Can Strain Dependent Inhibition of Cross-Bridge Binding Explain Shifts in Optimum Muscle Length?

Can Strain Dependent Inhibition of Cross-Bridge Binding Explain Shifts in Optimum Muscle Length? Abstract Skeletal muscle force is generated by cross-bridge interactions between the overlapping contractile proteins, actin and myosin. The geometry of this overlap gives us the force–length relationship in which maximum isometric force is generated at an intermediate, optimum, length. However, the force–length relationship is not constant; optimum length increases with decreasing muscle activation. This effect is not predicted from actin–myosin overlap. Here we present evidence that this activation-dependent shift in optimum length may be due to a series compliance within muscles. As muscles generate force during fixed-end contractions, fibers shorten against series compliance until forces equilibrate and they become isometric. Shortening against series-compliance is proportional to activation, and creates conditions under which shortening-induced force depression may suppress full force development. Greater shortening will result in greater force depression. Hence, optimum length may decrease as activation rises due to greater fiber shortening. We discuss explanations of such history dependence, giving a review of previously proposed processes and suggesting a novel mechanistic explanation for the most likely candidate process based on tropomyosin kinetics. We suggest this mechanism could change the relationship between actin–myosin overlap and cross-bridge binding potential, not only depressing force at any given length, but also altering the relationship between force and length. This would have major consequences for our understanding of in vivo muscle performance. Introduction Our understanding of skeletal muscle contraction is dominated by the cross-bridge and sliding filament theories. According to these theories, cyclical cross-bridge interactions between the overlapping contractile proteins actin and myosin generate force and act to slide the actin-containing thin filaments and myosin-containing thick filaments past one another (Huxley and Hanson 1954; Huxley and Niedergerke 1954; Huxley 1969, 1974). These theories accurately predict some features of muscle performance, such as the isometric force–length relationship (Gordon et al. 1966) and the isotonic force–velocity relationship (Hill 1938). However, they are entirely unable to explain phenomena such as the history dependence of muscle performance (Abbott and Aubert 1952a; Edman et al. 1982; Rassier and Herzog 2004; Herzog et al. 2006; Joumaa et al. 2012; Minozzo and Lira 2013; Nishikawa 2016) and the non-linear effects of muscle activation on the force–length and force–velocity relationships (Rack and Westbury 1969; Brown et al. 1999; Holt and Azizi 2014; Holt et al. 2014). Hence, these theories do not appear to provide a complete framework for understanding muscle contraction. Considerable attention has been devoted to understanding history dependence; it is well established that isometric force is enhanced following stretch (Edman et al. 1982; Herzog et al. 2006; Minozzo and Lira 2013; Nishikawa 2016), and depressed following shortening (Abbott and Aubert 1952a; Rassier and Herzog 2004; Joumaa et al. 2012). Mechanisms such as actin–titin interactions causing changes in titin stiffness (Edman et al. 1982; Powers et al. 2014; Schappacher-tilp et al. 2015; Nishikawa 2016) and length- and force-dependent changes in contractile and regulatory proteins causing changes in cross-bridge binding probability (Maréchal and Plaghki 1979; de Tombe et al. 2010; Corr and Herzog 2016) have been proposed to explain these phenomena. However, there is little agreement on the mechanism responsible (Schappacher-tilp et al. 2015; Corr and Herzog 2016). In contrast to the extensive study of history dependence, relatively little attention has been paid to the changes in the force–length relationship with activation level. Optimum length, the length at which a muscle can produce the most force, is known to be longer at lower activation levels than at higher activation levels (Rack and Westbury 1969). This activation-dependent shift in optimum length has been attributed to a length dependence of calcium sensitivity (Rack and Westbury 1969; Stephenson and Wendt 1984; Rassier et al. 1999). However, this mechanism does not seem to be entirely responsible for the effect (Holt and Azizi 2014). This study aims to explore potential mechanisms underpinning the relatively understudied activation-dependent shift in optimum length, in order to gain insights into the basic mechanisms of muscle contraction. We propose the novel hypothesis that series compliance within muscles, and the resultant fiber shortening during fixed-end contractions, may contribute to the activation-dependent shift in optimum length. Hence, we suggest that contractile history may affect optimum length, and that the same mechanism may contribute to both shortening depression and the activation-dependent shift in optimum length. Results and discussion Activation-dependent shifts in optimum length and shortening-induced force depression have been studied separately; this work presents a potential link between them (Supplementary Fig. S1). This study combines (1) analysis of existing and novel data in support of the hypothesis that series compliance and fiber shortening contribute to an activation-dependent shift in optimum length; (2) a review of potential mechanisms of shortening-induced force depression; (3) suggestion of a novel mechanism of force depression; and (4) an evaluation of the potential of this mechanism to explain not only force depression, but also the activation-dependent shift in optimum length. The ultimate goal of such a synthesis is to advance our understanding of the required additions to the cross-bridge and sliding filament theories in order to adequately capture muscle performance under dynamic or intact conditions. We suggest that the capacity of any proposed mechanism to explain multiple seemingly unrelated phenomena, such as force depression and the activation-dependent shift in optimum length, makes it a strong candidate for inclusion in our framework for understanding muscle contraction. The force–length relationship and muscle activation Skeletal muscle exhibits a force–length relationship. Maximally activated, isometrically contracting sarcomeres produce peak force at an intermediate, optimum, length. The existence of this force–length relationship was used as early support for the cross-bridge and sliding filament theories (Gordon et al. 1966); optimum length is thought to correspond to optimum actin–myosin overlap, and so to the maximum number of force-generating cross-bridges (Herzog et al. 2010). However, the force–length relationship is not as consistent as we would expect if it were dictated entirely by actin–myosin overlap. The force–length relationship is affected by muscle activation; optimum length is longer at lower activation levels than at higher activation levels (Rack and Westbury 1969; Close 1972; Stephenson and Wendt 1984; Balnave and Allen 1996). This activation-dependent shift has been commonly attributed to a length dependence of calcium sensitivity (Stephenson and Wendt 1984; Rassier et al. 1999), potentially related to changes in myofilament lattice spacing (Yang et al. 1998; Rassier et al. 1999; Fuchs and Smith 2001; MacIntosh 2017), thick and thin filament regulation (de Tombe et al. 2010), and the effect of sarcomeric regulatory proteins (de Tombe et al. 2010). If muscle is more sensitive to calcium at longer lengths, more force may be produced at these longer lengths when activation and calcium are low, despite the reduced actin–myosin overlap. However, while there almost certainly is a length dependence of calcium sensitivity (Stephenson and Wendt 1984; Rassier et al. 1999) that may contribute to the activation-dependent shift in optimum length (Stephenson and Wendt 1984; Rassier et al. 1999), it appears not to be the only mechanism involved. An activation-dependent shift is also thought to have been observed in the absence of calcium-based effects (Holt and Azizi 2014). A calcium-independent effect of activation level on optimum length Muscle is activated, and force is produced, in response to signals from the nervous system. Action potentials trigger the release of calcium from intracellular stores, causing a conformational change in the thin filament and initiating cross-bridge cycling. Muscle activation can be varied either by changing the frequency of action potentials being delivered to a given motor unit (Adrian and Bronk 1929), or by changing the number of motor units recruited (Liddell and Sherrington 1925). Altering either of these parameters will change total muscle force. However, only changing action potential frequency will change intracellular calcium concentrations ([Ca2+]i). In one study, the ability to change muscle activation by changing either action potential frequency or motor unit recruitment was used to explore whether the length dependence of calcium sensitivity was entirely responsible for the activation-dependent shift in optimum length. It was demonstrated that there was a large activation-dependent shift in optimum length independent of the mechanism used to alter activation (Holt and Azizi 2014). It is possible that the technique used here, namely sonomicrometry, overestimates the magnitude of the shift due to its failure to account for the curvature of the fibers in the muscle. However, these data are not the result of incorrect correction for passive force as recently suggested (MacIntosh 2017). The presence of an activation-dependent shift in optimum length independent of changes in [Ca2+]i strongly suggests that this phenomenon cannot be entirely due to a length dependence of calcium sensitivity. Our current understanding of muscle physiology does not readily offer another explanation of this phenomenon. However, based on these data (Holt and Azizi 2014) and previous studies (Street et al. 1966; Ichinose et al. 1997; de Brito Fontana and Herzog 2016), we suggest that series compliance within muscle and the resultant fiber shortening may play a role in this shift. Compliance may contribute to the activation-dependent shift in optimum muscle length The finding that there is a shift in optimum length independent of the method of changing activation level (Holt and Azizi 2014) presents a challenge to the theory that a length dependence of calcium sensitivity is entirely responsible for the activation-dependent shift in optimum length. Here we propose the relatively novel hypothesis that muscle compliance, and the resultant contractile history, could be responsible for this effect. In muscles with elements such as aponeuroses and tendons that provide significant series compliance, muscle fibers will shorten upon activation and stretch these elements during fixed-end contractions. When the passive force in the stretched series compliant elements balances that generated by muscle fibers, fibers will become isometric. At higher activation levels, where muscles produce more force, fibers will shorten further before isometry is reached. Optimum length has been shown to decrease when muscle shortens before contracting isometrically (Street 1966). Hence, in muscles with significant series compliance, optimum length may decrease with increasing activation level due to increased fiber shortening. This would result in an activation-dependent shift in optimum length. There is some evidence in support of this hypothesis from in vivo studies that examined the effect of activation level on the force–length properties of fiber in the human vastus lateralis (Ichinose et al. 1997; de Brito Fontana and Herzog 2016). In these studies, force and length were determined during contractions with ramped increases in activation. The in vivo nature of these studies meant that fibers were acting in series with compliance, and so shortened in proportion to activation and force. The findings from these studies show a shift in optimum length with activation level. However, this shift appeared to be driven by a decrease in optimum length at high activation levels, rather than an increase in optimum length at low activation levels as would be predicted by a length dependence of calcium sensitivity. The joint position, and so the starting fiber length, that gives peak force appears to be the same in all conditions. At higher activation levels, fibers shorten more against series compliance, seemingly resulting in a decrease in optimum length with increasing activation level. The presence of a calcium-independent effect of activation on optimum length (Holt and Azizi 2014), the effect of shortening on optimum length (Street et al. 1966), and the seeming decrease in optimum length at high activation levels (Ichinose et al. 1997; de Brito Fontana and Herzog 2016) leads us to propose that there may be two mechanisms driving an activation-dependent shift in optimum length: (1) a length dependence of calcium sensitivity that increases optimum length with decreasing [Ca2+]i; and (2) a decrease in optimum length with increased shortening, that in compliant muscles will manifest as a decrease in optimum length with increasing activation. In the remainder of this section of the paper we explore potential evidence to support the latter mechanism. Existing data suggest a role for compliance in the activation-dependent shift in optimum length Our initial evidence for compliance contributing to the activation-dependent shift in optimum length comes from a reanalysis of the data presented in Holt and Azizi (2014). In this reanalysis, we looked to confirm the in vivo findings that the length at which muscle was activated to produce maximum force was always constant (Ichinose et al. 1997; de Brito Fontana and Herzog 2016) under more controlled conditions. This would provide stronger evidence that in muscle with series compliance, optimum fiber length decreases with increased activation and shortening. In Holt and Azizi (2014), bullfrog plantaris muscles were connected between a force/length transducer and a fixed point, with sonomicrometry crystals implanted along the length of a fiber. Fixed-end contractions were elicited at a range of lengths sufficient to construct a force–length curve, using several different activation levels. During these fixed-end contractions, fibers shortened as they developed force and stretched the compliant aponeurosis, before reaching force equilibrium and isometry (Fig. 1A, B). The original analysis took force as the peak force that was produced (F) (Fig. 1A) and length as the final length of the fiber (Lfiber) once isometry had been reached (Fig. 1B). Force–length relationships were plotted, and peak force produced by the muscle (F0) and optimum fiber length determined (L0fiber) (Fig. 1C). In the new analysis presented here, we took length as total muscle length (Lmuscle) (Fig. 2B). This ignored any fiber shortening and reflected the initial length of the fibers (Fig. 2B). New force–length relationships were plotted, peak force (F0) and optimum length (L0muscle) determined (Fig. 2C), and linear mixed effects models (lme) used to determine whether there was any effect of activation level on optimum lengths. This analysis shows that while there is a significant effect of activation level on L0fiber (P < 0.001; lme) there is no significant effect of activation level on L0muscle (P = 0.065; lme) (Fig. 3). Hence, the length at which muscle fibers are activated to produce maximum force appears to be constant across activation conditions. This is consistent with our hypothesis that fiber shortening against series compliance may contribute to the activation-dependent shift in optimum length. Fig. 1 View largeDownload slide Sample force (A) and fiber length traces (B) for maximum (solid), and ∼40% activation (dotted), conditions. Force and fiber length values are determined as indicated. The schematic in B indicates the fiber lengths during the fixed-end contraction. Muscle fibers (gray) shortened against the compliant aponeurosis (white) and Lfiber was taken as the final length at which isometry was reached. This shortening is proportional to the muscle activation and force produced (A, B). The force and length values determined were plotted against one another (circles) for multiple fixed-end contractions under maximum (solid), ∼70% (dashed), and ∼40% (dotted) activation conditions, and a third-order polynomial fitted to each data set (C). F0 and L0fiber were then determined for each activation condition. Fig. 1 View largeDownload slide Sample force (A) and fiber length traces (B) for maximum (solid), and ∼40% activation (dotted), conditions. Force and fiber length values are determined as indicated. The schematic in B indicates the fiber lengths during the fixed-end contraction. Muscle fibers (gray) shortened against the compliant aponeurosis (white) and Lfiber was taken as the final length at which isometry was reached. This shortening is proportional to the muscle activation and force produced (A, B). The force and length values determined were plotted against one another (circles) for multiple fixed-end contractions under maximum (solid), ∼70% (dashed), and ∼40% (dotted) activation conditions, and a third-order polynomial fitted to each data set (C). F0 and L0fiber were then determined for each activation condition. Fig. 2 View largeDownload slide Results from the reanalysis of Holt and Azizi (2014). Sample force (A) and muscle length traces (B) for maximum (solid) and ∼40% activation (dotted) conditions. Force was determined as in Fig. 1 and muscle length was taken as the constant length shown in B. The schematic in B indicates how length is determined in this reanalysis compared with the original analysis (Fig. 1B). Length was taken as total muscle length. The force and length values determined were plotted against one another (circles) for multiple fixed-end contractions for maximum (solid), ∼70% (dashed), and ∼40% (dotted) activation conditions, and a third-order polynomial fitted to each data set (C). F0 and L0muscle were then determined. It should be noted that this is the same data as in Fig. 1, simply with length calculated as the muscle length (B) rather than the post-shortening fiber length (Fig. 1B). Fig. 2 View largeDownload slide Results from the reanalysis of Holt and Azizi (2014). Sample force (A) and muscle length traces (B) for maximum (solid) and ∼40% activation (dotted) conditions. Force was determined as in Fig. 1 and muscle length was taken as the constant length shown in B. The schematic in B indicates how length is determined in this reanalysis compared with the original analysis (Fig. 1B). Length was taken as total muscle length. The force and length values determined were plotted against one another (circles) for multiple fixed-end contractions for maximum (solid), ∼70% (dashed), and ∼40% (dotted) activation conditions, and a third-order polynomial fitted to each data set (C). F0 and L0muscle were then determined. It should be noted that this is the same data as in Fig. 1, simply with length calculated as the muscle length (B) rather than the post-shortening fiber length (Fig. 1B). Fig. 3 View largeDownload slide Summary L0fiber (solid bars) (data from Fig. 1) and L0muscle (open bars) (data from Fig. 2) optimum length data for all activation conditions. There is a significant effect of activation condition on optimum fiber length (P < 0.001; lme), but not on optimum muscle length (P = 0.065; lme). Hence, we suggest that the length at which fibers started a contraction (muscle length) at remained relatively constant across activation conditions, but that the increasing fiber shortening with increasing activation drove the observed decrease in the optimum fiber lengths. Fig. 3 View largeDownload slide Summary L0fiber (solid bars) (data from Fig. 1) and L0muscle (open bars) (data from Fig. 2) optimum length data for all activation conditions. There is a significant effect of activation condition on optimum fiber length (P < 0.001; lme), but not on optimum muscle length (P = 0.065; lme). Hence, we suggest that the length at which fibers started a contraction (muscle length) at remained relatively constant across activation conditions, but that the increasing fiber shortening with increasing activation drove the observed decrease in the optimum fiber lengths. Experimental manipulation of compliance affects the activation-dependent shift in optimum length In order to provide a more direct test of the role of compliance and fiber shortening in the activation-dependent shift in optimum length, we determined this shift in high and low compliance conditions. The data from Holt and Azizi (2014) provided a high compliance condition. The bullfrog plantaris muscle contains significant series compliance and fibers shortened against this during fixed-end contractions (Fig. 1B). A low compliance comparison was achieved by repeating this experiment in fiber bundles extracted from the plantaris muscle (Fig. 4) (see Supplementary Materials for full methods). No series compliance was present in this preparation, so unlike in the high compliance condition (Fig. 5B), fibers did not shorten during fixed-end contractions (Fig. 6B). The activation-dependent shift, the ratio of optimum length in the tetanic condition (L0Tet) (high activation) and optimum length in the twitch condition (L0Twi) (low activation), could then be determined for high and low compliance conditions. If fiber shortening against series compliance contributed to the activation-dependent shift, we would expect there to be a greater activation-dependent shift in the high-compliance condition than in the low compliance condition. This is well supported by our results. In the high compliance condition, fibers underwent a shortening strain of ∼30% in tetanic contractions (Fig. 5B), whereas fibers did not shorten in the low-compliance conditions (Fig. 6B). There was a significant interactive effect of activation and compliance on optimum length (P = 0.025; lme) (Fig. 7). In the low compliance condition, twitch optimum length was 8% longer than tetanic optimum length; in the high compliance condition it was 35% longer (see Supplementary Materials for complete results). Fig. 4 View largeDownload slide Schematic representation of high- (A) and low- (B) compliance conditions. The high-compliance condition is the intact bullfrog plantaris muscle in which fibers (gray) operate in series with a large, compliant aponeurosis (white). Fibers shorten, stretching the aponeurosis on activation. The low-compliance condition is fibers extracted from the bullfrog plantaris muscle. Fibers operate isolated from the compliant aponeurosis, and hence do not shorten upon activation. Fig. 4 View largeDownload slide Schematic representation of high- (A) and low- (B) compliance conditions. The high-compliance condition is the intact bullfrog plantaris muscle in which fibers (gray) operate in series with a large, compliant aponeurosis (white). Fibers shorten, stretching the aponeurosis on activation. The low-compliance condition is fibers extracted from the bullfrog plantaris muscle. Fibers operate isolated from the compliant aponeurosis, and hence do not shorten upon activation. Fig. 5 View largeDownload slide Sample force (A) and length traces (B) for high (solid) and low (dashed) activation conditions with high compliance. Force and length values were determined as previously (Fig. 1). The large fiber shortening reflects the high series compliance. These force–length points from multiple contractions were plotted against one another, and a third-order polynomials fitted to each data set (C). F0 and L0 were then determined. Fig. 5 View largeDownload slide Sample force (A) and length traces (B) for high (solid) and low (dashed) activation conditions with high compliance. Force and length values were determined as previously (Fig. 1). The large fiber shortening reflects the high series compliance. These force–length points from multiple contractions were plotted against one another, and a third-order polynomials fitted to each data set (C). F0 and L0 were then determined. Fig. 6 View largeDownload slide Sample force (A) and fiber length traces (B) for high (solid) and low (dashed) activation conditions with low compliance. Low compliance is reflected in the lack of fiber shortening (B). Force–length points from multiple contractions are plotted against one another, and a third-order polynomial fitted to each data set (C). F0 and L0 were then determined. Fig. 6 View largeDownload slide Sample force (A) and fiber length traces (B) for high (solid) and low (dashed) activation conditions with low compliance. Low compliance is reflected in the lack of fiber shortening (B). Force–length points from multiple contractions are plotted against one another, and a third-order polynomial fitted to each data set (C). F0 and L0 were then determined. Fig. 7 View largeDownload slide Summary data showing the activation-dependent shift (L0twi/L0tet) with high (data from Fig. 5) and low (data from Fig. 6) compliance. There was a significant interactive effect of activation and compliance on the activation-dependent shift (P = 0.025; lme), indicating that the shift increased with increasing compliance. Fig. 7 View largeDownload slide Summary data showing the activation-dependent shift (L0twi/L0tet) with high (data from Fig. 5) and low (data from Fig. 6) compliance. There was a significant interactive effect of activation and compliance on the activation-dependent shift (P = 0.025; lme), indicating that the shift increased with increasing compliance. The finding that the activation-dependent shift increases with increasing compliance is consistent with our hypothesis that, in compliant systems, a component of the activation-dependent shift is due to a decrease in optimum length with increasing activation levels as a result of greater fiber shortening. Stronger support for this would be provided by using a system in which series compliance could be artificially varied, and sarcomere length measured. The effect of varying activation by varying both calcium concentration and motor unit recruitment under different compliance conditions could then be determined and mapped to contractile protein overlap. While we advise some caution in the interpretation of these results, taken together with the shift in optimum length independent of changes in [Ca2+]i, the constancy of the length at which the muscle is activated in order to produce maximum force (Fig. 3) (Ichinose et al. 1997; de Brito Fontana and Herzog 2016), and the effect of shortening on optimum length (Street et al. 1966), there is certainly a case to be made that the contractile history of the fiber influences optimum length. It is well established that contractile history influences the magnitude of the force that can be produced (Abbott and Aubert 1952a; Edman et al. 1982; Rassier and Herzog 2004; Herzog et al. 2006; Joumaa et al. 2012; Minozzo and Lira 2013; Nishikawa 2016). However, what has previously been less clear is that contractile history could affect not only the force at any given length, but also the length at which maximum force could be produced. Hence, in the next part of this paper, we review these history-dependent effects, particularly shortening-induced force depression; explore the potential mechanisms underlying them; and discuss whether mechanisms responsible could also be responsible for the activation-dependent shift in optimum length. This will further the debate on potential amendments to cross-bridge and sliding filament theories. Force depression and optimum length The isometric force a muscle can produce following active shortening is well established to be lower than that during a solely isometric contraction at the same length (Abbott and Aubert 1952b; Rassier and Herzog 2004). This history dependence is not predicted by cross-bridge or sliding filament theories. There are multiple proposed mechanisms to explain this phenomenon; however, resolution has proved difficult (Maréchal and Plaghki 1979; Rode et al. 2009; de Tombe et al. 2010; Corr and Herzog 2016), so limiting our ability to extend cross-bridge and sliding filament theories. We suggest muscle fiber shortening against series compliance, interpreted through the phenomenon of shortening-induced force depression, potentially explains some of the activation-dependent shift in optimum length. Hence, the second part of this paper reviews the current state of our understanding of shortening-induced force depression, provides additions to proposed mechanisms, and evaluates their ability to contribute to an explanation of the activation-dependent shift in optimum length discussed in the previous section. Characteristics of force depression Shortening-induced force depression is the persistent reduction in the force a muscle or fiber produces following activation and the subsequent decrease in its length (Abbott and Aubert 1952b). See Rassier and Herzog (2004) for an excellent review of force depression. Force depression is characterized by (1) being long-lasting, (2) disappearing if activation is removed, even briefly, (3) increasing with the distance shortened, and (4) increasing with greater force during shortening (Abbott and Aubert 1952b; Herzog and Leonard 1997; Rassier and Herzog 2004). Force depression lasts several 10s of seconds, provided the muscle remains active and continues to bear force. However, it disappears upon the release of force (Abbott and Aubert 1952b; Herzog et al. 1998). The magnitude of force depression increases with increased shortening distance (Abbott and Aubert 1952b; de Ruiter et al. 1998), and when shortening more slowly and so generating more force (Hill 1938). These findings led to the idea that the magnitude of force depression is correlated to the work done, the product of the force the muscle generates, and the distanced shortened (Granzier and Pollack 1989). However, while these properties are well established, the mechanism responsible remains a matter of debate. Proposed mechanisms behind force depression: a brief review and elaboration Several mechanisms have been proposed, acting singly or in concert, to explain the phenomena of force depression. They have generally fallen into one of several categories: (a) sarcomere length non-uniformity, (b) fatigue product accumulation, or (c) cross-bridge inhibition. One of the earliest proposed mechanisms, non-uniformity of sarcomere length, posits that active shortening prompts stronger sarcomeres to slip over the top of the force–length curve to the short lengths of the ascending limb (Abbott and Aubert 1952b). This mechanism is now considered unlikely due to the presence of force depression on the stable ascending limb and the decrease in stiffness after shortening, as detailed in Rassier and Herzog (2004). Accumulation of fatigue products as a result of the work done during active shortening was suggested as an alternative to the non-uniformity mechanism (Granzier and Pollack 1989). Arguing against a fatigue product mechanism is the inconsistency between the slow clearing of fatigue products and the rapid recovery of full force production following relaxation to a no-tension state (characteristic 2 above) (Abbott and Aubert 1952b; Herzog and Leonard 1997). The remaining explanation for force depression is that cross-bridge formation is inhibited in the region of new filament overlap that arises as a result of active shortening. This explanation supposes that as muscle actively shortens, while the area of overlap between the thick and thin filaments is increasing, fewer myosin heads are able to link to the thin filament and undergo force-generating powerstrokes in the new overlap zone (Maréchal and Plaghki 1979). This mechanism is now considered the likeliest candidate (Rassier and Herzog 2004; Corr and Herzog 2016). However, it is only a partial explanation; there are many means by which cross-bridge formation may be inhibited. Current theories are dominated by the spatial effects of titin during shortening and mechanical effects on the thin filament. A proposed titin-based mechanism for decreased myosin attachment in the new overlap zone suggests that titin directly blocks the formation of new cross-bridges by being dragged between the thick and thin filaments (Rode et al. 2009). This explains the increase in force depression as shortening increases (characteristic 2 above), as greater shortening would allow titin to block a greater number of cross-bridges. Assuming the bound region detaches and recoils when activation is removed, this titin-based mechanism could explain the decrease in force depression upon deactivation (characteristic 3). However, this mechanism is inconsistent with increased force depression as greater force is generated during shortening, or as shortening occurs more slowly (Rassier and Herzog 2004). It has been proposed that myosin heads would detach titin from actin in a velocity dependent fashion, recovering shortening velocity dependence (Rode et al. 2009). This mechanism is inconsistent with the low Reynolds number intra-sarcomere environment: it implies an increase in the energetics of myosin-head–titin collisions as shortening velocity increases which, while intuitively plausible in the macroscopic world, isn’t present in the viscosity dominated regime within the sarcomere. However, this doesn’t rule out shortening-speed-dependent conformational changes in titin that may alter its cross-bridge interactions and recover a role for titin in force depression. We note here that while titin interference in the creation of new cross-bridges is not consistent with all properties of force depression, a single mechanism need not be responsible for the entirety of the force depression phenomenon. Alternatively, actin geometry could be altered by thin filament strain in the region of new overlap, preventing cross-bridge formation through a purely mechanical mechanism (Maréchal and Plaghki 1979; Rassier and Herzog 2004). This would be consistent with the characteristics of force depression listed above. Deformation of the thin filament would remain as long as force is maintained and return to an unstrained configuration when force is removed (characteristics 1 and 2 above). Increasing the distance shortened would increase the strained thin filament section encountered where new cross-bridges did not form (characteristic 3). Greater force during shortening would produce greater strain and thus greater inhibition (characteristic 4). Previous work has considered both axial and azimuthal deformation of actin geometry as a binding inhibition (Rassier and Herzog 2004), but we suggest a tropomyosin-mediated decrease in cross-bridge formation as a pathway to force depression. In this mechanism, decreased cross-bridge formation may come about not as a result of isolated G-actin deformation, but rather due to deformations in the thin filament leading to small changes in the energy landscape tropomyosin navigates as it transitions to its permissive open state. As the thin filament is subject to strain, so is the troponin/tropomyosin regulatory complex. The small actin deformations possible in the axial and azimuthal directions, on the level of disorder that has been shown to increase binding in isolation through breaking binding site symmetry (Daniel et al. 1998), may be sufficient to increase the energy barrier between tropomyosin’s closed or partially-closed and fully activated states such that cross-bridge formation is reduced. Such a reduction in cross-bridge formation would be amplified in areas of the thin filament with no initial overlap as tropomyosin in that region isn’t subject to the cooperative activation amplification that comes with formation of cross-bridges at adjacent sites (Tanner et al. 2012). Thus the effect of strain-dependent tropomyosin kinetics may be limited to the region of new overlap. A possible test of this mechanism could be performed by plating whole or unregulated thin filaments onto a deformable gel and tracking myosin motility under resting and deformed-gel conditions. A potential strain dependency of tropomyosin kinetics offers consistency with the known properties of force depression and is testable through further modeling and experiments. Strain-induced inhibition of cross-bridge binding may explain both force depression and the activation-dependent shift in optimum length The earlier hypothesis that compliance and fiber shortening, and therefore contractile history, could contribute to the activation-dependent shift in optimum length led us to consider whether the mechanisms proposed to explain force depression could also explain this phenomenon. Cross-bridge and sliding filament theories suggest that the variation in force seen across the force–length curve is due to changes in the overlap of actin and myosin filaments (Huxley and Hanson 1954; Huxley and Niedergerke 1954; Huxley 1969, 1974; Herzog et al. 2010). Changes in sarcomere length change overlap move the sarcomere to a new location on the force–length curve, and so result in a change in force. Under isometric, maximally activated, conditions these theories do seem to be adhered to; consistent force–length relationships are observed and changes in filament overlap are responsible for the majority of the variation in force (Gordon et al. 1966; Herzog et al. 2010; Williams et al. 2013). However, in fixed-end contractions, fibers contained within muscles with series compliance will shorten until a force balance is achieved at the length previously referred to as Lfiber. This shortening causes a change in actin–myosin overlap, but one that occurs under tension, hence providing the conditions necessary for force depression. If force depression is the result of strain-dependent inhibition of cross-bridge binding, then the potential for cross-bridge binding is not the same as if Lfiber had been arrived at passively. Active shortening may disrupt the relationship between actin–myosin overlap, and cross-bridge binding and force-generating potential. We suggest therefore that following active shortening, there is an absolute or real amount of filament overlap dictated by sarcomere length, but also an effective filament overlap dictated both by length and the altered cross-bridge binding probability in any new region of overlap (Fig. 8 and Supplementary Fig. S2). Effective filament overlap describes the number of motors generating force, rather than the number of motors with access to the thin filament, and offers a new view of the interaction between active shortening and the force–length curve. Fig. 8 View largeDownload slide Example of force depression induced shifts in location along the force–length curve due to the interaction of MTU series compliance and varying levels of activation. In the schematics the tendon is shown as a dark gray spring, the thick filament and myosin in red (mid-gray), the thin filament in blue (light gray), and the areas of initial and new overlap in green (pale gray) and red (dark grey), respectively. (A) Under high activation the contractile element of a MTU shortens from ℓIH to ℓFH due to series compliance, increasing the thick–thin overlap. The motors in this region have a decreased chance of forming new cross-bridges due to force depression. (B) The same MTU, at a different initial length, is subjected to low levels of activation that result in the contractile element shortening from ℓIL to ℓFL where the final length is the same as in panel A ( ℓFL=ℓFL). As less force was generated by lower levels of activation to reach this final CE length, there is less region of new overlap and thus fewer motors inhibited from forming new cross-bridges by force depression. Despite having identical final MTU lengths, the two cases have different effective overlaps (ϵH or ϵL), the fractions of the thick and thin filaments able to interact and generate force. This places the two cases at different effective points on the force–length curve, shifting the force–length curve in an activation dependent fashion for muscle subject to the series compliance found in the MTU. Fig. 8 View largeDownload slide Example of force depression induced shifts in location along the force–length curve due to the interaction of MTU series compliance and varying levels of activation. In the schematics the tendon is shown as a dark gray spring, the thick filament and myosin in red (mid-gray), the thin filament in blue (light gray), and the areas of initial and new overlap in green (pale gray) and red (dark grey), respectively. (A) Under high activation the contractile element of a MTU shortens from ℓIH to ℓFH due to series compliance, increasing the thick–thin overlap. The motors in this region have a decreased chance of forming new cross-bridges due to force depression. (B) The same MTU, at a different initial length, is subjected to low levels of activation that result in the contractile element shortening from ℓIL to ℓFL where the final length is the same as in panel A ( ℓFL=ℓFL). As less force was generated by lower levels of activation to reach this final CE length, there is less region of new overlap and thus fewer motors inhibited from forming new cross-bridges by force depression. Despite having identical final MTU lengths, the two cases have different effective overlaps (ϵH or ϵL), the fractions of the thick and thin filaments able to interact and generate force. This places the two cases at different effective points on the force–length curve, shifting the force–length curve in an activation dependent fashion for muscle subject to the series compliance found in the MTU. This effect has the potential to interact with muscle activation level. The level of overlap change and force depression both increase with increasing force generation, and are thus activation dependent. Holding series compliance constant and varying initial length and the level of activation, we can reach conditions where we have the same sarcomere length and the same absolute filament overlap, but different levels of force depression thus different levels of effective filament overlap (Fig. 8). This places our two sarcomeres at the same absolute point on the force–length curve but at different effective points. As sarcomere shortening increases monotonically with activation, so too does the gap between our effective overlap and our absolute overlap. This could provide the systematic leftward shift in the force–length curve at higher activation levels seen in Figs. 1–3. Hence, strain dependent inhibition of cross-bridge binding could be responsible not only for force-depression, but also contribute to the activation-dependent shift in optimum length. This ability of strain dependent inhibition of cross-bridge binding to explain both of these phenomena makes it a strong candidate for addition to cross-bridge and sliding filament theories, and its suggestion that compliance and activation could have interactive effects on the force–length relationship has major consequences for how we think about muscle function in vivo. Implications for understanding in vivo muscle performance The force–length relationship, based on cross-bridge and sliding filament theories, has become central to our prediction and interpretation of in vivo muscle performance. It is a major input into musculoskeletal models (Delp et al. 2007; Dick et al. 2017), and has been used to draw the broad conclusion that muscles tune their operating length to maximize some aspect of performance (Rome and Sosnicki 1991; Herzog et al. 1992; Burkholder et al. 2001; Tu 2004; Azizi and Roberts 2010; Rubenson et al. 2012; Arnold et al. 2013; Azizi 2014; Holt and Azizi 2016; Foster and Higham 2017; Nikolaidou et al. 2017). However, the variation in activation level with mechanical demand (Lieber and Brown 1992; Gorassini et al. 2000; Hodson-Tole and Wakeling 2007; Morris and Askew 2010; Vervuert et al. 2013; Seven et al. 2014), and the activation-dependent shift in optimum length (Rack and Westbury 1969; Close 1972; Stephenson and Wendt 1984; Balnave and Allen 1996; Holt and Azizi 2014), suggest these predictions and interpretations may be subject to systematic errors. An attempt to address this was made in Holt and Azizi (2016); in vivo function was mapped to activation-level specific force–length curves, leading to the more nuanced conclusion that optimum length is used in the most demanding behaviors, but that performance may be sacrificed, and sub-optimal lengths used, during less demanding activities. However, while this study attempted to account for the activation-dependent changes in optimum length, the finding that optimum length may depend not only on activation level, but also compliance and contractile history (Figs. 1–3, 5–8) (Street et al. 1966; Ichinose et al. 1997; de Brito Fontana and Herzog 2016), suggests that it did not fully capture the complexity of the situation. Not only does muscle activation level vary widely in vivo, but series compliance varies between species (Azizi 2014) and muscle morphologies (Wilson and Lichtwark 2011), and muscle shortening is a complex interaction between the level of activation of the muscle, its series compliance, and the external load (Griffiths 1991). Hence the “optimum” length for a muscle operating in vivo is a rich and situation-dependent landscape. Conclusions The cross-bridge and sliding filament theories that dominate our understanding of muscle contraction cannot explain many features of muscle performance under complex in vivo conditions, yet there remains a lack of consensus regarding appropriate amendment to these theories. Here we examined the unexplained and understudied activation-dependent shift in optimum length, as a means to advance this process. We suggest that shortening under tension against series compliance during fixed-end contractions may contribute to this phenomenon, and therefore that a common mechanism may be responsible for both the activation-dependent shift in optimum length and another major feature of muscle performance not explained by the cross-bridge and sliding filament theories: shortening-induced force depression. We suggest that strain-induced inhibition of cross-bridge binding in the new region of actin–myosin overlap, potentially mediated by tropomyosin deformation, decreases the force-generating potential of the muscle at its post-shortening length, so depressing force and altering the relationship between force and length in an activation-dependent manner. The ability of this mechanism to explain these seemingly unrelated phenomena makes it an appealing potential amendment to cross-bridge and sliding filament theories. 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Can Strain Dependent Inhibition of Cross-Bridge Binding Explain Shifts in Optimum Muscle Length?

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Abstract

Abstract Skeletal muscle force is generated by cross-bridge interactions between the overlapping contractile proteins, actin and myosin. The geometry of this overlap gives us the force–length relationship in which maximum isometric force is generated at an intermediate, optimum, length. However, the force–length relationship is not constant; optimum length increases with decreasing muscle activation. This effect is not predicted from actin–myosin overlap. Here we present evidence that this activation-dependent shift in optimum length may be due to a series compliance within muscles. As muscles generate force during fixed-end contractions, fibers shorten against series compliance until forces equilibrate and they become isometric. Shortening against series-compliance is proportional to activation, and creates conditions under which shortening-induced force depression may suppress full force development. Greater shortening will result in greater force depression. Hence, optimum length may decrease as activation rises due to greater fiber shortening. We discuss explanations of such history dependence, giving a review of previously proposed processes and suggesting a novel mechanistic explanation for the most likely candidate process based on tropomyosin kinetics. We suggest this mechanism could change the relationship between actin–myosin overlap and cross-bridge binding potential, not only depressing force at any given length, but also altering the relationship between force and length. This would have major consequences for our understanding of in vivo muscle performance. Introduction Our understanding of skeletal muscle contraction is dominated by the cross-bridge and sliding filament theories. According to these theories, cyclical cross-bridge interactions between the overlapping contractile proteins actin and myosin generate force and act to slide the actin-containing thin filaments and myosin-containing thick filaments past one another (Huxley and Hanson 1954; Huxley and Niedergerke 1954; Huxley 1969, 1974). These theories accurately predict some features of muscle performance, such as the isometric force–length relationship (Gordon et al. 1966) and the isotonic force–velocity relationship (Hill 1938). However, they are entirely unable to explain phenomena such as the history dependence of muscle performance (Abbott and Aubert 1952a; Edman et al. 1982; Rassier and Herzog 2004; Herzog et al. 2006; Joumaa et al. 2012; Minozzo and Lira 2013; Nishikawa 2016) and the non-linear effects of muscle activation on the force–length and force–velocity relationships (Rack and Westbury 1969; Brown et al. 1999; Holt and Azizi 2014; Holt et al. 2014). Hence, these theories do not appear to provide a complete framework for understanding muscle contraction. Considerable attention has been devoted to understanding history dependence; it is well established that isometric force is enhanced following stretch (Edman et al. 1982; Herzog et al. 2006; Minozzo and Lira 2013; Nishikawa 2016), and depressed following shortening (Abbott and Aubert 1952a; Rassier and Herzog 2004; Joumaa et al. 2012). Mechanisms such as actin–titin interactions causing changes in titin stiffness (Edman et al. 1982; Powers et al. 2014; Schappacher-tilp et al. 2015; Nishikawa 2016) and length- and force-dependent changes in contractile and regulatory proteins causing changes in cross-bridge binding probability (Maréchal and Plaghki 1979; de Tombe et al. 2010; Corr and Herzog 2016) have been proposed to explain these phenomena. However, there is little agreement on the mechanism responsible (Schappacher-tilp et al. 2015; Corr and Herzog 2016). In contrast to the extensive study of history dependence, relatively little attention has been paid to the changes in the force–length relationship with activation level. Optimum length, the length at which a muscle can produce the most force, is known to be longer at lower activation levels than at higher activation levels (Rack and Westbury 1969). This activation-dependent shift in optimum length has been attributed to a length dependence of calcium sensitivity (Rack and Westbury 1969; Stephenson and Wendt 1984; Rassier et al. 1999). However, this mechanism does not seem to be entirely responsible for the effect (Holt and Azizi 2014). This study aims to explore potential mechanisms underpinning the relatively understudied activation-dependent shift in optimum length, in order to gain insights into the basic mechanisms of muscle contraction. We propose the novel hypothesis that series compliance within muscles, and the resultant fiber shortening during fixed-end contractions, may contribute to the activation-dependent shift in optimum length. Hence, we suggest that contractile history may affect optimum length, and that the same mechanism may contribute to both shortening depression and the activation-dependent shift in optimum length. Results and discussion Activation-dependent shifts in optimum length and shortening-induced force depression have been studied separately; this work presents a potential link between them (Supplementary Fig. S1). This study combines (1) analysis of existing and novel data in support of the hypothesis that series compliance and fiber shortening contribute to an activation-dependent shift in optimum length; (2) a review of potential mechanisms of shortening-induced force depression; (3) suggestion of a novel mechanism of force depression; and (4) an evaluation of the potential of this mechanism to explain not only force depression, but also the activation-dependent shift in optimum length. The ultimate goal of such a synthesis is to advance our understanding of the required additions to the cross-bridge and sliding filament theories in order to adequately capture muscle performance under dynamic or intact conditions. We suggest that the capacity of any proposed mechanism to explain multiple seemingly unrelated phenomena, such as force depression and the activation-dependent shift in optimum length, makes it a strong candidate for inclusion in our framework for understanding muscle contraction. The force–length relationship and muscle activation Skeletal muscle exhibits a force–length relationship. Maximally activated, isometrically contracting sarcomeres produce peak force at an intermediate, optimum, length. The existence of this force–length relationship was used as early support for the cross-bridge and sliding filament theories (Gordon et al. 1966); optimum length is thought to correspond to optimum actin–myosin overlap, and so to the maximum number of force-generating cross-bridges (Herzog et al. 2010). However, the force–length relationship is not as consistent as we would expect if it were dictated entirely by actin–myosin overlap. The force–length relationship is affected by muscle activation; optimum length is longer at lower activation levels than at higher activation levels (Rack and Westbury 1969; Close 1972; Stephenson and Wendt 1984; Balnave and Allen 1996). This activation-dependent shift has been commonly attributed to a length dependence of calcium sensitivity (Stephenson and Wendt 1984; Rassier et al. 1999), potentially related to changes in myofilament lattice spacing (Yang et al. 1998; Rassier et al. 1999; Fuchs and Smith 2001; MacIntosh 2017), thick and thin filament regulation (de Tombe et al. 2010), and the effect of sarcomeric regulatory proteins (de Tombe et al. 2010). If muscle is more sensitive to calcium at longer lengths, more force may be produced at these longer lengths when activation and calcium are low, despite the reduced actin–myosin overlap. However, while there almost certainly is a length dependence of calcium sensitivity (Stephenson and Wendt 1984; Rassier et al. 1999) that may contribute to the activation-dependent shift in optimum length (Stephenson and Wendt 1984; Rassier et al. 1999), it appears not to be the only mechanism involved. An activation-dependent shift is also thought to have been observed in the absence of calcium-based effects (Holt and Azizi 2014). A calcium-independent effect of activation level on optimum length Muscle is activated, and force is produced, in response to signals from the nervous system. Action potentials trigger the release of calcium from intracellular stores, causing a conformational change in the thin filament and initiating cross-bridge cycling. Muscle activation can be varied either by changing the frequency of action potentials being delivered to a given motor unit (Adrian and Bronk 1929), or by changing the number of motor units recruited (Liddell and Sherrington 1925). Altering either of these parameters will change total muscle force. However, only changing action potential frequency will change intracellular calcium concentrations ([Ca2+]i). In one study, the ability to change muscle activation by changing either action potential frequency or motor unit recruitment was used to explore whether the length dependence of calcium sensitivity was entirely responsible for the activation-dependent shift in optimum length. It was demonstrated that there was a large activation-dependent shift in optimum length independent of the mechanism used to alter activation (Holt and Azizi 2014). It is possible that the technique used here, namely sonomicrometry, overestimates the magnitude of the shift due to its failure to account for the curvature of the fibers in the muscle. However, these data are not the result of incorrect correction for passive force as recently suggested (MacIntosh 2017). The presence of an activation-dependent shift in optimum length independent of changes in [Ca2+]i strongly suggests that this phenomenon cannot be entirely due to a length dependence of calcium sensitivity. Our current understanding of muscle physiology does not readily offer another explanation of this phenomenon. However, based on these data (Holt and Azizi 2014) and previous studies (Street et al. 1966; Ichinose et al. 1997; de Brito Fontana and Herzog 2016), we suggest that series compliance within muscle and the resultant fiber shortening may play a role in this shift. Compliance may contribute to the activation-dependent shift in optimum muscle length The finding that there is a shift in optimum length independent of the method of changing activation level (Holt and Azizi 2014) presents a challenge to the theory that a length dependence of calcium sensitivity is entirely responsible for the activation-dependent shift in optimum length. Here we propose the relatively novel hypothesis that muscle compliance, and the resultant contractile history, could be responsible for this effect. In muscles with elements such as aponeuroses and tendons that provide significant series compliance, muscle fibers will shorten upon activation and stretch these elements during fixed-end contractions. When the passive force in the stretched series compliant elements balances that generated by muscle fibers, fibers will become isometric. At higher activation levels, where muscles produce more force, fibers will shorten further before isometry is reached. Optimum length has been shown to decrease when muscle shortens before contracting isometrically (Street 1966). Hence, in muscles with significant series compliance, optimum length may decrease with increasing activation level due to increased fiber shortening. This would result in an activation-dependent shift in optimum length. There is some evidence in support of this hypothesis from in vivo studies that examined the effect of activation level on the force–length properties of fiber in the human vastus lateralis (Ichinose et al. 1997; de Brito Fontana and Herzog 2016). In these studies, force and length were determined during contractions with ramped increases in activation. The in vivo nature of these studies meant that fibers were acting in series with compliance, and so shortened in proportion to activation and force. The findings from these studies show a shift in optimum length with activation level. However, this shift appeared to be driven by a decrease in optimum length at high activation levels, rather than an increase in optimum length at low activation levels as would be predicted by a length dependence of calcium sensitivity. The joint position, and so the starting fiber length, that gives peak force appears to be the same in all conditions. At higher activation levels, fibers shorten more against series compliance, seemingly resulting in a decrease in optimum length with increasing activation level. The presence of a calcium-independent effect of activation on optimum length (Holt and Azizi 2014), the effect of shortening on optimum length (Street et al. 1966), and the seeming decrease in optimum length at high activation levels (Ichinose et al. 1997; de Brito Fontana and Herzog 2016) leads us to propose that there may be two mechanisms driving an activation-dependent shift in optimum length: (1) a length dependence of calcium sensitivity that increases optimum length with decreasing [Ca2+]i; and (2) a decrease in optimum length with increased shortening, that in compliant muscles will manifest as a decrease in optimum length with increasing activation. In the remainder of this section of the paper we explore potential evidence to support the latter mechanism. Existing data suggest a role for compliance in the activation-dependent shift in optimum length Our initial evidence for compliance contributing to the activation-dependent shift in optimum length comes from a reanalysis of the data presented in Holt and Azizi (2014). In this reanalysis, we looked to confirm the in vivo findings that the length at which muscle was activated to produce maximum force was always constant (Ichinose et al. 1997; de Brito Fontana and Herzog 2016) under more controlled conditions. This would provide stronger evidence that in muscle with series compliance, optimum fiber length decreases with increased activation and shortening. In Holt and Azizi (2014), bullfrog plantaris muscles were connected between a force/length transducer and a fixed point, with sonomicrometry crystals implanted along the length of a fiber. Fixed-end contractions were elicited at a range of lengths sufficient to construct a force–length curve, using several different activation levels. During these fixed-end contractions, fibers shortened as they developed force and stretched the compliant aponeurosis, before reaching force equilibrium and isometry (Fig. 1A, B). The original analysis took force as the peak force that was produced (F) (Fig. 1A) and length as the final length of the fiber (Lfiber) once isometry had been reached (Fig. 1B). Force–length relationships were plotted, and peak force produced by the muscle (F0) and optimum fiber length determined (L0fiber) (Fig. 1C). In the new analysis presented here, we took length as total muscle length (Lmuscle) (Fig. 2B). This ignored any fiber shortening and reflected the initial length of the fibers (Fig. 2B). New force–length relationships were plotted, peak force (F0) and optimum length (L0muscle) determined (Fig. 2C), and linear mixed effects models (lme) used to determine whether there was any effect of activation level on optimum lengths. This analysis shows that while there is a significant effect of activation level on L0fiber (P < 0.001; lme) there is no significant effect of activation level on L0muscle (P = 0.065; lme) (Fig. 3). Hence, the length at which muscle fibers are activated to produce maximum force appears to be constant across activation conditions. This is consistent with our hypothesis that fiber shortening against series compliance may contribute to the activation-dependent shift in optimum length. Fig. 1 View largeDownload slide Sample force (A) and fiber length traces (B) for maximum (solid), and ∼40% activation (dotted), conditions. Force and fiber length values are determined as indicated. The schematic in B indicates the fiber lengths during the fixed-end contraction. Muscle fibers (gray) shortened against the compliant aponeurosis (white) and Lfiber was taken as the final length at which isometry was reached. This shortening is proportional to the muscle activation and force produced (A, B). The force and length values determined were plotted against one another (circles) for multiple fixed-end contractions under maximum (solid), ∼70% (dashed), and ∼40% (dotted) activation conditions, and a third-order polynomial fitted to each data set (C). F0 and L0fiber were then determined for each activation condition. Fig. 1 View largeDownload slide Sample force (A) and fiber length traces (B) for maximum (solid), and ∼40% activation (dotted), conditions. Force and fiber length values are determined as indicated. The schematic in B indicates the fiber lengths during the fixed-end contraction. Muscle fibers (gray) shortened against the compliant aponeurosis (white) and Lfiber was taken as the final length at which isometry was reached. This shortening is proportional to the muscle activation and force produced (A, B). The force and length values determined were plotted against one another (circles) for multiple fixed-end contractions under maximum (solid), ∼70% (dashed), and ∼40% (dotted) activation conditions, and a third-order polynomial fitted to each data set (C). F0 and L0fiber were then determined for each activation condition. Fig. 2 View largeDownload slide Results from the reanalysis of Holt and Azizi (2014). Sample force (A) and muscle length traces (B) for maximum (solid) and ∼40% activation (dotted) conditions. Force was determined as in Fig. 1 and muscle length was taken as the constant length shown in B. The schematic in B indicates how length is determined in this reanalysis compared with the original analysis (Fig. 1B). Length was taken as total muscle length. The force and length values determined were plotted against one another (circles) for multiple fixed-end contractions for maximum (solid), ∼70% (dashed), and ∼40% (dotted) activation conditions, and a third-order polynomial fitted to each data set (C). F0 and L0muscle were then determined. It should be noted that this is the same data as in Fig. 1, simply with length calculated as the muscle length (B) rather than the post-shortening fiber length (Fig. 1B). Fig. 2 View largeDownload slide Results from the reanalysis of Holt and Azizi (2014). Sample force (A) and muscle length traces (B) for maximum (solid) and ∼40% activation (dotted) conditions. Force was determined as in Fig. 1 and muscle length was taken as the constant length shown in B. The schematic in B indicates how length is determined in this reanalysis compared with the original analysis (Fig. 1B). Length was taken as total muscle length. The force and length values determined were plotted against one another (circles) for multiple fixed-end contractions for maximum (solid), ∼70% (dashed), and ∼40% (dotted) activation conditions, and a third-order polynomial fitted to each data set (C). F0 and L0muscle were then determined. It should be noted that this is the same data as in Fig. 1, simply with length calculated as the muscle length (B) rather than the post-shortening fiber length (Fig. 1B). Fig. 3 View largeDownload slide Summary L0fiber (solid bars) (data from Fig. 1) and L0muscle (open bars) (data from Fig. 2) optimum length data for all activation conditions. There is a significant effect of activation condition on optimum fiber length (P < 0.001; lme), but not on optimum muscle length (P = 0.065; lme). Hence, we suggest that the length at which fibers started a contraction (muscle length) at remained relatively constant across activation conditions, but that the increasing fiber shortening with increasing activation drove the observed decrease in the optimum fiber lengths. Fig. 3 View largeDownload slide Summary L0fiber (solid bars) (data from Fig. 1) and L0muscle (open bars) (data from Fig. 2) optimum length data for all activation conditions. There is a significant effect of activation condition on optimum fiber length (P < 0.001; lme), but not on optimum muscle length (P = 0.065; lme). Hence, we suggest that the length at which fibers started a contraction (muscle length) at remained relatively constant across activation conditions, but that the increasing fiber shortening with increasing activation drove the observed decrease in the optimum fiber lengths. Experimental manipulation of compliance affects the activation-dependent shift in optimum length In order to provide a more direct test of the role of compliance and fiber shortening in the activation-dependent shift in optimum length, we determined this shift in high and low compliance conditions. The data from Holt and Azizi (2014) provided a high compliance condition. The bullfrog plantaris muscle contains significant series compliance and fibers shortened against this during fixed-end contractions (Fig. 1B). A low compliance comparison was achieved by repeating this experiment in fiber bundles extracted from the plantaris muscle (Fig. 4) (see Supplementary Materials for full methods). No series compliance was present in this preparation, so unlike in the high compliance condition (Fig. 5B), fibers did not shorten during fixed-end contractions (Fig. 6B). The activation-dependent shift, the ratio of optimum length in the tetanic condition (L0Tet) (high activation) and optimum length in the twitch condition (L0Twi) (low activation), could then be determined for high and low compliance conditions. If fiber shortening against series compliance contributed to the activation-dependent shift, we would expect there to be a greater activation-dependent shift in the high-compliance condition than in the low compliance condition. This is well supported by our results. In the high compliance condition, fibers underwent a shortening strain of ∼30% in tetanic contractions (Fig. 5B), whereas fibers did not shorten in the low-compliance conditions (Fig. 6B). There was a significant interactive effect of activation and compliance on optimum length (P = 0.025; lme) (Fig. 7). In the low compliance condition, twitch optimum length was 8% longer than tetanic optimum length; in the high compliance condition it was 35% longer (see Supplementary Materials for complete results). Fig. 4 View largeDownload slide Schematic representation of high- (A) and low- (B) compliance conditions. The high-compliance condition is the intact bullfrog plantaris muscle in which fibers (gray) operate in series with a large, compliant aponeurosis (white). Fibers shorten, stretching the aponeurosis on activation. The low-compliance condition is fibers extracted from the bullfrog plantaris muscle. Fibers operate isolated from the compliant aponeurosis, and hence do not shorten upon activation. Fig. 4 View largeDownload slide Schematic representation of high- (A) and low- (B) compliance conditions. The high-compliance condition is the intact bullfrog plantaris muscle in which fibers (gray) operate in series with a large, compliant aponeurosis (white). Fibers shorten, stretching the aponeurosis on activation. The low-compliance condition is fibers extracted from the bullfrog plantaris muscle. Fibers operate isolated from the compliant aponeurosis, and hence do not shorten upon activation. Fig. 5 View largeDownload slide Sample force (A) and length traces (B) for high (solid) and low (dashed) activation conditions with high compliance. Force and length values were determined as previously (Fig. 1). The large fiber shortening reflects the high series compliance. These force–length points from multiple contractions were plotted against one another, and a third-order polynomials fitted to each data set (C). F0 and L0 were then determined. Fig. 5 View largeDownload slide Sample force (A) and length traces (B) for high (solid) and low (dashed) activation conditions with high compliance. Force and length values were determined as previously (Fig. 1). The large fiber shortening reflects the high series compliance. These force–length points from multiple contractions were plotted against one another, and a third-order polynomials fitted to each data set (C). F0 and L0 were then determined. Fig. 6 View largeDownload slide Sample force (A) and fiber length traces (B) for high (solid) and low (dashed) activation conditions with low compliance. Low compliance is reflected in the lack of fiber shortening (B). Force–length points from multiple contractions are plotted against one another, and a third-order polynomial fitted to each data set (C). F0 and L0 were then determined. Fig. 6 View largeDownload slide Sample force (A) and fiber length traces (B) for high (solid) and low (dashed) activation conditions with low compliance. Low compliance is reflected in the lack of fiber shortening (B). Force–length points from multiple contractions are plotted against one another, and a third-order polynomial fitted to each data set (C). F0 and L0 were then determined. Fig. 7 View largeDownload slide Summary data showing the activation-dependent shift (L0twi/L0tet) with high (data from Fig. 5) and low (data from Fig. 6) compliance. There was a significant interactive effect of activation and compliance on the activation-dependent shift (P = 0.025; lme), indicating that the shift increased with increasing compliance. Fig. 7 View largeDownload slide Summary data showing the activation-dependent shift (L0twi/L0tet) with high (data from Fig. 5) and low (data from Fig. 6) compliance. There was a significant interactive effect of activation and compliance on the activation-dependent shift (P = 0.025; lme), indicating that the shift increased with increasing compliance. The finding that the activation-dependent shift increases with increasing compliance is consistent with our hypothesis that, in compliant systems, a component of the activation-dependent shift is due to a decrease in optimum length with increasing activation levels as a result of greater fiber shortening. Stronger support for this would be provided by using a system in which series compliance could be artificially varied, and sarcomere length measured. The effect of varying activation by varying both calcium concentration and motor unit recruitment under different compliance conditions could then be determined and mapped to contractile protein overlap. While we advise some caution in the interpretation of these results, taken together with the shift in optimum length independent of changes in [Ca2+]i, the constancy of the length at which the muscle is activated in order to produce maximum force (Fig. 3) (Ichinose et al. 1997; de Brito Fontana and Herzog 2016), and the effect of shortening on optimum length (Street et al. 1966), there is certainly a case to be made that the contractile history of the fiber influences optimum length. It is well established that contractile history influences the magnitude of the force that can be produced (Abbott and Aubert 1952a; Edman et al. 1982; Rassier and Herzog 2004; Herzog et al. 2006; Joumaa et al. 2012; Minozzo and Lira 2013; Nishikawa 2016). However, what has previously been less clear is that contractile history could affect not only the force at any given length, but also the length at which maximum force could be produced. Hence, in the next part of this paper, we review these history-dependent effects, particularly shortening-induced force depression; explore the potential mechanisms underlying them; and discuss whether mechanisms responsible could also be responsible for the activation-dependent shift in optimum length. This will further the debate on potential amendments to cross-bridge and sliding filament theories. Force depression and optimum length The isometric force a muscle can produce following active shortening is well established to be lower than that during a solely isometric contraction at the same length (Abbott and Aubert 1952b; Rassier and Herzog 2004). This history dependence is not predicted by cross-bridge or sliding filament theories. There are multiple proposed mechanisms to explain this phenomenon; however, resolution has proved difficult (Maréchal and Plaghki 1979; Rode et al. 2009; de Tombe et al. 2010; Corr and Herzog 2016), so limiting our ability to extend cross-bridge and sliding filament theories. We suggest muscle fiber shortening against series compliance, interpreted through the phenomenon of shortening-induced force depression, potentially explains some of the activation-dependent shift in optimum length. Hence, the second part of this paper reviews the current state of our understanding of shortening-induced force depression, provides additions to proposed mechanisms, and evaluates their ability to contribute to an explanation of the activation-dependent shift in optimum length discussed in the previous section. Characteristics of force depression Shortening-induced force depression is the persistent reduction in the force a muscle or fiber produces following activation and the subsequent decrease in its length (Abbott and Aubert 1952b). See Rassier and Herzog (2004) for an excellent review of force depression. Force depression is characterized by (1) being long-lasting, (2) disappearing if activation is removed, even briefly, (3) increasing with the distance shortened, and (4) increasing with greater force during shortening (Abbott and Aubert 1952b; Herzog and Leonard 1997; Rassier and Herzog 2004). Force depression lasts several 10s of seconds, provided the muscle remains active and continues to bear force. However, it disappears upon the release of force (Abbott and Aubert 1952b; Herzog et al. 1998). The magnitude of force depression increases with increased shortening distance (Abbott and Aubert 1952b; de Ruiter et al. 1998), and when shortening more slowly and so generating more force (Hill 1938). These findings led to the idea that the magnitude of force depression is correlated to the work done, the product of the force the muscle generates, and the distanced shortened (Granzier and Pollack 1989). However, while these properties are well established, the mechanism responsible remains a matter of debate. Proposed mechanisms behind force depression: a brief review and elaboration Several mechanisms have been proposed, acting singly or in concert, to explain the phenomena of force depression. They have generally fallen into one of several categories: (a) sarcomere length non-uniformity, (b) fatigue product accumulation, or (c) cross-bridge inhibition. One of the earliest proposed mechanisms, non-uniformity of sarcomere length, posits that active shortening prompts stronger sarcomeres to slip over the top of the force–length curve to the short lengths of the ascending limb (Abbott and Aubert 1952b). This mechanism is now considered unlikely due to the presence of force depression on the stable ascending limb and the decrease in stiffness after shortening, as detailed in Rassier and Herzog (2004). Accumulation of fatigue products as a result of the work done during active shortening was suggested as an alternative to the non-uniformity mechanism (Granzier and Pollack 1989). Arguing against a fatigue product mechanism is the inconsistency between the slow clearing of fatigue products and the rapid recovery of full force production following relaxation to a no-tension state (characteristic 2 above) (Abbott and Aubert 1952b; Herzog and Leonard 1997). The remaining explanation for force depression is that cross-bridge formation is inhibited in the region of new filament overlap that arises as a result of active shortening. This explanation supposes that as muscle actively shortens, while the area of overlap between the thick and thin filaments is increasing, fewer myosin heads are able to link to the thin filament and undergo force-generating powerstrokes in the new overlap zone (Maréchal and Plaghki 1979). This mechanism is now considered the likeliest candidate (Rassier and Herzog 2004; Corr and Herzog 2016). However, it is only a partial explanation; there are many means by which cross-bridge formation may be inhibited. Current theories are dominated by the spatial effects of titin during shortening and mechanical effects on the thin filament. A proposed titin-based mechanism for decreased myosin attachment in the new overlap zone suggests that titin directly blocks the formation of new cross-bridges by being dragged between the thick and thin filaments (Rode et al. 2009). This explains the increase in force depression as shortening increases (characteristic 2 above), as greater shortening would allow titin to block a greater number of cross-bridges. Assuming the bound region detaches and recoils when activation is removed, this titin-based mechanism could explain the decrease in force depression upon deactivation (characteristic 3). However, this mechanism is inconsistent with increased force depression as greater force is generated during shortening, or as shortening occurs more slowly (Rassier and Herzog 2004). It has been proposed that myosin heads would detach titin from actin in a velocity dependent fashion, recovering shortening velocity dependence (Rode et al. 2009). This mechanism is inconsistent with the low Reynolds number intra-sarcomere environment: it implies an increase in the energetics of myosin-head–titin collisions as shortening velocity increases which, while intuitively plausible in the macroscopic world, isn’t present in the viscosity dominated regime within the sarcomere. However, this doesn’t rule out shortening-speed-dependent conformational changes in titin that may alter its cross-bridge interactions and recover a role for titin in force depression. We note here that while titin interference in the creation of new cross-bridges is not consistent with all properties of force depression, a single mechanism need not be responsible for the entirety of the force depression phenomenon. Alternatively, actin geometry could be altered by thin filament strain in the region of new overlap, preventing cross-bridge formation through a purely mechanical mechanism (Maréchal and Plaghki 1979; Rassier and Herzog 2004). This would be consistent with the characteristics of force depression listed above. Deformation of the thin filament would remain as long as force is maintained and return to an unstrained configuration when force is removed (characteristics 1 and 2 above). Increasing the distance shortened would increase the strained thin filament section encountered where new cross-bridges did not form (characteristic 3). Greater force during shortening would produce greater strain and thus greater inhibition (characteristic 4). Previous work has considered both axial and azimuthal deformation of actin geometry as a binding inhibition (Rassier and Herzog 2004), but we suggest a tropomyosin-mediated decrease in cross-bridge formation as a pathway to force depression. In this mechanism, decreased cross-bridge formation may come about not as a result of isolated G-actin deformation, but rather due to deformations in the thin filament leading to small changes in the energy landscape tropomyosin navigates as it transitions to its permissive open state. As the thin filament is subject to strain, so is the troponin/tropomyosin regulatory complex. The small actin deformations possible in the axial and azimuthal directions, on the level of disorder that has been shown to increase binding in isolation through breaking binding site symmetry (Daniel et al. 1998), may be sufficient to increase the energy barrier between tropomyosin’s closed or partially-closed and fully activated states such that cross-bridge formation is reduced. Such a reduction in cross-bridge formation would be amplified in areas of the thin filament with no initial overlap as tropomyosin in that region isn’t subject to the cooperative activation amplification that comes with formation of cross-bridges at adjacent sites (Tanner et al. 2012). Thus the effect of strain-dependent tropomyosin kinetics may be limited to the region of new overlap. A possible test of this mechanism could be performed by plating whole or unregulated thin filaments onto a deformable gel and tracking myosin motility under resting and deformed-gel conditions. A potential strain dependency of tropomyosin kinetics offers consistency with the known properties of force depression and is testable through further modeling and experiments. Strain-induced inhibition of cross-bridge binding may explain both force depression and the activation-dependent shift in optimum length The earlier hypothesis that compliance and fiber shortening, and therefore contractile history, could contribute to the activation-dependent shift in optimum length led us to consider whether the mechanisms proposed to explain force depression could also explain this phenomenon. Cross-bridge and sliding filament theories suggest that the variation in force seen across the force–length curve is due to changes in the overlap of actin and myosin filaments (Huxley and Hanson 1954; Huxley and Niedergerke 1954; Huxley 1969, 1974; Herzog et al. 2010). Changes in sarcomere length change overlap move the sarcomere to a new location on the force–length curve, and so result in a change in force. Under isometric, maximally activated, conditions these theories do seem to be adhered to; consistent force–length relationships are observed and changes in filament overlap are responsible for the majority of the variation in force (Gordon et al. 1966; Herzog et al. 2010; Williams et al. 2013). However, in fixed-end contractions, fibers contained within muscles with series compliance will shorten until a force balance is achieved at the length previously referred to as Lfiber. This shortening causes a change in actin–myosin overlap, but one that occurs under tension, hence providing the conditions necessary for force depression. If force depression is the result of strain-dependent inhibition of cross-bridge binding, then the potential for cross-bridge binding is not the same as if Lfiber had been arrived at passively. Active shortening may disrupt the relationship between actin–myosin overlap, and cross-bridge binding and force-generating potential. We suggest therefore that following active shortening, there is an absolute or real amount of filament overlap dictated by sarcomere length, but also an effective filament overlap dictated both by length and the altered cross-bridge binding probability in any new region of overlap (Fig. 8 and Supplementary Fig. S2). Effective filament overlap describes the number of motors generating force, rather than the number of motors with access to the thin filament, and offers a new view of the interaction between active shortening and the force–length curve. Fig. 8 View largeDownload slide Example of force depression induced shifts in location along the force–length curve due to the interaction of MTU series compliance and varying levels of activation. In the schematics the tendon is shown as a dark gray spring, the thick filament and myosin in red (mid-gray), the thin filament in blue (light gray), and the areas of initial and new overlap in green (pale gray) and red (dark grey), respectively. (A) Under high activation the contractile element of a MTU shortens from ℓIH to ℓFH due to series compliance, increasing the thick–thin overlap. The motors in this region have a decreased chance of forming new cross-bridges due to force depression. (B) The same MTU, at a different initial length, is subjected to low levels of activation that result in the contractile element shortening from ℓIL to ℓFL where the final length is the same as in panel A ( ℓFL=ℓFL). As less force was generated by lower levels of activation to reach this final CE length, there is less region of new overlap and thus fewer motors inhibited from forming new cross-bridges by force depression. Despite having identical final MTU lengths, the two cases have different effective overlaps (ϵH or ϵL), the fractions of the thick and thin filaments able to interact and generate force. This places the two cases at different effective points on the force–length curve, shifting the force–length curve in an activation dependent fashion for muscle subject to the series compliance found in the MTU. Fig. 8 View largeDownload slide Example of force depression induced shifts in location along the force–length curve due to the interaction of MTU series compliance and varying levels of activation. In the schematics the tendon is shown as a dark gray spring, the thick filament and myosin in red (mid-gray), the thin filament in blue (light gray), and the areas of initial and new overlap in green (pale gray) and red (dark grey), respectively. (A) Under high activation the contractile element of a MTU shortens from ℓIH to ℓFH due to series compliance, increasing the thick–thin overlap. The motors in this region have a decreased chance of forming new cross-bridges due to force depression. (B) The same MTU, at a different initial length, is subjected to low levels of activation that result in the contractile element shortening from ℓIL to ℓFL where the final length is the same as in panel A ( ℓFL=ℓFL). As less force was generated by lower levels of activation to reach this final CE length, there is less region of new overlap and thus fewer motors inhibited from forming new cross-bridges by force depression. Despite having identical final MTU lengths, the two cases have different effective overlaps (ϵH or ϵL), the fractions of the thick and thin filaments able to interact and generate force. This places the two cases at different effective points on the force–length curve, shifting the force–length curve in an activation dependent fashion for muscle subject to the series compliance found in the MTU. This effect has the potential to interact with muscle activation level. The level of overlap change and force depression both increase with increasing force generation, and are thus activation dependent. Holding series compliance constant and varying initial length and the level of activation, we can reach conditions where we have the same sarcomere length and the same absolute filament overlap, but different levels of force depression thus different levels of effective filament overlap (Fig. 8). This places our two sarcomeres at the same absolute point on the force–length curve but at different effective points. As sarcomere shortening increases monotonically with activation, so too does the gap between our effective overlap and our absolute overlap. This could provide the systematic leftward shift in the force–length curve at higher activation levels seen in Figs. 1–3. Hence, strain dependent inhibition of cross-bridge binding could be responsible not only for force-depression, but also contribute to the activation-dependent shift in optimum length. This ability of strain dependent inhibition of cross-bridge binding to explain both of these phenomena makes it a strong candidate for addition to cross-bridge and sliding filament theories, and its suggestion that compliance and activation could have interactive effects on the force–length relationship has major consequences for how we think about muscle function in vivo. Implications for understanding in vivo muscle performance The force–length relationship, based on cross-bridge and sliding filament theories, has become central to our prediction and interpretation of in vivo muscle performance. It is a major input into musculoskeletal models (Delp et al. 2007; Dick et al. 2017), and has been used to draw the broad conclusion that muscles tune their operating length to maximize some aspect of performance (Rome and Sosnicki 1991; Herzog et al. 1992; Burkholder et al. 2001; Tu 2004; Azizi and Roberts 2010; Rubenson et al. 2012; Arnold et al. 2013; Azizi 2014; Holt and Azizi 2016; Foster and Higham 2017; Nikolaidou et al. 2017). However, the variation in activation level with mechanical demand (Lieber and Brown 1992; Gorassini et al. 2000; Hodson-Tole and Wakeling 2007; Morris and Askew 2010; Vervuert et al. 2013; Seven et al. 2014), and the activation-dependent shift in optimum length (Rack and Westbury 1969; Close 1972; Stephenson and Wendt 1984; Balnave and Allen 1996; Holt and Azizi 2014), suggest these predictions and interpretations may be subject to systematic errors. An attempt to address this was made in Holt and Azizi (2016); in vivo function was mapped to activation-level specific force–length curves, leading to the more nuanced conclusion that optimum length is used in the most demanding behaviors, but that performance may be sacrificed, and sub-optimal lengths used, during less demanding activities. However, while this study attempted to account for the activation-dependent changes in optimum length, the finding that optimum length may depend not only on activation level, but also compliance and contractile history (Figs. 1–3, 5–8) (Street et al. 1966; Ichinose et al. 1997; de Brito Fontana and Herzog 2016), suggests that it did not fully capture the complexity of the situation. Not only does muscle activation level vary widely in vivo, but series compliance varies between species (Azizi 2014) and muscle morphologies (Wilson and Lichtwark 2011), and muscle shortening is a complex interaction between the level of activation of the muscle, its series compliance, and the external load (Griffiths 1991). Hence the “optimum” length for a muscle operating in vivo is a rich and situation-dependent landscape. Conclusions The cross-bridge and sliding filament theories that dominate our understanding of muscle contraction cannot explain many features of muscle performance under complex in vivo conditions, yet there remains a lack of consensus regarding appropriate amendment to these theories. Here we examined the unexplained and understudied activation-dependent shift in optimum length, as a means to advance this process. We suggest that shortening under tension against series compliance during fixed-end contractions may contribute to this phenomenon, and therefore that a common mechanism may be responsible for both the activation-dependent shift in optimum length and another major feature of muscle performance not explained by the cross-bridge and sliding filament theories: shortening-induced force depression. We suggest that strain-induced inhibition of cross-bridge binding in the new region of actin–myosin overlap, potentially mediated by tropomyosin deformation, decreases the force-generating potential of the muscle at its post-shortening length, so depressing force and altering the relationship between force and length in an activation-dependent manner. The ability of this mechanism to explain these seemingly unrelated phenomena makes it an appealing potential amendment to cross-bridge and sliding filament theories. Acknowledgments We thank Manny Azizi and Kiisa Nishikawa for their experimental support and insightful comments, Tom Daniel and Mike Regnier for stimulating discussions, and Paul G. Allen, founder of the Allen Institute for Cell Science, for his vision, encouragement and support. Funding Funding was provided by NIH grant AR055295, NSF grant IIS-1110370, ARO Grant W911NF-13-1-0435, ARO Grant W911NF-14-1-0396, The Gordon and Betty Moore Foundation, The Alfred P Sloan Foundation, and The Washington Research Foundation Fund for Innovation in Data-Intensive Discovery. References Abbott BC , Aubert XM. 1952a . Changes of energy in a muscle during very slow stretches . J Physiol 117 : 104 – 17 . Google Scholar CrossRef Search ADS Abbott BC , Aubert XM. 1952b . The force exerted by active striated muscle during and after change of length . J Physiol 117 : 77 – 86 . Google Scholar CrossRef Search ADS Adrian E , Bronk DW. 1929 . The discharge of impulses in motor nerve fibers. Part II. 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Google Scholar CrossRef Search ADS PubMed © The Author(s) 2018. Published by Oxford University Press on behalf of the Society for Integrative and Comparative Biology. All rights reserved. For permissions please email: journals.permissions@oup.com. This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/about_us/legal/notices)

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Integrative and Comparative BiologyOxford University Press

Published: Aug 1, 2018

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