Cooperative Effect of Calcium Binding to Adjacent Troponin Molecules on the Thin Filament-Myosin Subfragment 1 MgATPase Rate*

The myosin subfragment 1 (S1) MgATPase rate was measured using thin filaments with known extents of Ca2+ binding controlled by varying the ratio of native cardiac troponin versus an inhibitory troponin with a mutation in the sole regulatory Ca2+ binding site of troponin C. Fractional MgATPase activation was less than the fraction of troponins that bound Ca2+, implying a cooperative effect of bound Ca2+ on cross-bridge cycling. Addition of phalloidin did not alter cooperative effects between bound Ca2+ molecules in the presence or absence of myosin S1. When the myosin S1 concentration was raised sufficiently to introduce cooperative myosin-myosin effects, lower Ca2+concentrations were needed to activate the MgATPase rate. MgATPase activation remained less than Ca2+ binding, implying a true, not just an apparent, increase in Ca2+ affinity. MgATPase activation by Ca2+ was more cooperative than could be explained by cooperativeness of overall Ca2+binding, the discrepancy between Ca2+ binding and MgATPase activation, or interactions between myosins. The results suggest the thin filament-myosin S1 MgATPase cycle requires calcium binding to adjacent troponin molecules and that this binding is cooperatively promoted by a single cycling cross-bridge. This mechanism is a potential explanation for Ca2+-mediated regulation of cross-bridge kinetics in muscle fibers.

The myosin subfragment 1 (S1) MgATPase rate was measured using thin filaments with known extents of Ca 2؉ binding controlled by varying the ratio of native cardiac troponin versus an inhibitory troponin with a mutation in the sole regulatory Ca 2؉ binding site of troponin C. Fractional MgATPase activation was less than the fraction of troponins that bound Ca 2؉ , implying a cooperative effect of bound Ca 2؉ on cross-bridge cycling. Addition of phalloidin did not alter cooperative effects between bound Ca 2؉ molecules in the presence or absence of myosin S1. When the myosin S1 concentration was raised sufficiently to introduce cooperative myosin-myosin effects, lower Ca 2؉ concentrations were needed to activate the MgATPase rate. MgATPase activation remained less than Ca 2؉ binding, implying a true, not just an apparent, increase in Ca 2؉ affinity. MgATPase activation by Ca 2؉ was more cooperative than could be explained by cooperativeness of overall Ca 2؉ binding, the discrepancy between Ca 2؉ binding and MgATPase activation, or interactions between myosins. The results suggest the thin filament-myosin S1 MgATPase cycle requires calcium binding to adjacent troponin molecules and that this binding is cooperatively promoted by a single cycling cross-bridge. This mechanism is a potential explanation for Ca 2؉ -mediated regulation of crossbridge kinetics in muscle fibers.
Just as isometric tension is cooperatively activated by Ca 2ϩ , so is the cardiac thin filament-myosin S1 1 MgATPase rate, even under conditions where there is no cooperativity in myosin S1 binding (1,2). A possible explanation for this behavior is that ATPase activation is proportional to Ca 2ϩ binding to the many TnCs on each thin filament and that this calcium binding is cooperative (3). We tested the idea that Ca 2ϩ binding and MgATPase activation are proportional and found to the contrary that they are not. Instead, fractional MgATPase activation was considerably less than fractional Ca 2ϩ binding and more closely paralleled the number of pairs of adjacent troponins with Ca 2ϩ bound to both.
To accomplish the above experiment, we employ a constitutively inhibitory form of cardiac troponin containing an inactivating mutation of the sole regulatory site of TnC, site II (4). This troponin, designated CBMII-Tn, results in a low thin filament-myosin S1 MgATPase rate that is not increased by the addition of Ca 2ϩ , analogous to previous results in which a similar TnC mutant was exchanged into myofibrils or muscle fibers (5)(6)(7). CBMII-Tn binds to actin-tropomyosin with an affinity identical to that of normal troponin in the absence of Ca 2ϩ . This binding, which is very tight for both normal troponin and for CBMII-Tn (4,8,9), permits the present report in which thin filaments are assembled with defined mixtures of normal troponin and CBMII-Tn. In the presence of saturating Ca 2ϩ concentrations, such thin filaments exhibit a fractional saturation of the TnC regulatory sites that is experimentally controllable by varying the relative concentrations of the two forms of troponin. This permits assessment of Ca 2ϩ -regulated myosin S1 MgATPase activity in a novel manner as a function of bound Ca 2ϩ rather than as a function of the free Ca 2ϩ .
In addition to varying the ratio of the two troponins, the myosin S1 and free Ca 2ϩ concentrations are also systematically varied in the present study. The results imply a previously unrecognized aspect of the cooperativity of muscle activation, that rapid cycling of an isolated cross-bridge depends on Ca 2ϩ binding to adjacent troponin molecules, and also suggest that cross-bridge cycling increases Ca 2ϩ affinity at least locally, regardless of the density of myosin on the thin filament. The relationship between the data and various models of thin filament structure and regulation are discussed.

EXPERIMENTAL PROCEDURES
Protein Preparation-Cardiac troponin and tropomyosin were purified from bovine heart using an ether powder technique (10). Rabbit fast skeletal muscle F-actin was obtained by the method of Spudich and Watt (11). Because bovine cardiac myosin S1 tends to precipitate at the concentrations used in many of the experiments, most of the data were obtained using rabbit fast skeletal muscle chymotryptic myosin S1 purified by ion exchange chromatography (12). Some of the experiments (see Fig. 1) were repeated using bovine cardiac myosin S1 purified as described previously (10). CBMII-Tn was prepared by reconstitution (13) of the ternary troponin complex from bovine cardiac TnI and TnT (13) and recombinant murine TnC mutant CBMII (4).
Assembly of Thin Filaments with Defined Fractional Saturation of TnC Regulatory Site II-F-actin, tropomyosin, and various mixtures of troponin and CBMII-Tn were combined in the indicated ratios under the ionic conditions used in the ATPase experiments. Since troponin binds to the thin filament with an affinity of 3-5 ϫ 10 8 M Ϫ1 (8,9) and the M amounts of the troponins included in the present experiments were stoichiometric or slightly sub-stoichiometric relative to the actin concentration, it was anticipated that essentially all of both added forms of troponin would be bound to the thin filament. This was tested by a sedimentation experiment. 15.5 M F-actin, 2 M tropomyosin, 1 M cardiac troponin (nonradioactive), and 1 M reconstituted CBMII-Tn labeled with iodo[ 14 C]acetic acid on TnT Cys-39 were combined in the presence of 100 M CaCl 2 , 5 mM Tris-HCl (pH 7.5), 3.5 mM MgCl 2 , 8 mM KCl, 1 mM dithiothreitol. The sample was sedimented in a TL100 centrifuge at 25°C for 20 min at 35,000 rpm. Quantitative SDS-polyacrylamide gel electrophoresis analysis by gel scanning and standard curve comparison (14) and liquid scintillation counting of samples indicated sedimentation of 94% of both troponins combined (assessed by * The costs of publication of this article were defrayed in part by the payment of page charges. This article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. Section 1734 solely to indicate this fact. ‡ To whom correspondence should be addressed. E-mail: larry-tobacman@uiowa.edu. 1 The abbreviations used are: myosin S1, myosin subfragment 1; TnC, TnT, TnI, troponin C, T, and I, respectively; CBMII, mouse troponin C mutant D65A/E66A; CBMII-Tn, troponin reconstituted from CBMII. SDS-polyacrylamide gel electrophoresis) and 92% of the labeled troponin (assessed by radioactivity). The fraction of actin pelleting was similar, 93%. All three values agreed within experimental error.
MgATPase Assays-The ATPase rate was measured by the release of radioactive phosphate from [␥-32 P]ATP (15) (NEN TM Life Science Products, 2-7 ϫ 10 7 cpm/mol) with five or more time points obtained at variable intervals of 20, 60, or 120 s, depending upon the ATPase rate. Conditions and protein concentrations were varied as described in each figure. The free Ca 2ϩ concentration was varied using 0.5 mM 1,2-bis-(2amino-5-bromo-phenoxy)ethane-N,N,NЈ,NЈ-tetraacetic acid as a Ca 2ϩ chelator and variable concentrations of CaCl 2 (3).
Calculation of the Fraction of Adjacent Troponin Pairs with Ca 2ϩ Bound to Both of Them-This fraction is equivalent to the fraction of adjacent troponin⅐troponin pairs in contrast to the other possible adjacent pairs: CBMII-Tn⅐CBMII-Tn, troponin⅐CBMII-Tn, and CBMII-Tn⅐troponin. The number of such boundaries depends upon two factors: (i) the relative amounts of the two troponins and (ii) the tendency of the two forms of troponin to bind in a random or nonrandom pattern. The fractional Ca 2ϩ saturation is ϭ troponin/(troponin ϩ CBMII-Tn). If binding were random, then the fraction of adjacent pairs with Ca 2ϩ bound on both elements of the pair would simply equal 2 . However, prior work shows that when the two forms of troponin are present in excess, they compete in a way that implies positive cooperativity and a small tendency for the two forms of troponin to segregate from each other (4). This same tendency must be assumed to exist in the present experiments, which differ in that the troponins are added in a stoichiometric amount relative to the sites on the actin filament.
For a closed linear filament including n troponins, with p ϭ n designated as the number of troponins with bound Ca 2ϩ , the fraction of tropomyosin⅐tropomyosin boundaries with Ca 2ϩ bound on both sides of the boundary can be shown to be as follows.
The sums are taken over j, which refers to the number of discrete regions of one or more consecutive Ca 2ϩ along the filament. By induction, j varies from 1 to p when p Ͻ n/2, and j varies from 1 to n Ϫ p when p Ͼ n/2 (exceptions are when p ϭ 0 or p ϭ n). Y is the cooperativity parameter, corresponding not only to the tendency for each type of troponin to segregate from the other (4) but also, in the case of a fully normal thin filament, to a Y-fold tighter Ca 2ϩ binding to a troponin that is adjacent to a troponin already with bound Ca 2ϩ (3). Consequently, each configuration has a statistical weighting factor of Y Ϫj , accounting for its free energy relative to configurations with different values for j. The number of adjacent, bound Ca 2ϩ ⅐Ca 2ϩ pairs equals p Ϫ j for any configuration. The other terms in the expression have to do with the number of ways of placing j regions containing pCa 2ϩ ions in n places. Simulations (not shown) with Equation 1 show it to be indistinguishable from f 22 ϭ 2 ϭ (p/n) 2 when Y ϭ 1 as expected because binding is random when Y has this value. Also, Equation 1 gives negligibly different results for n ϭ 30 and n ϭ 200 unless Y is much larger than is true for the present experiments. Finally, Equation 1 is numerically indistinguishable from the implicit relationship between f 22 and that arises from independent derivations of the functions f 22 (Ca 2ϩ ) and (Ca 2ϩ ) (16). Fig. 1 shows the effect of altering the fractional Ca 2ϩ saturation of the thin filament by varying the relative concentrations of troponin and CBMII-Tn. The normalized results of six experiments are shown, and it is clear that the relationship between Ca 2ϩ binding and MgATPase rate activation is not a linear one (straight line). Rather, activation lags behind Ca 2ϩ binding. When 50% of the troponins bind Ca 2ϩ and 50% do not, the fractional MgATPase rate activation is only 30 -35% that of the maximal stimulation observed for full Ca 2ϩ saturation. The solid line, which does not fit the data, is the result expected if MgATPase activation were proportional to Ca 2ϩ binding regardless of whether Ca 2ϩ binding is cooperative.

Relationship between Ca 2ϩ Binding to the Thin Filament Regulatory Sites and Normalized MgATPase Rate in the Presence of a Low Myosin S1:Actin Ratio-
The dashed lines in Fig. 1 are theoretical curves for the fraction of troponin⅐troponin boundaries with Ca 2ϩ bound on both sides, which depends in part upon the degree of cooperativity in the binding of the two forms of troponin to the thin filament. The equilibrium constant Y governs the tendency of troponin and CBMII-Tn to separately cluster on the thin filament rather than bind randomly (3,4). Y also dictates the cooperativity of Ca 2ϩ binding to a thin filament containing only normal troponin, with Y Ͼ 1 indicating positive cooperativity. The experimentally determined value of Y is approximately 1.5 (4), and the long dashes in Fig. 1 correspond to this value. A slightly better fit is found with Y ϭ 4, as indicated by the theoretical curve represented with shorter dashes. This might suggest that the true value for Y is greater than the previously measured value of about 1.5. A more likely explanation is that the degree of MgATPase rate activation does not precisely correspond to the fraction of troponin⅐troponin pairs that have Ca 2ϩ on both sides. In either case, the deviation from linearity in Fig. 1 indicates that Ca 2ϩ binding to more than one troponin is required for full actin activation of ATP hydrolysis at any given thin filament site.
An important aspect of the cooperative process illustrated in Fig. 1 is that it is not due to interactions between myosin S1 molecules. The myosin S1 concentration was only 1% that of the actin concentration, making myosin⅐myosin cooperativity unlikely. To confirm this experimentally, the MgATPase rate was shown to be linear with the myosin S1 concentration over a 16-fold range (0.25-4% that of the actin concentration). Linearity with myosin S1 concentration was shown both at pCa 5 and at pCa 5.89 (10 -15% activation) for thin filaments with troponin and no CBMII-Tn and at pCa 5 for thin filaments with 50% troponin and 50% CBMII-Tn (data not shown).
The curvature in Fig. 1 is not attributable to hyperbolic dependence of the MgATPase rate on the regulated actin concentration in the presence of saturating Ca 2ϩ concentrations (10,17). Any such tendency would work in the reverse direction, producing a convex relationship or else tending to straighten a concave curve such as shown. This is not a major factor in Fig. 1 in any case because the MgATPase rates for the Ca 2ϩ -saturated thin filaments are about one-fourth to onethird the V max observed with saturating thin filament concentrations (data not shown). The actin concentration for the data sets in the figure are below the actin K app , which diminishes the importance of this consideration.
Phalloidin Does Not Alter Cooperative Interactions between Troponin Molecules-The polymerization ability of the troponin⅐tropomyosin complex (18 -20) and atomic models of actin⅐actin contacts in F-actin (4,21) suggest that longitudinal contacts along the thin filament are the most likely source of cooperativity. However, this does not exclude the possibility that cooperativity occurs across rather than along the actin filament. To test this, we added phalloidin, which binds near the thin filament axis with an orientation that is invariant with thin filament conformation (22) and both decreases thin filament flexibility and alters strand-strand interactions (23)(24)(25). Any cooperativity that was dependent upon such interactions might be changed by the addition of phalloidin. The Fig. 2 inset shows that the cooperative effect of bound Ca 2ϩ on MgATPase rate activation was similar to results found in the absence of phalloidin. The results are indistinguishable from Fig. 1.
The main portion of Fig. 2 provides a measurement of Ca 2ϩdependent cooperative interactions between troponin molecules on the thin filament in the absence of myosin, again in the presence of phalloidin. This experiment differs from the ATPase data in that the sum of the troponin and CBMII-Tn concentrations is in constant excess relative to the sites on the thin filament. The two forms of troponin compete for binding sites on actin-tropomyosin, and the pattern of this competition implies that these binding sites (for troponin) interact in a manner sensitive to Ca 2ϩ . This experiment measures the value of the cooperativity parameter and equilibrium constant Y, which is found to be 1.7 Ϯ 0.4 in the presence of phalloidin. This result implies weak Ca 2ϩ -sensitive interactions of a strength indistinguishable from that found previously in the absence of phalloidin (4). Curve-fitting of the data also results in a value for K R , the fold-increase in the affinity of troponin for actin⅐tropomyosin that results from Ca 2ϩ dissociation from site II. K R is 2.4 Ϯ 0.2 in the presence of phalloidin, which is indistinguishable from K R in the absence of phalloidin, 2.2 Ϯ 0.1 (4).
Effects of Ca 2ϩ Concentration and Myosin S1 Concentration on the Thin Filament-Myosin S1 MgATPase Rate-The experiment in Fig. 1 employed a mixture of normal troponin and CBMII-Tn. An extrapolation of these results suggests that for a thin filament with normal troponin only, the MgATPase rate will not increase in proportion to Ca 2ϩ binding as the free Ca 2ϩ concentration is increased. Fig. 3A shows the normalized MgATPase rate as a function of the free Ca 2ϩ concentration in the presence of either low myosin S1 concentrations as were also present in Fig. 1 (Fig. 3, Ⅺ) or in the presence of much higher myosin S1 concentrations (Fig. 3,ϫ). The rightmost two curves show the difference between Ca 2ϩ binding (short dashes) and adjacent Ca 2ϩ pair binding (solid curve, fit to MgATPase data (Ⅺ)) according to Fig. 1 under conditions where myosin⅐myosin cooperativity is precluded by low myosin S1 concentrations. The K app from the ATPase curve underestimates the true binding constant, but this discrepancy is small, 3.7 versus 2.4 ϫ 10 5 M Ϫ1 for K binding versus K app . The relationship between these curves is determined by Fig. 1; if the short dash curve in Fig. 3A is set as the independent variable and the solid curve as the dependent variable, then a graph describing the data in Fig. 1 is the result. However, the lines actually were obtained by a best fit of Equation 1 to the experimental data (Ⅺ). Assuming the MgATPase rate is proportional to the fraction of adjacent troponin pairs with Ca 2ϩ on both sides, then the best fit regulatory site Ca 2ϩ affinity is 3.7 Ϯ 0.6 ϫ 10 5 M Ϫ1 and the cooperativity parameter Y ϭ 3.4 Ϯ 0.9. The mean value for Y from 10 such experiments was 5 Ϯ 1, corresponding to a Hill coefficient of 2.2 (3) and similar to the level of cooperativity reported previously (1,3,13,26,27).
The analysis in Fig. 3A indicates that there is little difference in the cooperative shapes for Ca 2ϩ binding and for Ca 2ϩ pair binding (the solid and short dash curves are equally steep). Similarly, if Y is set at a noncooperative value of 1, both curves are less steep but they remain parallel, and the relationship between them is still consistent with Fig. 1 (not shown). This indicates that Fig. 1 is consistent with the data in Fig. 3A, but only if overall Ca 2ϩ binding to the thin filament regulatory sites is cooperative. Since this process is known to have little cooperativity for reconstituted thin filaments (3,4,28,29), some other explanation will be needed to rationalize the larger cooperativity observed for MgATPase activation versus the free Ca 2ϩ concentration ( Fig. 3A and Ref. 1).
Another source of cooperativity in MgATPase assays is effects of myosin S1 on the thin filament. Careful studies of Weber and co-workers (30) using skeletal muscle regulatory proteins have shown increased MgATPase rates, increased Ca 2ϩ affinity, and apparent Ca 2ϩ affinity (32,33). These effects are observed when the myosin concentration is high relative to actin or when conditions favor strong actin-myosin bond formation (34). Fig. 3B shows the potentiating effect of high myosin S1 concentrations on the thin filament-myosin S1 MgATPase rate using cardiac regulatory proteins. For an actin concentration of 5 M, the MgATPase rate deviated from linearity when the myosin S1 concentration was Ͼ3 M. This deviation correlated with a shift in the Ca 2ϩ K app in MgATPase versus pCa experiments. There was no shift for 3 M myosin S1 (not shown), a small shift for 5 M myosin S1 (not shown), and a 2.5-fold shift in K app in the presence of 10 M myosin S1 (Fig.  3A, ϫ). The apparent Ca 2ϩ affinity from these and other titrations was 2.4 Ϯ 0.2 ϫ 10 5 M Ϫ1 in the presence of 0.3 M myosin S1 and 6.0 Ϯ 0.9 ϫ 10 5 M Ϫ1 in the presence of 10 M myosin S1.
Comparison among the three curves in Fig. 3A shows that MgATPase rate activation of 10 M myosin S1 (ϫ) occurs at even lower Ca 2ϩ concentrations than the calculated Ca 2ϩ saturation of the regulatory sites (short dashes) when the myosin FIG. 3. Cooperative relationships among free Ca 2؉ , bound Ca 2؉ , thin filament-myosin S1 MgATPase rate, and myosin S1 concentration. A, conditions: 25°C, 20 mM imidazole (pH 7.3), 3.5 mM MgCl 2 , 7 mM KCl, 1 mM ATP, 5 M skeletal muscle F-actin, 1.5 M cardiac tropomyosin, 1 M troponin, and either 0.3 M (Ⅺ) or 10 M (ϫ) rabbit skeletal muscle myosin S1. The apparent Ca 2ϩ affinity is increased 2.5-fold in the presence of the higher myosin S1 concentration, from K app ϭ 2.4 Ϯ 0.2 ϫ 10 5 M Ϫ1 to K app ϭ 6 Ϯ 1 ϫ , fractional saturation of the thin filament with Ca 2ϩ when the myosin S1 concentration is low (short dashes, calculated from Equation 34.83 in Ref. 16 using the parameters (K, Y) derived from the previously mentioned fit). B, MgATPase rate is cooperatively increased by myosin S1 concentrations Ͼ3 M. In the presence of myosin S1 concentrations approaching the actin concentration, the MgATPase rate was higher than the rate found with low myosin S1:actin ratios (dashed line). The pCa was 4.8, a saturating Ca 2ϩ concentration. C, relationship between fractional Ca 2ϩ binding and normalized MgATPase rate activation in the presence of a relatively high myosin S1 concentration. The data are similar to what is found with low myosin S1 concentrations (Fig. 1), implying that the shift observed in panel A involves a myosin-induced change in Ca 2ϩ binding. Conditions were as above, with 0.1 mM CaCl 2 and protein concentrations of 10 M myosin S1, 5 M actin, 1 M tropomyosin, and 0.7 M troponin and CBMII-Tn. concentration is low. If the high myosin S1 (ϫ) versus low myosin S1 (Ⅺ) MgATPase rate shift had occurred without any change in true Ca 2ϩ binding, MgATPase activation would precede fractional Ca 2ϩ binding. This would be the opposite of the relationship in Fig. 1, a convex rather than a concave curve. This possibility is evaluated and excluded by Fig. 3C, which shows fractional MgATPase activation as a function of bound Ca 2ϩ . Even in the presence of high myosin S1 concentrations that "potentiate" the thin filament, Ca 2ϩ binding precedes fractional activation. Therefore, the shift seen in Fig. 3A (ϫ versus  Ⅺ) involves a myosin-induced increase in Ca 2ϩ affinity. However, there may also be some change in the precise relationship between fractional Ca 2ϩ binding and fractional MgATPase rate activation; Fig. 3C appears to show less deviation from linearity than does Fig. 1. In this regard, it should be noted that strongly bound cross-bridges can activate the thin filament under appropriate conditions, even in the absence of any Ca 2ϩ binding (31).

MgATPase Activation as a Function of the Free Ca 2ϩ Concentration for Thin Filaments Containing Mostly CBMII-Tn-
When only 25% of the troponins on the thin filament are capable of binding Ca 2ϩ , i.e. 75% of the troponin is of the form CBMII-Tn, a gradual increase in the Ca 2ϩ concentration produces a small level of activation that is shown in Fig. 4. The figure is a normalized composite of four experiments, and in all of them the noise precluded any assessment of Y. The data is noisy because a 25:75 ratio of troponin:CBMII-Tn produces only a low MgATPase rate (Fig. 1); the average Ca 2ϩ -saturated rate is twice the EGTA rate for these data sets. The solid curve is a noncooperative binding isotherm. Comparison of the data points to this theoretical curve suggests that cooperativity may actually be present (the data deviates from the curve), but this may be an artifact of the normalization of each data set.
The K app could be measured with enough precision, 4.3 Ϯ 1.2 ϫ 10 5 M Ϫ1 , to permit comparison to the value found for thin filaments with fully normal troponin, 2.4 Ϯ 0.2 ϫ 10 5 M Ϫ1 (n ϭ 10, with representative data shown in Fig. 3A, Ⅺ). This was unexpected, since the CBMII-Tn might have cooperatively interacted with adjacent troponin molecules to decrease Ca 2ϩ affinity. It is unclear why a modest increase in apparent affinity occurred instead, but the effect is small in any case. DISCUSSION The thin filament has at least three conformations: an inhibited state in the presence of EGTA, a Ca 2ϩ -induced state, and an active state observed in the presence of strongly binding myosin cross-bridges (35)(36)(37). These structures have been compared with three-dimensional reconstructions of myosin S1decorated thin filaments (38,39), leading to the conclusion that tropomyosin interferes with the binding site for myosin S1 in the inhibited state and (to a lesser extent) in the Ca 2ϩ state but not in the active state. Solution studies of cross-bridge thin filament binding support this conclusion (40) even though the initial stage of myosin S1⅐ATP binding to the thin filament is Ca 2ϩ -insensitive (10,41). The structural data strongly suggest that completion of the MgATPase cycle requires a local conformational change in the thin filament from the Ca 2ϩ -induced state to a myosin-induced state. Otherwise, tropomyosin would prevent tight actin⅐myosin binding that is part of the cycle. To explain the deviation from linearity in Fig. 1, we now suggest that this single cross-bridge-induced conformational change requires Ca 2ϩ binding to more than one troponin positioned on adjacent tropomyosin molecules along the thin filament. This interpretation parallels implications drawn from the very cooperative equilibrium binding of myosin S1 to the thin filament (42)(43)(44). Such binding is so cooperative that theoretical models (45,46) explaining it invoked a myosin-promoted conformational change for the tropomyosin⅐troponin-7 actin unit that strongly depended upon the same myosin-induced conformational change in adjacent units. The kinetics of tight thin filament-myosin S1 binding suggest a similar conclusion (47). Longitudinal cooperativity of this type is also implied by studies of muscle fibers subjected to partial extraction of TnC (48 -51). In fact, the nonlinear relationship in Fig. 1 is very similar to tension versus TnC data in skeletal muscle fibers (50). We suggest that this type of cooperative interaction between adjacent regulatory units also occurs during the MgATPase cycle, even for single, isolated myosin heads along the thin filament.
Strongly binding cross-bridges increase the affinity of Ca 2ϩ for the thin filament (28,31,(52)(53)(54)(55). This process has been invoked to explain the leftward shift in MgATPase versus pCa curves that occurs with high myosin S1 concentrations or low ATP concentrations (32,33) using skeletal muscle proteins. The most direct explanation for the shift would be a true change in Ca 2ϩ affinity at the regulatory site(s) of TnC. The present data show that this shift also occurs when cardiac troponin⅐tropomyosin is used and, more importantly, confirms the previous interpretation. By studying the relationship between MgATPase rate and bound Ca 2ϩ using CBMII-Tn, Fig. 3 demonstrates that this shift is caused primarily by a myosininduced increase in affinity and not primarily by a change in the relationship between Ca 2ϩ binding and activation.
Perhaps the greatest significance of the above conclusion is that it suggests a mechanism for the problematic cooperativity of MgATPase activation in the presence of low myosin S1 concentrations. This cooperativity is difficult to explain because Ca 2ϩ binding per se is much less cooperative (4, 56) and because the newly described cooperative effects of bound Ca 2ϩ (Fig. 1) fail to provide an explanation. This is demonstrated in Fig. 3A, which shows the relationship between cooperative activation by Ca 2ϩ of the MgATPase rate under low myosin S1 conditions (Ⅺ, solid line) and implied fractional Ca 2ϩ binding under the same conditions (short dashes). The theoretical curves in Fig. 3A are based upon the nearest neighbor analysis described under "Experimental Procedures," but the general shape of the curves is dictated by the data, not the equations. The steepness of the solid curve is chosen to match the empirical, model-independent MgATPase observations (Ⅺ), and the short dash curve matches the corresponding extent of Ca 2ϩ binding implied by Fig. 1. The fact that these curves are parallel shows that cooperative Ca 2ϩ binding is implied by the MgATPase data. If instead, one begins with the assumption that Ca 2ϩ binding is not cooperative, then a similar analysis (curves not shown) demonstrates that MgATPase activation would have little cooperativity, which is contrary to experimental results. These comparisons indicate that either overall Ca 2ϩ binding is more cooperative than has been found in most reports (reviewed in Ref. 56; see also Ref. 4) or, more likely, that the MgATPase rate cooperativity involves a local (i.e. near the cross-bridge binding site on the thin filament) myosin-induced increase in Ca 2ϩ binding. More specifically, we suggest that at intermediate Ca 2ϩ concentrations there is myosin-induced binding of additional Ca 2ϩ to the thin filament site(s) adjacent to the cycling cross-bridge and that this phenomenon occurs for isolated cross-bridges that are not acting in concert.
In a schematic representation of this model (Fig. 5), three categories of actin monomers within the thin filament are distinguished: actin sites with one nearby Ca 2ϩ -troponin, sites with several successive Ca 2ϩ -troponins, and sites with a strongly bound myosin S1 nearby. (To retain their separate characteristics, the sites may need greater separation from each other than shown in the figure.) All the troponins have the potential to bind Ca 2ϩ in the figure (no CBMII is present), but the free Ca 2ϩ concentration is subsaturating. The nonlinear results in Fig. 3B and the myosin S1-induced shift in K app in Fig. 3A are examples of the third potentiated class of sites producing faster myosin cycling than at other sites, as described previously by Weber and co-workers (30,32). To explain the cooperative effect of the Ca 2ϩ concentration on the Mg-ATPase rate in the absence of potentiated sites, it is now proposed that Ca 2ϩ binding is locally cooperative at the first class of sites, where single myosins cycle, and that this crossbridge-induced binding of additional Ca 2ϩ is important for completion of the cycle.
A long-standing issue in muscle regulation is whether and to what extent the Ca 2ϩ concentration alters cross-bridge kinetics (58) as opposed to controlling the recruitment of a variable number of cross-bridges, all with the same kinetics (59). More recent analyses of cross-bridge function have established that several processes have a graded response to the Ca 2ϩ concentration (49, 60 -67). The present data pertain to this problem. For example, force development occurs in several steps, including at least one transition before phosphate release (49,63,65,68). Most studies indicate that the rate of force development is very sensitive to the Ca 2ϩ concentration (49,60,61,65,66). These observations can be explained if the MgATPase model in Fig. 5 is also applicable in muscle fibers. An early kinetic step producing strong myosin binding can be expected to alter the position of the tropomyosin strand and raise the Ca 2ϩ affinity of adjacent troponin(s) (36,56). The Ca 2ϩ dependence of force generation kinetics can be explained if the concentration-dependent binding of additional Ca 2ϩ to adjacent troponin(s) alters the rate constants for completion of the power stroke and/or reversal of the early transition. This suggested mechanism for graded activation is an additional aspect of regulation, compatible with an additional control point dependent upon the density of bound cross-bridges (sites 2 and 3 in Fig. 5 have different properties) and with either steric or allosteric effects on recruitment (61, 69 -71). Mechanical studies will be needed to explore this proposal. Investigation of the transient and steady state properties of muscle fibers in which native TnC has been partially replaced by CBMII may prove a useful approach.