Mechanisms of residual force enhancement in skeletal muscle: insights from experiments and mathematical models

Abstract

A skeletal muscle that is stretched while contracting will produce more force at steady state than if it is stretched passively and then stimulated to contract. This phenomenon is known as residual force enhancement and has been widely studied since its description more than 60 years ago. The idea that the mechanical properties of a muscle are governed not just by its present length but also by its length at earlier time points has far reaching implications since muscles stretch and shorten routinely in normal use. In this review, we present the experimental and theoretical advances that have been made toward understanding the mechanisms that underlie residual force enhancement. In the past 10 years, experiments and models have focused on essentially three candidate mechanisms for residual force enhancement: (half-) sarcomere inhomogeneity, activity of so-called ‘passive’ mechanical elements in the sarcomere (titin), and the intrinsic properties of myosin crossbridges. Evidence, both computational and experimental, is accumulating for each of these mechanisms such that a final description of the phenomenon seems attainable in the near future. We conclude that computational models that incorporate more than one putative mechanism may ultimately facilitate reconciliation of the growing number of ideas and experimental data in this field.

Introduction

According to textbook descriptions, the contractile force of skeletal muscle is diminished if the muscle is stretched to the point at which the overlap of thick and thin filaments in its sarcomeres is reduced. However, if the muscle is stretched to a long length during active contraction, a very different response is observed. Force increases rapidly during the stretch and remains elevated above the expected level (even after the stretch-induced transient dies away) for the remainder of the activation. This muscle property is commonly known as residual force enhancement after stretch. A typical force enhancement experiment is depicted in Fig. 1.

Fig. 1.

Schematic diagram of a typical residual force enhancement protocol. The protocol consists of two contractions, one in which the muscle is stretched (red traces) and another in which the muscle is held at constant length (black traces). In stretch experiments, the muscle is initially held at a constant length and activated, whether by electrical stimulus (intact preparations) or by placing it in a Ca²⁺-containing solution (permeabilized fibers or isolated myofibrils). Following activation, the muscle is stretched from the initial length (L ₁) to a new length (L ₂) and held for several seconds until steady-state force is achieved (P _str). In the isometric reference case, the muscle length is set to L ₂ prior to activation, and held constant until a steady-state force (P _iso) is attained. The residual force after stretch is determined by the difference between steady-state forces obtained in the stretch and isometric contractions (ΔP _iso-str)

Since its initial description by Abbott and Aubert (1952), residual force enhancement has been studied widely as an example of history dependent mechanical behavior in skeletal muscle. A definitive explanation of force enhancement has eluded investigators for decades, and remains the object of intensive research (Herzog et al. 2008; Edman 2010; Telley and Denoth 2007). In this review, we first consider a set of generally accepted features that define residual force enhancement at present, and use these as a basis to discuss several putative mechanisms and accompanying insights that have been provided by computational models. Finally, we consider the next logical steps to be made in both models and experiments in order to arrive at a more complete explanation of force enhancement.

Essential experimental features

We have identified six properties of force enhancement that have each been observed by at least two separate groups:

1. The magnitude of force enhancement is proportional to stretch magnitude

2. The magnitude of enhancement is independent of stretch velocity

3. At least some enhancement can be observed at muscle and sarcomere lengths corresponding to the ascending limb, plateau, and descending limb of the length–tension relationship

4. Enhanced force following stretch can exceed isometric force at optimal muscle or sarcomere length

5. Muscle segments, sarcomeres, and half-sarcomeres exhibit slow length changes and non-uniform lengths before and after active stretch

6. A portion of the enhanced force after stretch persists even after muscle deactivation

Edman and co-workers (1982; 1978) along with Julian and Morgan (1979a) established Properties 1 and 2 in systematic studies of residual force enhancement in frog skeletal muscle fibers. Since then, studies have confirmed these findings in a wide variety of muscle preparations and conditions (Fig. 2). Although Property 1 is violated for very large stretches of intact muscle (Hisey et al. 2009), stretches of ~40% in myofibrils (Joumaa et al. 2008a) and single mechanically-isolated sarcomeres (Leonard et al. 2010) resulted in extremely high levels of relative enhancement, suggesting that even large stretches can support proportional increases in force enhancement in certain preparations.

Fig. 2.

Relationship of residual force enhancement to stretch magnitude for various preparations. Residual force enhancement (as a percentage of P _iso; Fig. 1) is plotted against stretch magnitude (as a percentage of L ₁) sampled from published studies. These include studies of intact frog skeletal muscle fibers (Edman et al. 1982, blue symbols; Rassier and Herzog 2004a, orange), in situ cat soleus muscle (Morgan et al. 2000, magenta), single rabbit psoas myofibrils (Joumaa et al. 2008a, red), and single mechanically isolated sarcomeres of rabbit psoas myofibrils (Leonard et al. 2010, green)

Force enhancement on the ascending limb, and enhanced forces that exceed isometric force at optimal length (Properties 3 and 4) were originally more controversial (Morgan and Proske 2007; Herzog and Leonard 2006). Early studies did not report enhancement when stretch occurred on the ascending limb or plateau (Edman et al. 1978, 1982; Julian and Morgan 1979a), but more deliberate studies in intact frog fibers (Peterson et al. 2004) and single rabbit psoas myofibrils (Pun et al. 2010) revealed small but statistically significant levels of enhanced force on these length intervals. Since they occur in at least two distinct types of preparation, Properties 3 and 4 thus seem to be general characteristics of residual force enhancement.

Experiments tracking the lengths of segments, sarcomeres, and even half-sarcomeres along actively contracting muscle preparations have consistently shown that lengths are non-uniform and subject to slow, internal movement even when total length of the preparation is held constant (Property 5). Early observations of non-uniformities were made by Julian and Morgan (1979a, b) and Edman et al. (1982) in segments of contracting frog muscle fibers. Measureable amounts of segment length dispersion were seen in both isometric contractions and following stretch of active muscle, but Edman et al. (1982) reported that segment lengths were significantly more uniform after active stretch.

Several recent studies of residual force enhancement have been performed in single myofibrils. In these preparations, it is possible to track the length of each sarcomere or half sarcomere as it is activated with Ca²⁺ and subjected to imposed stretches (Joumaa et al. 2008a; Pun et al. 2010; Telley et al. 2006a, b). Without exception, these studies showed dispersion of half-sarcomere lengths and internal motion of half-sarcomeres throughout stretch protocols, just as was shown in intact fibers. Rassier (2008) reported that stretch increases dispersion of half-sarcomere lengths, which contrasts with an earlier report of reduced dispersion after stretch (Telley et al. 2006b). The latest experimental data in this area lie between these measurements and indicate that dispersion of half-sarcomere lengths remains constant (Pun et al. 2010), a finding that is consistent with the observations of Edman et al. (1982) in frog skeletal fibers.

Significantly, none of these studies in myofibrils showed any evidence of sarcomere ‘popping’, or rapid, uncontrolled lengthening of sarcomeres during stretch. Popping had been observed previously in electron micrographs of intact frog fibers fixed after an active stretch protocol, and was used to explain the presence of enhanced residual force (Brown and Hill 1991). Following publication of the first relevant myofibril study (Telley et al. 2006b), Morgan and Proske (2006) argued that the absence of popping in that case could be explained if the applied stretch had occurred on the plateau of the myofibril’s length–tension curve, rather than on the descending limb. This remained a possibility because Telley et al. (2006b) did not specifically demonstrate decreased isometric tension at the post-stretch myofibril length. However, subsequent work by another group showed convincing evidence that myofibrils stretched to points on the descending limb exhibit large residual force enhancement without sarcomere popping (Joumaa et al. 2008a).

One final property of residual force enhancement is that a substantial portion of the enhanced force after stretch remains after muscle preparations are deactivated (Property 6). Herzog and Leonard (2002) were the first to note that enhanced force persisted after the end of contraction in an actively-stretched fiber (Fig. 3). This ‘passive force enhancement’ was greater than the steady-state force achieved when the same fiber is stretched at rest. Passive force enhancement reportedly persists until the muscle is shortened, and, just like active force enhancement, its magnitude is proportional to stretch. The same study found that passive enhancement ranged from ~10% of active enhancement for small stretches to ~80% for larger ones. Passive force enhancement was subsequently documented by the Herzog group in other preparations, including single intact muscle fibers (Rassier and Herzog 2004a, b) and isolated myofibrils (Joumaa et al. 2007, 2008b; Leonard and Herzog 2010). This implies that the fundamental mechanism underlying passive force enhancement must lie within or perhaps between (half-) sarcomeres, and cannot be solely attributed to macroscopic behavior such as tendon compliance.

Fig. 3.

Schematic diagram of experiments showing properties of passive force enhancement. This schematic depicts the observation by Herzog and Leonard (2002) that enhanced force persists even after muscle activation ceases, an effect they termed ‘passive force enhancement.’ To determine whether this was caused by a passive stiffness element in sarcomeres that is engaged at the moment of activation, Rassier and Herzog (2004b) activated muscles at the post-stretch length (L ₂) and then shortening them to L ₁ for the usual stretch. When the shortening directly preceded stretch (green traces), force enhancement during and after activation was reduced (though substantial active enhancement remained). Passive and active enhancement were not significantly affected when shortening preceded stretch by ~1 s (Edman et al. 1982), suggesting that formation of the parallel stiffness depends on both length and time

Mechanisms of residual force enhancement

Some have suggested that force enhancement could be a molecular-scale effect caused by by long-lasting stretch-induced alterations to individual myosin crossbridges (Herzog et al. 2006; Herzog and Leonard 2000). This idea was explored in different mathematical models of crossbridges by Walcott and Herzog (2008), who concluded that traditional crossbridge models were incapable of producing force enhancement in this way. They postulated that the existence of a second crossbridge cycle, induced by external stretch, would be required to produce force enhancement in single crossbridges. Studies using a three-bead laser trap show that, for smooth muscle myosins, the duration of actin binding events is increased by applied stretch (Veigel et al. 2003). However, similar experiments with skeletal muscle myosin (Mehta and Herzog 2008) failed to show any increase in dwell time after stretch, as would be required by the ‘stuck crossbridges’ of Walcott and Herzog. On the other hand, Bianco et al. (2007) have presented data suggesting that steric coupling between associated pairs of myosin heads may be responsive to stretch. They used x-ray diffraction of intact, contracting fibers to show that stretch promotes binding of additional myosins in a manner consistent with enhanced attachment of partner heads. Clearly this recruitment mechanism could contribute to enhanced force during and after stretch.

Stepping up the spatial scale, many workers have hypothesized that the sarcomeric protein titin contributes to residual force enhancement by producing a ‘passive’ or at least ‘non-crossbridge’ force response (Property 6 and Fig. 3). One of the earliest studies was performed by Edman et al. (1982) who tested the hypothesis that force enhancement could be due to a non-crossbridge elastic structure that engaged at the time of activation. They did this through an altered force enhancement protocol wherein a single frog muscle fiber was initially held at the post-stretch length, stimulated to contract, shortened, and then stretched in the usual way (similar to Fig. 3, blue traces). They reasoned that if a parallel elastic element were engaged during activation at long length, it would slacken during shortening and therefore upon stretch contribute no residual tension. However, enhancement persisted in spite of the maneuver, suggesting that the element, if it existed at all, realigned itself with the shorter length during the ~1 s elapsed between shortening and stretch. Much later, Herzog and Leonard (2000) showed that enhancement could in fact be blunted with a shortening-stretch cycle during isometric tetanus if shortening directly preceded stretch (Fig. 3, green traces; see also Rassier and Herzog 2004b).

Evidence of a different sort has also implicated passive structures such as titin in residual force enhancement. Bagni and co-workers studied tension responses to fast stretches that were applied to single frog muscle fibers at various stages of activation (Bagni et al. 1994, 2002, 2004; Colombini et al. 2007). The speed and magnitude of these stretches were made large enough to exceed levels required to forcibly detach crossbridges, such that any remaining tension rise during stretch after crossbridge rupture could be attributed to ‘passive’ components. Using this method, they identified a ‘static’ stiffness that was independent of total tension, preceded the development of twitch tension, and had a time course during twitch that closely resembled that of intracellular Ca²⁺ (Bagni et al. 1994). Static stiffness is only 2–5% of total stiffness at maximum muscle activation (Bagni et al. 2002; Colombini et al. 2007), but it is sufficiently large to figure into force enhancement responses and compares favorably with the magnitude of passive force enhancement as reported by Herzog and Leonard (2002). The magnitude of static stiffness was not influenced by crossbridge inhibition (BDM), but factors that reduced the intracellular Ca²⁺ concentration also reduced static stiffness (Bagni et al. 2004). Similarly, Rassier and Herzog (2004a) saw no effect of BDM on the magnitude of the residual force following stretch, in spite of lowered active tension.

Titin is considered an excellent candidate to account for ‘passive’ aspects of force enhancement because its stiffness has been demonstrated to increase in a Ca²⁺-dependent manner (Labeit et al. 2003), which is consistent with the Ca²⁺ dependence of static stiffness (Colombini et al. 2007). Furthermore, the PEVK region of titin has been shown to bind with actin (Kellermayer and Granzier 1996), which could form the basis of a passive stiffness that ‘adjusts’ to sarcomere length in a time-dependent manner (Rassier and Herzog 2004b; Bianco et al. 2007). Indeed, recent studies in myofibrils, showing that passive force enhancement exists at extreme sarcomere lengths beyond thick/thin filament overlap (Leonard and Herzog 2010) and is abolished by titin degradation (Joumaa et al. 2008b), offer direct support for the idea that titin is the main structural component of passive force enhancement. While these hypotheses are promising, it is not yet clear exactly how a titin-based increase in passive stiffness that is initially engaged by Ca²⁺ remains in place after Ca²⁺ is withdrawn and active contraction ceases.

Titin-based mechanisms proposed by several experimentalists are incorporated in recent theoretical work by Rode et al. (2009), who developed a detailed mathematical model of titin-based residual force enhancement that considers known in vitro properties of titin stiffness and simple geometric relationships between titin and thick and thin filaments. They suggest that Ca²⁺ activation of the thin filament promotes titin–actin interactions, ‘pinning’ the PEVK region of titin to a location on the actin filament that varies according to half-sarcomere length at the time of activation. This model is appealing in many ways as it is consistent with the static stiffness described by Bagni and co-workers (1994, 2002, 2004) as well the time-dependent formation of parallel active tension observed in shortening-stretch cycles (Rassier and Herzog 2004b). Although promising in many ways, one point that must be addressed before accepting titin–actin interactions as contributing to residual force enhancement is the fact that PEVK fragments from skeletal muscle titin were not found to interact strongly with actin at physiologic ionic strength (Yamasaki et al. 2001).

The other remaining putative mechanisms of residual force enhancement do not involve structures found within sarcomeres, but rather interactions among many sarcomeres. The most widely studied of these is the idea of sarcomere popping, originally proposed and modeled by Morgan (1990, 1994). This theory considers a muscle fiber at an initial length on the descending limb of the length–tension curve, where all sarcomeres lose thick/thin filament overlap (and hence active tension) with increasing length. It also supposes that one or more sarcomeres are weaker or more compliant than others. When a stretch is imposed during active contraction, the weakest sarcomeres would take up most of the length change for the fiber, reach a yield point, and lengthen rapidly (‘pop’) until caught by passive structural elements (Morgan 1990). This would lead to enhanced force after stretch since yielding of one or more weak sarcomeres would allow the others to remain at shorter lengths and therefore higher points on the length–tension curve. Morgan tested this idea in computational simulations of series-connected sarcomeres with Hill-type force–velocity relationships and small (2–10%) variations in sarcomere active contraction strength (Morgan 1990). He found that the model exhibited sarcomere popping during stretch as anticipated, along with residual force enhancement following stretch.

The popping sarcomere hypothesis as originally described is at odds with experiments showing enhanced forces on the ascending limb and exceeding isometric tension at optimal length (see Herzog et al. 2006 for review, and additional discussion in Herzog and Leonard 2006; Morgan and Proske 2007), casting doubt on its being the principal mechanism of residual force enhancement. More direct evidence against the theory is given by the conspicuous absence of sarcomere popping in stretched myofibrillar preparations (Telley et al. 2006b; Joumaa et al. 2008a; Pun et al. 2010). In spite of mounting evidence that disproves the idea of popping sarcomeres as essential to the force enhancement response, it should be acknowledged that the work of Morgan has advanced the field by focusing experiments on critical aspects of the response (i.e. Properties 3 and 4) and by initiating investigations into the quantitative effects of heterogeneity.

The presence of sarcomere length heterogeneity in general (separate from the extreme case of popping) has long been suggested as a contributor to residual force enhancement (Julian and Morgan 1979a), and has been the focus of numerous computational studies (Stoecker et al. 2009; Telley and Denoth 2007; Telley et al. 2003; Walcott and Sun 2009; Givli 2010), including our own (Campbell et al. 2011). While most of the experiments focusing on heterogeneity have considered lengths of sarcomeres, most of these computational models, including our own, have analyzed the behavior of linked half-sarcomeres. The distinction between whole and half-sarcomeres, though subtle, is potentially important. Since thick filaments do not have to be centered in sarcomeres, half-sarcomere heterogeneity could influence residual force enhancement even in fibers that exhibit regular z-line spacing.

Each of the models published to date differ in varying degrees as to their formulation, assumptions, and purpose. Several types of half-sarcomere variability are used in the models, including variation in the number of myosin crossbridges (or contractile force) per half-sarcomere (Telley et al. 2003; Campbell et al. 2011), perturbations to initial velocity (Walcott and Sun 2009), and variation in passive tension (Campbell et al. 2011). It is interesting to note that in each case these models exhibit residual force enhancement, regardless of the particular details of their implementation (Telley et al. 2003; Walcott and Sun 2009; Givli 2010; Campbell et al. 2011). The common thread in each, aside from non-uniform half-sarcomere properties, is the presence of a history-dependent mechanical component in the model. The implication is that dynamic interactions among half-sarcomeres that result from non-uniformities can contribute to force enhancement.

Our own model (Campbell et al. 2011) depicts half-sarcomeres in series, with length-dependent active tension in each modeled as a population of cycling crossbridges after the manner of Huxley (1957). After showing that small, random variations in the number of crossbridges per half-sarcomere were capable of producing residual force enhancement, we conducted a series of simulations to show that heterogeneity can explain several basic properties of the response (Properties 1–4 above). We view this as strong evidence that half-sarcomere dynamics can play a key role in experimentally observed residual force enhancement, contrary to some claims in the experimental literature (Joumaa et al. 2008a; Pun et al. 2010).

We subsequently used a simple model of just two half-sarcomeres in series to demonstrate how dissimilar half-sarcomeres can produce residual force enhancement after stretch (Campbell et al. 2011) (see Fig. 4). One of the two half-sarcomeres is assumed to be ‘weaker’, having fewer myosin crossbridges than the other. With stretch, force in both half-sarcomeres exceeds the amount predicted by their respective steady-state length–force curves, since the stretch places higher-than-normal strain in their attached crossbridges (Fig. 4, Time 1). When stretch ends, force drops rapidly as crossbridges in both half-sarcomeres cycle and re-attach at normal strain levels. However, force soon stabilizes at an enhanced level as the behaviors of the two half-sarcomeres diverge. The stronger half-sarcomere reaches a point where its force level lies below its steady-state curve, while the weaker one continues to hold excess force (Fig. 4, Times 2 and 3). This configuration prevents force from changing except at a very slow rate: as crossbridges with excess strain detach in the weaker half-sarcomere, the stronger half-sarcomere shortens, the weaker half-sarcomere lengthens, excess strain is re-introduced into its crossbridges, and force remains nearly constant.

Fig. 4.

Schematic diagram of residual force enhancement caused by dynamically interacting dissimilar half-sarcomeres. Two half-sarcomeres in series are considered, one having more crossbridges (red traces and symbols) than the other (blue traces and symbols). Instantaneous length–force points are shown for each half-sarcomere at three times during a residual force enhancement protocol, overlaid on their steady-state length–force curves for comparison. In order to more easily understand the dynamic behavior of the half-sarcomeres, arrows for each point indicate the direction of length or force change that would occur if the the half-sarcomere were suddenly isolated and its force or length clamped, respectively. As the system is stretched, force rises above the levels of both steady-state curves (Time 1). At the end of stretch, force drops quickly until it falls below the level of the stronger half-sarcomere’s steady-state curve (Time 2). After that point, force remains nearly constant at an enhanced level as half-sarcomere lengths drift (Time 3). See text for further details. (Adapted from our previous work, Campbell et al. 2011)

These dynamic interactions among half-sarcomeres are evident in the model as slow half-sarcomere length changes that occur both before and after stretch (Stoecker et al. 2009; Telley et al. 2003; Campbell et al. 2011), and are very similar in appearance to slow creep of half-sarcomere lengths measured in experiments (e.g., Telley et al. 2006a). This behavior partially matches Property 5 (above), but interestingly none of the models predicts a stabilizing effect of stretch on the range of segment/half-sarcomere lengths seen in experiments (Edman et al. 1982; Telley et al. 2006b); in fact they predict ranges of half-sarcomere length that are somewhat larger than those reported in activated myofibrils [~600 nm in models (Telley et al. 2003; Campbell et al. 2011) vs. ~400 nm in experiments (Telley et al. 2006b; Joumaa et al. 2008a)]. This discrepancy suggests that the models could be improved by including the Ca²⁺-sensitive parallel elastic element (which is needed anyway to match Property 6), or perhaps by including movement-enhanced binding of myosins as implied by some work in single molecules (Iwaki et al. 2009) and experiments combining x-ray diffraction and fiber mechanics (Brunello et al. 2007).

Future insights from models and experiments

In reviewing the most recent experimental and modeling advances in the field of residual force enhancement, some immediate next steps are apparent. For more than 15 years, experiments have suggested the presence of both active and passive components of the stretch response, and yet no model to date has considered the two components simultaneously. Representing both half-sarcomere dynamics and a parallel stiffness that can potentially be augmented during activation may help explain many experimental observations. We have identified three such instances.

If half-sarcomere variation were simply due to non-uniformity of either passive or active tension capacity, we could expect that shorter (stiffer) half-sarcomeres in isometric contraction would lengthen less during stretch than longer (compliant) ones. However, Joumaa et al. (2008a) and Telley et al. (2006a, b) showed that in myofibrillar preparations the length of individual (half-) sarcomeres prior to stretch does not always predict the degree of lengthening during stretch, with some initially shorter half-sarcomeres lengthening more and ending up at greater lengths than ones that were initially longer. We show in Fig. 5 that this behavior can actually occur in our published model if we include variability in both the active and passive components.

Fig. 5.

Instances of half-sarcomere length ‘crossover’ during stretch in a computer model. We used our recently published model (Campbell et al. 2011) to simulate a single myofibril consisting of ten half-sarcomeres in series undergoing a 4% stretch over 0.8 s (a). In this case, both passive and active properties of half-sarcomeres were varied at random by ±5%. b Under these circumstances half-sarcomere length prior to stretch did not predict relative length after stretch, as apparently stronger (shorter) half-sarcomeres stretched further than apparently weaker (longer) half-sarcomeres in some cases (Telley et al. 2006b; Joumaa et al. 2008a). Time course of half-sarcomere length is shown for half-sarcomeres exhibiting this ‘crossover’ behavior. The curved shape of the time courses during stretch is caused by the weakest half-sarcomere (not shown) that elongated substantially more than the others during the applied stretch

Certain components of tension responses during and after stretch that appear to arise from passive elements may actually originate from half-sarcomere heterogeneity. Sorting out the relative contributions of active (crossbridge-based) and passive components in experimental records is a challenge to which models are well suited. For instance, Pinniger et al. (2006) noted three phases of the force response during stretch; two steep initial phases which they attributed to attached crossbridges, and a third, shallower force rise that they assigned to a parallel elastic element. In fact, several computational models have shown that both phases can be obtained without need of a special parallel elastic element simply by assuming some variability in the active tension capacity of half-sarcomeres (Telley et al. 2003; Campbell et al. 2011; Morgan 1990; Campbell 2009).

Finally, whereas most models predict that half-sarcomere heterogeneity will increase during a stretch (Fig. 5), some experiments suggest that the heterogeneity actually decreases (Edman et al. 1982; Telley et al. 2006b). An exciting prospect is that heterogeneity in combination with a Ca²⁺-activated parallel elastic component could explain the reduced half-sarcomere length dispersion following stretch. This can be seen by reconsidering the simple system of two dissimilar half-sarcomeres, this time where each has a steep passive tension that was engaged at the time of Ca²⁺ activation (Fig. 6). Now, as the weaker half-sarcomere absorbs more of the applied stretch, it is stabilized by the parallel passive element, such that at the conclusion of stretch it falls below its length–tension curve, while tension in the stronger, shorter half-sarcomere is maintained above its length–tension curve. From here, residual force enhancement will occur as previously, with the important distinction that in this case the weaker half-sarcomere shortens while the stronger one lengthens—pushing the two back toward one another and hence tending to homogenize lengths.

Fig. 6.

Possible effects of Ca²⁺-induced titin–actin interactions on dynamically interacting dissimilar half-sarcomeres. This schematic shows the same processes described in Fig. 4, but considers the addition of Ca²⁺-induced titin–actin interactions that form a stiff passive mechanical element in parallel with each half-sarcomere as envisioned by Rode et al. (2009). The effect of titin–actin interactions on the steady-state length–force curves are shown as dashed lines. Activation disperses half-sarcomere lengths as before, but the applied stretch is more evenly distributed due to the weaker half-sarcomere being supported by the parallel stiffness (Time 1). Following the end of stretch (Time 2) force will fall as before but, critically, it is the weaker half-sarcomere that falls below its steady-state length–force curve, while the stronger half-sarcomere stays above (Time 3). Once again this ‘one over, one under’ condition prevents rapid adjustment of force to isometric levels, since the two half-sarcomeres attempt to adjust their force in opposite directions (resulting in little net change). With the addition of titin–actin interactions, this causes the half-sarcomere lengths to normalize toward each other rather than diverge (as in Fig. 4)

Just as experiments have suggested new avenues for computational models to explore, the reverse is also true. For example, our recent model predicts that residual force enhancement is highly sensitive to the amount of half-sarcomere heterogeneity in a simulated myofibril (Campbell et al. 2011). While experiments have tried to eliminate homogeneity using length-clamped fiber segments (Edman et al. 1982) or by mechanically isolating single sarcomeres (Leonard et al. 2010), it occurred to us that increasing heterogeneity, as we did computationally, could be revealing. One potential way of doing this would be to subject a permeabilized muscle fiber to a localized jet of buffer containing a crossbridge inhibitor such as BDM. In principle, blocking formation of strong crossbridges in one region would increase heterogeneity of half-sarcomeres and the effect of this on enhancement could then be determined. As models are used to simulate other experimental protocols such as tension clamps during enhancement (Edman and Tsuchiya 1996), more testable hypotheses are sure to emerge.

Conclusions

The most recent decade of research into the mechanisms of residual force enhancement has not produced a consensus view of the situation. Nonetheless, critical gains have been made in the form of many new studies, both experimental and computational. In our view, much of the controversy in the field can be resolved by considering a combination of mechanisms. We have argued here that in fact some key observations are difficult to explain without this integrative view. In the near future, computational models could be developed that (1) combine putative enhancement mechanisms with quantitative accuracy and (2) successfully recapitulate responses to a wide spectrum of experimental protocols. We predict that ultimately models that include heterogeneity of active and passive properties of half-sarcomeres along with properly represented Ca²⁺-dependent behavior of titin stiffness will reconcile the many experimental observations present in the literature, and lead to a clearer understanding of the mechanics of contracting muscle.