Proc. Nati. Acad. Sci. USAVol. 85, pp. 3265-3269, May 1988Physiological Sciences

Effect of Ca2+ on cross-bridge turnover kinetics in skinned singlerabbit psoas fibers: Implications for regulation ofmuscle contraction

(Ca2+ regulation/forcepCa (-log[Ca2+ ) relation/Ca2+ sensitivity/cooperativity/modulation of force)

BERNHARD BRENNERInstitute of Physiology II, University of Tubingen, D-7400, Tubingen, Federal Republic of Germany

Communicated by W. F. Harrington, December 30, 1987 (received for review October 12, 1987)

ABSTRACT The effect of Ca2" upon the rate constant offorce redevelopment following a period of isotonic shorteningwith immediate restretch to the starting sarcomere length wasstudied in rabbit psoas fibers at 5TC. Control experimentssupport the assumption that the rate constant of force rede-velopment represents isometric cross-bridge turnover kinetics.(fapp + gmp), where fapp and gapp are the rate constantscharacterizing the transitions from the non-force-generatingstates to the force-generating states and back to the non-force-generating states, respectively. Parallel measurements of therate constant of force redevelopment and of force, stiffness,and fiber ATPase during isometric contraction allow the effectof Ca2" uponfapp and gpp to be determined. Analysis revealsthat Ca2" has a marked effect upon fappq while g,,pp remainsapproximately unchanged. Furthermore, in the range above25-30% of maximum Ca2 activation, regulation of force,stiffness, and ATPase is mediated through changes in fppyBelow this range, however, it cannot be ruled out that, inaddition, cross-bridges are also switched in and out of theturnover process ("recruitment"). As a consequence of regu-lation through turnover kinetics, both Ca2" sensitivity and theslope of force-pCa (-log[Ca2+i) relations are shown to beaffected by the ratio fa,,/gapp, which may represent animportant mechanism of modulation of contractile function inaddition to modulation through changes within the regulatoryprotein system.

It is generally believed that muscle contraction occurs uponcyclic interaction between the myosin- and the actin-containing filaments, mediated by parts of the myosin mol-ecules, termed cross-bridges. According to the cross-bridgemodel of Huxley (1), while hydrolyzing ATP, cross-bridgescycle between two states: a force-generating state in whichcross-bridges are attached to actin and a non-force-generating state in which cross-bridges are detached fromactin. In vertebrate striated muscle, cross-bridge action iscontrolled by Ca2 + through the regulatory proteins troponinand tropomyosin, which are located within the actin fila-ments (2). According to the cross-bridge model of Huxley(1), the increase in isometric force with Ca2" (e.g., ref. 3) isattributed to the increase in number of cross-bridges at-tached to actin in the force-generating state, whereas relax-ation occurs when cross-bridges accumulate in the non-force-generating state. Two mechanisms were proposed forCa2 + activation to change the number of cross-bridges in theforce-generating state. According to Podolsky and Teichholz(4), Ca2+ acts in an all-or-none fashion. Depending on thestate of the regulated actin units, cross-bridges may either becycling with fixed turnover kinetics or may not cycle at all;i.e., cross-bridges are switched in and out of the turnover

process (simple "on/off switch" or "recruitment"). In thismodel, the increase in isometric force directly reflects theincrease in the number of turning-over cross-bridges. Alter-natively, Julian (5) proposed that Ca2+ changes cross-bridgeturnover kinetics in a graded way, while the number ofcross-bridges involved in active cycling remains constant.So far there have been few experiments that can distin-

guish these two possibilities. Previous measurements of theforce-velocity relation only provided inconclusive results asto whether Ca2" affects cross-bridge turnover kinetics (4, 6).X-ray diffraction studies of changes in the tropomyosinreflections upon activation (7-10) do not distinguish whetherthe indicated movement of tropomyosin upon activationcontrols the number of turning-over cross-bridges or, alter-natively, controls cross-bridge turnover kinetics.

In the present study, experiments were designed such thatturnover kinetics and the number of turning-over cross-bridges can be determined simultaneously. Measurements ofthe rate constant of force redevelopment after a period ofisotonic shortening (11, 12) are complemented by measure-ments of force, stiffness, and ATPase activity during isomet-ric contraction. With some simplifying assumptions, equa-tions relating these four measured parameters to turnoverkinetics and to the number of turning-over cross-bridges canbe solved exactly (13), and hence possible effects of Ca2+concentration upon turnover kinetics can be distinguishedfrom possible effects upon the number of turning-over cross-bridges.

It is found that isometric force, stiffness, and ATPaseactivity are regulated by Ca2+ through changes in cross-bridge turnover kinetics, whereas the total number of cross-bridges involved in active cycling appears unchanged withinexperimental accuracy. However, for Ca2 activation levelsbelow 25-30%o, changes in the number of cross-bridges in-volved in active cycling, in addition to the observed changesin turnover kinetics, cannot be ruled out. The major impli-cations of Ca2+ regulation through turnover kinetics con-cern the interpretation of force-pCa (- log[Ca2 + ]) relations.The characteristics of force-pCa relations (e.g., position andslope) can no longer be attributed directly to Ca2+ sensitiv-ity and cooperativity of the regulatory protein system. Apreliminary account of this work was previously presented(13).

THEORYCorrelations between physiological parameters and cross-bridge turnover kinetics can be derived in analogy to theproposal of Huxley (1), where a two-state model, includingone force-generating state and one non-force-generatingstate, was assumed and where two rate constants (f and g)were defined to describe cross-bridge turnover between

Abbreviation: pCa, -log[Ca2+].

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these two states. Although according to recent biochemicalstudies (e.g., refs. 14-16) each of these two states has to bereplaced by a group of states, turnover of cross-bridgesbetween these two groups of states can still be described bytwo rate constants, the apparent rate constantsf, and gcorresponding to the rate constants f and g of Huxley (1).The transition from the non-force-generating states to theforce-generating states is characterized byfapp, whereas gappdescribes the return to the non-force-generating states bymeans of ADP release and rebinding of ATP.Assuming that the apparent rate constants for the reverse

transitions,fa4pp and g - can be neglected under the condi-tions of the present stud', the following relations apply. Thesteady-state fraction of turning-over cross-bridges in theforce generating 4tates (aFS) is given by

aPFS = fapp/(app + gapp), [1]

and isometric force F and isometric stiffness S of an exam-ined fiber are given by

F = n Ffapp/(fapp + gapp) [2a]S = n Sfapp/(fapp + gapp), [2b]

where n is the number of turning-over cross-bridges per halfsarcomere of tho examined fiber, F is the mean forceproduced by a cross-bridge in the force-generating states,and Y is the mean stiffness of the cross-bridge in theforce-generating states.Assuming hydrolysis of one molecule of ATP per cycle,

isometric ATPase of an examined fiber, measured as theproduction of Pi or ADP, is

ATPase = n bfappgapp/(fapp + gapp), [3]

where n is defined as in Eq. 2 and b is the number of halfsarcomeres within this fiber.Force redevelopment subsequent to isotohic shortening

represents relaxation from the isotohic to the isometricsteady-state distribution of cross-bridges between the twogroups of states (12). Thus, the rate constant of reapproach-ing isometric steady state force (krerev) is given by (17)

kredev = fapp + gapp. [4]Note that kre4fv is independent of n.

MATERIALS AND METHODSFiber Preparation and Solutions. All experiments were

performed on skinned single-fiber segments of the rabbitmusculus psoas major. The skinning procedure and theisolhtion and mounting of skinned single-fiber segments werepreviously described (18, 19).

Preactivating and activating solutions contained 10 mMimridazole, 1 mM MgATP, 2 mM MgCl2, 10 mM creatinephosphate, 200-250 units of creatine phosphokinase (Sigma)per ml and 1 mM EGTA or 1 mM CaEGTA, respectively,and were adjusted to pH 7.0 at the experimental temperature(5WC)'. Ionic strength was set to 0.17 M by adding KCl.Solution temperature was thermistor-regulated to about0.1OC. For ATPase measurements, creatine phosphate andcreatine phosphokinase were replaced by phosphoenolpyru-vate (10 mM) sAnd pyruvate kinase (5000 units/ml), and lacticdehydrogenase (500 units/ml) and NADH (0.5 mM) wereadded to follow ADP production of the contractile system bythe change in light absorbance at 365 nm (20). For inhibitionof myokinase, 0.5 mM P',P5-di(adenosine-5')pentaphos-phate was added during ATPase measurements. To allow for

diffusion of enzymes into the contractile system prior toCa2" activation, fibers were incubated for at least 15 min inthe corresponding preactivating solution, which differedfrom the activating solution only in the free Ca2+ concen-tration.

Mechanical Apparatus. Details of the mechanical appara-tus, including laser light diffraction for measurement andservo-control of sarcomere length, were previously de-scribed (12, 18).

Experimental Protocol. Isometric force, stiffness, andATPase activity were measured while fibers were'eycledbetween isometric contraction and short periods of lightlyloaded isotonic shortening. This procedure ensured excel-lent preservation of cross-striation even during prolongedfull Ca2" activation (19). This proved essential not onlylforsarcomere length measurement by laser light diffraction butalso for ATPase measurements, where, otherwise, ATPaseactivity became impaired at high levels of Ca2" activation,leading to a nonlinear relation between force and ATPase.

Fiber stiffness was measured during length changes (re-leases or stretches) applied to one end of the fibers. Appar-ent fiber stiffness was defined as the resulting slope whenforce is plotted vs. the change in sarcomere length recordedduring the applied length changes. Changes in sarcomerelength were measured by laser light diffraction. To avoiddetection of possible cross-bridge attachment in the non-force-generating states, the speed of length changes wasadjusted to 1000-5000 nm per half sarcomere per second(12).To optimize the signal for ATPase measurements, seg-

ments of single fibers, =20 mm long, were incubated introughs of 120 1.d. ADP production was usually assayed afterincubation periods of 5-10 min, depending on the level ofactivation. Active ATPase was corrected for ATPase duringrelaxation, which after treatment with Triton X-100 andaddition of 10 mM sodium azide to the experimental solutionwas less than the scatter of ATPase measurements at fullCa2" activation (

Proc. Natl. Acad. Sci. USA 85 (1988) 3267

Bkredev (s)

k4 (s-1)0

0

p.J

o-

0 Ss

C1 .0 -

0

U:-E

* U

0.1-

0.5 1.0

force (norm.)

S

X 005- ~w

Ix98~~~~~

I ----I0 1 0 1 2

time after restretch (s)

FIG. 1. Force redevelopment at various levels of Ca2" activation. (A) Protocol for measurement of force redevelopment at three differentlevels of Ca2+ activation. The top trace sarcomere length measured by laser light diffraction. For clarity, only the trace for maximum activationis shown. The bottom traces represent force. During force redevelopment, sarcomere length of the center part of the fiber was held constantby activating a servo system. (B) Effect of Ca2+ on the rate constant of force redevelopment (kede,) and the apparent rate constant of phase4 (14). kredev is plotted vs. isometric steady-state force (e) recorded immediately before the period of isotonic shortening. The apparent rateconstant of phase 4 (k4) is plotted vs. force recorded prior to the stepwise release inducing the isometric transients. k4 is defined as l/T, whereX is the time required for recovery of force to (1 l/e) of total recovery during phase 4. (C) Effect of load during isotonic shortening on forceredevelopment, illustrated for two different levels of Ca2' activation. (Left) Maximum Ca2' activation. (Right) Partial Ca2+ activation resultingin 45% of isometric force at full activation. Load during isotonic shortening is expressed as fraction of isometric force: *, 0.05; o, 0.34; x, 0.65.

bridges (ftpp + gapp see Eq. 4b) change with the level ofCa2+ activation. This assumption and the resulting conclu-sion are substantiated by the following control experiments.

First, at two different levels of Ca2" activationi, the timecourse of force redevelopment subsequent to shortening atdifferent loads was analyzed (Fig. 1C). Although the rateconstant of force redevelopment is different at the two levelsof Ca2+ activation (compare the different slopes of the twoplots in Fig. 1C), at both activation levels the time course offorce redevelopment is not affected by the load -duringisotonic shortening (compare the different symbols in eachplot), even though the number of cross-bridges attached inthe force-generating states increases with load (21). Thismakes it unlikely that kredev is affected by possible changesin the activation level during the period of shortening due toa low number of cross-bridges attached in the force-generating states.

Second, during isotonic shortening, cross-bridges mightaccumulate in states not populated under isometric steady-state conditions such that the time course of force redevel-opment is affected by transitions back to the isometricpathway. Comparison of the time course of force redevel-opment after a period of isotonic shortening with the timecourse of phase 4 of the isometric transients (22), however,revealed that both the magnitude and the Ca2+ sensitivity of

0~~~~~~~~~~~

A lo0- aE ~ ~~~~~~~//

:0.5 eX 11

0-0.5 1.0force (norm.)

B 1.0-

Eso

r,,0.IAl

0 0.5 i.0force (norm.)

FIG. 2. (A) Effect of Ca2I activation on isometric ATPaseactivity. ATPase is plotted vs. isometric force recorded whileATPase was measured. Data of five fibers are summarized. FiberATPase is expressed as the fraction of the ATPase measured atmaximum Ca2+ activation. The line represents the linear least-squares fit to the data points. Within experimental error no system-atic deviation from a straight line was detected. ATPase per myosinhead, determined according to ref. 23, was 0.98 ± 0.18 s-I (mean ±SEM) at full Ca2l activation. (B) Effect of Ca2' concentration onapparent fiber stiffness and isometric force. Data from four fibersare grouped together. Bars represent SEM. Each group representstwo to five measurements.

the apparent rate constant of phase 4 (Fig. 1B, open circles)are comparable to those of force redevelopment followingisotonic shortening (Fig. 1B, closed circles), although thetime course of phase 4 cannot adequately be described by asingle exponential function (ref. 22 and unpublished results).This similarity makes it unlikely that two totally differentprocesses determine kredev and phase 4, supporting theassumption that both are mainly determined by isometricturnover kinetics. These control experiments thus furtherstrengthen the conclusion that the effect of Ca2+ upon kredevreflects the effect of Ca2" upon cross-bridge turnover kinet-iCs (fapp + gapp).To separate effects of Ca2+ activation upon fapp from

effects upon gapp, isometric fiber ATPase and isometricforce were recorded simultaneously (Fig. 2A). According toEqs. 2a and 3, the ratio of isometric ATPase to isometricforce equals b-gapp/F or the ratio of isometric ATPase to iso-metric stiffness equals b-g.pp/IS. The ratio of isometric stiff-ness to isometric force, which equals S/F (Eqs. 2a and 2b),remains approximately constant with the degree of Ca2+activation (Fig. 2B). It is unlikely that the effects of Ca2+activation on Si and F exactly compensate each other. Thelinear relationship shown in Fig. 2B thus suggests that both Sand F are not affected by Ca2t activation. On this basis, theapproximately linear relation between ATPase and force (Fig.2A)* indicates that within experimental accuracy gapp is notsignificantly affected by Ca2+ activation. Thus, the changesin k1(dcV directly reflect the changes in fapp (Eq. 1).

In an earlier series of experiments analyzing the effect ofMgATP concentration on isometric force, isometric stiff-ness, isometric ATPase, and kredev (e.g., ref. 7), we obtainedan estimate for gapp at maximum Ca2 + activation. In severalrunsgapp was 1.05 ± 0.18 s-1 (mean ± SEM; 3 runs). Thisvalue, together with the ratio of ATPase to force, allows usto derive gapp at all levels of Ca2+ activation (see Fig. 3A)without assumptions about the absolute number of turning-over cross-bridges. Since at the different activation levelsthe ratio of ATPase to force remains constant within exper-

*Previously, nonlinear relations between ATPase and force werereported (24, 25). In these studies, in addition to the lack of ATPbackup, no precautions were taken to avoid progressive structuraldisorder with increasing Ca2l activation. Both factors impairATPase more than force (unpublished observations) resulting innonlinearity in the ATPase-force relation. That the nonlinearity, atleast in part, arises from these factors is further supported by thelinear relation reported for the structurally more stable and slowrabbit soleus muscle (26).

A

eng +> -0 2upm

25xC 5N

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0 B

00 0,.k-as"1- -~~~~~~~~

0.5 1.)force (norm.)

E 1.0-

",

0. Se0

0.

0.5

0. 0

f (om-.P 0 0.5 1.0

force (norm.)

C

1.5 0

E 1.0- * * ;* 4

0SC *C 0.5- *

0

0 0.5 1.0

force (norm.)

FIG. 3. (A) Effect of Ca2l activation onfa and g . *, g.pp vs. isometric force observed at the corresponding Ca2" activation. gpp wasobtained from the value of gapp observed at fufi Ca2+ activation (1.05 ± 0.18 s-1) and the ratio of ATPase to force at the different levels ofCa2+ activation expressed as the fraction of ATPase to force at maximum Ca2+ activation. -, Linear least-squares fit to the data. o,fa =kmdev - gapp kredev was taken from data shown in Fig. lB. gapp was taken from the solid line of this figure. ---, 4th-order polynomial fitted bythe least-squpres method to the open circles. (B) The steady-state fraction of cycling cross-bridges in the force-generating states Lfapp/(fapp +plotted vs. isometric force observed at the different levels of Ca2+ activation.f.p and gapp were taken from A (gapp was taken from the solid line).(C) nF vs. isometric force, illustrating possible effects of Ca2+ activation upon the number of cross-bridges involved in active cycling (n).

imental error, the value of gapp is independent of Ca2"activation. With gapp thus derived, values offapp (Fig. 3A,open circles) can be determined at the various levels of Ca2 +activation using the data on krpdev of Fig. 1B. As evidencedin Fig. 3A, Ca2" concentration has a marked effect uponfapp, which approaches zero at'low levels of Ca2 + activation.Thus' Ca2+ regulation of isometric force, stiffness, andATPase, at least in part, is mediated through changes infapp(Eqs. 2 and 3). To assess additional contribution of changesin n (Eqs. 2 and 3) to Ca2+ regulation of force, stiffness, andATPase (e.g., in a dual mechanism of regulation), first thesteady-state fraction of cycling cross-bridges in the force-generating states (aFS; Eq. 1) was calculated from the knownvalues offapp and gap (Fig. 3B). The ratio of isometric force(Eq. 2a) over aFS (Eq. 1) equals nF. Since we can assumethat F is unaffected by Ca2" (see Fig. 2B), this ratioprovides information about possible changes in with Cat+concentration. In Fig. 3C, for the various levels of Ca2+activation, this ratio is plotted against the observed isometricforce. The increase in isometric force with Ca2+ activationat force levels above 25-30% obviously is only due tochanges in cross-bridge turnover kinetics (fpp) while nremains approximately constant. At force levels below25-30%, due to increasing scatter when plotting the ratio ofprogressively smaller values, it cannot be ruled out that, inaddition to the changes in fapp, changes in the number ofcross-bridges involved in active cycling could also contrib-ute to the variation in isometric force, stiffness, and ATPase.

DISCUSSIONThe major finding of the present study is the observation thatCa2+ affects the rate constant of force redevelopment(kredev) following a period of isotonic shortening with imme-diate restretch to the original filament overlap. The de-scribed control experiments indicate that kTedev is not sig-nificantly affected by possible changes in the activation levelduring isotonic shortening or by a possible different turn-over pathway during the isotonic period. It is thereforesuggested that the time course of force redevelopmentmainly reflects the reapproach of the isometric steady-statedistribution of cross-bridges between the force-generatingstates and the non-force-generating states. On this basis,kredev reflects isometric cross-bridge turnover kinetics [i.e.,fapp + gapp (Eq. 4)]. The conclusion that the observed effects+2+pof`Ca2+ upon kredev represent effects of Cae+ on cross-bridge turnover kinetics is further substantiated by theobservation that Ca2+ has a similar effect on the apparentrate constant of phase 4 of the isometric transients (Fig. 1B),another parameter thought to be sensitive to changes incross-bridge turnover kinetics (22). Such effects of Ca2+ oncross-bridge turnover kinetics, however, are unexpected on

the basis of an all-or-none switching action of Ca2+, whichonly changes the number of cross-bridges involved in activecycling but not their turnover kinetics (4).tThe effect of Ca2+ on cross-bridge turnover kinetics is

further specified using isometric steady-state parameters(i.e., tension, stiffness, arid ATPase activity) and theirrelations to cross-bridge turnover kinetics. The major impli-cation of this further analysis is that Ca2" regulation of theisometric steady-state parameters is mediated throughchanges in turnover kinetics (fpp) while the number ofcross-bridges involved in active cycling (n in Eqs. 2 and 3)remains approximately unchanged within experimental er-ror. Due to increasing scatter when taking the ratio ofprogressively smaller data values, it is impossible to derivethe behavior of n for activation levels below 25-30% ofmaximum activation. Thus, an on/off switching control(e.g., as one part of a dual mechanism of Ca2+ regulation)cannot be completely ruled out. However, it had to berestricted to at most the low levels of Ca2+ activation.

Regulation of muscle contraction through turnover kinet-ics does not require unusual kinetic schemes. It is rather anatural consequence of some recent kinetic schemes of theactomyosin ATPase cycle (e.g., ref. 28) if it is assumed that(i) the "turned on" and "turned off " forms of the regulatedactin units are in a dynamic equilibrium with fast rateconstants compared to cross-bridge turnover, (ii) this equi-librium is affected by Ca2+, and (iii) Pi release only occurswhen cross-bridges are attached to the turned on form ofactin. In terms of such kinetic schemes, it can be shown thatif rate constants for the turning on and off of regulated actinunits or for Ca2 + binding to troponin C are slow compared tocross-bridge turnover, the all-or-none recruitment mecha-nism will predominate. In contrast, fast rate constants willresult in modulation of turnover kinetics.The importance of distinguishing between Ca2 + regulation

through recruitment and regulation through turnover kinet-ics is illustrated in Fig. 4, where isometric force and thecorresponding values of fapp are replotted against Ca2+concentration (Fig. 4A). The nonsymmetric sigmoidal force-pCa relation is characterized by a steeper slope and a higher"Ca2+-sensitivity" (pCa at half maximum) than the fapp7

tUsing sinusoidal length perturbations, Kawai et al. (27) concludedthat Ca2' acts through a simple on/off switching action, with nosignificant effect on cross-bridge kinetics'. For curve-fitting pur-poses, these authors assumed no change in the characteristicfrequencies with Ca2+ (i.e., no change in cross-bridge kinetics).However, inspection of their figure 2A reveals a shift of thelow-frequency maximum of the dynamic modulus from about 5 Hzdown to about 1 Hz when Ca2' is reduced. This shape changeclearly indicates a Ca2+ effect on cross-bridge kinetics. Furtheranalysis is needed to determine whether this shift corresponds tothe Ca2+ effect on kredev and k4 reported in this paper.

Afapp (Is1) 5.

9app (s41)

0-

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Proc. Natl. Acad. Sci. USA 85 (1988) 3269

Aforce (norm )

fapp (norm.)1 .o-

0.5-

0-

B10force

fapp(norm)0.5

C

force (norm

fCPP(norm)

60 5.5pCa

Pco

FIG. 4. (A) Isometric force (e) andfapp (o) plotted vs. pCa. Bothparameters are expressed as a fraction of the value at maximumCa2+ activation. Note the difference in Ca2" sensitivity and theslope of the two plots. gpp is approximately independent of Ca2+activation (see solid line in Fig. 3A). (B) Calculated effects ofchanges in turnover kinetics upon force-pCa relations. The dashedline represents fta -pCa relation (taken from A). Three solid linesrepresent force-p&a relations, calculated by assuming Lpplg-m atmaximum Ca2+ activation to be (from top to bottom) 12:1, 6:1, and3:1. gapp is assumed to be independent of Ca2 . Isometric force Wascalculated according to Eq. i with n and F arbitrarily set to 1. (C)Force-pCa relations in B normalized to force at maximum Ca2"activation.

pCa relation. This suggests that with regulation throughturnover kinetics Ca2'-sensitivity and slope of force-pCarelations do not directly correlate with Ca2' binding to theregulatory proteins. This is further confirmed in Fig. 4 B andC where force-pCa relations were calculated on the basis ofthe known fapp-pCa relation plotted in Fig. 4A. Assuming aconstant normalized fapp-pCa relation, changes in fapp orgapp (i.e., changes in the fapp/gapp ratio) have characteristiceffects on foree-pCa relations; increasing this ratio leads toan increase in apparent Ca2+ sensitivity and in slope (ap-parent cooperativity) although Ca2+ binding to regulatoryproteins, as evidenced by the normalized fa- -pCa relation,was not changed; Calculation shows that tese effects arethe result of the nonsymmetricfapp-pCa relation and of Ca2`affecting cross-bridge turnover kinetics.The demonstrated possibility of modulating force-pCa

relations through changes infapp/gapp has several importantimplications. First, as it was already pointed out, force-pCarelations do not allow direct conclusions about affinity andcooperativity of Ca2+ binding to troponin C. Second, a highHill coefficient of force-pCa relations does not necessarilyindicate long-range cooperativity between a large number ofCa2'-binding sites located along the thin filaments but maysimply be the result of a large fapp/gapp ratio. Third, theapparent inconsistency between the force- or ATPase-pCarelations and the Ca2+ binding to the low-affinity sites oftroponin C (e.g., refs. 29 and 30) may be explained on thebasis of Fig. 4 B and C). Most importantly, however, anyphysiological or pharmacological intervention that affectsthefap Igapp ratio can modulate the force-pCa relation in theway ilustrated in Fig. 4 B and C while Ca2+ binding totrdpotnin C is unchanged. For example, preliminary experi-

ments suggest that the effect of myosin light chain-2 phos-phorylation upon force-pCa relations (31) is due to anincrease in fapp with phosphorylation. This illustrates thatchanges in turnover kinetics (fa /gapp) may represent afurther mechanism, in addition to cganges in Ca2l binding totroponin C, for physiological or pharmacological modulationof the force-pCa relation (i.e., for modulation of contractilefunction). Modulation of contractile function by this mech-anism may be of special significance in myocardium andsmooth muscle.

I would like to thank my colleagues at the National Institutes ofHealth, especially Drs. E. Eisenberg and L. C. Yu, and also Prof.Dr. R. Jacob of the University of Tubingen for many stimulatingdiscussions and comments on the manuscript. The work was sup-ported by the Deutsche Forschungsgemeinschaft (Br 849/1-2).

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