Kinetics of association of myosin subfragment-1 to unlabeled and pyrenyl-labeled actin.

The kinetics of reaction of myosin subfragment-1 (S) with F-actin have been monitored by the changes in light scattering and in pyrenyl-actin fluorescence at 20°C, pH 7.5, and physiological ionic strength. The association rate constant of S to F-actin decreases about 10-fold as the molar ratio of bound S increases from 0 to 1. This decrease in k is most likely due to the steric hindrance of available binding sites by initially bound S. The apparent rate constant for association of S to bare filaments is 9 μM s, a value 1 order of magnitude higher than the one previously estimated from experiments in which S was in excess over F-actin. The anticooperative binding kinetics of S to F-actin are consistent with the negative cooperativity displayed in the equilibrium binding curves of S to pyrenyl-F-actin. Fluorescence titration curves of partially labeled pyrenyl-F-actin by S are sigmoidal, consistent with a 4-fold higher affinity of S for unlabeled than for labeled actin. This conclusion is strengthened by kinetic data of S binding to partially labeled F-actin, which exhibit a biphasic behavior due to the slower dissociation of S from unlabeled than from labeled actin.

The kinetics of reaction of myosin subfragment-1 (S 1 ) with F-actin have been monitored by the changes in light scattering and in pyrenyl-actin fluorescence at 20°C, pH 7.5, and physiological ionic strength. The association rate constant of S 1 to F-actin decreases about 10-fold as the molar ratio of bound S 1 increases from 0 to 1. This decrease in k ؉ is most likely due to the steric hindrance of available binding sites by initially bound S 1 . The apparent rate constant for association of S 1 to bare filaments is 9 M ؊1 s ؊1 , a value 1 order of magnitude higher than the one previously estimated from experiments in which S 1 was in excess over F-actin. The anticooperative binding kinetics of S 1 to F-actin are consistent with the negative cooperativity displayed in the equilibrium binding curves of S 1 to pyrenyl-F-actin.
Fluorescence titration curves of partially labeled pyrenyl-F-actin by S 1 are sigmoidal, consistent with a 4-fold higher affinity of S 1 for unlabeled than for labeled actin. This conclusion is strengthened by kinetic data of S 1 binding to partially labeled F-actin, which exhibit a biphasic behavior due to the slower dissociation of S 1 from unlabeled than from labeled actin.
The interaction of the myosin head (myosin subfragment-1, S 1 ) 1 with the actin filament (F-actin) is central to the ATPdriven vectorial movement of myosin along the filament and the resulting production of force by muscle. The mechanism of interaction of S 1 with F-actin, the nature of the different complexes formed with or without nucleotide bound to S 1 and their relative stabilities have been analyzed by a variety of kinetic and equilibrium methods. In the absence of ATP or ADP, S 1 forms a high affinity "rigor" complex (K ϭ 10 7 M Ϫ1 ) with Factin, with a maximal binding stoichiometry of one S 1 per F-actin subunit (1)(2)(3)(4). Formation of the F-actin-S 1 rigor complex can be monitored by the associated increase in light scattering or by the 80% quenching of fluorescence of the pyrenyl probe covalently bound to Cys-374 of actin. The kinetic analysis of increase in light scattering (5,7) or of decrease in pyrenyl fluorescence (6 -11) provided a description of the mechanism of actin-S 1 complex formation in terms of 2 consecutive reactions, a rapid bimolecular reaction followed by an isomerization process as follows, where S and A represent myosin subfragment 1 and the actin subunit in the filament.
AS* SCHEME I The kinetic data, however, were sometimes more complex than expected within Scheme I, and simplifying assumptions were often made in the interpretation of the data, or conditions were chosen under which the experimental complexity appeared lower, allowing a satisfactory description of the data by Scheme I. For instance, biphasic fluorescence changes (8) or deviations from simple exponentials (11) have been reported under conditions where a pseudo-first order process was expected, which could not be explained by S 1 microheterogeneity; agreement has not been reached on the values found for the rate constants involved in Scheme I, under similar ionic conditions (7,8,10); non-linearity has been observed in binding measurements of heavy meromyosin to F-actin (12). Other experiments showed that functional parameters, e.g. the K m of ATP in the actomyosin ATPase (13) and the protection afforded by actin against proteolytic degradation of S 1 (14,15) were dependent on the saturation of the filament by S 1 . It was proposed recently (11) that the existence of two rigor complexes in which S 1 would interact either with one or with two adjacent F-actin subunits in the filament would account for many of the aforementioned deviations from Scheme I. This model uses the facts that the myosin head can form a ternary complex with two G-actin molecules in a low ionic strength buffer where actin alone remains monomeric (16 -18) and that the reconstructions of the actin-myosin interface (19) using the crystallized structures of G-actin (20) and of S 1 (21) and the atomic model of the filament (22) indicate that S 1 can make rigor contacts with two actin subunits interacting with each other via longitudinal bonds along the long pitch helix.
In the present work, we have used light scattering and pyrenyl fluorescence to monitor the kinetics of interaction of S 1 with F-actin in rigor. Experiments have been performed at different actin/S 1 molar ratios, and both light scattering and fluorescence kinetic data have been analyzed in a comprehensive fashion. We find that the rate constant for association of S 1 to F-actin depends on the extent of saturation of the filament by S 1 . Our results also show evidence for a ϳ5-fold lower affinity of S 1 for pyrenyl-labeled actin than for unlabeled actin.
Proteins were kept on ice at a concentration of 50 -80 M and used within 2 weeks following purification.
Static Fluorescence Measurements-Equilibrium binding of S 1 to Factin in the rigor state was monitored by the quenching of fluorescence of pyrenyl-F-actin upon binding S 1 . CaATP-G-actin 1:1 complex (20 M) containing known proportions of pyrenyl G-actin was polymerized by addition of 0.1 M KCl and 2 mM MgCl 2 , followed by the addition of 1.5 molar equivalent phalloidin. Samples were prepared by diluting the stock F-actin-phalloidin 10-fold in F 0 buffer (5 mM Tris-Cl Ϫ , pH 7.8, 0.1 mM CaCl 2 , 0.2 mM DTT, 2 mM MgCl 2 , 0.1 M KCl) containing different amounts of S 1 (A 1 ) in the range (0 -3 M). The fluorescence of pyrenyl-F-actin was monitored at 20°C in a Spex fluorolog spectrofluorimeter with excitation and emission wavelengths set at 366 and 387 nm, respectively, after incubation of samples at room temperature for a few minutes.
Rapid Kinetics-The kinetics of interaction of F-actin with S 1 (A 1 ) in the absence of ATP was monitored in the stopped-flow apparatus (DX.17 MV, Applied Photophysics) at 20°C. Light scattering was monitored at 90°to the incident light at a wavelength of 400 nm, and slits of 1 mm. Pyrenyl fluorescence was monitored using an excitation wavelength of 366 nm (slit 0.5 mm) and placing a KV380 Schott filter on the emission beam. One of the drive syringes contained F-actin stabilized by phalloidin in F 0 buffer, prepared as described above. The other drive syringe contained S 1 (A 1 ) also in F 0 buffer. In light scattering measurements, either unlabeled or fully (90%) pyrenyl-labeled F-actin was used. In fluorescence measurements, fully labeled pyrenyl-F-actin was mainly used, except in the experiment described in Fig. 5.
In all kinetic measurements, four to six consecutive shots were performed and the traces were averaged before being analyzed. Kinetic data were routinely analyzed within first order processes, when appropriate, using the software attached to the instrument. When the data deviated from the exponential behavior, a model was proposed based on the qualitative examination of the trend shown by the data upon changing concentrations of either actin or S 1 , and the kinetic curves were analyzed within the equation describing the new model. In the absence of analytical expression of the time dependence of the intensity of scattered light or of fluorescence, simulation of the kinetic curves was carried out using the HOPKINSIM simulation program, and appropriate values of the rate constants were adjusted by hand to obtain a satisfactory superimposition of a large number of experimental curves obtained at series of concentrations of actin and S 1 onto the corresponding theoretical kinetic curves.

Light Scattering Measurements of the Kinetics of Interaction of F-actin with S 1 (A 1 ): The Process Shows Negative
Cooperativity-The time course of increase in light scattering linked to the formation of the F-actin-S 1 rigor complex was measured either at constant F-actin and increasing S 1 , or at constant S 1 and increasing F-actin. The increase in intensity of scattered light was a simple first order process only when F-actin was in excess over S 1 , conditions leading to partially decorated actin filaments, with final molar ratios of bound S 1 per F-actin of Յ0. 30. In contrast, when S 1 was equal to or in excess over F-actin, conditions under which the molar ratio x of bound S 1 increased from 0 to 1 during the binding process, the reaction was not first order. In addition, the extent of increase in the intensity of scattered light per unit of F-actin-S 1 complex formed at the end of the reaction, was ϳ10% higher when fully decorated filaments were formed (S 1 in excess over F-actin) than when partially decorated filaments were formed (F-actin in excess over S 1 ). Typical traces obtained for the formation of 2 M F-actin-S 1 from either 2 M S 1 and 6 M F-actin, or from 2 M F-actin and 6 M S 1 , are shown in Fig. 1 and are clearly not superimposable. The overall reaction was slower when S 1 was in excess over F-actin, and the fit to a monoexponential was poor (variance ϭ 2.10 Ϫ3 ) as compared to the good fit (variance ϭ 6.10 Ϫ4 ) obtained when F-actin was in excess over S 1 (Fig. 1, insets). The pseudo first order rate constant obtained when F-actin was in excess over S 1 varied linearly with F-actin in the range 0 -15 M, as shown in Fig. 2. A value of 9 M Ϫ1 s Ϫ1 for the apparent bimolecular association rate constant of S 1 to F-actin was derived from the data. The value of the dissociation rate constant (intercept) was too low (Ͻ Ͻ1 s Ϫ1 ) to be determined with accuracy.
The large difference between the two time courses shown in Fig. 1 arises from the two following points.
First, the increase in intensity of light scattered at 90°, R 90 , is not a linear function of x, as described by the following equation (28,29).
K is a constant proportional to the square of the refractive index increment, P x is the shape factor of a filament containing on average x molecules of S 1 bound/F-actin subunit, M A and M S are the molecular masses of actin (42 kDa) and S 1 (130 kDa), respectively, and F 0 is the total concentration of F-actin subunits. Because the intensity of scattered light increases cooperatively with x, as is apparent in titration curves (5), the final increase in light scattering when F-actin is in excess over S 1 (curve a in Fig. 1) is lower than when S 1 is in excess over F-actin (curve b in Fig. 1). When F-actin is in excess over S 1 , x remains small (0 Յ x Յ 0.2) during 80% of the whole reaction and R 90 can be assumed to be proportional to x, but this assumption is no longer valid when S 1 is in excess over F-actin. Second, comparison of curves a and b indicates that the rate of association of S 1 to F-actin declines as the density of bound S 1 increases. Hence, the rate constant derived from experiments in which F-actin is in excess over S 1 (Fig. 2) refers to the association rate constant k ϩ 0 of S 1 to undecorated or poorly decorated filaments (x Ͻ 0.2). To analyze the kinetic curves obtained when F-actin was reacted with an excess of S 1 , we took into account the non-linearity of R 90 with x, and we tentatively assumed a simple linear decrease of k ϩ from k ϩ 0 to k ϩ 1 as x increases from 0 to 1, as follows. Assuming the binding of S 1 to F-actin to be essentially irreversible, and S 1 to be in excess over F-actin, where [S 0 ] represents the total concentration of S 1 , which leads to Equation 4.
The time dependence of x described by Equation 4 was combined with Equation 1 to derive the time dependence of the intensity of scattered light upon binding of S 1 to F-actin under conditions where x varies from 0 to 1 during the reaction. Fig. 3 shows the good fit of the simulated time courses to experimental traces at three sets of F-actin and S 1 concentrations. The value of k ϩ 0 was imposed equal to 9 M Ϫ1 s Ϫ1 as derived from Fig. 2; the fit was good for all curves using the same value of 0.5 Ϯ 0.5 M Ϫ1 s Ϫ1 for k ϩ 1 . Values of k ϩ 1 higher than 1 M Ϫ1 s Ϫ1 did not fit the data. Hence, the results are consistent with the view that, as the actin filament becomes increasingly saturated by S 1 , steric hindrance slows down further binding of S 1 to available sites.
Kinetics of the Change in Pyrenyl-F-actin Fluorescence upon Binding S 1 -The same experiments as the ones described in the previous section were carried out using the change in pyrenyl-actin fluorescence to monitor the binding of S 1 to Factin. A preliminary experiment showed that the kinetics of change in light scattering were identical, using 90 -100% labeled pyrenyl-actin, to the one observed with unlabeled actin.
The changes in fluorescence recorded upon adding 5 M S 1 (A 2 ) to 1 M pyrenyl-F-actin, or 5 M pyrenyl-F-actin to 1 M S 1 (A 2 ) are shown in Fig. 4. Again the two time courses corresponding to the formation of 1 M pyrenyl-F-actin-S 1 were not superimposable. In agreement with the light scattering results described above, the reaction was slower when S 1 was in excess over F-actin, and the time courses could be well analyzed within a first order process only when F-actin was in excess over S 1 . The pseudo first order rate constant then varied linearly with pyrenyl-F-actin concentration and the data points superimposed with those derived from light scattering kinetics (Fig. 2), indicating that the same rate-limiting process was monitored by either light scattering or pyrenyl fluorescence. In order to appreciate whether the length of the filaments affected the kinetics of interaction with S 1 , gelsolin was added to Factin in a proportion varying between 1:500 to 1:50. The kinetics were unaffected by the length of the filaments in this range.
When S 1 was in excess over pyrenyl-F-actin, the data were tentatively analyzed using Equation 2 and expressing that the observed fluorescence is the weighted sum of the fluorescences of F-actin and F-actin-S 1 complex, as follows, where [F 0 ] is the total concentration of F-actin, and f 0 and f 1 the intrinsic fluorescences of F-actin (taken as 1 M Ϫ1 by convention) and of F-actin-S 1 , respectively.
Fluorescence titration curves of 100% labeled pyrenyl-F-actin by S 1 showed that the fluorescence was 85% quenched at saturation by S 1 (f 1 ϭ 0.15). Simulated fluorescence time courses using Equations 4 and 5 were generated under conditions where S 1 is in excess over F-actin. The fit to experimental data (Fig. 5) was good using the same values of rate parameters k ϩ 0 and k ϩ 1 as in the light scattering experiments, at different F-actin/S 1 ratios in the range 3-6. Note that the model within which the rate constant for S 1 association to F-actin varies with the saturation of filaments by S 1 accommodates both the light scattering kinetics (in which the intensity of scattered light is not a linear function of bound S 1 ) and the fluorescence kinetics (in which the change in fluorescence is a linear function of bound S 1 , as was verified by titration curves).
Equilibrium Binding of S 1 to F-actin Also Shows Negative Cooperativity-Since the kinetics of binding of S 1 to F-actin show negative cooperativity, equilibrium binding measurements should also display an anticooperative behavior. The fluorescence titration curves of pyrenyl-F-actin by S 1 have routinely been satisfactorily analyzed assuming a simple hyperbolic binding isotherm; however, recent sedimentation assays have revealed anticooperativity (11). Equilibrium binding of S 1 to pyrenyl-F-actin has therefore been reexamined under physiological ionic conditions identical to those used in kinetic experiments. Fluorescent titration curves are shown in Fig. 6 at 2 and 0.4 M pyrenyl-F-actin. Although the difference between the theoretical best fits provided by a regular hyperbolic and an anticooperative binding schemes is small, clearly the data are accurate enough to demonstrate that the anticooperative binding scheme provides a better fit to the experimental curve. The theoretical hyperbolic best fit (K ϭ 0.02 M) indeed crosses the experimental curve twice. The anticooperative binding curve was calculated using the following classical equation (30), where K o and K x represent the equilibrium dissociation constants for binding of S 1 to an undecorated filament and to a filament containing a proportion, x, of F-actin-S 1 subunits.
The nature of the function (x) being unknown, the simple linear function (x) ϭ ␣x was tried. The best fit (shown in Fig.  6) was obtained for K o ϭ 0.01 M and ␣ ϭ -2.3 Ϯ 0.2. Note that this value of ␣ also corresponds to a 10-fold decrease in affinity of S 1 for F-actin as the saturation of the filament by myosin heads increases from 0 to 1. Other experiments done at higher ionic strength (0.3 M KCl) yielded binding curves that were consistent with a lower overall affinity, but showed no increased anticooperativity.
All the above experiments have been unable to detect an appreciable difference between unlabeled and labeled actin in reacting with S 1 . However, the affinity of S 1 for both actins is so high that a 10-fold difference in the apparent rate constants of dissociation of S 1 from F-actin or pyrenyl-F-actin would not have been detected.
Reaction of S 1 with Mixtures of Unlabeled and Pyrenyl-labeled F-actin-Samples of 2 M F-actin containing different percentages (5-90%) of pyrenyl-labeled actin were made and fluorescence titration curves of F-actin by S 1 were carried out in the spectrofluorimeter. The curves, shown in Fig. 7, were not superimposable. When F-actin was Ն90% pyrenyl-labeled, the fluorescence decreased linearly upon addition of S 1 and the minimum was reached at 2 M S 1 , consistent with a very high affinity of S 1 for F-actin, as reported previously (8). When pyrenyl-actin represented a lower molar fraction of total actin, the titration curve was sigmoidal. The sigmoidicity was more pronounced, and the equivalence point (2 M S 1 ) was reached more abruptly, consistent with a lower affinity of S 1 for pyrenyl-F-actin than for unlabeled actin. The data were identical when the partially labeled F-actin samples were made up either by polymerizing a partially labeled G-actin solution (leading to hybrid filaments made of labeled and unlabeled subunits), or by gently mixing unlabeled F-actin and fully labeled F-actin that had been prepolymerized separately. This piece of data shows that the observed quenching of fluorescence of a given pyrenyl-labeled F-actin subunit upon binding S 1 is independent of the nature (labeled or unlabeled) of the neighboring subunits in the filament. The data shown in Fig. 7 were therefore analyzed within the following simple scheme, where F and F* represent unlabeled and pyrenyl-labeled F-actin, respectively, and FS and *FS their complexes with S 1 .

SCHEME II
The relative change in pyrenyl fluorescence, Y, reflected the binding of S 1 to pyrenyl-F-actin, as described by the following equation, where (0), (S 0 ) and (ϱ) represent the fluorescence intensity observed in the absence of S 1 , or at a given total concentration S 0 or at a saturating concentration of S 1 .
The combined equations for mass conservation lead to: hence, only the ratio K b /K a can actually be derived from these measurements. The value of K b /K a had to be in the range of 4 -5 to provide a satisfactory fit to all the curves taken together. The 2 value increased when values of 3.5 or 5.5 were used for In conclusion, equilibrium binding measurements of S 1 to mixtures of unlabeled and pyrenyl-labeled F-actin demonstrate that the affinity of S 1 for labeled actin is 4-fold lower than for unlabeled actin. In conclusion, derivatizing Cys-374 alters the interaction of S 1 with F-actin. Kinetic evidence for a lower affinity of S 1 for pyrenyl-actin was provided by observing the time course of pyrenyl fluorescence upon mixing S 1 with partially labeled F-actin; Fig. 8A shows that when S 1 was substoichiometric with respect to total F-actin, a biphasic change was observed, a rapid quenching of fluorescence being followed by a slower recovery to a higher final fluorescence intensity. This biphasic change indicates that the rates of S 1 association to labeled or unlabeled F-actin both are very fast, but the rate of dissociation of the F-actin-S 1 complex is lower for unlabeled actin. The extent of fluorescence recovery was lower when the molar ratio of S 1 to F-actin increased; no fluorescence recovery was observed when the concentration of S 1 was sufficient to bind to both labeled and unlabeled actin. As shown in Fig. 8B, the amplitude of the rapid transient decrease in fluorescence varied linearly with S 1 until the 1:1 molar ratio to total F-actin was reached, then remained constant, giving a high affinity titration curve within which pyrenyl-F-actin would react with S 1 with the same affinity as unlabeled F-actin. The final fluorescence in turn confirmed the sigmoidal titration curve displayed in equilibrium measurements (Fig. 7). The slow recov-ery of fluorescence was a first order process of rate constant k ϭ 0.16 s Ϫ1 , which is the rate-limiting process in the dissociation of S 1 from pyrenyl-F-actin.
Temperature Dependence of the Kinetics of Binding of S 1 to F-actin-All kinetic data obtained thus far at 20°C under physiological conditions (Fig. 2) show no evidence for a higher limit in the pseudo first order rate constant for the reaction of S 1 with F-actin. A limit of 200 s Ϫ1 has been measured by others at 20°C and low ionic strength (10). At physiological ionic strength, no rate-limiting step could be measured (8), in agreement with the present data. However, the existence of a conformational change of F-actin-S 1 following the formation of a rapid equilibrium collision complex was inferred from two observations. First, the value found for the apparent second order rate constant for association of S 1 to F-actin at physiological ionic strength was typically Յ10 6 M Ϫ1 s Ϫ1 at 20°C, which is lower than expected for a diffusion-controlled reaction (5,8,30). Second, the binding reaction had a very strong temperature dependence, indicative of the involvement of a conformation change in the measured process (31). It should be noted, however, that the low value found in previous works (5,8,31) for the apparent second order rate constant under conditions similar to ours (physiological pH and ionic strength, 20°C) was derived from the analysis of turbidity or pyrene fluorescence changes, in excess of S 1 over F-actin, within a simple first order process. As we have seen above (Figs. 1 and 4), this analysis is not correct and leads to underestimated values of the association rate constant. The value of 9 M Ϫ1 s Ϫ1 derived from Fig. 2 when F-actin is in excess over S 1 is 1 order of magnitude higher than previously reported numbers and falls in the range of values expected for a diffusion-controlled process between Factin and S 1 .
To get a more precise idea of the mechanism of F-actin-S 1 interaction, the temperature dependence of the binding reaction was studied under physiological ionic strength.
In a temperature range of 4 -25°C, S 1 was mixed with an excess of pyrenyl-labeled F-actin. The concentration of F-actin was varied in the range 2-10 M. In this range k obs varied linearly with F-actin at all temperatures, as shown in Fig. 9. The apparent second order rate constant k ϩ showed a strong temperature dependence (Fig. 9, inset). The slope of the plot of ln k ϩ versus 1/T decreased somewhat at higher temperatures, as the diffusion became rate-limiting. A value of 27 kcal/mol was derived from the maximum slope of the plot (measured below 10°C). This value is in agreement with previous determinations derived from pressure relaxation studies of F-actin-S 1 -ADP and, as proposed by Geeves and Gutfreund (31), reflects the involvement of a conformational change in the measured reaction. This conformation change may occur on the rapidly preformed F-actin-S 1 complex, or on one of the two reacting proteins prior to the diffusion-controlled reaction. The first model described by Scheme I has thus far been favored in the field. Within Scheme I, the present data should be accommodated as follows. The value of k Ϫ1 /k ϩ1 should be at least 40 M, to account for the strictly linear dependence of k obs on F-actin in the range 0 -15 M. The value of k ϩ1 should be appreciably faster than the observed apparent k ϩ , i.e. k ϩ1 Ͼ Ͼ 10 M Ϫ1 s Ϫ1 ; hence, k Ϫ1 should be larger than 400 s Ϫ1 . The value of k Ϫ2 , which corresponds to the rate-limiting process observed in the slow fluorescence recovery shown in Fig. 7, is 0.16 s Ϫ1 for pyrenyl-actin and should be 4-fold lower (0.04 s Ϫ1 ) for unlabeled actin. The apparent second order rate constant represents k ϩ2 /K 1 , and has a value of 9 M Ϫ1 s Ϫ1 at 20°C; hence, k ϩ2 should have a value of at least 400 s Ϫ1 . The isomerization of the collision complex would therefore increase the affinity of S 1 for unlabeled F-actin by 400/0.04 ϭ 10 4 -fold, leading to a value of 4 nM for the global equilibrium dissociation constant of the rigor complex. Note that this value refers to the higher affinity binding of S 1 to the undecorated filament. As the filament becomes increasingly decorated, the affinity of S 1 may decrease up to 10-fold, according to the above data. DISCUSSION The equilibrium and kinetics of binding of myosin subfragment-1 to F-actin have been investigated with the aim to understand whether the binding was a reaction described by a single equilibrium dissociation constant, as concluded from earlier works (1)(2)(3)(4)(5), or whether the equilibrium binding might be non-Michaelian, and associated to more complex kinetics, as suggested by more recent works, which pointed to the nonlinear dependence of physical parameters monitoring F-actin-S 1 interaction on the extent of S 1 bound fo F-actin (11,(32)(33)(34)(35). We find that the rate of S 1 binding to F-actin depends on the extent of bound S 1 , steric hindrance of the first bound S 1 molecules preventing further binding of S 1 to the partially decorated filament. Both the kinetics of changes in light scattering or pyrenyl-actin fluorescence (Figs. 1 and 3-5) and the equilibrium binding measurements (Fig. 6) can be quantitatively accounted for, assuming that the apparent bimolecular association rate constant decreases linearly with the molar ratio of bound S 1 . A good fit to the data was obtained assuming a 10-fold decrease in the association rate constant k ϩ of S 1 to F-actin as the saturation of the filament by myosin heads increases from 0 to 1. The corresponding 10-fold decrease in affinity of S 1 for F-actin, over the whole titration curve, was confirmed in the analysis of the equilibrium binding data (Fig. 6).
Evidence for negative cooperativity in binding kinetics was provided by the observation that the F-actin-S 1 complex is formed faster when F-actin is in excess over S 1 than in the opposite situation, and that the process is not first order when S 1 is in excess over F-actin. Such non-exponential reactions have been noticed in the past (8,11). One of the main points of the present paper is to propose a quantitative analysis of this phenomenon. The kinetic data were analyzed assuming a linear dependence of k ϩ on the stoichiometry x of S 1 bound per F-actin subunit. This linear dependence was chosen, in the absence of any experimental evidence for a defined binding scheme, because it provided a simple analytical solution (Equation 4) that allowed simulation and fitting of the non-exponential binding kinetics. A more physically reasonable analysis of the data, taking into account the polarity of the filament, would imply to define different rate constants for association of S 1 to F-actin subunits having either no neighboring bound S 1 , or a neighboring bound S 1 either on the barbed or on the pointed end side of the subunit to which S 1 is binding. Consideration of the helical nature of the filament, in which each actin subunit has in fact four neighbors, would introduce further splitting of the different association rate constants. A complete analysis would also require a statistical analysis of the distribution of S 1 bound at each value of x, and Monte Carlo methods would be necessary to model the binding scheme adequately. However, we are not sure that fitting such a complex model to the present data would provide an unambiguous set of association rate constants.
Our results and conclusions are in partial agreement and in partial disagreement with the recent work of Andreev et al. (11). While the equilibrium and kinetic data of S 1 association to F-actin are in agreement with the anticooperativity in S 1 binding demonstrated by the sedimentation data of Andreev et al. (11), our kinetic observations and our interpretation differ. Andreev et al. reported that the time course of quenching in pyrene fluorescence upon binding S 1 was biphasic when F-actin was in excess over S 1 , which was interpreted as a two-step binding of S 1 to one F-actin subunit, followed by an isomerization consistent with binding of S 1 to a second actin subunit in the filament. Our observations are somewhat different, since a simple exponential binding process was recorded when F-actin was in excess over S 1 (Figs. 1 and 4), and the reaction deviated from a monoexponential when S 1 was in excess over F-actin. The data presented by Andreev et al. with S 1 in excess over F-actin ( Fig. 2A in Ref. 11) actually show a very poor fit to a monoexponential (in agreement with our data), the experimental trace intersecting the exponential best fit several times. Although our data do not support the two-step binding model proposed by Andreev et al., clearly the basic observations of nonlinear kinetics and anticooperative equilibrium binding are confirmed in the present work, with a different interpretation.
In most kinetic studies of S 1 binding carried out thus far, the concentration of S 1 was varied, in excess over F-actin. The data therefore referred to a high molar ratio of bound S 1 , and were assumed to be well described by monoexponentials, which now appears to be incorrect (see curves b, residuals, in Figs. 1 and  4). Within our interpretation, the rate constant derived from such kinetic data, analyzed within a single first order process, reflects the average rate of binding of S 1 to a partially decorated filament, and is therefore lower than the rate constant for association of S 1 to a bare filament, which we find equal to 9 M Ϫ1 s Ϫ1 at 20°C under physiological conditions. This value is within the range expected for a diffusion-controlled reaction; however, the high activation energy of the reaction strongly suggests that a structural change of the F-actin S 1 complex is involved, in agreement with previous works, and also with the model derived from the kinetics of interaction of G-actin with S 1 (18).
The present work gives the first quantitative estimate of the difference in affinity of S 1 for pyrenyl-labeled and unlabeled actin. This difference was previously thought to be insignificant, because S 1 reacts at apparently identical rates with either unlabeled or fully labeled F-actin. However, the reaction of S 1 with mixtures of labeled and unlabeled actins clearly shows evidence, both in the equilibrium binding curves and in the kinetics, for a 4 -5-fold difference in affinity, most likely accounted for by a difference in the rate constant k Ϫ2 in Scheme I.
The conclusion that labeling of Cys-374 on actin interferes with S 1 binding is in agreement with predictions made in the reconstruction of the actin-myosin head interface from the crystallized structures of actin and S 1 (19) and using the atomic model of the actin filament (22). It also agrees with the fact that in F-actin, Cys-374 can be N, NЈ-paraphenylenedimaleimide cross-linked to Lys-191 of the adjacent subunit along the short pitch helix (36), and this cross-link is inhibited by S 1 binding (37), indicating S 1 binding to F-actin perturbs the region of the C terminus of F-actin. It should be noted that the difference in affinity of labeled and unlabeled actin for S 1 is not displayed by G-actin (16,18). Accordingly S 1 can be cross-linked to Cys-374 in the G-actin-S 1 complexes, not in the F-actin-S 1 complex (38).
Finally our results suggest that the cross-bridges might op-erate differently when the thin filaments are poorly or fully saturated by myosin heads, a conclusion in agreement with biochemical data (13) and structural evidence (39) for the interaction between adjacent heads bound to the filament at high S 1 /actin ratios.