of myosin filaments. We propose that the coalescence of crosslinked actin clusters could be key for sarcomeric pattern formation. In our simulations, sarcomere spacing is set by filament length prompting tight length control already at early stages of pattern formation. The proposed mechanism could be generic and apply both to premyofibrils and nascent myofibrils in developing muscle cells as well as possibly to striated stress-fibers in non-muscle cells.     

The N terminus of myosin-binding protein C extends toward actin filaments in intact cardiac muscle

Myosin and actin filaments are highly organized within muscle sarcomeres. Myosin-binding protein C (MyBP-C) is a flexible, rod-like protein located within the C-zone of the sarcomere. The C-terminal domain of MyBP-C is tethered to the myosin filament backbone, and the N-terminal domains are postulated to interact with actin and/or the myosin head to modulate filament sliding. To define where the N-terminal domains of MyBP-C are localized in the sarcomere of active and relaxed mouse myocardium, the relative positions of the N terminus of MyBP-C and actin were imaged in fixed muscle samples using super-resolution fluorescence microscopy. The resolution of the imaging was enhanced by particle averaging. The images demonstrate that the position of the N terminus of MyBP-C is biased toward the actin filaments in both active and relaxed muscle preparations. Comparison of the experimental images with images generated in silico, accounting for known binding partner interactions, suggests that the N-terminal domains of MyBP-C may bind to actin and possibly the myosin head but only when the myosin head is in the proximity of an actin filament. These physiologically relevant images help define the molecular mechanism by which the N-terminal domains of MyBP-C may search for, and capture, molecular binding partners to tune cardiac contractility.     

Irregular patterns of actin and myosin filaments in human skeletal muscle cells

Summary The ultrastructural study of cross sections of normal skeletal muscle cells showed the existence of irregular patterns of actin filaments in connection with the hexagonal pattern of the myosin filaments. The actin filaments surrounding each myosin filament vary in number from 6 to 11. The most frequent relationship is 9 to 1, followed by 10 to 1 and 8 to 1. The hexagonal pattern of actin filaments was observed only in the 6 to 1 arrays; as the actin filaments increase in number, they tend to form different polygons or circles around the myosin filaments. All described patterns may occur in each sarcomere. The actin to myosin filament ratio varies from 3 to 4 within each individual myofibril. The described variability of the actin filaments arrays leads to several difficulties in an explanation of the mechanism of muscular contraction.Director, Chief of Section, Histology. Profesor Agregado de Embriología e HistologíaProfesor Adjunto de Embriología e HistologíaResidente de Anatomía Patol'ogica de la Ciudad Sanitaria La Paz     

X-ray diffraction studies of the contractile mechanism in single muscle fibres

The molecular mechanism of muscle contraction was investigated in intact muscle fibres by X-ray diffraction. Changes in the intensities of the axial X-ray reflections produced by imposing rapid changes in fibre length establish the average conformation of the myosin heads during active isometric contraction, and show that the heads tilt during the elastic response to a change in fibre length and during the elementary force generating process: the working stroke. X-ray interference between the two arrays of myosin heads in each filament allows the axial motions of the heads following a sudden drop in force from the isometric level to be measured in situ with unprecedented precision. At low load, the average working stroke is 12 nm, which is consistent with crystallographic studies. The working stroke is smaller and slower at a higher load. The compliance of the actin and myosin filaments was also determined from the change in the axial spacings of the X-ray reflections following a force step, and shown to be responsible for most of the sarcomere compliance. The mechanical properties of the sarcomere depend on both the motor actions of the myosin heads and the compliance of the myosin and actin filaments.     

The length–tension curve in muscle depends on lattice spacing

Classic interpretations of the striated muscle length–tension curve focus on how force varies with overlap of thin (actin) and thick (myosin) filaments. New models of sarcomere geometry and experiments with skinned synchronous insect flight muscle suggest that changes in the radial distance between the actin and myosin filaments, the filament lattice spacing, are responsible for between 20% and 50% of the change in force seen between sarcomere lengths of 1.4 and 3.4 µm. Thus, lattice spacing is a significant force regulator, increasing the slope of muscle''s force–length dependence.     

Light chain phosphorylation regulates the movement of smooth muscle myosin on actin filaments

In smooth muscles there is no organized sarcomere structure wherein the relative movement of myosin filaments and actin filaments has been documented during contraction. Using the recently developed in vitro assay for myosin-coated bead movement (Sheetz, M.P., and J.A. Spudich, 1983, Nature (Lond.)., 303:31-35), we were able to quantitate the rate of movement of both phosphorylated and unphosphorylated smooth muscle myosin on ordered actin filaments derived from the giant alga, Nitella. We found that movement of turkey gizzard smooth muscle myosin on actin filaments depended upon the phosphorylation of the 20-kD myosin light chains. About 95% of the beads coated with phosphorylated myosin moved at velocities between 0.15 and 0.4 micron/s, depending upon the preparation. With unphosphorylated myosin, only 3% of the beads moved and then at a velocity of only approximately 0.01-0.04 micron/s. The effects of phosphorylation were fully reversible after dephosphorylation with a phosphatase prepared from smooth muscle. Analysis of the velocity of movement as a function of phosphorylation level indicated that phosphorylation of both heads of a myosin molecule was required for movement and that unphosphorylated myosin appears to decrease the rate of movement of phosphorylated myosin. Mixing of phosphorylated smooth muscle myosin with skeletal muscle myosin which moves at 2 microns/s resulted in a decreased rate of bead movement, suggesting that the more slowly cycling smooth muscle myosin is primarily determining the velocity of movement in such mixtures.     

Myosin filament sliding through the Z-disc relates striated muscle fibre structure to function

Striated muscle contraction requires intricate interactions of microstructures. The classic textbook assumption that myosin filaments are compressed at the meshed Z-disc during striated muscle fibre contraction conflicts with experimental evidence. For example, myosin filaments are too stiff to be compressed sufficiently by the muscular force, and, unlike compressed springs, the muscle fibres do not restore their resting length after contractions to short lengths. Further, the dependence of a fibre''s maximum contraction velocity on sarcomere length is unexplained to date. In this paper, we present a structurally consistent model of sarcomere contraction that reconciles these findings with the well-accepted sliding filament and crossbridge theories. The few required model parameters are taken from the literature or obtained from reasoning based on structural arguments. In our model, the transition from hexagonal to tetragonal actin filament arrangement near the Z-disc together with a thoughtful titin arrangement enables myosin filament sliding through the Z-disc. This sliding leads to swivelled crossbridges in the adjacent half-sarcomere that dampen contraction. With no fitting of parameters required, the model predicts straightforwardly the fibre''s entire force–length behaviour and the dependence of the maximum contraction velocity on sarcomere length. Our model enables a structurally and functionally consistent view of the contractile machinery of the striated fibre with possible implications for muscle diseases and evolution.     

A mechanism for sarcomere breathing: volume change and advective flow within the myofilament lattice

During muscle contraction, myosin motors anchored to thick filaments bind to and slide actin thin filaments. These motors rely on energy derived from ATP, supplied, in part, by diffusion from the sarcoplasm to the interior of the lattice of actin and myosin filaments. The radial spacing of filaments in this lattice may change or remain constant during contraction. If the lattice is isovolumetric, it must expand when the muscle shortens. If, however, the spacing is constant or has a different pattern of axial and radial motion, then the lattice changes volume during contraction, driving fluid motion and assisting in the transport of molecules between the contractile lattice and the surrounding intracellular space. We first create an advective-diffusive-reaction flow model and show that the flow into and out of the sarcomere lattice would be significant in the absence of lattice expansion. Advective transport coupled to diffusion has the potential to substantially enhance metabolite exchange within the crowded sarcomere. Using time-resolved x-ray diffraction of contracting muscle, we next show that the contractile lattice is neither isovolumetric nor constant in spacing. Instead, lattice spacing is time varying, depends on activation, and can manifest as an effective time-varying Poisson ratio. The resulting fluid flow in the sarcomere lattice of synchronous insect flight muscles is even greater than expected for constant lattice spacing conditions. Lattice spacing depends on a variety of factors that produce radial force, including cross-bridges, titin-like molecules, and other structural proteins. Volume change and advective transport varies with the phase of muscle stimulation during periodic contraction but remains significant at all conditions. Although varying in magnitude, advective transport will occur in all cases in which the sarcomere is not isovolumetric. Akin to “breathing,” advective-diffusive transport in sarcomeres is sufficient to promote metabolite exchange and may play a role in the regulation of contraction itself.     

Donnan potentials from striated muscle liquid crystals. Sarcomere length dependence.

Donnan potentials from A-bands and I-bands were measured as a function of sarcomere length in skinned long-tonic muscle fibers of the crayfish. These measurements were made using standard electrophysiological technique. Simultaneously, the relative cross-sectional area of the fibers was determined. Lattice plane spacings and hence unit-cell volumes were determined by low-angle x-ray diffraction. At a sarcomere length at which the myosin filaments and actin filaments nominally do not overlap, measurements of potential, relative cross-sectional area, and unit-cell volume were used in conjunction with Donnan equilibrium theory to calculate the effective linear charge densities along the myosin filament (6.6 X 10(4) e-/mu) and actin filament (6.8 X 10(3) e-/mu). Using these linear charge densities, unit-cell volumes and Donnan equilibrium theory, an algorithm was developed to predict A-band and I-band potentials at any sarcomere length. Over the range of sarcomere lengths investigated, the predicted values coincide with the experimental data. The ability of the model to predict the data demonstrates the applicability of Donnan equilibrium theory to measurements of electrochemical potential from liquid-crystalline systems.     

Sarcomere structures in the rabbit psoas muscle as revealed by fluorescent analogs of phalloidin

Summary The fluorescent analogs of phalloidin (rhodamine-and fluorescein-phalloidin) bind tightly to the skinned fibres of rabbit psoas muscle at essentially the same sites as phalloidin and mainly stain the known regions of actin localization in the sarcomere: the thin filaments and Z bands. On both sides of the Z bands, unstained zones were observed, suggesting the presence of proteins tightly bound to the thin filaments. In myofibrils which are stretched to such an extent that the actin and myosin filaments do not overlap, stained bands could also be seen at the myosin-band border, which suggests the localization of actin at these sites.     

Cloning, expression, and protein interaction of human nebulin fragments composed of varying numbers of sequence modules.

Nebulin, a family of giant myofibrillar proteins of 600-900 kDa, contains a large number of highly conserved sequence repeats of 31-38 amino acids. To investigate the significance of this repeat, human skeletal muscle nebulin cDNA fragments encoding two, six, seven, eight, or fifteen repeat modules were expressed in high yield as nonfusion proteins in Escherichia coli with the pET3d plasmid vector. F-actin cosedimentation and solid phase binding assays demonstrated that all nebulin fragments, except the smallest two-module 67-mer, bound to muscle actin with high affinity under physiological ionic conditions. Solid phase binding assays also revealed that a six-module fragment, NB5, binds to myosin and C-terminal protein but fails to bind to tropomyosin, troponin, and tubulin. Furthermore, the binding of NB5 to actin was inhibited by both tropomyosin and troponin. Immunoelectron microscopic localization of NB5 indicated that this N-terminal region fragment is situated near the distal end of thin filaments in the sarcomere. These results indicate that nebulin is a giant protein with an unprecedently large number of actin-binding sites along its length and is anchored at the C terminus to the Z line in the sarcomere. Nebulin may function as a multifunctional template protein that regulates the length of thin filaments and participates in muscle activities by interacting with actin and myosin filaments in the sarcomere of skeletal muscles.     

The stiffness of skeletal muscle in isometric contraction and rigor: the fraction of myosin heads bound to actin.

Step changes in length (between -3 and +5 nm per half-sarcomere) were imposed on isolated muscle fibers at the plateau of an isometric tetanus (tension T0) and on the same fibers in rigor after permeabilization of the sarcolemma, to determine stiffness of the half-sarcomere in the two conditions. To identify the contribution of actin filaments to the total half-sarcomere compliance (C), measurements were made at sarcomere lengths between 2.00 and 2.15 microm, where the number of myosin cross-bridges in the region of overlap between the myosin filament and the actin filament remains constant, and only the length of the nonoverlapped region of the actin filament changes with sarcomere length. At 2.1 microm sarcomere length, C was 3.9 nm T0(-1) in active isometric contraction and 2.6 nm T0(-1) in rigor. The actin filament compliance, estimated from the slope of the relation between C and sarcomere length, was 2.3 nm microm(-1) T0(-1). Recent x-ray diffraction experiments suggest that the myosin filament compliance is 1.3 nm microm(-1) T0(-1). With these values for filament compliance, the difference in half-sarcomere compliance between isometric contraction and rigor indicates that the fraction of myosin cross-bridges attached to actin in isometric contraction is not larger than 0.43, assuming that cross-bridge elasticity is the same in isometric contraction and rigor.