Modulation of Contractile Activation in Skeletal Muscle by a Calcium-insensitive Troponin C Mutant*

Calcium controls the level of muscle activation via interactions with the troponin complex. Replacement of the native, skeletal calcium-binding subunit of troponin, troponin C, with mixtures of functional cardiac and mutant cardiac troponin C insensitive to calcium and permanently inactive provides a novel method to alter the number of myosin cross-bridges capable of binding to the actin filament. Extraction of skeletal troponin C and replacement with functional and mutant cardiac troponin C were used to evaluate the relationship between the extent of thin filament activation (fractional calcium binding), isometric force, and the rate of force generation in muscle fibers independent of the calcium concentration. The experiments showed a direct, linear relationship between force and the number of cross-bridges attaching to the thin filament. Further, above 35% maximal isometric activation, following partial replacement with mixtures of cardiac and mutant troponin C, the rate of force generation was independent of the number of actin sites available for cross-bridge interaction at saturating calcium concentrations. This contrasts with the marked decrease in the rate of force generation when force was reduced by decreasing the calcium concentration. The results are consistent with hypotheses proposing that calcium controls the transition between weakly and strongly bound cross-bridge states.

The cyclic interaction of myosin and actin produces force and shortening in contractile cells. In muscle fibers, actin and myosin interaction is regulated by the intracellular calcium concentration acting through the thin filament regulatory proteins, the troponin complex, and tropomyosin. Until calcium binds to the troponin complex, the muscle fiber remains relaxed with Ͼ95% of the myosin cross-bridges detached (1,2). The influx of calcium into the filament lattice of muscle fibers stimulates the association of actin and myosin enabling the production of force or shortening and accelerating the actomyosin ATPase rate by Ͼ100-fold during isometric contraction.
Several models have been postulated to account for this control. The steric-blocking model of cross-bridge regulation asserts that tropomyosin/troponin (Tm/Tn) 1 prevents cross-bridge attachment in the absence of calcium by "blocking" cross-bridge access to binding sites on the thin filament (3,4). Alternatively, the kinetic regulation model assumes that the cross-bridges can, under all conditions, bind weakly to the thin filament, and calcium controls the kinetics of cross-bridge turnover via changes in the weakly bound to strongly bound crossbridge transition (5,6). More recently, three-dimensional reconstructions of electron micrographs have identified three distinct structural states of the thin filament (7)(8)(9). In the absence of calcium, tropomyosin blocks strong myosin binding sites on actin. Following Ca 2ϩ binding to the troponin complex, the tropomyosin shifts away from the myosin binding sites but does not completely expose all the putative strong binding sites on actin. Further movement of the tropomyosin requires strong cross-bridge binding to fully expose the myosin binding sites. Full activation of the thin filament requires Ca 2ϩ binding to the troponin complex and subsequent strong cross-bridge binding to the thin filament.
Brenner and Eisenberg (10) developed a method to measure the kinetics of cross-bridge transitions from weakly bound to force-producing states in activated muscle fibers and found the rate of tension development (k tr ) to be calcium-sensitive (6). This result is inconsistent with the steric-blocking mechanism, and Brenner (6) suggested that Tm/Tn controlled the rate of P i release. However, subsequent work showed that the kinetics of P i release are independent of [Ca 2ϩ ] (11)(12)(13)(14). Further, others have found the kinetics of cross-bridge cycling to be unaffected by compounds that affect thin filament dynamics (15). Taken together, these studies indicate that calcium is regulating muscle activation by control of cross-bridge access. To reconcile these observations with Brenner's data and hypothesis it was proposed that [Ca 2ϩ ] controls the transition from weak to strong cross-bridge binding preceding the generation of force (11,16).
To this point, studies have investigated calcium regulation of muscle contraction by adding various compounds, removing proteins, or adjusting the free calcium concentration. These investigations have left several issues unresolved. In particular, when the free calcium concentration rises, it is unclear how this increases the rate of force generation. Does the effect require a relatively high density of myosin binding to actin, which tends to activate the thin filament, or does calcium binding to troponin have a more direct effect on k tr ?
In the present study we describe a method to control the fraction of troponin complexes to which calcium is bound, thereby also controlling the fraction of the thin filament available for myosin binding while maintaining the free calcium concentration constant at a high level. The native skeletal TnC was extracted from thin filaments of skinned muscle fibers and replaced with variable combinations of cardiac TnC and an inactive cardiac mutant TnC, CBMIITnC (17,18). Cardiac TnC has a single regulatory calcium binding site (site II) as site I does not contain the necessary charged residues to bind calcium (19,20). Mutation of two negatively charged residues (Asp-65 and Glu-66) to neutral alanine residues prevents calcium binding to site II (18) and blocks activation of thin filament-myosin S1 MgATPase activity by Ca 2ϩ (21). The CBMII TnC can be exchanged onto skinned muscle fiber thin filaments following extraction of the native skeletal TnC, effectively removing any calcium-dependent activation of the muscle fiber. Complete replacement of endogenous TnC with a similar CB-MII TnC completely relaxed skeletal muscle fibers and rendered them insensitive to [Ca 2ϩ ] (17). In this way, we have been able to investigate isometric force and the rate of force generation as functions of the fraction of the thin filament that is able to bind myosin under conditions where calcium binding to troponin is fixed at a level determined by the CBMII TnC content.
Isometric tension and the rate of force redevelopment were measured in muscle fibers following TnC replacement with ratios of cTnC and CBMII TnC. The data show that isometric force is directly proportional to the number of active thin filament units (A 7 TmTn) at saturating calcium, and the rate of force redevelopment is unaffected by a reduction in crossbridge number. The results suggest that k tr is primarily controlled by calcium binding to troponin rather than the density of cross-bridges binding to the actin filament. Together, the results suggest that calcium controls cross-bridge access to the thin filament by regulating an equilibrium between weakly (non-productively) and strongly (productively) bound, but nonforce-bearing cross-bridges. The present work accounts for the discrepancies between the opposing kinetic and steric-blocking models of thin filament regulation. A preliminary report of this work was published previously (22).
Fiber Preparation and Mechanical Apparatus-Psoas muscle fibers were dissected from female New Zealand white rabbits, glycerinated, and stored as described by Millar and Homsher (11). Single fibers were dissected, and the ends were fixed by flowing 1% glutaraldehyde in 50% glycerol over each end of the fiber. Aluminum T-clips were then attached to the ends, and the fibers were mounted between a force transducer (SensoNor AE801 strain gauge) and a shaker motor (Ling 100A). The sarcomere length was set to 2.6 m/sarcomere (measured by He:Ne laser diffraction), and the fiber width and total length were measured. The rate of tension redevelopment (k tr ) was measured by activating the fiber, and after reaching steady-state isometric tension, abruptly shortening the fiber by ϳ20%, reducing force to zero. The fiber was held at this length for 25 ms while the fiber shortened at maximal velocity, and then the fiber was rapidly restretched to its original length (6). All mechanical measurements were performed at 15°C.
Data Acquisition and Curve Fitting-Tension and displacement signals were recorded and the digitized records were analyzed using KFIT (11) and SigmaPlot 4.0 (SPSS Inc., Chicago, IL). The k tr records were fit by a single exponential equation of the form P ϭ P o ϩ ⌬P (1 Ϫ e Ϫkt ), where P is the tension, P o is the initial force, ⌬P is the amplitude of the redeveloped tension, and k is the k tr . Force-pCa curves were fit to the Hill Equation in the form P ϭ P o /(1 ϩ 10 nH(pKϪpCa) where P o is maximal force produced (pCa 4.5), pK (pCa 50 ) is the calcium concentration yielding 0.5 P o and n H is the Hill coefficient. Significance was determined using Student's t test and the confidence level was set at p Ͻ 0.05. The data were reported as mean Ϯ S.E. with (n) the number of fibers analyzed.
Modeling Equations-The steady-state solutions for the fraction of the cross-bridges in the weakly bound (W o ), strongly bound (S o ), and force-exerting (F o ) states shown in Scheme 1 are given by the equation, The solution for the fraction of cross-bridges in the force-exerting state F(t) at anytime, t, after changing any of the rate constants from a steady-state value is given by the equation, TnC Extraction/Reconstitution-To extract the endogenous skeletal TnC from the fibers, the sarcomere length was increased to Ͼ3.0 m in REL (23), and the fibers were transferred to a solution containing 5 mM EDTA, 20 mM Tris⅐HCl (pH 7.2), and 0.5 mM trifluoperazine dihydrochloride at 15-17°C (24). The fibers were incubated until the Ca 2ϩactivated isometric tension fell below 10% P o , generally within 10 min. The fibers were reconstituted by incubation in REL containing 0.5 mg/ml TnC for 1 min followed by a wash in REL for 30 s. This was repeated until Ca 2ϩ -dependent tension reached a maximal, constant value. Skeletal TnC was kindly provided by Marion Greaser (University of Wisconsin) and purified as described by Greaser and Gergely (25). Cardiac TnC (cTnC and CBMIITnC) were isolated as described (18).
SDS-PAGE-The extraction and reconstitution of the troponin C was quantified using SDS-PAGE. Muscle fibers containing sTnC, cTnC, or ratios of CBMII TnC:cTnC were placed in sample buffer, heated to 95°C for 3 min, and sonicated to denature and solubilize the muscle fiber. The samples were loaded onto a 12% Tris⅐HCl separating gel. Following electrophoresis, the gels were stained using the silver stain technique of Guilian et al. (26) with minor modifications. After staining, the gels were dried and scanned. The apparent molecular mass of CBMIITnC is 1 kDa Ͻ purified cTnC, allowing quantitative separation of the two proteins by their mobilities. Gels were analyzed using a GS-700 scanning densitometer (Bio-Rad) normalizing bands to the area under the myosin light chain 1 peak. The relative content of cTnC and CBMII TnC was determined from the ratio of cTnC:(cTnC ϩ CBMII TnC) after accounting for differences in staining intensity and a background peak at the same position as CBMII TnC.

RESULTS
Extraction/Replacement of TnC-To investigate the effects of cardiac and CBMII TnC replacement on muscle function, it was necessary to effectively remove the native sTnC from the muscle fibers. Extraction reduced the Ca 2ϩ -dependent force to 8.1% (Ϯ1.1) of the maximal force obtained at pCa 4.5 (P o ϭ 146.7 (Ϯ3.8) kN/m 2 , n ϭ 45). Extraction of the sTnC and subsequent reconstitution with purified sTnC returned maximal Ca 2ϩ -dependent force to 125.5 Ϯ 10.3 kN/m 2 or 86 Ϯ 3% P o (n ϭ 5). Reconstitution with cTnC returned maximal isometric force to 96.9 (Ϯ4.7) kN/m 2 (n ϭ 17) or 65.4% (Ϯ 5.0) of that observed prior to extraction.
It was important to determine whether the cardiac TnC or CBMII TnC preferentially bound to the fiber thin filaments under the extraction/replacement procedure. The difference in the apparent molecular weights of cTnC and CBMII TnC made quantification possible. Fig. 1 shows two lanes (A) and the associated profiles (B) of a gel containing fiber segments reconstituted with 100% cTnC and 50% cTnC:50% CBMII TnC. Determination of the relative proportion of cTnC in fibers reconstituted with various ratios of cTnC and CBMII TnC as a function of the relative amount of cTnC added to the reconstitution solution is shown in Fig. 2. The results demonstrate that the binding of cTnC and CBMII TnC to the thin filaments of fibers is similar under the reconstitution conditions used (pCa 9.0), which is consistent with the binding studies performed using cTn and CBMII Tn complexes (18). Steady-state Isometric Tension-As CBMII TnC is a cardiac TnC mutant, it was necessary to determine the effects on the isometric tension and the calcium sensitivity in fibers after extraction of native TnC and replacement with cTnC. The difference in calcium sensitivity between the control fibers containing native sTnC and those after cTnC replacement is shown in Fig. 3. Following replacement of sTnC with cTnC the Hill coefficient was reduced from 2.83 (Ϯ0.11) to 1.95 (Ϯ0.13) and the pCa 50 shifted from 6.72 (Ϯ0.01) to 6.64 (Ϯ0.02); both changes are significant (p Ͻ 0.05). The reduction in the Hill coefficient and shift in pCa 50 after replacement with cTnC have been reported previously (27,28). Extraction of endogenous sTnC and replacement with purified sTnC produced no signif-icant differences (p Ͻ 0.01) in the Ca 2ϩ sensitivity (n H ϭ 2.87 (Ϯ0.43); pCa 50 ϭ 6.74 (Ϯ0.03); n ϭ 5).
Isometric force is believed to be dependent on the number of cross-bridges attached to the thin filament (29 -31). To determine whether reduction in the level of thin filament activation directly correlates with a decrease in the number of thin filament sites, fibers were reconstituted with various ratios of CBMII:cTnC, and the isometric tension was measured. Fig. 4 demonstrates that tension fell in direct proportion to the reduction in cTnC content of the fiber. Because the cTnC content added correlated well with the amount of cTnC bound to the thin filaments (Fig. 2), the cTnC content is given as the ratio of cTnC added. The tension measured after replacement with CBMII:cTnC was normalized against the average maximal force produced after replacement with 100% cTnC (65.4% of sTnC; n ϭ 17). Regression analysis indicated that the slope was not different from 1 (p Ͼ 0.3) and the y intercept was not different from 0 (p Ͼ 0.2). The direct proportionality of the force reduction to the fractional content of cTnC suggests that there is little cooperativity between the steady-state isometric force and the number of attached cross-bridges at saturating [Ca 2ϩ ].
The Rate of Tension Redevelopment (k tr )-The rate of tension redevelopment (6) is controlled by [Ca 2ϩ ], which implies regulation of a cross-bridge transition involving force generation (11,12,32). Representative traces of k tr as a function of pCa are shown in Fig. 5A. It is evident that the Ca 2ϩ sensitivity of force redevelopment is affected by the troponin C isoform. These differences are seen more clearly in Fig. 5B, which shows k tr as a function of the relative tension. Skeletal TnC produced a greater rate of force redevelopment at high calcium concentrations (18.1 Ϯ 0.46 s Ϫ1 , n ϭ 45 at pCa 4.5) but was reduced almost 10-fold at low [Ca 2ϩ ]. At pCa 6.0, the force was still ϳ98% of that at pCa 4.5 but k tr fell to only 14.8 (Ϯ1.2) s Ϫ1 (n ϭ 10). The k tr fell to near minimal values (2.89 Ϯ 0.4 s Ϫ1 ; n ϭ 10) at pCa 6.7 even though isometric force was still ϳ50% P o .
Replacement with cTnC reduced the maximal k tr at pCa 4.5 to 10.8 (Ϯ0.5) s Ϫ1 (n ϭ 17). However, reducing the calcium concentration from pCa 4.5 to pCa 7.0 caused the rate of tension redevelopment to fall to 3.32 (Ϯ0.36) s Ϫ1 (n ϭ 7), only a 3-fold reduction. These results indicate that the rate of tension redevelopment is markedly affected by [Ca 2ϩ ] regardless of TnC isoform. The TnC isoform bound to the thin filament, however, modulates the rate of tension redevelopment in active muscle fibers (33).
Although k tr is highly sensitive to variations in the [Ca 2ϩ ], it is unclear whether the effect is caused by [Ca 2ϩ ]-dependent limits on cross-bridge cycling, cross-bridge number, or both. To evaluate the effects of reducing the cross-bridge number independent of [Ca 2ϩ ], k tr was measured after extraction and replacement with different ratios of CBMII:cTnC at constant, saturating [Ca 2ϩ ]. Representative traces of k tr with different ratios of cTnC and CBMII TnC are shown in Fig. 6A. It is apparent that k tr was largely unaffected by the reduction in the number of actin monomers available for myosin cross-bridge interaction. Isometric tension fell in direct proportion to the addition of CBMII TnC (Fig. 4) whereas k tr remained unchanged and near maximal for all conditions except for 75% CBMII TnC (7.24 Ϯ 0.21 s Ϫ1 ; n ϭ 7). Fig. 6B plots k tr as a function of relative tension in fibers containing 100% cTnC with tension varied by altering the pCa of the activating solution and fibers containing various ratios of CBMII:cTnC. The results indicate that k tr is greatly affected by changes in [Ca 2ϩ ] but is not sensitive to reductions in the fraction of the thin filament that can be activated and bind cross-bridges until tension is reduced to Ͻ35% P o . DISCUSSION Huxley's (29) two-state model of muscle contraction assumes that isometric force is produced by S1 interactions with the thin filament and therefore is dependent on the number of attached cross-bridges. Gordon et al. (30) and Edman (31) found that isometric force is proportional to the degree of thickthin filament overlap and hence the number of cross-bridges. In this study, we have shown that isometric tension under saturating calcium conditions declines in proportion to calcium binding to troponin.
Prior studies have reduced the fraction of the thin filament that can be activated by partial extraction of the endogenous TnC (23, 27, 34, 35). The removal of TnC from the troponin complex prevents calcium binding and inactivates the thin filament in the regions containing the partial troponin complexes (TnI-TnT-Tm units). Because TnC and TnI interact with each other in undefined ways (36), TnC extraction may alter several of these interactions and affect other aspects of the thin filament regulatory mechanism (37). Replacement of sTnC by either cTnC or CBMII TnC avoids this complication by maintaining a full complement of TnC on the thin filament while reducing the number of potentially active regulatory units.
The force-pCa plot of fibers containing sTnC and cTnC (Fig.  3) is similar to those obtained previously (11,27). The force is reduced cooperatively as the calcium concentration is lowered in fibers containing either sTnC or cTnC. The reduction in cooperativity (n H ) upon reconstitution with cTnC is qualitatively consistent with the presence of only one Ca 2ϩ binding site in cTnC. However, the force-pCa relationship in skinned fibers is more cooperative than can be explained by Ca 2ϩ binding to individual troponin subunits. If this were the only cooperative mechanism involved, the maximal n H would be 1.0 for cardiac and 2.0 for skeletal TnC (19). Adjacent troponin subunits are connected by tropomyosin, and enhancement of Ca 2ϩ binding via cooperative strong cross-bridge binding may influence the degree of thin filament activation and therefore the mechanism of regulation (21,38,39).
The results obtained here present a clearer picture of what occurs during force generation in isometrically contracting skinned muscle fibers. The most straightforward, although not the only, interpretation of Fig. 4 is that under conditions of saturating calcium (pCa 4.5), CBMII TnC replacement of cTnC limits the number of cross-bridges capable of binding to the thin filament, and the isometric force decreases in direct proportion to the reduction in active thin filament regulatory units. Isometric force, a steady-state measurement, is linearly dependent on the number of thin filament actin monomers available for cross-bridge interaction. This suggests that reduction in the number of active thin filament units, by either lowering [Ca 2ϩ ] or increasing the proportion of CBMIITnC, directly limits the number of cross-bridges capable of binding to the thin filament.
The relationship between force and Ca 2ϩ binding is linear (Fig. 4) and therefore seems unaffected by cooperative interac- tions between adjacent regulatory units along the length of the thin filament. Either a concave or a convex relationship between the fractional occupancy and the isometric force would be evidence for such cooperativity but was not observed. The absence of this behavior does not prove, however, that force generation is unaffected by longitudinal cooperativity along the thin filament. One reason for this is that the linear behavior in Fig. 4 may reflect a balance of compensating cooperative effects in which regulatory units with calcium induce myosin binding on adjacent units and units without calcium restrict myosin binding on adjacent units. Each phenomenon has been reported in other types of experiments with partial extraction of either whole troponin (40) or TnC (35). Further, any non-linearity between the fractional Ca 2ϩ binding and isometric force would be difficult to detect for thin filament occupancies less than 25% and therefore cannot be excluded with the present data. Finally, the use of CBMII TnC precludes a possible source of cooperativity that can occur in normal thin filaments; cross-bridge binding to a regulatory unit where Ca 2ϩ is bound may induce Ca 2ϩ binding on adjacent regulatory units (21). Despite these caveats about an underlying complexity in the system, the linear results in Fig. 4 imply that Ca 2ϩ is controlling the steady-state isometric force by limiting cross-bridge access to the thin filament.
Is a similar mechanism at work during transient events (e.g. k tr ) in the muscle fiber? In this study, we tested whether k tr is dependent on [Ca 2ϩ ] or the number of cross-bridges attached to the thin filament. Brenner (6) showed that the rate of tension redevelopment is highly dependent on the [Ca 2ϩ ] with a nonlinear decline in k tr as calcium levels were reduced and suggested that regulation occurred during a weak to strong transition. If Ca 2ϩ specifically controls the transition from a weakly bound to a strongly bound, force-generating state, then k tr and k P i , the rate of the tension decline following photogeneration of P i from caged-P i , would be the same. However, measurements of k P i revealed little or no Ca 2ϩ dependence (11,12,14,16) even though k tr measured in the same preparation exhibited a strong dependence on [Ca 2ϩ ]. In the present study, decreasing the number of force-generating cross-bridges reduced steady-state tension but did not greatly affect the rate  6. A, representative traces of k tr measurements after replacement with cardiac TnC or ratios of CBMII and cardiac TnC. For the 25% CBMII and 75% CBMII records, the force has been normalized to 65% of the maximal force obtained at pCa 4.5 in the fiber prior to extraction/ replacement with CBMII. The traces are taken from three separate fibers. B, k tr as a function of relative tension. Tension was reduced by decreasing the [Ca 2ϩ ] for cTnC fibers (E) and by varying the cTnC: CBMII TnC ratio for the CBMII TnC fibers (q). All data are shown as mean Ϯ S.E. with n Ͼ 5 for each point. The solid line through the cTnC corresponds to the prediction from the equations below for cTnC. The CBMII TnC data are fit to a hyperbolic equation. of force generation until the level of thin filament activation was less than ϳ35%. These results are consistent with the hypothesis that [Ca 2ϩ ] controls a cross-bridge transition preceding force generation, proposed to be a transition from a weakly bound to a strongly bound, non-force-bearing crossbridge state (11,14,16,41).
If k tr actually measures a two-step process as suggested, (i.e. a weakly to strongly bound transition followed by the forcegenerating isomerization or P i release), the differing effects of CBMII or decreasing the calcium concentration on cross-bridge function can be explained by the model described below.
In this model, cross-bridges are detached or weakly attached (W), strongly attached but not generating force (S), or strongly attached and generating forcing (F). The weakly attached states (M⅐ATP, M⅐ADP⅐P i , or A-M⅐ADP⅐P i ) attach and detach from the actin filament rapidly and do not sustain significant force (the hyphen indicates a weakly attached state). The strongly attached state (AM⅐ADP⅐P i ) does not generate force. Entry into the strongly bound state (S) involves a thin filament isomerization controlled by troponin and tropomyosin with the forward rate k ϩ1 (increased by elevations in [Ca 2ϩ ]) and the reverse rate k Ϫ1 . The strongly attached and force-exerting state (F), AM⅐ADP (and its isomers), are generated by an isomerization and/or the release of P i from the strongly bound AM⅐ADP⅐P i state controlled by k ϩ2 and k Ϫ2 . The rigor crossbridge, AM, is also a strongly bound, force-exerting crossbridge. The rate of force-generating cross-bridge detachment to the detached/weakly attached cross-bridges (W) is defined by k 3 and is slow (2-4 s Ϫ1 ) under isometric conditions as estimated from the steady-state isometric ATPase rate (6). At low [P i ] (ϳ1 mM), k ϩ2 and k Ϫ2 are ϳ20 s Ϫ1 and 3 s Ϫ1 , respectively (11,12,16). We assume that the TnC isoform and pCa have no direct effect on k ϩ2 , k Ϫ2 , or k 3 as these rates should only depend on the myosin and actin present, neither of which changed during our experiments. We also assume that addition of calcium and/or replacement of regulatory proteins changes only k ϩ1 and/or k Ϫ1 . Assuming that k Ϫ1 , k ϩ2 , k Ϫ2 , and k 3 are constant in skeletal muscle and that k Ϫ1 is 4 s Ϫ1 , then steady-state isometric force, F o , is a hyperbolic function of k ϩ1 defined by The analytical expressions used to determine the steadystate isometric force, F o (assumed to be proportional to the fraction of cross-bridges attached in the force-generating states) and the time course of force production, F(t), either from rest or after a period of rapid shortening and re-stretch are given under "Experimental Procedures." The time course of force generation is dominated by the A exp( 1 t) term as B is insignificant compared with A at relative forces of Ͻ90%. At larger values of k ϩ1 (Ͼ15 s Ϫ1 when F o Ͼ 90%), the difference in magnitude of 1 and 2 produces a slowing of k tr to a value Ͻ15% different from that predicted by 1 alone. The consequence of this behavior is that the rise of force subsequent to rapid shortening (during which k 3 is Ͼ100 s Ϫ1 ) is well fit by a single exponential term (R 2 Ͼ 0.95).
The overall cross-bridge cycling rate is slow and limited by an irreversible isomerization step preceding ADP release defined by k 3 (42). The measurement of the rate of tension redevelopment isolates the force-generating step (k ϩ2 and k Ϫ2 ) and the preceding equilibrium (k ϩ1 /k Ϫ1 ) from the overall crossbridge cycle. Thus when k ϩ1 and k Ϫ1 are both small, k tr approaches k 3 as a limit. Modeling of this mechanism suggests that Ca 2ϩ activates the muscle by increasing k ϩ1 while not affecting k Ϫ1 . By varying k ϩ1 from 0 to 20 s Ϫ1 to simulate changes in the free calcium concentration, the model correctly defines the observed non-linear behavior of k tr as a function of relative isometric force in fibers containing sTnC (Fig. 5B, solid line labeled sTnC). The changes in k tr produced by replacing sTnC with cTnC (Figs. 5B and 6B, solid lines labeled cTnC) can be produced by raising k Ϫ1 from 4 s Ϫ1 to 10 s Ϫ1 and allowing k ϩ1 to vary from 0 to 13 s Ϫ1 as the calcium concentration is raised. Therefore, the model suggests that the rate detached or weakly bound cross-bridges productively bind to the thin filament determines the rate of force generation.
In the present experiments, as [Ca 2ϩ ] was reduced in fibers containing sTnC, k tr fell from ϳ18 to 2 s Ϫ1 . After extraction of endogenous sTnC and replacement with cTnC, the Ca 2ϩ -induced reduction in k tr was smaller, from ϳ11 to 3 s Ϫ1 . Cardiac muscle exhibits a smaller, 3-6-fold increase in k tr as [Ca 2ϩ ] is raised from submaximal to maximal levels (43,44). Although k tr depends on [Ca 2ϩ ] and the myosin isoform (6,32), it is significant that substitution of TnC alone causes large changes in the sensitivity and rates associated with force generation. Because incorporation of different TnC isoforms should not alter the cross-bridge structure or the intrinsic cross-bridge cycling rate of myosin, the changes must be caused by TnC-dependent effects. The proposed model correctly accounts for the observed differences of fibers containing sTnC or cTnC in k tr as a function of relative force. As shown in Fig. 5B, the cTnC data are well fit by simply increasing k Ϫ1 from 4 s Ϫ1 to 10 s Ϫ1 and reducing the maximal rate of k ϩ1 to 13 s Ϫ1 leaving the other rate constants unchanged.
Why is k tr reduced in the presence of cTnC compared with sTnC? The most likely reason is that the TnC interaction with Ca 2ϩ and signaling to the other Tn subunits, Tm and actin, play a role in controlling the state of the thin filament activation, which is a complex and incompletely understood process (45)(46)(47). Although differences in the Ca 2ϩ affinity between the two TnC isoforms may contribute to this behavior, it is more likely that the changes are due to the markedly different structure of the cardiac TnC stalk and regulatory domain from that of skeletal TnC. The structural differences may alter the ability of the TnC to interact with TnI and effect sequential changes in tropomyosin position that influence the rate of cross-bridge attachment. Also, NMR studies reveal that the cTnC structure shows a more closed conformation than sTnC and that the Ca 2ϩ binding and dissociation produces slower conformational changes in cTnC (48,49). Thus, the rate at which TnC can undergo the required conformational changes to affect the inherent properties of the thin filament are TnC isoform-dependent and therefore alter fiber function. Such changes have physiological implications because cardiac muscle does not require the rapid and complete activation necessary for normal physiological function in skeletal muscle.
The presence of CBMII TnC affects only the regulatory units containing the mutant TnC while the other regulatory units are all potentially fully active. If the [Ca 2ϩ ] is saturating, most of the native TnC molecules will be in their calcium-bound state and the tropomyosin will be oscillating primarily over the Ca 2ϩ -induced position providing cross-bridge access to the actin binding sites. Thus, k tr will not be limited by calcium binding and the weak to strong cross-bridge transition. The mechanism described in Scheme 1 successfully predicts the behavior shown in Fig. 5B in which k tr changes little until the steady-state isometric force rises to values greater than ϳ50% maximal. This way of thinking about the regulatory mechanism indicates that regulation involves kinetic regula-Scheme I tion of the transition from a weakly bound to a strongly bound state as first suggested by Brenner (6). It also suggests that kinetic and steric mechanisms are not truly separate because steric effects from tropomyosin positioning on the thin filament affect the weak to strong cross-bridge transition.
Evidence for a potential role of strongly bound cross-bridges contributing to thin filament activation at lower thin filament occupancies is given by the data in Fig. 6. k tr is plotted as a function of the isometric force at pCa 4.5 in fibers containing various fractions of CBMII TnC. At forces Ͼ25% of maximal, k tr is independent of isometric force. However, at 25% isometric force k tr is markedly reduced even though [Ca 2ϩ ] is saturating. This could occur if reduction in the level of thin filament activation is accelerated by the decline in productively attached cross-bridges. The results imply that Ca 2ϩ controls the steadystate isometric force over the range of 25-100% force by limiting cross-bridge access to the thin filament.
How do these results relate to biochemical and structural mechanisms thought to underlie the regulation of muscle contraction? Biochemical investigations have revealed the presence of three thin filament states: blocked, closed, and open, (45) while more recent cryo-electron microscopy studies have identified three structural states of the thin filament: off, Ca 2ϩinduced, and myosin-induced (7-9) that may correspond to the biochemical states. The initiation of contraction involves calcium binding to the low affinity site(s) of the troponin C subunit (site I in cTnC and both sites I and II in sTnC) on the troponin complex. The binding of calcium initiates a structural change in the interaction between TnC and TnI resulting in a relaxation of the TnI-based inhibition. The tropomyosin is now able to shift from the blocked or "off" position to the closed or "Ca 2ϩ -induced", intermediate position closer to the groove between the actin strands. This shift in the average position of the Tm molecule opens myosin binding sites on the actin filaments important for strong, stereo-specific cross-bridge attachment. Cross-bridges will be able to bind productively, i.e. proceed to a strong binding conformation and continue through the actomyosin ATPase cycle unimpeded by the presence or absence of calcium. The attachment of strongly bound cross-bridges is associated with a further shift in the average position of tropomyosin to the open or "myosin-induced" state, increasing the probability of other cross-bridges binding to the thin filament. For each state, tropomyosin position is probably a dynamic oscillation in which Tm does not occupy a single static position on the actin surface but continually shifts back and forth over the actin surface. As [Ca 2ϩ ] falls, the probability that Tm is in the open position falls, reducing the rate of strong, stereospecific cross-bridge binding. Strong cross-bridge binding likely stabilizes the Tm position that makes available further myosin binding sites on nearby actin monomers (8,47). Scheme 1 quantifies these relationships and together with the structural interpretation of regulation described above leads to explanations for the curvilinear relationship between force and k tr and for the decline of k tr at high concentrations of CBMII. At thin filament occupancy Ͼ25% of maximal there is sufficient cooperativity along the thin filament that Ca 2ϩ simply controls the rate of attachment (k ϩ1 ) of myosin to the thin filament and k tr is varied with Ca 2ϩ . At low thin filament occupancy by cross-bridges, the proportion of time Tm spends in the closed or blocked positions will be greater and the thin filament will be partially inactivated. This will reduce the cooperativity of calcium binding to the thin filament and the spread of activation along the thin filament. It will therefore produce a decline in k tr (even at saturating Ca 2ϩ ) (Fig. 6B, open  circles).