Theoretical kinetic studies of models for binding myosin subfragment-1 to regulated actin: Hill model versus Geeves model

It was previously shown that a one-dimensional Ising model could successfully simulate the equilibrium binding of myosin S1 to regulated actin filaments (T. L. Hill, E. Eisenberg and L. Greene, Proc. Natl. Acad. Sci. U.S.A. 77:3186-3190, 1980). However, the time course of myosin S1 binding to regulated actin was thought to be incompatible with this model, and a three-state model was subsequently developed (D. F. McKillop and M. A. Geeves, Biophys. J. 65:693-701, 1993). A quantitative analysis of the predicted time course of myosin S1 binding to regulated actin, however, was never done for either model. Here we present the procedure for the theoretical evaluation of the time course of myosin S1 binding for both models and then show that 1) the Hill model can predict the "lag" in the binding of myosin S1 to regulated actin that is observed in the absence of Ca++ when S1 is in excess of actin, and 2) both models generate very similar families of binding curves when [S1]/[actin] is varied. This result shows that, just based on the equilibrium and pre-steady-state kinetic binding data alone, it is not possible to differentiate between the two models. Thus, the model of Hill et al. cannot be ruled out on the basis of existing pre-steady-state and equilibrium binding data. Physical mechanisms underlying the generation of the lag in the Hill model are discussed.     

A new model of cooperative myosin-thin filament binding

Cooperative myosin binding to the thin filament is critical to regulation of cardiac and skeletal muscle contraction. This report delineates and fits to experimental data a new model of this process, in which specific tropomyosin-actin interactions are important, the tropomyosin-tropomyosin polymer is continuous rather than disjointed, and tropomyosin affects myosin-actin binding by shifting among three positions as in recent structural studies. A myosin- and tropomyosin-induced conformational change in actin is proposed, rationalizing the approximately 10,000-fold strengthening effect of myosin on tropomyosin-actin binding. Also, myosin S1 binding to regulated filaments containing mutant tropomyosins with internal deletions exhibited exaggerated cooperativity, implying an allosteric effect of tropomyosin on actin and allowing the effect's measurement. Comparisons among the mutants suggest the change in actin is promoted much more strongly by the middle of tropomyosin than by its ends. Regardless of calcium binding to troponin, this change in actin facilitates the shift in tropomyosin position to the actin inner domain, which is required for tight myosin-actin association. It also increases myosin-actin affinity 7-fold compared with the absence of troponin-tropomyosin. Finally, initiation of a shift in tropomyosin position is 100-fold more difficult than is its extension from one actin to the next, producing the myosin binding cooperativity that underlies cooperative activation of muscle contraction.     

Kinetic model of regulation of muscle protein activity.

The widely accepted steric model of calcium regulation of actin-myosin interactions in vertebrate muscles has to be completed to fit the kinetic data. It should be supposed that: (1) the thin filaments consist of functionally independent units, containing seven actin sites regulated by one troponin-tropomyosin complex; (2) actin sites become available for myosin heads only due to fluctuations of tropomyosin position; (3) binding of calcium to troponin results either in the shift of the tropomyosin equilibrium position or in the weakening of its interactions with actin strand so that the probability of effective fluctuations increases; (4) link formation between myosin head and some of the available actin site fixates the tropomyosin in such a position that the other six actin sites of the same functional unit become available for myosin too.The model gives linear kinetic scheme for the transitions of a functional unit between nine states (a “turned off” state, and eight “turned on” ones with different occupancy by myosin heads). The dependences of the apparent rate constants of actomyosin formation and dissociation upon the myosin head and substrate concentrations are obtained from the Lymn-Taylor scheme. The frequency of the actomyosin complexes dissociation is assumed to give the ATPase rate.The model fits the kinetic data on the ATP hydrolysis by myosin subfragment-1 with regulated or unregulated actin as a cofactor under various conditions. It shows a sharp dependence of activation upon the apparent affinity of the actin and myosin sites. Therefore, the model appears to be applicable to myosin controlled systems.     

Cooperative regulation of myosin-actin interactions by a continuous flexible chain II: actin-tropomyosin-troponin and regulation by calcium

The model of myosin regulation by a continuous tropomyosin chain is generalized to a chain of tropomyosin-troponin units. Myosin binding to regulated actin is cooperative and initially inhibited by the chain as before. In the absence of calcium, myosin is further inhibited by the binding of troponin-I to actin, which through the whole of troponin pins the tropomyosin chain in a blocking position; myosin and TnI compete for actin and induce oppositely-directed chain kinks. The model predicts equilibrium binding curves for myosin-S1 and TnI as a function of their first-order affinities K(S1) and L(TI). Myosin is detached by the actin binding of TnI, but TnI is more efficiently detached by myosin when the kink size (typically nine to ten actin sites) spans the seven-site spacing between adjacent TnI molecules. An allosteric mechanism is used for coupling the detachment of TnI to calcium binding by TnC. With thermally activated TnI kinks (kink energy B approximately k(B)T), TnI also binds cooperatively to actin, producing cooperative detachment of myosin and biphasic myosin-calcium Hill plots, with Hill coefficients of 2 at high calcium and 4-6 at low calcium as observed in striated muscle. The theory also predicts the cooperative effects observed in the calcium loading of TnC.     

Cooperative regulation of myosin-S1 binding to actin filaments by a continuous flexible Tm–Tn chain

The regulation of striated muscle contraction involves cooperative interactions between actin filaments, myosin-S1 (S1), tropomyosin (Tm), troponin (Tn), and calcium. These interactions are modeled by treating overlapping tropomyosins as a continuous flexible chain (CFC), weakly confined by electrostatic interactions with actin. The CFC is displaced locally in opposite directions on the actin surface by the binding of either S1 or Troponin I (TnI) to actin. The apparent rate constants for myosin and TnI binding to and detachment from actin are then intrinsically coupled via the CFC model to the presence of neighboring bound S1s and TnIs. Monte Carlo simulations at prescribed values of the CFC stiffness, the CFC??s degree of azimuthal confinement, and the angular displacements caused by the bound proteins were able to predict the stopped-flow transients of S1 binding to regulated F-actin. The transients collected over a large range of calcium concentrations could be well described by adjusting a single calcium-dependent parameter, the rate constant of TnI detachment from actin, k _?I. The resulting equilibrium constant $ K_{\text{B}} \equiv 1/K_{\text{I}} $ varied sigmoidally with the free calcium, increasing from 0.12 at low calcium (pCa >7) to 12 at high calcium (pCa <5.5) with a Hill coefficient of ~2.15. The similarity of the curves for excess-actin and excess-myosin data confirms their allosteric relationship. The spatially explicit calculations confirmed variable sizes for the cooperative units and clustering of bound myosins at low calcium concentrations. Moreover, inclusion of negative cooperativity between myosin units predicted the observed slowing of myosin binding at excess-myosin concentrations.     

A Mechanistic Model of Ca Regulation of Thin Filaments in Cardiac Muscle

We present a model of Ca-regulated thin filaments in cardiac muscle where tropomyosin is treated as a continuous elastic chain confined in the closed position on the actin helix by electrostatic forces. The main distinction from previous works is that the intrinsic stress-free helical shape of the tropomyosin chain was taken into account explicitly. This results in the appearance of a new, to our knowledge, tension-like term in the energy functional and the equilibrium equation. The competitive binding of calcium and the mobile segment of troponin-I to troponin-C were described by a simple kinetic scheme. The values of dimensionless model parameters were estimated from published data. A stochastic Monte Carlo simulation of calcium curves has been performed and its results were compared to published data. The model explains the high cooperativity of calcium response of the regulated thin filaments even in the absence of myosin heads. The binding of myosin heads to actin increases the calcium sensitivity while not affecting its cooperativity significantly. When the presence of calcium-insensitive troponin-C was simulated in the model, both calcium sensitivity and cooperativity decreased. All these features were previously observed experimentally.     

A Mechanistic Model of Ca Regulation of Thin Filaments in Cardiac Muscle

We present a model of Ca-regulated thin filaments in cardiac muscle where tropomyosin is treated as a continuous elastic chain confined in the closed position on the actin helix by electrostatic forces. The main distinction from previous works is that the intrinsic stress-free helical shape of the tropomyosin chain was taken into account explicitly. This results in the appearance of a new, to our knowledge, tension-like term in the energy functional and the equilibrium equation. The competitive binding of calcium and the mobile segment of troponin-I to troponin-C were described by a simple kinetic scheme. The values of dimensionless model parameters were estimated from published data. A stochastic Monte Carlo simulation of calcium curves has been performed and its results were compared to published data. The model explains the high cooperativity of calcium response of the regulated thin filaments even in the absence of myosin heads. The binding of myosin heads to actin increases the calcium sensitivity while not affecting its cooperativity significantly. When the presence of calcium-insensitive troponin-C was simulated in the model, both calcium sensitivity and cooperativity decreased. All these features were previously observed experimentally.     

Striated Muscle Regulation of Isometric Tension by Multiple Equilibria

Cooperative activation of striated muscle by calcium is based on the movement of tropomyosin described by the steric blocking theory of muscle contraction. Presently, the Hill model stands alone in reproducing both myosin binding data and a sigmoidal-shaped curve characteristic of calcium activation (Hill TL (1983) Two elementary models for the regulation of skeletal muscle contraction by calcium. Biophys J 44: 383–396.). However, the free myosin is assumed to be fixed by the muscle lattice and the cooperative mechanism is based on calcium-dependent interactions between nearest neighbor tropomyosin subunits, which has yet to be validated. As a result, no comprehensive model has been shown capable of fitting actual tension data from striated muscle. We show how variable free myosin is a selective advantage for activating the muscle and describe a mechanism by which a conformational change in tropomyosin propagates free myosin given constant total myosin. This mechanism requires actin, tropomyosin, and filamentous myosin but is independent of troponin. Hence, it will work equally well with striated, smooth and non-muscle contractile systems. Results of simulations with and without data are consistent with a strand of tropomyosin composed of ∼20 subunits being moved by the concerted action of 3–5 myosin heads, which compares favorably with the predicted length of tropomyosin in the overlap region of thick and thin filaments. We demonstrate that our model fits both equilibrium myosin binding data and steady-state calcium-dependent tension data and show how both the steepness of the response and the sensitivity to calcium can be regulated by the actin-troponin interaction. The model simulates non-cooperative calcium binding both in the presence and absence of strong binding myosin as has been observed. Thus, a comprehensive model based on three well-described interactions with actin, namely, actin-troponin, actin-tropomyosin, and actin-myosin can explain the cooperative calcium activation of striated muscle.     

A mosaic multiple-binding model for the binding of caldesmon and myosin subfragment-1 to actin.

Binding of caldesmon to actin causes a decrease in the quantity of bound myosin and results in a reduction in the rate of actin-activated adenosine triphosphate hydrolysis. It is generally assumed that the binding of caldesmon and myosin to actin is a pure competitive interaction. However, recent binding studies of enzyme digested caldesmon subfragments directed at mapping the actin binding site of caldesmon have shown that a small 8-kD fragment around the COOH-terminal can compete directly with the myosin subfragment 1 (S-1) binding to actin; at least one other fragment that binds to actin does not inhibit the actin-activated adenosine triphosphate activity of myosin. That is, only a part of the caldesmon sequence may be responsible for directly blocking the binding of S-1 to actin. This prompts us to question the actual mode of binding of intact caldesmon and myosin S-1 to actin: whether the entire intact caldesmon molecule is competing with S-1 binding (pure competitive model) or just a small part of it (mosaic multiple-binding model). To answer this question, we measured the amount of myosin S-1 and caldesmon bound per actin monomer as a function of the total concentration of S-1 added to the system at constant concentrations of actin and caldesmon. A formalism for calculating the titration data based on the pure competitive model and a mosaic multiple-binding model was then developed. When compared with theoretical calculations, it is found that the binding of caldesmon and S-1 to actin cannot be pure competitive if no cooperativity exists between S-1 and caldesmon.(ABSTRACT TRUNCATED AT 250 WORDS)     

Relationship between regulated actomyosin ATPase activity and cooperative binding of myosin to regulated actin

The protein complex, troponin-tropomyosin, which is bound to the thin actin filament, regulates muscle contraction and relaxation. In the absence of Ca²⁺ the troponin-tropomyosin complex causes muscle to relax, whereas in the presence of Ca²⁺, contraction occurs. Biochemical studies have shown that the troponin-tropomyosin complex has a dual effect on the interaction of the myosin cross-bridge with actin. In the presence of ATP, troponin-tropomyosin strongly inhibits the actomyosin ATPase activity, whereas in the absence of ATP, troponin-tropomyosin confers positive cooperativity on the binding of myosin to actin. We have proposed a simple model [Hill, T. L., Greene, L. E., and Eisenberg, E. (1980)Proc. Natl. Acad. Sci. USA 77, 3186–3190] that accounts for these biochemical observations by postulating that the troponin-tropomyosin-actin complex (regulated actin) can occur in two forms, a turned-on form and a turned-off form. This model defines several cooperativity parameters that describe the behavior of regulated actin. In previous studies we have determined the values of these parameters by studying the cooperative binding of myosin to regulated actin in the absence of ATP. In the present study we also used ATPase and fluorescence measurements to determine these cooperativity parameters. Assuming that the fluorescence change occurs only when two adjacent tropomyosin units shift into the turned-on form, our results show that all three methods give the same values for the cooperativity parameters. These results confirm the prediction of our model that a regulated actin unit that is turned off not only binds S-1 weakly but is also unable to activate the actomyosin ATPase activity.     

Assessment of the mechanical activity of cardiac isomyosins V1 and V3 by the in vitro motility assay with regulated thin filament

The dependences of thin filament sliding velocity on the calcium concentration in solution (pCa 5 to 8) for rabbit cardiac myosin isoforms V1 and V3 were determined in a set of experiments using an in vitro motility assay with a reconstructed thin filament. The constructed pCa-versus-velocity curves had a sigmoid shape. It was demonstrated that the sliding velocity of regulated thin filament at the saturating calcium concentration (pCa 5) did not differ from the actin sliding velocity for each isoform. The determined values of Hill’s cooperativity coefficient for isomyosins V1 and V3 were 1.04 and 0.75, respectively. It was demonstrated that isomyosin V3 was more sensitive to calcium as compared with isomyosin V1. Using the same assay, the dependence of thin filament sliding velocity on the concentration of the actin-binding protein α-actinin (analog of a force-velocity dependence) was determined at the saturating calcium concentration for each myosin isoform (V1 and V3). The results suggest that the calcium regulation of V1 and V3 contractile activity follows different mechanisms.     

Removal of tropomyosin overlap modifies cooperative binding of myosin S-1 to reconstituted thin filaments of rabbit striated muscle

Cooperative binding of myosin S-1.ADP to regulated F-actin was previously reported and has been interpreted by a two-state model in which an important source of cooperativity is nearest neighbor interactions between the 7-actin.tropomyosin (TM).troponin units (functional units) (Hill, T.L., Eisenberg, E., and Greene, L. (1980) Proc. Natl. Acad. Sci. U.S.A. 77, 3186-3190). It has been postulated that the head-to-tail overlap between adjacent TM molecules is the structural basis of the nearest neighbor interactions. We tested the hypothesis by examining S-1.ADP binding to reconstituted regulated F-actin containing either intact TM or nonpolymerizable TM from which the COOH-terminal 11 residues were removed. In the absence of Ca2+, substitution of nonpolymerizable TM for TM reduced significantly the slope of the steeply rising phase of the sigmoidal S-1.ADP binding curve. Nevertheless, considerable residual cooperativity remained. Analysis of the data using the two-state model of Hill et al. suggests that removal of TM overlap abolishes nearest neighbor interactions, while the concerted change of the state of 7 actins in a functional unit can account for the residual cooperativity.     

Electrostatic contributions to the binding of myosin and myosin-MgADP to F-actin in solution

The ionic strength dependence of skeletal myosin subfragment 1 (S1) binding to unregulated F-actin was measured in solutions containing from 0 to 0.50 M added lithium acetate (LiOAc) in the absence and presence of MgADP. The data were analyzed by using a theory based on an ion interaction model that is rigorous for high ionic strength solutions [Pitzer, K. S. (1973) J. Phys. Chem. 77, 268-277] in order to obtain values for K, the equilibrium association constant when the ionic strength is zero, and for [zMzA[, the absolute value of the product of the net electric charges of the actin binding site on myosin (zM) and the myosin binding site on actin (zA). The presence of MgADP reduced K by a factor of 10, as expected, and reduced [zMzA[ by about 1 esu2. Because the presence of MgADP is not likely to change the net charge of the myosin binding site on actin, these data are consistent with a model in which MgADP binding to S1 reduces its affinity for actin by a mechanism that reduces the net electric charge of the acting binding site on S1. The value of [zMzA[ in the absence of ADP was 8.1 +/- 0.9 esu2, which, if one uses integer values, suggests that zM and zA are in the 8+ to 1+ esu and 1- to 8- esu ranges, respectively. ADP binding then reduces zM to the 7+ to 0.88+ esu range.     

The regulation of myosin binding to actin filaments by Lethocerus troponin

Lethocerus indirect flight muscle has two isoforms of troponin C, TnC-F1 and F2, which are unusual in having only a single C-terminal calcium binding site (site IV, isoform F1) or one C-terminal and one N-terminal site (sites IV and II, isoform F2). We show here that thin filaments assembled from rabbit actin and Lethocerus tropomyosin (Tm) and troponin (Tn) regulate the binding of rabbit myosin to rabbit actin in much the same way as the mammalian regulatory proteins. The removal of calcium reduces the rate constant for S1 binding to regulated actin about threefold, independent of which TmTn is used. This is consistent with calcium removal causing the TmTn to occupy the B or blocked state to about 70% of the total. The mid point pCa for the switch differed for TnC-F1 and F2 (pCa 6.9 and 6.0, respectively) consistent with the reported calcium affinities for the two TnCs. Equilibrium titration of S1 binding to regulated actin filaments confirms calcium regulated binding of S1 to actin and shows that in the absence of calcium the three actin filaments (TnC-F1, TnC-F2 and mammalian control) are almost indistinguishable in terms of occupancy of the B and C states of the filament. In the presence of calcium TnC-F2 is very similar to the control with approximately 80% of the filament in the C-state and 10-15% in the fully on M-State while TnC-F1 has almost 50% in each of the C and M states. This higher occupancy of the M-state for TnC-F1, which occurs above pCa 6.9, is consistent with this isoform being involved in the calcium activation of stretch activation. However, it leaves unanswered how a C-terminal calcium binding site of TnC can activate the thin filament.     

Activation of actin-cardiac myosin subfragment 1 MgATPase rate by Ca2+ shows cooperativity intrinsic to the thin filament

The magnesium adenosinetriphosphatase (MgATPase) rate of cardiac myosin subfragment 1 (S-1) was studied in the presence of regulated actin in order to investigate the mechanism by which Ca2+ cooperatively induces cardiac muscle contraction. The MgATPase rate increased cooperatively with Ca2+, exhibiting a Hill coefficient of 1.8 and 50% activation at pCa 5.75. This cooperative response occurred despite an experimental design excluding several potential sources of cooperativity. First, to exclude spurious cooperativity due to erroneous calculation of pCa at low ionic strength, the affinities of Ca2+ and Mg2+ for [ethylenebis(oxyethylenenitrilo)]tetraacetic acid (EGTA) were measured by a novel method using Quin 2. At pH 7.06, 25 degrees C, and mu = 30 mM, the KD was 140 nM for CaEGTA and 2.7 mM for MgEGTA. Second, the cooperativity was not produced by actin-myosin S-1 binding; myosin S-1 was bound to only 1 of every 300 actin promoters, and earlier work [Tobacman, L. S., & Adelstein, R. S. (1986) Biochemistry 25, 798-802] had shown that cardiac myosin S-1 binds with equal affinity to the thin filament at very low Ca2+ and at saturating Ca2+ concentrations. Furthermore, the adenosine 5'-triphosphate turnover rate of the myosin S-1 was independent of enzyme concentration at low, intermediate, and saturating Ca2+ concentrations. Finally, since cardiac troponin has only one regulatory Ca2+-specific site, cooperative interactions between such sites could not occur. These data suggest that part of the cooperativity conferred by interaction between adjacent troponin-tropomyosin complexes is intrinsic to the thin filament and independent of myosin.     

Cooperative binding to the Ca2+-specific sites of troponin C in regulated actin and actomyosin

The Ca2+-binding component of troponin (TnC) and its proteolytic fragments containing Ca2+-binding sites I-III (TH1) or sites III and IV (TR2C) have been labeled with the fluorescent probes dansylaziridine (DANZ) at methionine 25 or 5-(iodoacetamidoethyl)amino-naphthalene-1-sulfonic acid (AEDANS) at cysteine-98. These probes report binding of Ca2+ to the low and high affinity sites, respectively. Fluorescence changes as a function of [Ca2+] were measured for the free peptides, their complexes with troponin I + troponin T, and these complexes bound to actin-tropomyosin in the presence of Mg2+ and ATP with and without myosin. An apparent Hill coefficient of 1.0-1.1 has been obtained for the Ca2+-induced fluorescence changes in TnC, its fragments, and their ternary complexes regardless of the label used. When a ternary complex containing appropriately labeled TnC or its fragment is bound to the actin-tropomyosin complex, the Hill coefficient for the titration of the low affinity sites increases to 1.5-1.6 and further increases to greater than 2 in the presence of myosin. To interpret the apparent Hill coefficients, we used a model containing two binding sites and a single reporter of the conformational change. Hill coefficients between 1.0 and 1.2 can be obtained for the fluorescence change without true cooperativity in metal binding, depending on the mechanism of the fluorescence change; i.e. the contribution of the singly or doubly occupied species to the fluorescence change. A Hill coefficient between 1.2 and 2, however, always indicates cooperativity in binding independently of the mechanism. Thus, our finding that fluorescence titrations of Ca2+ binding to TnCDANZ bound to actin-tropomyosin exhibit a Hill coefficient of 1.5 in the absence of myosin and 2.4 in its presence indicates the existence of true positive cooperativity in metal binding to sites I and II. No cooperativity was observed for AEDANS-labeled complexes that reflect Ca2+-binding to the high affinity sites. Plots of the Ca2+ dependence of myosin ATPase activity activated by actin-tropomyosin in the presence of any of the troponin complexes used had apparent Hill coefficients of approximately 4. The higher value suggests cooperative interactions in the activation of ATPase beyond those involved in Ca2+-binding to the Ca2+-specific sites.     

Theoretical studies on competitive binding of caldesmon and myosin S1 to actin: prediction of apparent cooperativity in equilibrium and slow-down in kinetics of S1 binding by caldesmon

Several laboratories have reported cooperative binding of S1 to actin in the presence of caldesmon. This cooperative binding has been interpreted with a model similar to that proposed for the binding of S1 to regulated actins in which the binding affinity of S1 is controlled by the position of the tropomyosin filaments. In a recent paper [Sen, A., Chen, Y., Yan, B., and Chalovich, J. M. (2001) Biochemistry 40, 5757-64], we showed qualitatively that S1 binding resulted in rapid dissociation of caldesmon from actin or actin-tropomyosin. This suggests that the cooperativity observed in the case of caldesmon is not due to a conformational change in actin-caldesmon but to the displacement of caldesmon. We show in this paper that the pure competitive binding model, in which both S1 and caldesmon are competing for the same binding sites on actin, can simulate quantitatively the effect of caldesmon on both the equilibrium and the kinetics of S1 binding to actin. This model successfully predicts an apparent cooperativity for the binding of S1 to actin-caldesmon without the need to assume multiple actin-caldesmon structures and produces a decreased rate of S1 binding to actin in the presence of caldesmon. This suggests that the inhibitory action of caldesmon on the actin-activated ATPase activity of myosin in solution and on the generation of active force in a contracting muscle may be simply due to the blocking of myosin binding sites on actin by caldesmon.     

Kinetic model for the interaction of myosin subfragment 1 with regulated actin

A one-dimensional kinetic Ising model is developed to describe the binding of myosin subfragment 1 (SF-1) to regulated actin. The model allows for cooperative interactions between individual actin sites with bound SF-1 ligands rather than assuming that groups of actin monomer sites change their state in a cooperative fashion. With the triplet closure approximation, the model yields a set of 16 independent differential (master) equations which may be solved numerically to yield the extent of binding as a function of time. The predictions of the model are compared with experiments on the transient binding of SF-1 to regulated actin in the presence of Ca2+ and in the absence of Ca2+ with varying amounts of SF-1 prebound to the actin filament and on the equilibrium binding of SF-1 X ADP to regulated actin in the absence of Ca2+. In all cases, the calculations fit the data to within the experimental errors. In the case of SF-1 X ADP, the results suggest that a repulsive interaction exists between adjacently bound SF-1 at the ends of two neighboring seven-site actin units.     

The thermodynamics of calcium binding to thermolysin

Calcium binding isotherms were determined for thermolysin in the range pH 5.6-10.5, and from 5 to 45 degrees C. An extensive statistical analysis of the binding data suggests that at least two of the four binding sites bind Ca2+ with complete positive cooperativity and independently of the other two. Nonlinear regression analysis of the binding data was used to calculate cooperative (K1) and independent (K2) binding constants for the four calcium sites. Thermodynamic parameters obtained from a van't Hoff analysis indicate that calcium binding to both cooperative and independent sites is an entropy-driven process. At pH 7.0, delta H1 = 90.4 kJ/mol; delta H2 = 97.5 kJ/mol; delta S1 = 456 J K-1 mol-1; delta S2 = 262 J K-1 mol-1. These results are compared to those obtained for other calcium-binding proteins. An analysis of the pH dependence of the calcium binding constants indicates that the binding of four protons at the cooperative site and one to two protons at the independent sites, modulates the calcium affinity. This confirms an earlier structural assignment of the double-site as the locus of the two cooperatively binding Ca2+. Calcium binding to thermolysin is enhanced in the presence of an active site directed inhibitor, suggesting that there may be positive cooperativity between substrate and calcium binding.     

Kinetics of actin-myosin binding. I. An exactly soluble one-variable model.

To treat the kinetics of actin-myosin binding as simply as possible, a one-variable model is developed and the notion of effectivity factors is introduced. An effectivity factor is a ratio of the reaction rate in the presence of cooperativity to that in the noncooperative case and is calculated by averaging cooperativity factors over all sites belonging to one seven-site actin unit. The technique is applicable to a variety of models involving cooperative association and dissociation processes. This averaging assumes the equivalence of all regulated actin units. The model may be solved exactly for arbitrary degrees of "preloading" of subfragment 1 (S1) on the regulated actin.