Cardiac sarcomeres produce greater active force in response to stretch, forming the basis of the Frank-Starling mechanism of the heart. The purpose of this study was to provide the systematic understanding of length-dependent activation by investigating experimentally and mathematically how the thin filament “on–off” switching mechanism is involved in its regulation. Porcine left ventricular muscles were skinned, and force measurements were performed at short (1.9 µm) and long (2.3 µm) sarcomere lengths. We found that 3 mM MgADP increased Ca2+ sensitivity of force and the rate of rise of active force, consistent with the increase in thin filament cooperative activation. MgADP attenuated length-dependent activation with and without thin filament reconstitution with the fast skeletal troponin complex (sTn). Conversely, 20 mM of inorganic phosphate (Pi) decreased Ca2+ sensitivity of force and the rate of rise of active force, consistent with the decrease in thin filament cooperative activation. Pi enhanced length-dependent activation with and without sTn reconstitution. Linear regression analysis revealed that the magnitude of length-dependent activation was inversely correlated with the rate of rise of active force. These results were quantitatively simulated by a model that incorporates the Ca2+-dependent on–off switching of the thin filament state and interfilament lattice spacing modulation. Our model analysis revealed that the cooperativity of the thin filament on–off switching, but not the Ca2+-binding ability, determines the magnitude of the Frank-Starling effect. These findings demonstrate that the Frank-Starling relation is strongly influenced by thin filament cooperative activation.

At the turn of the 20th century, Frank and Starling discovered that cardiac pump function is enhanced as ventricular filling is increased (i.e., the Frank-Starling law of the heart; see Katz, 2002 and references therein). The “law” forms the fundamental principle of the heart in cardiovascular physiology, defining the relation between the diastolic and systolic performances of cardiac chambers. It is widely accepted that the length dependence of Ca2+-based myofibrillar activation (i.e., expressed as “Ca2+ sensitivity of force”) largely underlies the law (e.g., Allen and Kurihara, 1982; Allen and Kentish, 1985; Kentish et al., 1986); however, the molecular mechanism of this seemingly simple phenomenon still remains elusive and warrants an in-depth investigation.

The cross-bridge formation is a stochastic process in the striated muscle sarcomere (e.g., Huxley, 1957). Therefore, it has been proposed that the binding of myosin to actin is enhanced upon the reduction in the distance between the thick and thin filaments (i.e., interfilament lattice spacing), resulting in an increase in active force production and, apparently, Ca2+ sensitivity of force (Ishiwata and Oosawa, 1974; McDonald and Moss, 1995; Fuchs and Wang, 1996; Fukuda et al., 2000). Indeed, studies with synchrotron x ray revealed that passive force due to extension of the giant elastic protein titin (also known as connectin) modulates the lattice spacing within the physiological sarcomere length (SL) range in cardiac muscle (Cazorla et al., 2001; Fukuda et al., 2003, 2005). Konhilas et al. (2002b), however, challenged this proposal, demonstrating that the lattice spacing and Ca2+ sensitivity of force are not well correlated. Therefore, factors other than the titin-based lattice spacing modulation are likely at play in the regulation of length-dependent activation.

As has been reported, multiple cooperative processes are involved in active force generation in striated muscle (e.g., Brandt et al., 1982, 1987, 1990; Moss et al., 1985); i.e., cooperative binding of Ca2+ to troponin (Tn) C (TnC), cooperative binding of myosin to the thin filaments, and synergistic interactions between myosin binding to actin and Ca2+ binding to TnC (e.g., Bremel et al., 1973; Güth and Potter, 1987; Hoar et al., 1987; Zot and Potter, 1989; Swartz and Moss, 1992). Likewise, it is widely accepted that the formation of strongly bound cross-bridges enhances cooperative recruitment of neighboring myosin to the thin filaments. Bremel and Weber (1972) were the first to demonstrate in solution that an increase in the fraction of rigor cross-bridges (rigor myosin subfragment 1) cooperatively activates myosin ATPase, as with the increased Ca2+ concentration, indicating that Ca2+ and strongly bound cross-bridges synergistically regulate the “on–off” equilibrium of the thin filament state. Later, the group of Moss provided evidence in skinned muscle fibers that actomyosin interaction is indeed promoted in the presence of the strong-binding cross-bridge analogue N-ethylmaleimide myosin subfragment 1 (NEM-S1) via activation of the thin filaments, as manifested by the increased rate of contraction (Swartz and Moss, 1992, 2001; Fitzsimons et al., 2001a,b) and the increased Ca2+ sensitivity of force (Fitzsimons and Moss, 1998; Fitzsimons et al., 2001a,b). Similarly, we previously reported that the application of MgADP, i.e., the ensuing formation of the actomyosin–ADP complex, cooperatively enhances cross-bridge recruitment and that of inorganic phosphate (Pi) does the opposite in skinned fibers of cardiac and skeletal muscles (Shimizu et al., 1992; Fukuda et al., 1998, 2000, 2001a).

Earlier studies have suggested that the thin filament–based on–off switching mechanism is involved in the regulation of length-dependent activation in cardiac muscle. Indeed, Fitzsimons and Moss (1998) reported that length-dependent activation is attenuated in the presence of NEM-S1. Fukuda et al. (2000) confirmed this notion by providing evidence that the length dependence becomes smaller in the presence of MgADP (hence the actomyosin–ADP complex). It has also been reported that direct modulation of thin filament regulatory proteins, e.g., Tn isoform changes (Arteaga et al., 2000; Terui et al., 2008) or TnI point mutation (Tachampa et al., 2007), markedly affects length-dependent activation. Therefore, it is likely that length-dependent activation depends on the state of the thin filaments, either modulated directly by regulatory protein isoform switching or indirectly by strongly bound cross-bridges. However, it is still unknown how Ca2+-dependent on–off switching of the thin filament state and interfilament lattice spacing coordinate to regulate myocardial length-dependent activation.

Accordingly, this study was undertaken to systematically uncover the molecular basis of length-dependent activation in cardiac muscle, focusing on the role of thin filament cooperative activation. We varied the level of thin filament cooperative activation in skinned porcine left ventricular muscle (PLV) directly by Tn exchange or indirectly by the application of MgADP or Pi, or the combination of both. Our analysis revealed that the magnitude of length-dependent activation is inversely related to the rate of rise of active force, highlighting a pivotal role of thin filament cooperative activation in the regulation of the Frank-Starling relation. Furthermore, our mathematical model analyses revealed the relationship between the characteristics of thin filament activation and length-dependent activation, and led us to conclude that length-dependent activation is under the strong control of thin filament cooperative activation.

All experiments performed in this study conform to the Guide for the Care and Use of Laboratory Animals (1996. National Academy of Sciences, Washington D.C.). For expanded materials and methods, please see the supplemental material.

### Preparation of skinned muscle

Skinned muscles were prepared according to the method in our recent studies (Terui et al., 2008; Matsuba et al., 2009). In brief, porcine hearts (from ∼1.0-yr-old animals) were obtained from a local slaughterhouse. Muscle strips (1–2 mm in diameter and ∼10 mm in length) were dissected from the papillary muscle of the left ventricle in Ca2+-free Tyrode’s solution (see Fukuda et al., 2001a for composition) containing 30 mM 2,3-butanedione monoxime (BDM).

### Muscle mechanics

Isometric force was measured according to the method in our recent studies (Terui et al., 2008; Udaka et al., 2008; Matsuba et al., 2009). In brief, PLVs were skinned in relaxing solution (5 mM MgATP, 40 mM BES, 1 mM Mg2+, 10 mM EGTA, 1 mM dithiothreitol, 15 mM phosphocreatine, 15 U/ml creatine phosphokinase (CPK), and 180 mM ionic strength [adjusted by K-propionate], pH 7.0), containing 1% (wt/vol) Triton X-100 and 10 mM BDM overnight at ∼3°C (Fukuda et al., 2003, 2005). Muscles were stored for up to 3 wk at −20°C in relaxing solution containing 50% (vol/vol) glycerol. All solutions contained protease inhibitors (0.5 mM PMSF, 0.04 mM leupeptin, and 0.01 mM E64).

Small thin preparations (∼100 µm in diameter and ∼2 mm in length) were dissected from the porcine ventricular strips for force measurement. SL was measured by laser diffraction during relaxation, and active and passive forces were measured at 15°C (pCa adjusted by Ca2+/EGTA based on a computer program by Fabiato, 1988). MgADP (up to 10 mM) or Pi (up to 20 mM) was added to the individual activating solutions in accordance with our previous studies (Fukuda et al., 2000, 2001a), while maintaining ionic strength at 180 mM. When MgADP was present, 0.1 mM P1,P5-di(adenosine-5′)pentaphosphate was added to both activating and relaxing solutions, with no CPK to maintain the ADP/ATP ratio (Fukuda et al., 1998, 2000). We also used pimobendan (provided by Nippon Boehringer Ingelheim) to increase the affinity of TnC for Ca2+ (Fukuda et al., 2000). Pimobendan was initially dissolved in DMSO and diluted with the individual solutions (Fukuda et al., 2000). The final concentration of DMSO was 1%, having no effect on active or passive force, as observed in our previous study (Fukuda et al., 2000).

The muscle preparation was first immersed in relaxing solution, and SL was set at 1.9 µm. Active and passive forces were measured at 1.9 µm and then at 2.3 µm, as described in our previous studies (rundown <10% for active and passive forces; Fukuda et al., 2003, 2005; Terui et al., 2008). Active force data were fitted to the Hill equation (Fukuda et al., 2000), and the difference between the values of the midpoint of the force–pCa curve (i.e., pCa50) at SL 1.9 and 2.3 µm was used as an index of the SL dependence of Ca2+ sensitivity of force (expressed as ΔpCa50). The steepness of the force–pCa curve was expressed as the Hill coefficient (nH).

The rate of rise of active force was assessed according to the method in our previous work (Fukuda et al., 2001b), using the preparations used for the steady-state isometric force measurement. In brief, SL was set at 1.9 µm in relaxing solution. The preparation was then immersed in low EGTA (0.5 mM) relaxing solution for 1 min and transferred to the control activating solution (5 mM MgATP, 40 mM BES, 1 mM Mg2+, 10 mM EGTA, 1 mM dithiothreitol, 15 mM phosphocreatine, 15 U/ml CPK, and 180 mM ionic strength (adjusted by K-propionate), pH 7.0, pCa 4.5), without MgADP or Pi, followed by relaxation. The procedure was then repeated in the presence of MgADP or Pi (or after Tn reconstitution), and the time to half-maximal activation was compared with that obtained in the preceding contraction in the same preparation, hence minimizing the effect of diffusion that is dependent on the muscle thickness. The ratio of the time to half-maximal activation, defined as t1/2, was used as an index of cooperativity of cross-bridge recruitment.

In some experiments, the velocity of isometric force development was obtained at half-maximal activation in the presence of MgADP (up to 10 mM) or Pi (up to 20 mM), and the value was compared with that in the preceding contraction in the same preparation with no MgADP or Pi (pCa 4.5 and SL, 1.9 µm). The ratio of the velocity of isometric force development was defined as V1/2.

### Tn exchange

The fast skeletal Tn complex (sTn) was extracted from rabbit fast skeletal muscle, and Tn exchange (2 mg/ml for 1 h) was performed on PLV, according to our previously published procedure (see Terui et al., 2008; Matsuba et al., 2009). As detailed in these previous reports, our protocol allowed for only a small increase (∼10%) in the band intensity of each Tn subunit upon sTn reconstitution.

Control experiments showed that the treatment of PLV with the Tn complex (6 mg/ml for 1 h under the same condition) from the porcine ventricle does not alter the steady-state active force or t1/2 at various Ca2+ concentrations (see Matsuba et al., 2009; not depicted). Also, due to a relatively greater magnitude of rundown in steady-state active force (i.e., ∼30%; not depicted), we did not conduct mechanical experiments in the present study using PLV that had been incorporated with the cardiac Tn complex after sTn reconstitution.

### Model analysis

The model calculates the active isometric force at a given SL and at a given Ca2+ concentration, based on the SL-dependent change in the lattice spacing and the Ca2+-based on–off switching of the thin filament state. The on–off state was defined according to the lateral fluctuation of the thin filaments in the myofilament lattice (Ishiwata and Fujime, 1972; Umazume and Fujime, 1975; Yoshino et al., 1978; Yanagida et al., 1984; see Fig. S1); however, the lateral fluctuation does not necessarily represent the physical thermal fluctuation of the thin filaments, but rather, it mathematically portrays the equilibrium of the thin filament state between “off” and “on,” depending on the Ca2+ concentration (Solaro and Rarick, 1998).

In our model, the overlap length between the thick and thin filaments at a SL of L is given by:

$12(L0−L),$
(1)

where L0 (3.8 µm) is the maximal SL at no filament overlap. Thick and thin filament length was assumed to be 1.6 and 1.1 µm, respectively (Sosa et al., 1994). In this study, the overlap length was set to be constant (0.75 µm), independent of SL because in the SL range between 1.9 and 2.3 µm (where the experiments were performed), the number of myosin heads in the overlap region reportedly remains constant based on the thick filament geometry (Sosa et al., 1994).

Next, we assumed that the lattice spacing, d, decreases upon the increase in SL under the constant lattice volume, V, as has been observed in x-ray diffraction studies (Cazorla et al., 2001; Fukuda et al., 2003, 2005), according to the following equation:

$V=d2⋅L.$
(2)

Based on the value of d10 (i.e., the d10 lattice spacing) of 43 nm at SL 2.0 µm (see Fukuda et al., 2003 for the d10 value of sarcomeres expressing both N2B and N2BA titins at similar levels, as in PLV; Terui et al., 2008), the lattice spacing is estimated to be 28.7 nm, and thereby the lattice volume, V, was set to be 0.0016 µm3.

Next, we described the position-dependent probability of actomyosin interaction by the Gaussian distribution (Ishiwata and Oosawa, 1974):

$P(q)=1πσA2exp (−q2/σA2),$
(3)

where q is a lateral coordinate perpendicular to the filament long axis, and σA is the width of the Gaussian distribution (variance, σA2/2). To take into account the Ca2+-dependent change of the actomyosin interaction, the degree of σA was changed in accordance with the Ca2+ concentration based on the Hill equation (see Ishiwata and Oosawa, 1974 and Figs. S1 and S2):

$σA(pCa)=σmax11+10−nH_actin(pCa50_actin−pCa),$
(4)

where σmax is the maximal width of the Gaussian distribution (21 nm), determining the maximal interaction probability at the saturating Ca2+ concentration (pCa 4.5). The parameter nH_actin represents the cooperativity of thin filament activation in the model calculation, and pCa50_actin represents the sensitivity of the thin filaments to Ca2+.

Here, we considered that the actomyosin interaction takes place over the region where the lateral coordinate, q, exceeds a certain distance, da, because myosin heads are located apart from the thick filament backbone up to a distance a (24 nm). The cumulative interaction probability with respect to the unit overlap length, I, is therefore given by:

$I=∫d−a∞P(q)dq.$
(5)

Finally, the active isometric force at a given SL and pCa is expressed as the product of overlap length and interaction probability:

$F=F0⋅12(L0−L)⋅1π⋅1σA(pCa)∫d−a∞exp (−q2/σA2(pCa))dq,$
(6)

where F0 (45 and 36, for control and sTn-reconstituted PLV, respectively) is the fitting parameter to quantitatively simulate the experimental results.

### Statistics

Significant differences were assigned using the paired or unpaired Student’s t test as appropriate. Data are expressed as mean ± SEM, with n representing the number of muscles. Linear regression analyses were performed in accordance with the method described in previous studies (Fukuda et al., 2001b; Terui et al., 2008). Statistical significance was assumed to be P < 0.05. NS indicates P > 0.05.

### Online supplemental material

The supplemental material provides an expanded description of our model analysis. In addition, Fig. S1 shows a schematic illustration of our model used to simulate the present experimental data. Fig. S2 provides characteristics of our model, showing how Ca2+ sensitivity of force is changed in response to a change in thin filament cooperative activation. Fig. S3 shows the relation of Ca2+ binding to the thin filaments or thin filament cooperative activation versus length-dependent activation in our model. Fig. S4 shows the experimentally observed effect of MgADP on the rate of active force redevelopment, ktr (overall cross-bridge cycling rate; see Discussion), at varying activation levels. Fig. S5 shows the experimentally obtained relation between SL and active force at various Ca2+ concentrations (converted from force–pCa curves) and the simulation by our model. Fig. S6 shows the model simulation showing the effect of an increase in the average length of myosin heads on Ca2+ sensitivity of force and length-dependent activation.

### Effect of MgADP or Pi on the rate of rise of active force

First, we investigated the effect of various concentrations of MgADP or Pi on the rate of rise of active force under the control condition without sTn reconstitution. We found that MgADP significantly decreased t1/2 at low concentrations (1 and 3 mM) but increased it at a high concentration (10 mM) (Fig. 1 A), and Pi exerted apparently similar effects at low (1, 3, and 5 mM) and high (10 and 20 mM) concentrations (Fig. 1 B). The accelerating effect of MgADP was maximal at 3 mM, whereas the decelerating effect of Pi reached a quasi-plateau at 10 mM. Therefore, based on the previous studies indicating that the rate of contraction is modulated by a change in the fraction of strongly bound cross-bridges via alteration of the on–off equilibrium of the thin filament state (Swartz and Moss, 1992, 2001; Fitzsimons et al., 2001a,b), we regarded 3 mM MgADP as the amount to enhance thin filament cooperative activation and 20 mM Pi to reduce it and used them in the following experiments.

Fig. 1 C shows the relationship between t1/2 and V1/2. We found that a significant linear relationship with a similar slope value existed between the parameters in the presence of varying concentrations of MgADP (Fig. 1 C, left; slope 1.07) or Pi (right, slope 1.24), indicating that t1/2 reflects the rate of rise of active force.

### Effect of MgADP or Pi on length-dependent activation with and without sTn reconstitution

Next, we investigated how the SL-dependent increase in Ca2+ sensitivity of force responds to alteration of thin filament cooperative activation. In this series of experiments, we performed sTn reconstitution to directly enhance thin filament cooperative activation, as demonstrated in our previous study (Terui et al., 2008), with and without MgADP or Pi.

We found that 3 mM MgADP or sTn reconstitution similarly shifted the force–pCa curve leftward, to a greater magnitude at SL 1.9 µm than at 2.3 µm, and consequently decreased ΔpCa50 (Fig. 2, A and B). Ca2+ sensitivity of force was synergistically increased by MgADP in sTn-reconstituted PLV, accompanied by a marked attenuation of length-dependent activation (Fig. 2 B). As shown in Fig. 2 C, the rate of rise of active force was increased by MgADP or sTn reconstitution by a similar magnitude. Similar to the finding on Ca2+ sensitivity of force, the rate of rise of active force was increased by MgADP in sTn-reconstituted PLV (Fig. 2 C).

We then tested the effect of Pi on length-dependent activation. Without sTn reconstitution, 20 mM Pi shifted the force–pCa curve rightward to a greater magnitude at SL 1.9 µm than at 2.3 µm, and consequently increased ΔpCa50 (Fig. 3 A). Pi increased ΔpCa50 also in sTn-reconstituted PLV (Fig. 3 B). In contrast to MgADP, Pi retarded the rate of rise of active force in both control and sTn-reconstituted PLV (Fig. 3 C). The values of pCa50, nH, and maximal force at SL 1.9 and 2.3 µm under various conditions are summarized in Table I.

Fig. 4 summarizes the relationship between t1/2, pCa50 and ΔpCa50 obtained with MgADP or Pi in control and sTn-reconstituted PLV. We found that pCa50 and ΔpCa50 were linearly correlated with each other (Fig. 4 A), and that pCa50 and ΔpCa50 were a linear function of t1/2 (Fig. 4, B and C). As reported previously (Dobesh et al., 2002), however, no significant correlation was found between nH and pCa50 or ΔpCa50 (Fig. 5).

### Effect of pimobendan on length-dependent activation with and without sTn reconstitution

The observed relationship of Ca2+ sensitivity of force and length-dependent activation may be a consequence associated with the leftward shift of the force–pCa curve (Hanft et al., 2008). We therefore tested the effect of pimobendan on length-dependent activation, with passive force carefully controlled (which was not performed in our previous study on rat ventricular trabeculae; Fukuda et al., 2000), because the compound has been reported to specifically increase the affinity for Ca2+ of the low-affinity site of TnC (see Hagemeijer, 1993 and references therein).

Pimobendan (2 × 10−4 M) shifted the force–pCa curve leftward to a magnitude similar to that by MgADP or sTn reconstitution at SL 1.9 µm (i.e., ∼0.2 pCa units; see Fig. 2), with no effect on passive force (Fig. 6 A). However, unlike MgADP or sTn reconstitution, pimobendan exerted no effect on ΔpCa50 (see Fukuda et al., 2000). The Ca2+-sensitizing effect of pimobendan was markedly diminished after sTn reconstitution, with no significant increase in Ca2+ sensitivity of force at either SL (Fig. 6 B). Likewise, pimobendan did not affect the rate of rise of active force in control or sTn-reconstituted PLV (Fig. 6 C).

The values of pCa50, nH, and maximal force obtained with and without pimobendan are summarized in Table II.

### Simulation of length-dependent activation

Finally, we analyzed the experimental findings based on a mathematical model (Ishiwata and Oosawa, 1974; Shimamoto et al., 2007; refer to Materials and methods and supplemental material for details). In this model, active isometric force is given by the interaction probability between the thick and thin filaments, and the probability depends on two factors: (1) Ca2+ concentration and (2) interfilament lattice spacing (Fig. S1). The equilibrium of the thin filament state between “off” and “on” (see Solaro and Rarick, 1998 and references therein) was assumed to change with the Ca2+ concentration based on the Hill equation, and expressed as lateral fluctuation in the myofilament lattice (Figs. S1 and S2). We performed experiments within the SL range (i.e., 1.9–2.3 µm); the overlap length between the thick and thin filaments is considered not to change significantly (see Moss and Fitzsimons, 2002 and references therein), but the lattice spacing does, due to titin extension, as revealed by previous studies with muscles expressing both N2B and N2BA titins (Fukuda et al., 2003), as in PLV (Terui et al., 2008).

Fig. 7 A shows the force–pCa curves simulated by our model for the experimental data with and without sTn reconstitution in PLV. The model parameters are nH_actin and pCa50_actin, representing the characteristics of thin filament on–off switching in response to Ca2+. Based on Eq. 6, we simulated the force–pCa curves of PLV with and without sTn reconstitution, and thereby the pCa50 and ΔpCa50 values were calculated (for optimization of fitting, see supplemental material).

Under the control condition without sTn reconstitution, a reduction in the lattice spacing due to an increase in SL from 1.9 to 2.3 µm (refer to Materials and methods) increased maximal Ca2+-activated force and shifted the force–pCa curve leftward, resulting in ΔpCa50 of 0.24 pCa units. As shown in Fig. S2, an increase in nH_actin decreased ΔpCa50 and concomitantly shifted the midpoint of the force–pCa curve rightward. On the other hand, an increase in pCa50_actin linearly shifted the force–pCa curve leftward. To reproduce the experimental data after sTn reconstitution, we increased both nH_actin and pCa50_actin; as a result, the attenuation of length-dependent activation was well simulated, accompanied by appropriate pCa50 values for both SLs (as in Fig. 7 A; compare Figs. 2 and 3, and Terui et al., 2008).

Finally, we systematically investigated how varying the values of nH_actin and pCa50_actin affects length-dependent activation by constructing a 3-D graph consisting of nH_actin, pCa50_actin, and ΔpCa50 (Fig. 7 B). We found that ΔpCa50 was strongly influenced by nH_actin; however, the contribution of pCa50_actin to ΔpCa50 was minimal throughout the range we examined (see also Fig. S3). We plotted the pairs of the values of nH_actin and pCa50_actin that fulfill the linear relationship between pCa50 and ΔpCa50 (i.e., red points in Fig. 7 B), which was experimentally obtained in Fig. 4 A.

We demonstrated in this study that the Frank-Starling relation is strongly influenced by thin filament cooperative activation. The SL-dependent increase in Ca2+ sensitivity of force was inversely related to the rate of rise of active force, suggesting that length-dependent activation is tuned via on–off switching of the thin filament state in cardiac muscle. Further, our model analysis revealed that thin filament cooperative activation, but not the affinity for Ca2+, determines the magnitude of length-dependent activation, coupled with lattice spacing modulation. Here, we discuss the present findings, focusing on the role of thin filament cooperative activation in the regulation of length-dependent activation in cardiac muscle.

As is well established, strongly bound cross-bridges cooperatively activate the thin filaments (e.g., Bremel and Weber, 1972), resulting in the promotion of actomyosin interaction (refer to Introduction). Previous studies showing that NEM-S1 increases the speed of contraction in skinned fibers (Swartz and Moss, 1992, 2001; Fitzsimons et al., 2001a,b) support the notion that strongly bound cross-bridges accelerate the recruitment of neighboring myosin to the thin filaments via enhanced thin filament cooperative activation. In the present study, MgADP decreased t1/2 at low concentrations (1 and 3 mM) but increased it at a high concentration (10 mM), and Pi exerted apparently similar effects at low (1, 3, and 5 mM) and high (10 and 20 mM) concentrations (Fig. 1; see Kentish, 1986). However, the underlying molecular mechanisms for the modulation of t1/2, i.e., the rate of rise of active force, should differ for MgADP and Pi. At low MgADP concentrations (e.g., 3 mM used in the present study), the actomyosin–ADP complex may exhibit its promoting effect on actomyosin interaction by enhancing thin filament cooperative activation, but at high MgADP concentrations, the presence of large fractions of the complex may cause deceleration of contraction due to its slow cycling rate, as demonstrated previously in experiments measuring the shortening of the velocity at zero load (e.g., Cooke and Pate, 1985; Metzger, 1996), despite the highly activated state of the thin filaments. On the other hand, the binding of Pi to the actomyosin complex is reportedly enhanced upon the increase in the strain of the complex (Webb et al., 1986; Metzger, 1996). Therefore, at low concentrations, Pi may preferentially bind to the slowly cycling actomyosin–ADP complex, resulting in an increase in the rate of rise of active force, as demonstrated previously in experiments measuring kinetics following flash photolysis (Lu et al., 1993; Araujo and Walker, 1996), the shortening of the velocity at zero load (Metzger, 1996) and the rate of force redevelopment (ktr) (Tesi et al., 2000). However, at high concentrations (e.g., 20 mM used in the present study), Pi may decrease the fraction of the actomyosin–ADP complex to a level where neighboring myosin cannot be effectively recruited to actin, resulting in a decrease in the rate of rise of active force. Therefore, although the alteration of t1/2 includes processes other than thin filament cooperative activation, the present findings allow us to consider that it at least in part reflects a change in thin filament cooperative activation.

MgADP at 3 mM increased the rate of rise of active force and, concomitantly, left-shifted the force–pCa curve (Figs. 1 and 2). The magnitude of the change was similar to that observed upon sTn reconstitution (i.e., direct modulation of regulatory proteins to enhance thin filament cooperative activation; see Terui et al., 2008) for both t1/2 and Ca2+ sensitivity of force (Fig. 2). These findings suggest that, albeit modulated via different pathways, i.e., either indirectly or directly, thin filament cooperative activation is enhanced by a similar magnitude with 3 mM MgADP and sTn reconstitution. Interestingly, 3 mM MgADP increased both Ca2+ sensitivity of force and the rate of rise of active force in sTn-reconstituted PLV, accompanied by a marked depression of length-dependent activation (Fig. 2). These additive effects of MgADP suggest that thin filament cooperative activation can be synergistically modulated via strong-binding cross-bridge formation and regulatory protein isoform switching. In contrast, 20 mM Pi exerted effects opposite to those of 3 mM MgADP, with and without sTn reconstitution, by decreasing Ca2+ sensitivity of force and slowing the rate of rise of active force (Fig. 3). Therefore, the observed effect of 20 mM Pi on length-dependent activation likely results from the reduced thin filament cooperative activation. Furthermore, pimobendan did not affect length-dependent activation, indicating that Ca2+ binding to TnC is not the parameter determining the magnitude of this phenomenon. Therefore, given the close relationship between t1/2 (or pCa50) and ΔpCa50 (Fig. 4), we consider that thin filament cooperative activation plays a pivotal role in setting the magnitude of length-dependent activation.

It has been reported in various experimental settings that a positive feedback mechanism exists between Ca2+ binding to TnC and cross-bridge formation in the sarcomere (Allen and Kurihara, 1982; Güth and Potter, 1987; Hoar et al., 1987; Hofmann and Fuchs, 1988; Zot and Potter, 1989). Therefore, the linear relationship of t1/2 versus pCa50 observed in the present study (Fig. 4) likely reflects the positive feedback effect on Ca2+ binding to TnC via cross-bridge formation due to enhanced thin filament cooperative activation.

One may point out that ktr, i.e., the sum of the apparent rate of cross-bridge attachment (fapp) and detachment (gapp) (Brenner, 1988; Swartz and Moss, 1992; Fitzsimons et al., 2001a,b, and references therein; Terui et al., 2008), more suitably represents thin filament cooperative activation in muscle mechanics than t1/2. Indeed, ktr is reportedly increased upon enhanced thin filament cooperative activation, when modulated directly (e.g., sTn reconstitution; Terui et al., 2008) or indirectly (NEM-S1 application; Swartz and Moss, 1992; Fitzsimons et al., 2001a,b). In the present study, 3 mM MgADP decreased ktr at both maximal and submaximal activations (Fig. S4), in agreement with the result of previous studies with rabbit skeletal muscle (Lu et al., 1993; Tesi et al., 2000). MgADP is known to inhibit the release of ADP from the actomyosin complex at the end of the cross-bridge cycle (see Fukuda et al., 1998, 2000, and references therein). Therefore, the inhibitory effect of MgADP on gapp may overshadow its accelerating effect on cross-bridge formation (fapp), resulting in a decrease in ktr. However, the observed increase in the rate of rise of active force, regardless of the Tn isoform, suggests that MgADP at low concentrations (such as 3 mM in the present experimental setting; see Figs. 1 and 2) accelerates fapp via enhancement of thin filament cooperative activation.

Pimobendan did not significantly increase Ca2+ sensitivity of force after sTn reconstitution in PLV (Fig. 6). It has, however, been reported that pimobendan increases Ca2+ sensitivity of force in amphibian skeletal muscles (Piazzesi et al., 1987; Wakisaka et al., 2000). The absence of Ca2+ sensitization observed in the present study may indicate the compound’s specificity regarding the site of action; namely, in mammals, the binding affinity of pimobendan for (fast) skeletal TnC may be lower than that for cardiac TnC, producing a minimal effect on (fast) skeletal muscle. It should also be pointed out that the rate of rise of active force was unaltered by pimobendan (Fig. 6), confirming our view that the increase in the rate of rise of active force by MgADP or sTn reconstitution results not from the increase in the affinity of TnC for Ca2+, but from enhanced thin filament cooperative activation.

Dobesh et al. (2002) reported that the nH of the force–pCa curve is not correlated with ΔpCa50, leading them to conclude that thin filament cooperative activation plays no significant role in determining the magnitude of length-dependent activation. Consistent with this finding, we observed no significant correlation between the nH of the force–pCa curve and ΔpCa50 (Fig. 5). However, it is unclear to what extent the steady-state nH reflects thin filament cooperative activation. For instance, NEM-S1 or MgADP has been used to enhance thin filament cooperative activation; however, both NEM-S1 (Swartz and Moss, 1992, 2001; Fitzsimons and Moss, 1998; Fitzsimons et al., 2001a,b) and MgADP (at high concentrations: Fukuda et al., 1998, 2000) reportedly decrease the nH of the force–pCa curve, resulting presumably from enhanced recruitment of neighboring cross-bridges, especially at low Ca2+ concentrations (see Fukuda et al., 1998 for ADP contraction occurring in the absence of Ca2+). Therefore, in the present study, we regarded the rate of rise of active force as an index of thin filament cooperative activation (as in, e.g., Swartz and Moss, 1992, 2001; Fitzsimons et al., 2001a,b), rather than the nH of the force–pCa curve.

Earlier, we discussed that at high activation states (i.e., high Ca2+ concentrations, MgADP application, or sTn reconstitution), cross-bridge recruitment upon SL elongation becomes less pronounced due to a decrease in the fraction of recruitable cross-bridges (that can potentially generate active force), resulting in the attenuation of length-dependent activation (see Fukuda et al., 2009 and references therein). The present model calculation provides a mechanistic insight into this interpretation. Namely, the SL elongation (i.e., lattice reduction) –induced increase in the probability of cross-bridge formation becomes less pronounced upon the increase in thin filament cooperative activation. This is because the acceleration of Ca2+-dependent widening of the Gaussian distribution, of which the magnitude depends on nH_actin (see Eq. 6 and Fig. S1), diminishes the lattice spacing–dependent change of the actomyosin interaction (determined by da). On the other hand, pCa50_actin had little effect on the lattice spacing dependence; i.e., length-dependent activation. Indeed, the attenuation of length-dependent activation upon sTn reconstitution was quantitatively simulated by our model (Fig. 7 A) by increasing nH_actin with appropriate pCa50 values for both short and long SLs as a result of an increase in pCa50_actin. In addition, our model could quantitatively simulate the relationship of SL versus active force at various Ca2+ concentrations, converted from the force–pCa curves obtained under the control condition in Figs. 2 and 3 (i.e., shallower at high Ca2+ concentrations; Kentish et al., 1986; Fukuda et al., 2001b; see Fig. S5), emphasizing the adequacy of our model to analyze the molecular mechanism of length-dependent activation.

However, it is important to discuss limitations of the present study. First, we noted a mismatch between the experimental data and the simulated curves; namely, an increase in nH_actin increased the steepness of the force–pCa curve at both SLs (compare Fig. 7 A), whereas the nH of the force–pCa curve was not increased upon sTn reconstitution or MgADP application at either SL (Table I). We consider that this mismatch reflects the limitation of the experiments with skinned myocardial fibers. For example, internal sarcomere shortening that presumably occurs during isometric contraction (Fukuda et al., 2001b) may decrease active force production by a greater magnitude at high Ca2+ concentrations, resulting in an underestimation of the steepness of the force–pCa curve (as discussed in Fukuda et al., 2005). It should also be stressed that the mismatch reflects the limitation of the use of t1/2 as an index of thin filament cooperative activation; namely, MgADP, Pi, or sTn reconstitution may alter the cross-bridge kinetics via a pathway that is not coupled with thin filament cooperative activation. Clearly, future studies with various techniques are needed to clarify this issue. Second, the decrease in the intermolecular distance, i.e., da upon the addition of MgADP (earlier assumed to represent the lattice spacing modulation via cross-bridge formation in fast skeletal muscle; Shimamoto et al., 2007) enhanced length-dependent activation (Fig. S6) in contrast to the experimental result (Fig. 2). This apparent discrepancy may suggest that a change in thin filament cooperative activation has a greater impact on length-dependent activation, masking the effect of a cross-bridge–dependent lattice spacing change.

The 3-D graph obtained in the present model calculation (Fig. 7 B) suggests the role of the thin filaments in the regulation of length-dependent activation; namely, the magnitude of this phenomenon depends only slightly on the Ca2+-binding ability of TnC, as confirmed by our experimental analysis with pimobendan (Fig. 6 and Fukuda et al., 2000), but rather strongly on the cooperativity of the thin filament on–off switching (Figs. S1 and S2). Here, it is worthwhile noting that pCa50_actin needed to be varied to quantitatively simulate the experimentally obtained relationship of pCa50 versus ΔpCa50 (Fig. 7 B). This may be due to a coupling between thin filament cooperative activation and cross-bridge formation, and to the ensuing feedback effect that enhances Ca2+ binding to TnC (Güth and Potter, 1987; Kurihara and Komukai, 1995). Therefore, the inverse relationship between pCa50 and ΔpCa50 (Fig. 4 A) may be an apparent phenomenon resulting from enhanced Ca2+ binding to TnC, coupled with acceleration of thin filament cooperative activation.

However, we admit that the present modeling is not suitable to account for the differing magnitudes of length-dependent activation in fast skeletal muscle versus slow skeletal muscle. Indeed, Konhilas et al. (2002a) reported that length-dependent activation is less in slow skeletal muscle, despite a lesser magnitude of thin filament cooperative activation. It is therefore likely that the difference in the magnitude of length-dependent activation between fast skeletal muscle and slow skeletal muscle results from factors that do not involve thin filament cooperative activation, such as isoform variance of thin filament– and thick filament–based proteins. It is an area of future research to clarify this issue by using various skeletal muscle tissues.

Considering that Ca2+ sensitivity of force varies depending on the type of heart disease (for review see Ohtsuki and Morimoto, 2008), it is likely that thin filament cooperative activation is altered in disease. The findings of the present study suggest that length-dependent activation is modulated via mutation occurring in the thin filaments and/or breakdown of ATP and the ensuing elevations in ADP and Pi in the vicinity of cross-bridges. Indeed, it has been reported that the Frank-Starling mechanism is depressed in skinned left ventricular muscles from patients with terminal heart failure (Schwinger et al., 1994; Brixius et al., 2003). It would be interesting to simulate, based on the 3-D state diagram (Fig. 7 B), how the Frank-Starling relation is altered by the occurrence of a mutation in a regulatory protein and/or the breakdown of ATP in various types of heart disease in various animal species, including humans.

In conclusion, thin filament cooperative activation plays a central role in the regulation of the Frank-Starling mechanism of the heart.

We thank Ms. Naoko Tomizawa (The Jikei University School of Medicine, Tokyo, Japan) for technical assistance.

Pimobendan was kindly donated by Nippon Boehringer Ingelheim. The work of the authors was supported in part by Grants-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science and Technology of Japan (to N. Fukuda and S. Kurihara) and by grants from the Japan Science and Technology Agency (Core Research for Evolutional Science and Technology; to N. Fukuda) and the Institute of Seizon and Life Sciences (to S. Kurihara).

Richard L. Moss served as editor.

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Abbreviations used in this paper:
NEM-S1

N-ethylmaleimide myosin subfragment 1

nH

Hill coefficient

Pi

inorganic phosphate

PLV

porcine left ventricular muscle

SL

sarcomere length

sTn

fast skeletal troponin complex

Tn

troponin

## Author notes

T. Terui and Y. Shimamoto contributed equally to this paper.

Y. Shimamoto’s present address is Laboratory of Chemistry and Cell Biology, The Rockefeller University, 1230 York Avenue, New York, NY 10065.