The authors have declared that no competing interests exist.
Conceived and designed the experiments: CDW MR TLD. Performed the experiments: CDW. Analyzed the data: CDW TLD. Contributed reagents/materials/analysis tools: CDW TLD. Wrote the paper: CDW MR TLD.
We most often consider muscle as a motor generating force in the direction of shortening, but less often consider its roles as a spring or a brake. Here we develop a fully three-dimensional spatially explicit model of muscle to isolate the locations of forces and energies that are difficult to separate experimentally. We show the strain energy in the thick and thin filaments is less than one third the strain energy in attached cross-bridges. This result suggests the cross-bridges act as springs, storing energy within muscle in addition to generating the force which powers muscle. Comparing model estimates of energy consumed to elastic energy stored, we show that the ratio of these two properties changes with sarcomere length. The model predicts storage of a greater fraction of energy at short sarcomere lengths, suggesting a mechanism by which muscle function shifts as force production declines, from motor to spring. Additionally, we investigate the force that muscle produces in the radial or transverse direction, orthogonal to the direction of shortening. We confirm prior experimental estimates that place radial forces on the same order of magnitude as axial forces, although we find that radial forces and axial forces vary differently with changes in sarcomere length.
Locomotion requires energy. Very fast locomotion requires a larger amount of energy than muscle can produce in such a short time period, thus muscle must use energy that it previously produced and stored as elastic deformation. Cyclical or repeated movements can be directly powered by muscle, but energy may be conserved in such cases through elastic energy storage. Traditionally we've looked primarily at tendons, insect exoskeletons, and bones as locations where this energy is stored. However, a small but growing body of literature has recently suggested the backbone filament proteins in muscle act as elastic storage locations. We suggest that the myosin motors themselves are capable of storing more energy than the filaments, energy that may be released to power very fast movements or reduce the cost of cyclical movements. We further suggest that this energy is stored in the radial deformations of myosin motors, in a direction that is perpendicular to the axis of muscle shortening.
Strain energy storage in muscle systems is most often associated with stretched tendons or other elastic supporting materials
However, recent work suggests that in certain situations the cross-bridges may be locked onto muscle' thin filaments, frozen into a lattice that can act to store energy
Cross-bridges are more often thought of as force generators than energy storage sites. The force generated by individual myosin heads arises from deformations as they form cross-bridges between the thick and thin filaments and undergo a rotation about a lever arm
The one-dimensional cross-bridge model shown in (A) produces force and exists only in the axial direction. The two-dimensional cross-bridge model shown in (B) produces both axial and radial forces, and responds to changes in lattice spacing. A multi-filament model using one-dimensional cross-bridge, shown in (C), is diagrammed as a three-dimensional system but is insensitive to changes in lattice spacing and unable to explore radial force produced during contraction. Using two-dimensional cross-bridges in the same model geometry, in (D), allows the recording of radial forces and altered force dynamics with altered lattice spacing.
Radial force was observed during contraction in intact muscle fiber experiments dating back to the 1950s
Radial force may have functional implications. The internally generated radial force is a partial determinant of fiber radial compliance
In addition to the more commonly analyzed axial forces, the model presented here addresses both radial force generation and the strain energy in the filaments and cross-bridges of the contractile lattice. These phenomena are linked, and are results of deformation of cross-bridges in the axial and radial directions. The interdependence of these properties is uniquely addressable using spatially explicit models of muscle contraction with lever-arm myosin geometries (
Below we present results for simulations at the level of a half-sarcomere, the smallest fully-regulated component of muscle. Our half-sarcomere is composed of springs representing four myosin (thick) and eight actin (thin) filaments, arranged with boundary conditions which provide a semi-infinite lattice (
Radial forces produced within the half sarcomere are both large and correlated with energy storage. Our model monitors radial forces produced by lever arm cross-bridge models composed of an angular (or torsional) and an extensional (linear) spring (
In the fully activated conditions of our simulations, both the axial and radial forces quickly rise to an asymptotic maximum (
The mean (lines) and standard deviations (shaded regions) of axial and radial forces as they develop at a sarcomere length of
The radial force, at all sarcomere lengths, is of the same order of magnitude as the axial force (
Asymptotic maxima of 10 runs at each sarcomere length with standard deviation. Radial and axial forces obey similar scaling trends across the sarcomere lengths and lattice spacings of a classic length-tension curve. The level of radial force varies from 2.4 times the level of axial force at extremely short sarcomere lengths to 0.9 times the axial force at the longest sarcomere lengths. The radial force plateau ends at a shorter sarcomere length than does axial force plateau.
The majority of strain energy stored in the complete contractile lattice of filaments and cross-bridges is partitioned in the cross-bridges (
The energy stored in the springs comprising the cross-bridges and filaments changes, much as force does, with sarcomere length. (A) As sarcomere length increases, the energy stored across all cross-bridges rises and falls more steeply than does the energy stored in the filaments. (B) At all lengths, the energy stored in the cross-bridges comprises more than 3/4 of the sarcomere's strain based energy. (C) The energy stored in the thick and thin filaments is approximately equal, while the extensional spring of the cross-bridges stores the major share of the energy at all sarcomere lengths.
The elastic energy storage may be more finely parsed: into the components located in each of the two springs constituting every cross-bridge and the components in each of the two filament types (
Energy stored in the cross-bridges follows the radial force produced by the system (
As the sarcomere shortens below
All energy present in the isometrically contracting half-sarcomere derives from the hydrolysis of ATP. This permits a direct comparison of the energy input to the system, as measured by the consumption of ATP, to the energy stored across all filaments and cross-bridges. The fraction of energy stored is shown to change as sarcomere length drops below
The role that muscle' radial geometry plays in determining its functioning is still poorly understood. It is difficult to experimentally measure the forces muscle generates in the radial direction and the strain and energies which result from such forces. The studies that have attempted to measure radial forces have all done so indirectly, through back-calculating from changes in radial stiffness or lattice spacing changes on activation
The elements of the sacomere's contractile lattice, cross-bridges as well as thick and filaments, are storing a substantial amount of energy. At peak energy levels where the 16% of bound cross-bridges in our model store
This strain energy is primarily stored in the cross-bridges—rather than in the thick and thin filaments—despite the turnover and energy dissipation inherent in the our model of cross-bridge kinetics
Radial force' role in muscle remains unclear. Radial force may simply be a byproduct of the motor and filament geometry which has evolved to generate force or it may produce a useful effect. The high correlation between radial force and strain energy stored in the cross-bridges may indicate that radial force and distortion act as an energy storage mechanism which permits the cross-bridges to store more strain based energy than the thick and thin filaments. It is possible that radially associated energy could then be redirected to produce axial force, much as happens when energy is stored in the deformation of elastic solids. Such a mechanism would provide a means to store the energy powering after-stretch transient shortening, the shortening of recently stretched muscle against a load equal to its maximal isometric force
However, radial strain based energy storage will not necessarily register as force at the filament ends. As such, it may be difficult to address in experiments, although radial stiffness observations suggest a means by which such tension and energy storage could be quantified
The variable energy retention efficiencies shown in
The non-constant storage of input energy means that changing the degree of filament overlap or lattice spacing affects the amount of energy stored in muscle and thus the amount of energy which may be released to power contraction. Thus it is possible that operating at different sarcomere lengths could change the efficiency with which muscle retains energy or dissipates it, driving the muscle towards functioning as a spring or a break.
Energy storage in muscle is still a relatively unexplored field. The highly structured and three-dimensional nature of muscle makes it likely that, historically, we have overlooked forms of energy storage and efficiency regulation. This work points towards cross-bridges as a site where energy can be stored, either in preparation for use in rapid movements or to reduce the energy requirements of cyclical movements. Further investigation of how energy is partitioned between sub-sarcomeric structures and of how radial force is generated will continue to expand our understanding of these new mechanisms. Particularly, this work may help us to understand cases where energy is stored as deformations in an axis orthogonal to that of the direction of muscle shortening, such as in a proposed mechanism by which the heart stores elastic strain introduced into transverse fibers during filling
Our spatially-explicit model of the half-sarcomere represents the thick and thin filaments as chains of springs connecting each myosin crown or actin-binding site
Four thick and eight thin filaments are arranged in an evenly spaced hexagonal lattice with toroidal boundary conditions. As shown in
Lattice spacing changes with sarcomere length to maintain a constant lattice volume. Thus lattice spacing separating the faces of adjacent thick and thin filaments (
Along each thick filament are 60 myosin crowns, with three myosin heads per crown. The myosin heads on a given crown are azimuthally rotated by 120 degrees from their neighbors. The crowns are grouped into a three crown, 43 nm repeating pattern
Each thin filament is made up of two actin strands. Each strand hosts 45 actin binding-sites giving a whole filament 90 actin binding-sites
Our cross-bridges are comprised of one torsional spring and one extensional spring
Inefficiency in converting, through ATP hydrolysis, chemical to mechanical energy during state transitions is accounted for as distortions of the cross-bridge. This inefficiency is manifest as heat. Additionally as we suggest below, mechanical strain energy which drives motion may also be returned as recoil of cross-bridges or filament backbones.
The binding of an individual myosin head is determined by the distance to the nearest available binding site and energy landscape created by the properties of the head' constituent springs. The process is one of perturbation, distance calculation, and stochastic attachment. A myosin head is perturbed with a random Boltzmann distributed energy, providing a new myosin tip location
The thick and thin filaments are coupled together by the cross-bridges. Each bound cross-bridge both generates and transmits force. This coupling yields a three dimensional network of springs.
We solve for the root location of our spring-network at each time-step. The root is the set of locations of actin binding-sites and myosin crowns that provides no net axial force at any internal point in the spring-network. A modified form of the Powell hybrid method allows the actin and myosin locations to iteratively settle into their solution values
At each time-step, actin and myosin locations are allowed to settle in the axial dimension while being held rigidly in the radial dimension. The total axial force (
Future models may treat the filaments as radially-deformable axially-tensioned beams subject to filament persistence length, electrostatic effects, and viscous stresses and thus be able to permit radial movement. Radial bending or deformation of the thick and thin filaments could potentially reduce the level of radial force within the lattice by increasing lattice spacing disorder. Reduced radial force has the potential to affect the partitioning of energy between the cross-bridges and filaments, shifting energy stored in cross-bridge deformation to a newly-created radial deformation component of the thick and thin filaments' energies. Radial bending of the filaments and subsequent changes in the distribution of axial and radial forces will be resisted by the highly constrained nature of the sarcomere lattice as well as the inherent stiffnesses of the filaments themselves.
The energy in a cross-bridge or filament is the sum of the energy in every spring in that cross-bridge or filament. Thus the energy of a single cross-bridge (
A simulated contraction follows the course described in the diagram shown in
The model was allowed to complete 10 contractions (starting from unbound cross-bridges) for every set of input parameters, each continuing for 400 ms (400 time-steps at 1 ms resolution). The asymptotically developed forces and energies were calculated as the mean of the force produced over the last 50 ms.
These simulations took place on a dynamically created cluster of spot-priced machine instances in Amazon's EC2 service (
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The authors would like to thank Simon Sponberg for statistical advice and Nicole George for helpful discussions.