Force velocity relationship | S&C Research
When our muscles are stretched to the ideal length, it can maximize UExcel Weather and Climate: Study Guide & Test Prep Therefore, the muscle is stretched to its resting length within the body. The effect of resting fiber length on muscular contraction is referred to as the length-tension relationship. generation and power output vary with changes in body temperature. impaired at low muscle temperatures [5, 6] and voluntary muscular force . also layered with vinyl sheet and lay flush with the walkway. Gait Protocol. .. fatigue on eccentric and concentric force-velocity relationships in human muscle,”. J Appl Physiol Respir Environ Exerc Physiol. Apr;42(4) Temperature and force-velocity relationship of human muscles. Binkhorst RA, Hoofd L.
Computational models that relate these relationships to the underlying anatomy provide a means to validate theories of muscle physiology. Computational models that accurately capture these relationships are an essential component of complete models of musculoskeletal systems see Musculoskeletal Mechanics and Modeling.
Neural control of muscles arises from a combination of commands from the brain as well as feedback from the periphery that are integrated by interneuronal circuits in the spinal cord.
Muscle itself is a major source of the peripheral feedback, as it possesses many specialized mechanoreceptors that are sensitive to stretch and tension see Proprioceptors and Models of Transduction.
Both the stretch and tension experienced by muscle depend on the motion of the skeletal segments to which they attach; for a detailed explanation of the interactions among muscles, the skeleton, and environmental objects, see Musculoskeletal Mechanics and Modeling. The physiological properties of muscle can change over time.
In the short term, force generation may increase or decrease as a result of chemical changes associated with potentiation and fatigue, respectively. In the longer term, muscle morphology and physiology may change as a result of trophic responses to patterns of use. All of these effects tend to be specific to the various muscle fiber types, making it important to develop muscle models that reflect the subpopulations of fiber types and the relative recruitment of the motoneurons that control them.
Relevant physiology Each muscle is controlled by a group of motoneurons known as a motor pool or motor nucleus. All motoneurons in such a pool generally receive the same drive signals, although there are some exceptions Loeb and Richmond, Each motoneuron along with all of its muscle fibers is defined as a motor unit, whose recruitment and firing rate in response to the same drive varies substantially depending on the size and impedance of the motoneuron see Frequency-recruitment.
Larger motoneurons tend to innervate larger numbers of larger diameter muscle fibers. The force production and corresponding energy consumption of each motor unit depend mainly on its firing rate, muscle fiber length and velocity. Because muscle fibers within a motor unit generally have the same contractile properties, they can be lumped into one mathematical entity whose force generating capacity is proportional to the total cross-sectional area of its fibers and whose energy consumption is proportional to their volume.
All of the muscle fibers in all of the motor units of a given muscle tend to move together, experiencing the same sarcomere lengths and velocities. Because of this homogeneity, most of the experimental phenomena related to contraction of whole muscle can be explained by processes occurring at the sarcomere level. The sarcomere is the basic unit of the contractile apparatus. It is demarcated at its ends by thin Z-plates from which a matrix of thin filaments of actin project in each direction.
These interdigitate with a matrix of thick filaments of myosin that are held in the center of the sarcomere by strands of the highly elastic connectin titin filaments that tether the thick filaments to the Z-plates. In order for muscles to contract, protrusions from myosin called myosin heads must first be cocked to a relatively high-strain configuration while bound weakly to actin and then attached more tightly to neighboring binding sites on the actin.
The resulting cross-bridges act like springs that pull on the thin filaments. The metabolic energy required to cock the myosin head is provided directly by ATP molecules that are present in the sarcoplasm. In order for myosin heads to attach to neighboring binding sites, a regulatory protein called tropomyosin that normally occludes actin binding sites must undergo a conformational change.
This occurs indirectly through binding of calcium to troponin, a relatively smaller molecule that is bound to tropomyosin at regular intervals along its length. When calcium binds to troponin, a local conformational change is induced that exposes nearby actin binding sites so that the cocked myosin heads can attach to form cross-bridges. The normally relaxed state of inactive muscle is achieved by ATP-powered pumping of the calcium out of the sarcoplasm and into a network of vesicles in the sarcoplasmic reticulum called longitudinal tubules, preventing cross-bridge formation.
When the muscle is activated, calcium is released from cisterns in these longitudinal tubules in response to action potentials elicited in the muscle fibers as a result of a chemical synapse with the motor axons. These action potentials propagate along the cell membrane and its invaginations deep into the muscle fiber called transverse tubules. Some phosphate groups are hydrolyzed from the regenerated ATP and bond with creatine to form phosphocreatine PCr through the creatine kinase reaction.
PCr functions as an additional energy reservoir for contractions and is also thought to facilitate the transport of high energy phosphates from the mitochondria to the myofilaments. The model of muscle presented here attempts to represent each of these structures and processes as explicit terms in the set of equations that comprise the model. Structure of the model The two major classes of muscle models are Hill-type and Huxley models that differ mainly in their representation of the contractile element.
Classical Hill-type models are phenomenological, employing arbitrary mathematical functions that relate experimental conditions i. Phenomenological models simply reflect the data set used to build them so they are less likely to be valid for untested conditions. Data for classical Hill models originate from maximally stimulated muscle and therefore do not account well for muscle force under physiological firing rates Perreault et al.
By contrast, Huxley-type models Huxley,Zahalak,Ma and Zahalak, are mechanistic,; the mathematical functions that comprise them are derived directly from a hypothesized mechanism of cross-bridge dynamics.
The extent to which these types of models generalize depends on the validity of the hypothesized mechanisms. Furthermore, these models tend to be complex, computationally expensive, and contain parameters that are difficult to obtain experimentally.
- Length-Tension Relationship in Skeletal Muscle
- Force velocity relationship
The structure of the muscle model presented here, called Virtual Muscle, has both mechanistic and phenomenological elements to maximize generality and computational efficiency. The model is an assembly of phenomenological models, each characterizing a major physiological process underlying muscle contraction Figure 1.
Modeling physiological processes independently as opposed to the aggregate behavior avoids over-fitting data to specific preparations and therefore improves the likelihood that the model will be valid under untested conditions. The data driving the model originate from experiments on mammalian muscle that were designed to isolate specific processes that were then quantified. These include a wide variety of preparations with various muscle fiber architectures, but the model parameters have been normalized to dimensionless variables that depend on the constant structures of the sarcomeres, which are highly similar across all mammalian skeletal muscles.
Specific models of muscle can be created by expressing their inputs in terms of these dimensionless variables Zajac, and converting the outputs back to physical units using architectural and other muscle specific parameters see Parameters needed to model specific muscles. The model estimates force and rate of metabolic energy consumption as a function of time in response to neural excitation, muscle length and velocity, across the full range of physiological conditions. For an example of force predictions and corresponding validation analyses, see Figures 10 and 11 in Brown and Loeb ; for energy predictions, see Figure 7 in Tsianos et al.
Independently modeled physiological processes and their contribution to muscle force production and energy consumption. The model accounts for the differences among physiological processes of different fiber types, which enables simulating behavior of muscle with arbitrary fiber compositions.
To model the force generating properties of individual fiber types, studies were performed on whole muscles that happen to be composed almost entirely of one fiber type. The feline soleus muscle was used for the slow-twitch model Joyce et al. The experiments were performed on anaesthetized cats at 37 degrees C with perfusion and innervation intact.
Obtaining energetics data from whole muscle in vivo is challenging, so these experiments are performed typically on isolated fiber bundles.
Poorly Understood Aspects of Striated Muscle Contraction
The energetics model was therefore based on in vitro studies performed on mouse soleus slow-twitch and extensor digitorum longus EDL; fast-twitch fiber bundles Barclay,Barclay et al.
These experiments were carried out at 25 degrees C because muscle fibers in these types of preparations are prone to damage at higher temperatures. Increasing the temperature has only a modest effect on energetics Barclay et al. The membrane potential induced by the input current depends on cell resistivity, which is larger for smaller motoneurons that innervate fewer muscle fibers. For this reason, as the drive increases, small motor units are recruited first, followed by larger motor units having higher recruitment thresholds Henneman and Mendell, Increasing the drive past the threshold of a given motoneuron results in higher firing rates until the drive saturates, at which point firing rate also saturates Monster and Chan, This monotonic relationship can be approximated by a line whose slope depends on the recruitment threshold of the motor unit see Figure 2.
Motor Unit Recruitment and Modulation. The plot top emphasizes the relatively higher recruitment threshold of larger motor units e. The schematic bottom provides a mechanistic explanation for this phenomenon, namely the size principle. The same synaptic current produces a smaller excitatory post synaptic potential EPSP in the larger motor neuron because its input impedance is lower. Therefore, larger input currents are necessary for overcoming the threshold for action potential generation and motor unit recruitment.
Principles of Neural Science. Calcium kinetics are modulated by other binding and buffering agents found in the muscle fiber and its sarcoplasmic reticulum, particularly parvalbumin and calsequestrin Cannell and Allen,but these processes have not been incorporated into the model presented here. Other factors that influence the finer details of the sigmoidal relationship are outlined in Length dependency of cross-bridge dynamics.
At low firing rates, the calcium released by each action potential is completely cleared before the next pulse, resulting in small, discrete twitches.
Temperature and force-velocity relationship of human muscles.
As firing rate increases, the calcium released by each pulse starts to accumulate to higher concentrations depending on the rate of calcium reuptake, allowing the calcium to diffuse further and expose more cross-bridge binding sites.
Eventually, the calcium concentration in all parts of the muscle fiber and throughout the entire excitation interval becomes sufficient to expose all binding sites and activation plateaus even if firing rate continues to increase. This is called a tetanic contraction. Calcium Kinetics and Cross-bridge Activation. The effect of firing rate on isometric force output of muscle is shown from zero to tetanic activation. The trace is a best-fit curve for data from whole cat caudofemoralis muscle at 37 degrees C Brown et al.
Blood supply was intact and the muscle was activated via asynchronous stimulation of five bundles of motoneuron axons. Overlaid graphics highlight underlying processes at the sarcomere level that give rise to the behavior. Only a small portion of the sarcomere is shown here due to symmetry.
Thick horizontal bars represent thick filaments and thin horizontal lines represent thin filaments. Vertical bars correspond to Z-disks.
The small red spheres are calcium ions whose bond with actin sites is indicated by short black lines that link them and a small red dot overlaid on top of the actin grey dots indicate inactive binding sites. If a cross-bridge is formed then the small ovals in the figure representing myosin heads are colored blue and are in a cocked configuration large angle with respect to their neck region.
The equation below captures the effects of firing rate on cross-bridge activation Brown et al. Y corresponds to yielding behavior exhibited by slow-twitch fibers and S corresponds to sag observed in fast-twitch fibers. Both phenomena relate to cross-bridge activation through hypothesized mechanisms discussed in the corresponding sections see Sag and Yield.
See Tsianos et al. Time constant parameters were obtained from data and analyses performed by Brown and Loeb, This is thought to occur due to an increase in the rate of calcium reuptake, which effectively reduces the concentration of calcium in the sarcoplasm and hence the number of crossbridges that can form see Relevant Physiology. Experimental data depicting the sag phenomenon in response to constant stimulation for several frequencies.
The traces depict actual data recorded from whole cat caudofemoralis muscle at 37 degrees C Brown et al. The graphics on the bottom portion illustrate the effect of increasing the rate of calcium reuptake on calcium concentration and cross-bridge formation. It produces large forces at intermediate lengths and relatively smaller forces at either shorter or longer lengths Gordon et al. At some intermediate length, typically referred to as optimal length, overlap between actin and myosin filaments is maximal.
Therefore, the number of cross-bridges that could form and the contractile force are also maximal. Myofilament overlap is less at longer lengths so the number of potential cross-bridges and force are also smaller. Muscle force decreases at incrementally longer lengths until the point at which no more cross-bridges can form and muscle can no longer produce active force.
At relatively shorter lengths, actin filaments start to slide passed each other i. At lengths less than 0. This passive component of the force-length relationship is modeled separately and described in Thick filament compression.
This passive component is also modeled separately and described in Parallel elastic element. Muscle force dependence on fascicle length is shown for tetanically stimulated muscle.
The trace is a best-fit curve for data from whole cat soleus muscle at 37 degrees C Brown et al. Nerve and blood supply was intact and the muscle was activated via cuff electrodes placed around the tibial nerve. Overlaid graphics depict myofilament overlap and effects on cross-bridge formation for different lengths.
Note that the FL relationship does not include the steeper decline at the shortest sarcomere lengths that arises from thick-filament compression, as described below. The force corresponding to optimal length is by definition maximal so normalizing all force profiles to maximal isometric force under tetanic conditions provides a useful reference point. The equation capturing the normalized relationship is shown below. Shortening muscle is accompanied by motion of the attached cross-bridges toward relatively less strained configurations Ford et al.
Larger rates of shortening increase the probability that a given cross-bridge will be in a low-strain configuration; therefore, the force generated by the entire population is lower overall.
By contrast, cross-bridges in lengthening muscle become relatively more stretched Piazzesi et al. The graphics illustrate the hypothesized mechanisms for force enhancement relative to isometric in the lengthening case and depression in the shortening case. Note that in reality myosin heads do not move synchronously as shown in the figure because they are subject to thermodynamic noise and other stochastic events; the graphic illustrates their configuration on average.
The equation below is included in the model to account for this phenomenon. Note that the force-velocity relationship also depends on muscle length. See Potentiation for a detailed description of this phenomenon as well as a mechanistic explanation. The difference in tetanic force relative to isometric for shortening and lengthening conditions may also be attributed to some extent to a change in the number of crossbridges attached.Explanation of the Force Velocity Relationship - Meaning and Implications
In fact, it has recently been concluded that the force-velocity relationship can be explained almost entirely by a change in cross-bridge number rather than cross-bridge strain Brunello et al. This interpretation, however, is based on estimates of fiber stiffness that may be inaccurate.
Muscle fiber stiffness is measured by applying length perturbations to activated muscle Ford et al. Stiffness is then simply calculated as the magnitude of the force change divided by the magnitude of the imposed length change. To estimate the contribution of crossbridge stiffness, which is proportional to the number of crossbridges attached, it is often assumed that myofilament stiffness can be modeled as a linear spring in series with another spring representing the lumped influence of all crossbridges on stiffness.
However, it is likely that myofilament stiffness is nonlinear Irving et al. Crossbridge stiffness has also been shown to be highly nonlinear from measurements performed at both the molecular Kaya and Higuchi, and fiber level Nocella et al. It is plausible that stiffness changes during shortening and lengthening contractions reflect a change in cross-bridge stiffness resulting from change in strain rather than a change in the number of cross-bridges attached.
It is also worth noting that these interpretations of the Force-Velocity relationship are based on experiments on frog muscle held at a temperature near 0 degrees C, which is substantially lower than physiological temperature. The Force-Velocity relation is affected significantly by temperature Rall and Woledge, A given increase in shortening velocity leads to a smaller decrease in force for higher temperatures.
This could reflect a smaller effect of contractile velocity on the number of crossbridges attached at this temperature, which can be explained by an increase in the rate of crossbridge attachment.
It may also reflect cross-bridges being in a relatively stiffer configuration, given the larger strain experienced by the cross-bridges at higher temperatures Ford et al. In either case, the underlying mechanism will not affect the validity of the force model because the force-velocity relation is obtained directly from empirical data rather than derived from a hypothesized mechanism.
The number of crossbridges formed does affect the energetics of contraction, however. For submaximal levels of activation, slow-twitch muscle could experience yielding where the number of cross-bridges decreases substantially for both the shortening and lengthening condition. This phenomenon and model have been described in Yield. Interactions among length, velocity and activation Length dependency of cross-bridge dynamics Cross-bridge activation increases with length.
Theoretically, this can result from either an increase in the number of cross-bridges attached or an increase in cross-bridge strain, but cross-bridge strain has been shown to be similar across sarcomere lengths Gordon et al. The number of cross-bridges increases with higher rate of cross-bridge attachment, lower rate of cross-bridge detachment, or higher concentration of calcium in the sarcoplasm.
Calcium concentration, however, has been shown to be independent of length Balnave and Allen, Changing length of the sarcomeres has effects on the rate of attachment and detachment of cross-bridges to activated and exposed binding sites on the actin. Because a muscle fiber does not gain or lose volume as it changes length, the fiber must have a larger diameter when it is at a shorter length.
The hexagonally packed lattice of myofilaments in each sarcomere will then be more widely spaced, changing the distance between the myosin heads and the thin filaments where they must bind to form cross-bridges. The heads are located on the ends of hinged arms containing myosin light chains, which are canted away from the longitudinal axis of the thick filament.
When the sarcomere is at a long length, the myosin heads barely fit between the thick and thin filaments, so they are close to and can bind rapidly to form cross-bridges. The small distance between the myosin heads and actin binding sites also increases the strength of the actomyosin bond, which would reduce the probability of cross-bridge detachment. At short lengths, the myosin heads are less favorably disposed and their binding may also be affected adversely by the double-overlap of the thin filaments.
These effects are particularly large at lower levels of activation, where activated binding sites are more scarce on the thin filaments. The net result is shown in Figure 7, which is based on data and models first described in Brown et al.
Length Dependence of Cross-bridge Activation. Cross-bridge activation is measured as the percentage of maximal force generated for a given amount of myofilament overlap i. Fascicle length was fixed at 0.
In general, a larger portion of the available contractile machinery is activated for the same firing rate when the muscle is stretched. The precise slope of this relationship, i. A low firing rate activates only a small portion of the binding sites on the actin filaments. An incremental increase in the firing rate results in a relatively larger increase in the number of cross-bridges formed. This is probably due to a relatively larger amount of exposed binding sites per active troponin molecule; this mechanism is known as cooperativity Gordon et al.
Activation of troponin induces a local conformational change in tropomyosin, exposing neighboring binding sites on the actin. When more troponin molecules are activated along tropomyosin, a more global conformational change could be occurring, thus, freeing up more actin binding sites.
Crossbridge formation itself may also facilitate exposure of adjacent binding sites by inducing relative motion between actin and tropomyosin. Length dependency of calcium kinetics Changing length of the sarcomeres has effects on the calcium kinetics that govern activation. The cisterns from which the calcium is released appear to be tethered to the Z-plates at a location that is near the middle of the actin-myosin overlap when the muscle is at optimal length Brown et al.
At longer lengths, calcium has to diffuse over longer distances to get to the actin binding sites, which takes more time. Thus the rise times of the activation are longer see Brown et al. As mentioned in the previous section Length dependency of cross-bridge dynamicsit is likely that cross-bridge attachment rate increases and detachment rate decreases with length and would contribute to the decrease in muscle relaxation time observed at longer lengths Brown and Loeb, aRassier and Macintosh, Potentiation It was originally observed that the force generated by a single twitch of a fast-twitch muscle was much larger after a brief tetanic contraction than before Hughes, In fact, the potentiated state tends to prevail for minutes after a few, brief trains of stimulation at physiological rates, suggesting that the normal operating state of fast-twitch muscle is the potentiated state and that the conditions observed after a long period of quiescence reflect a dispotentiated state Brown and Loeb, Furthermore, the dispotentiated state exhibits rather odd behavior such as a pronounced rightward shift in the shape of the force-length curve at subtetanic frequencies.
For these reasons, the model of the fast-twitch muscle fibers presented here captures their behavior in the fully potentiated state. Acta Physiologica Scandinavica, 2 Intended rather than actual movement velocity determines velocity-specific training response.
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Muscle Physiology and Modeling - Scholarpedia
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