The molecular basis of the steep force-calcium relation in heart muscle.

Contraction of heart muscle is regulated by binding of Ca(2+) ions to troponin in the muscle thin filaments, causing a change in filament structure that allows myosin binding and force generation. The steady-state relationship between force and Ca(2+) concentration in demembranated ventricular trabeculae is well described by the Hill equation, with parameters EC(50), the Ca(2+) concentration that gives half the maximum force, and n(H), the Hill coefficient describing the steepness of the Ca(2)(+) dependence. Although each troponin molecule has a single regulatory Ca(2+) site, n(H) is typically around 3, indicating co-operativity in the regulatory mechanism. This review focuses on the molecular basis of this co-operativity, and in particular on the popular hypothesis that force-generating myosin cross-bridges are responsible for the effect. Although cross-bridges can switch on thin filaments at low MgATP concentrations, we argue that the evidence from contracting heart muscle cells shows that this mechanism does not operate in more physiological conditions, and would not play a significant role in the intact heart. Interventions that alter maximum force and EC(50) do not in general produce a significant change in n(H). Complete abolition of force generation by myosin inhibitors does not affect the n(H) values for either Ca(2+) binding to the thin filaments or changes in troponin structure, and both values match that for force generation in the absence of inhibitors. These results provide strong evidence that the co-operative mechanism underlying the high value of n(H) is not due to force-generating cross-bridges but is rather an intrinsic property of the thin filaments. Copyright (c) 2009 Elsevier Ltd. All rights reserved.

Introduction

Contraction of heart muscle cells is triggered on a beat-to-beat basis by a transient increase in intracellular free Ca2+ ion concentration, [Ca2+]i. The Ca2+ ions bind to troponin in the actin-containing thin filaments of the muscle sarcomere, producing a change in the structure of the thin filaments that allows a force-generating interaction with the adjacent myosin-containing thick filaments. Force is very sensitive to [Ca2+]i, and small changes in [Ca2+]i can produce functionally significant changes in cardiac output. The steep force–calcium relationship facilitates rapid and synchronous activation of the contractile filaments, and minimises the metabolic cost of Ca2+ cycling. Longer term control of cardiac output is achieved through independent signalling pathways that modulate both the calcium transient and the force–calcium relationship. The fundamental importance of the latter relationship for the normal function of the heart, for the impaired function of the failing heart, and for the design of drugs that could potentially correct such defects has led to considerable interest in the molecular mechanisms that underlie and modulate the force–calcium relationship [1-3]. Here we focus on the molecular basis of the steepness of the relationship, and re-examine some of the evidence that led to the widespread hypothesis that the steep response is primarily due to co-operative activation of the calcium regulatory system by force-generating myosin cross-bridges [4-6]. We conclude that the evidence from studies of heart muscle cells contracting in close to physiological conditions does not support that hypothesis and that co-operative activation is likely to be an intrinsic property of the actin-containing thin filaments.

The steady-state relationship between isometric force and intracellular free calcium concentration

The steep Ca2+ dependence of isometric force generation has been extensively studied in heart muscle cells in which the surface membrane has been permeabilized, so that free [Ca2+] can be controlled using EGTA buffers. A typical result from our own experiments of this type [7] is shown in Fig. 1. Free [Ca2+] is expressed on a logarithmic scale, as pCa = − log10 [Ca2+]. The relationship between the normalised steady force and pCa is almost symmetrical around the value (pCa50) corresponding to half-maximum force and is quite accurately described by the Hill equation:where the Hill coefficient (nH) is a convenient measure of the steepness of the Ca2+-dependence. In demembranated ventricular trabeculae pCa50 is typically in the range 5.5 to 6.0, corresponding to a Ca2+ concentration (EC50) in the low micromolar range, matching the dissociation constant of the regulatory Ca2+ binding site on troponin in the thin filament [8,9]. nH in demembranated trabeculae is typically 3–4, much higher than expected from a simple binding model in which Ca-free troponin is associated with the relaxed (zero force) state and Ca-bound troponin is associated with the active force-generating state, which would give nH = 1. The high values of nH observed in demembranated trabeculae therefore indicate some kind of co-operative mechanism, and even higher values of nH have been reported for intact heart muscle cells [10]. Here we focus on studies using demembranated cardiac muscle preparations, in which the concentrations of free Ca2+, MgATP and other small molecules can be accurately controlled, Ca2+ binding to regulatory proteins can be measured directly, and myofilament protein components can be replaced, modified or labelled for mechanistic studies.

Fig. 1

Steady-state force–calcium relation from demembranated trabeculae of rat ventricle; mean ± SE for 5 trabeculae, data from ‘Before reconstitution of TnC-BR55–62’ in Table 1 of Sun et al. [7]. The line is the Hill equation with pCa50 = 5.60, nH = 3.51.

Modulation of the steady-state force–calcium relation

Many factors affect the steady-state force–calcium relation in demembranated trabeculae (Table 1), probably through several distinct mechanisms. Interventions that increase the maximum force measured at saturating [Ca2+], like increasing sarcomere length, phosphorylation of myosin regulatory light chain, compression of the filament lattice and addition of MgADP, tend to increase calcium sensitivity (pCa50), i.e., shift the force–calcium relation to the left. Conversely, interventions that reduce maximum force, like decreased pH, or addition of phosphate, vanadate or blebbistatin, decrease calcium sensitivity, i.e., shift the force–calcium relation to the right. The changes in pCa50 are typically about 0.2 pCa units, corresponding to a ca. 60% change in the linear [Ca2+] parameter EC50, and can be measured quite precisely. The correlation between increased force and increased pCa50 suggests that there might be a general effect of force-generating myosin cross-bridges to increase the calcium sensitivity. However, this effect is not accompanied by an increase in the steepness of the force–calcium relationship as measured by nH. In the one example in Table 1 in which a significant change in nH was observed, for RLC phosphorylation, increased force was associated with a decrease in the steepness of the force–calcium relation. Although the absolute value of nH differs substantially between the various studies, presumably due to methodological differences, the force–calcium relation is in general modulated by via changes in pCa50 and maximum force, not by changes in its steepness.

Table 1

Factors that modulate the steady-state force–calcium relationship in demembranated cardiac muscle cells, characterized by the change (Δ, %) in maximum isometric force at saturating [Ca2+], and in the Hill equation parameters pCa50 and nH defined in the text. One example reference is given for each factor.

Factor	Change	Max. force Δ (%)	ΔpCa50	nH	Reference
Sarcomere length	1.9 → 2.3 μm	+ 45	+ 0.14	7.4 → 6.9 (ns)	[60]
Lattice spacing	0 → 6% dextran	+ 12	+ 0.17	6.00 → 5.33 (ns)	[61]
MgATP	5 → 0.5 mM	ns	+ 0.15	5.1 → 5.5	[62]
MgADP	0 → 1 mM	+ 45	+ 0.4	4.8 → 4.7 (ns)	[63]
Caffeine	0 → 20 mM	− 16	+ 0.31	2.72 → 2.64 (ns)	[64]
TnI phosphorylation	by PKA	0	− 0.15	—	[65]
TnI phosphorylation	by PKD	− 9	− 0.21	—	[66]
RLC phosphorylation	by MLCK	+ 30	+ 0.12	3.8→ 3.2 (P < 0.05)	[67]
pH	7.0 → 6.2	− 50	− 1.3	5.48 → 6.02 (ns)	[68]
Inorganic phosphate	0 → 20 mM	− 57	− 0.19	2.72 → 2.84 (ns)	[64]
Vanadate	0 → 0.3 mM	− 80	− 0.14	5.5 → 4.9 (ns)	[69]
Blebbistatin	0 → 0.01 mM	− 73	− 0.34	ns	[70]

PKA, protein kinase A; PKD, protein kinase D; MLCK, myosin light chain kinase; ns, not statistically significant.

The dynamic relationship between force and free [Ca2+] following action potential stimulation

The steady-state force–calcium relation in Fig. 1 does not hold on the timescale of the isometric twitch of an intact muscle cell elicited by action potential stimulation, or in the intact heart. In these conditions, the increase in intracellular free [Ca2+] is faster than force development during the rising phase of the twitch, and the recovery of [Ca2+]i, although not as fast as its upstroke, still precedes mechanical relaxation [11,12]. Thus the instantaneous force–calcium relation varies during the time course of the twitch, reflecting the kinetics of the signalling pathway linking free [Ca2+] to changes in myofilament structure and force generation. Peak [Ca2+]i in a twitch is typically 0.5–1.0 μM, so the [Ca2+]i transient is both too small and too brief to fully activate the contractile filaments. In general, the time course of the twitch, for a given [Ca2+]i transient, will depend on both the steep steady-state force–calcium relation (Fig. 1) and the dynamics of the signalling pathway between [Ca2+]i and force. The dynamic force–calcium relation may also be affected by muscle shortening [13], as would occur during the ejection phase of the cardiac cycle. A complete description of the molecular mechanisms underlying the relationship between calcium and contractility in the working heart depends on multiple pathways and interactions that are as yet poorly understood, although detailed models of the system have been developed in an attempt to discover how specific mechanistic hypotheses would affect cardiac performance [14-16]. The molecular basis of the steep force–calcium relation and the effect of force-generating myosin cross-bridges on this relation are key features of such models. Here we focus on these relationships as determined during steady isometric contraction as a necessary first step towards a more complete dynamic description.

Molecular components and interactions in calcium regulation

The primary signalling pathway underlying calcium regulation of cardiac output is embodied in the multiprotein complex of the thin filament, containing actin, tropomyosin, and troponin. Actin forms the core of the filament, in a double-stranded helix with seven monomers per strand in each ca. 38 nm axial repeat. One tropomyosin and one troponin molecule bind to these seven actins, and this 1:1:7 complex is often referred to as a regulatory unit. The structures of actin [17,18], parts of tropomyosin [19], and parts of troponin [20,21] have been described at high resolution, but key regions of protein–protein interaction within and between regulatory units remain to be characterized. Thus, the mechanism of the thin filament signalling pathway cannot be deduced from current high-resolution structural data. Structural studies at lower resolution by electron microscopy and X-ray diffraction, combined with other biophysical and biochemical data, led to the steric blocking model of calcium regulation, in which Ca2+ binding to troponin leads to an azimuthal movement of tropomyosin around the thin filament to uncover the myosin binding sites on actin [22-25]. In the original steric blocking model, the tropomyosin position was considered to control binding of myosin to actin; later developments of the model [26] emphasised tropomyosin control of ‘strong’ or productive binding to actin via modulation of the kinetics of a transition between weakly and strongly bound myosin states. The details of the structural and biochemical intermediates involved in calcium regulation of contractility are not considered further here. Instead, we focus on functional states of the regulatory system defined by the presence or absence of Ca2+ in the TnC regulatory site and the active force.

A simple four-state model of calcium regulation

Four functional states of a regulatory unit that might be populated sequentially during a heart beat are diagrammed in Fig. 2. The best characterised state is the diastolic or relaxed state (top left) in which the TnC regulatory site is empty (Apo state) and there is no active force. Since the rising phase of [Ca2+]i is faster than that of force, there is presumably a transient ‘pre-force’ state (top right) in which Ca2+ is bound to the regulatory site of TnC but force has not yet been generated. The structure of the contractile filaments in the pre-force state in heart muscle is unknown but, by extrapolation from X-ray studies of skeletal muscle [27,28], tropomyosin may already have moved from a blocking to a ‘clear’ conformation in which it no longer impedes myosin–actin interaction. Adopting this hypothesis about the structure of the pre-force state, force generation can be represented by the classical force-generating cross-bridge cycle characterised by an effective attachment rate constant f1 and detachment rate constant g1 [29], leading to the active/systole state. In this scheme the pre-force state is an ‘ON’ state of the regulatory system, and activation and force generation are separate sequential processes, with effective forward and backward rate constants k+ 1, k− 1 and f1, g1, respectively. This separation is independent of the simplifications implied by lumping a series of structural, biochemical, or mechanical transitions into a single step in each case [30,31].

Fig. 2

Four-state model for calcium regulation of contraction in heart muscle cells.

The sequence of state transitions during relaxation is not simply the reversal of those that occur during activation; the function and mechanisms are asymmetric. In general, a regulatory unit containing an actin-bound and force-generating myosin molecule could respond to the falling phase of the [Ca2+]i transient by one of two pathways: either Ca2+ dissociates first from troponin or myosin dissociates first from actin. The first possibility implies an additional state of the regulatory system – a ‘pre-relax’ state (Fig. 2, lower left) – in which the regulatory site of TnC is empty although myosin cross-bridges are still generating force, and may hold tropomyosin in the clear position. Detachment of myosin from actin in the pre-relax state, with rate constant g2, would lead to the normal relaxed/diastole state. Alternatively, if myosin dissociates from actin via the g1 pathway before Ca2+ dissociates from troponin, relaxation would be completed via reversal of the normal activation pathway, with rate constant k− 1. The relative significance of these two relaxation pathways may depend on mechanical conditions, via an effect on the detachment (g) rate constants, as well as on the kinetics of the Ca2+ transient.

The role of force-generating myosin cross-bridges in the force-calcium relation

In the scheme in Fig. 2, the steepness of the force–calcium relation could be reproduced by co-operativity in the activation/deactivation steps (K1 and K2). This scheme is fundamentally different from models in which force-generating myosin cross-bridges contribute to switching on the thin filaments [4,31], which would require an additional positive feedback loop. The concept of myosin switching on the regulatory system is widely accepted, and has an intuitive appeal in the context of the original steric blocking model. If tropomyosin movement is required for myosin binding to actin, actin-bound myosin might be expected to hold tropomyosin in the clear or ON position (the states on the bottom line of Fig. 2). The thin filament signalling pathway that couples Ca2+ binding to tropomysin movement might then operate in reverse, so that a troponin conformation with higher Ca2+ affinity might be favoured when tropomyosin is held in the clear or ON position. Alternatively, or additionally, there may be coupling between the tropomyosin molecules in adjacent regulatory units along the thin filament [1,32-34], so that the clear position of a tropomyosin in a regulatory unit with Ca2+ bound to its troponin can spread to adjacent regulatory units with Apo troponins, allowing myosin binding in those adjacent units. Such a co-operative mechanism might be mediated by myosin head binding, or it might be intrinsic to the thin filament. The former possibility has received more attention, perhaps because there is a well-established biochemical model in which myosin binding switches on the regulatory system, at low concentrations of MgATP, i.e., in rigor-like conditions [35-37]. In these conditions, actin–myosin ATPase activity can be observed at nanomolar [Ca2+], and the Ca2+ affinity of troponin is increased by roughly an order of magnitude compared with that observed at physiological [MgATP]. It has been widely assumed that similar effects are produced in the intact sarcomere by force-generating myosin cross-bridges at physiological MgATP concentrations and that these effects are responsible for the steepness of the force–calcium relation. The next two sections review some of the experimental evidence that might be used to test these assumptions.

Effect of force-generating myosin cross-bridges on Ca2+ binding to thin filaments

Studies of the effect of force-generating myosin cross-bridges on the extent of Ca2+ binding to troponin as a function of free [Ca2+] in skinned trabeculae [38,39] are often quoted as evidence for a significant role of force-generating cross-bridges in the regulation of cardiac contraction [2,5,6]. The relevant results from Wang and Fuchs [39] are replotted in Fig. 3. The force–calcium relation (squares) can be fitted by the Hill equation with pCa50 5.8 and nH 2.8 (short dashed line). The amount of bound Ca2+ (circles) is well described by the sum of two Hill equations (continuous line), reflecting the presence of high- and low-affinity sites in a 2:1 ratio, as expected for the two Ca2+/Mg2+ sites and one Ca2+ regulatory site in each troponin molecule [8,38,39]. Vanadate (1 mM) almost completely abolished the isometric force (not shown) and decreased the amount of bound calcium (filled circles, long-dashed line) at each pCa, in comparison with control activations (open circles, continuous line). It also decreased pCa50 for the low-affinity bound calcium component by 0.2 pCa units, starting from a control value of 5.8 that matched that for active force. nH for the low-affinity bound calcium component was 3.9 in control conditions and 3.1 after the addition of vanadate. The reduction of total calcium binding by about 20% in the presence of vanadate might be related to its propensity to extract TnC from skinned muscle cells in activating or relaxing solutions [40,41], and this effect might be responsible for the reduction of nH from 3.9 to 3.1, since extraction of even a small fraction of TnC reduces the steepness of the force–calcium relation [33]. In any case, nH for Ca2+ binding to the regulatory calcium sites after active force has been abolished by vanadate is still close to 3, and within the precision of the measurements matches nH for control force in the same preparation and conditions. We conclude that calcium binding studies on demembranated trabeculae [38,39] do not support the hypothesis that force-generating myosin cross-bridges are responsible for the steepness of the steady-state force–calcium relation in heart muscle.

Fig. 3

Force (squares) and bound Ca (circles) as a function of pCa, replotted from Figs. 2A and 3A of Wang and Fuchs [39]. The open circles denote control activations and the closed circles in the presence of 1 mM vanadate, which almost completely eliminated active force. The force data are fitted by a standard Hill curve with pCa50 5.8 and nH 2.8. The bound Ca data are fitted by the sum of two Hill curves assuming a 2:1 ratio of high-affinity sites (pCa50 6.5 and nH 0.7 in control; pCa50 6.7 and nH 0.7 in 1 mM vanadate) to low-affinity sites (pCa50 5.8 and nH 3.9 in control; pCa50 5.6 and nH 3.1 in 1 mM vanadate).

Fluorescent probes of troponin structure in skinned trabeculae

Native troponin subunits in skinned trabeculae can be readily replaced by subunits labelled with fluorescent probes at specific sites, so it is possible to measure structural changes in troponin during calcium activation. For example, we have replaced TnC by an expressed protein in which the native cysteine residues had been removed, and residues 55 and 62, along the C helix of TnC adjacent to the regulatory Ca2+ binding site, had been substituted by cysteines that were cross-linked by a bifunctional rhodamine [7]. The absorption and emission dipoles of the rhodamine are aligned with the C helix [42]. After the labelled TnC has been incorporated into ventricular trabeculae, the orientation of its C helix with respect to the trabecular long axis can be measured by polarised fluorescence [43]. Such trabeculae exhibit a normal force–calcium relation (Fig. 4, squares) that is well fit by the Hill equation (short dashed line) with pCa50 5.28 ± 0.05 and nH 3.10 ± 0.24 (mean ± SE, n = 5). The orientation of the C helix of TnC, measured simultaneously in the same trabeculae (circles), had essentially the same Ca dependence, with Hill parameters (continuous line) pCa50 5.23 ± 0.04 and nH 3.01 ± 0.13. Thus, there is a simple relation between TnC structure and force in the steady state; Ca2+ binding to the regulatory site induces a structural change in TnC, and force is proportional to the fraction of TnC molecules in the ON state. Complete inhibition of active force by the specific myosin inhibitor blebbistatin (25 μM) (data not shown) had almost no effect on the Ca dependence of the orientation of the C helix of TnC (Fig. 4, triangles), which had Hill parameters pCa50 5.14 ± 0.04 and nH 2.86 ± 0.27 (long-dashed line). Thus force-generating cross-bridges have almost no effect on the structural change in the C helix of TnC induced by Ca2+ binding. Since this structural change has the same [Ca2+] dependence as force under control conditions, we conclude that force-generating cross-bridges make a negligible contribution to the steepness of the force–calcium relation.

Fig. 4

Force (squares) and the orientation of a bifunctional rhodamine probe on the C helix of TnC, θ (circles, triangles) as a function of pCa, replotted from Figs. 1B and 2A of Sun et al. [7]. The circles denote control activations and the triangles in the presence of 25 μM blebbistatin, which almost completely eliminated active force. The Hill parameters for force (short dashed line) are pCa50 5.28, nH 3.10; for control orientation (continuous line), pCa50 5.23, nH 3.01; for blebbistatin (dashed line), pCa50 5.14, nH 2.86.

This conclusion is supported by studies in which probes were inserted at other locations on TnC, although the structural changes reported by such probes depend on both probe location and probe chemistry. An IAANS probe on Cys 84 in ventricular trabeculae showed a fluorescence intensity change with the same pCa50 as active force [44]. This signal was unaffected by force inhibition by vanadate, in agreement with the results from the C-helix bifunctional rhodamine probe (Fig. 4). Polarised fluorescence from an IATR probe on Cys 84 [45] revealed a biphasic dependence of probe angle on free [Ca2+], and a small right shift in the orientation–pCa relation on force inhibition with 1 mM vanadate. Cys 84 is on the end of the D helix of TnC, immediately adjacent to the linker between the N- and C-terminal lobes of TnC, and near the TnI switch peptide in the ternary troponin complex [21], where it may be sensitive to structural changes in both lobes.

Two probe sites on the E helix of TnC, in its C-terminal lobe, have been investigated using dichroism from an IATR probe on Cys 98 [46] and polarised fluorescence from a bifunctional rhodamine probe cross-linking residues 95 and 102 [7]. As in the case of the Cys 84 probes, the polarised fluorescence signal revealed a small additional structural change at sub-threshold [Ca2+] that was not detected by dichroism, and this complicates the Hill fits. However, pCa50 for the major structural change was similar to that of active force. Inhibition of active force by 1 mM vanadate [46] or 25 μM blebbistatin [7] shifted pCa50 to the right by ca 0.2 pCa units with no significant change in nH in either case.

These results from fluorescent probes at different sites on troponin in skinned trabeculae support the quantitative conclusions from the analysis of calcium binding measurements in Fig. 3. To a first approximation, steady-state force is proportional to the fraction of troponin molecules in the Ca bound or ON state, and this fraction has a steep [Ca2+] dependence, typically characterised by a Hill coefficient (nH) of about 3. Inhibition of active force shifts the relationship between troponin structure and [Ca2+] to the right by 0.1 to 0.2 pCa units without changing its steepness (nH). Thus, both the calcium binding and fluorescent probe studies show that the steepness of the force–calcium relationship is not due to force-generating cross-bridges. It might be argued that this conclusion depends on the detailed mechanism of force inhibition by vanadate or blebbistatin and that myosin cross-bridges might still be able to switch on the thin filaments in the presence of these inhibitors. However, biochemical and structural studies of the mechanism of force inhibition seem to rule out such an explanation. Blebbistatin, for example, binds to a specific site in the actin-binding cleft of the myosin head [47], blocking the myosin ATPase cycle in an actin-detached state [48].

Limitations of biochemical models of the effect of myosin cross-bridges on calcium regulation

The studies on demembranated trabeculae described above do not support the widely accepted hypothesis that force-generating myosin cross-bridges are responsible for the steepness of the force–calcium relation. That hypothesis was to a large extent based on an extrapolation from biochemical studies on isolated proteins and protein complexes, and there are many differences between trabecular studies in near-physiological conditions and biochemical studies on a subset of protein components in which the native lattice relationship between myosin- and actin-containing filaments is not preserved. Force cannot be measured in the biochemical studies, so the extent of activation is generally determined from the rate of MgATP utilisation or signals related to the MgATPase. Low [MgATP] or rigor-like conditions are often employed to mimic the effects of a strongly bound myosin-actin complex. The effect of rigor-like complexes to displace tropomyosin and switch on the thin filaments is well established [49] and can also be observed in demembranated trabeculae in the absence of MgATP [7] or mimicked by myosin head fragments that have been chemically modified so that they are trapped in a rigor-like state [50]. In demembranated trabeculae these rigor-like interactions increase the Ca2+ affinity of troponin by an order of magnitude, but the steepness of the calcium sensitivity of force and structural changes in troponin is reduced to levels consistent with the absence of co-operativity. The Ca2+ dependence of the orientation of the C helix of TnC, for example, has an nH value of 1.05 ± 0.12 (mean ± SE, n = 5) in the presence of rigor cross-bridges, compared with 3.01 ± 0.13 at near-physiological [MgATP]. There is no sign of the rigor-like calcium dependence in trabeculae contracting at physiological [MgATP], as described in the previous two sections. We conclude that the rigor-like interaction of myosin heads with actin is not a useful model for the effect of the effect of force-generating heads on the Ca2+ regulatory system under physiological conditions.

The molecular basis of the steep force–calcium relationship

If force-generating cross-bridges do not determine the steepness of the force–calcium relationship, the underlying mechanism must be intrinsic to the thin filament. The most likely mechanism is coupling between adjacent regulatory units along the filament, so that Ca2+ binding to one regulatory unit promotes Ca2+ binding to its neighbours, an effect which may be mediated by the end-to-end coupling between adjacent tropomyosin molecules [1,34,51]. Such a mechanism would be consistent with the separation between regulatory and mechanical transitions in the four-state model of Fig. 2, in which the co-operative steps responsible for the steep force–calcium relationship are ascribed to the horizontal transitions with equilibrium constants K1 and K2 and with the observation that many different factors can modulate force without affecting the steepness of its Ca2+ dependence as measured by nH (Table 1).

The explanation of the steepness of the force–calcium relationship on the basis of thin filament co-operativity is also compatible with the low fraction of myosin molecules that are likely to be actin-attached and force-generating at systole. Although it is difficult to measure this fraction in heart muscle cells, comparison of force per myofibrillar cross-sectional area [52] and in vitro force per myosin [53,54] in heart and skeletal muscles suggests that the fraction of myosins attached to actin at maximal activation in heart muscle is similar or slightly lower than that in skeletal muscle, which is about 20–30% [55-57]. The fraction attached at systole would be roughly half of that at maximal activation [58], or about 10%. The molar ratio of actin monomers to myosin motor domains is roughly two, so only about 5% of actin monomers will have a myosin motor domain attached at systole, and most regulatory units will have no attached myosins at systole. Myosin-driven co-operativity does not seem a likely mechanism under these conditions.

Effect of force-generating myosin cross-bridges on the Ca2+ affinity of the thin filaments

In contrast with the absence of an effect of force-generating myosin cross-bridges on the steepness of the force–Ca relationship, measured by nH, many studies have provided evidence that force-generating cross-bridges produce a small increase in the Ca2+ affinity of troponin, as measured by a increase in pCa50. Most interventions that reduce the number of force-generating cross-bridges in near-physiological conditions decrease pCa50 by 0.1–0.2 units (Table 1; Figs. 3 and 4). Although several different mechanisms are likely to be involved in these interventions, many of which remain obscure, the correlation suggests the presence of a general mechanism linking pCa50 to maximum force. For example, force-generating cross-bridges might hold tropomyosin in the clear position, feeding back to the Ca2+ affinity of troponin. This would correspond to a small difference between the equilibrium constants K1 and K2 in the scheme in Fig. 2. A change in pCa50 of only 0.1 units can have a significant effect on the level of activation when free [Ca2+] is close to the concentration required to half-saturate the troponin regulatory sites, as at systole. Thus, for example, with nH = 3, such a change in pCa50 would produce a change in troponin-bound Ca2+ of 17%. This effect may be responsible for the ‘bumps’ in free [Ca2+] that have been observed when muscle cells are allowed to shorten during relaxation [13,59], since this manoeuvre is expected to decrease the number of force-generating cross-bridges, decreasing the Ca2+ affinity of troponin and leading to Ca2+ dissociation.

The steepness of the dynamic force–calcium relation

The detailed evidence and arguments presented above relate primarily to the steady-state force–calcium relation in demembranated trabeculae, and it is important to consider whether the mechanistic conclusions can be reliably extrapolated to dynamic activation of intact heart muscle on the timescale of the heartbeat. It is clear that the force–calcium relationship is steep in both conditions (e.g., [10]), and the simplest explanation of this would be that the same mechanism, which we have argued is intrinsic to the thin filament, operates in both cases. The alternative hypothesis, that myosin cross-bridges can transiently activate thin filaments during the rising phase of the twitch despite the absence of an effect in the steady state, seems unlikely a priori, and also seems incompatible with the view that thin filament activation is fast compared with force generation (Fig. 2). Although there is no direct evidence on this point from intact heart muscle cells, the thin filament structural change induced by the Ca2+ transient in intact skeletal muscle is already complete when force has risen to only about 20% of its maximal value [27,28]. In summary, it seems highly unlikely that force-generating cross-bridges influence the rate of activation in the intact heart. The above arguments do not, however, exclude the possibility that force-generating cross-bridges influence the kinetics of relaxation by delaying the return of tropomyosin to the OFF state, as discussed above in relation to alternative pathways of relaxation (Fig. 2). This phenomenon may be functionally significant during relaxation of the intact heart.

Summary and conclusion

A review of studies of the Ca2+ dependence of force generation, Ca2+ binding and structural changes in troponin in demembranated trabeculae does not support the popular hypothesis that force-generating myosin cross-bridges play an important role in the steepness of this relationship. Although force-generating myosin cross-bridges have a small but functionally significant effect on the calcium affinity of troponin, they do not affect the steepness of the force–calcium relationship, which is an intrinsic property of the muscle thin filament.