Rate of binding of tropomyosin to actin filaments

The decrease of the rate of actin polymerization by tropomyosin molecules which bind near the ends of actin filaments was analyzed in terms of the rate of binding of tropomyosin to actin filaments. Monomeric actin was polymerized onto actin filaments in the presence of various concentrations of tropomyosin. At high concentrations of monomeric actin (c1) and low tropomyosin concentrations (ct) (c1/ct greater than 10), actin polymerization was not retarded by tropomyosin because actin polymerization was faster than binding of tropomyosin to actin filaments. At low actin concentrations and high tropomyosin concentrations (c1/ct less than 5), the rate of elongation of actin filaments was decreased because actin polymerization was slower than binding of tropomyosin at the ends of actin filaments. The results were quantitatively analyzed by a model in which it was assumed that actin-bound tropomyosin molecules which extend beyond the ends of actin filaments retard association of actin monomers with filament ends. Under the experimental conditions (100 mM KCl, 1 mM MgCl2, pH 7.5, 25 degrees C), the rate constant for binding of tropomyosin to actin filaments turned out to be about 2.5 X 10(6) to 4 X 10(6) M-1 S-1.     

The Effect of Tropomyosin Mutations on Actin-Tropomyosin Binding: In Search of Lost Time

The initial binding of tropomyosin onto actin filaments and then its polymerization into continuous cables on the filament surface must be precisely tuned to overall thin-filament structure, function, and performance. Low-affinity interaction of tropomyosin with actin has to be sufficiently strong to localize the tropomyosin on actin, yet not so tight that regulatory movement on filaments is curtailed. Likewise, head-to-tail association of tropomyosin molecules must be favorable enough to promote tropomyosin cable formation but not so tenacious that polymerization precedes filament binding. Arguably, little molecular detail on early tropomyosin binding steps has been revealed since Wegner’s seminal studies on filament assembly almost 40 years ago. Thus, interpretation of mutation-based actin-tropomyosin binding anomalies leading to cardiomyopathies cannot be described fully. In vitro, tropomyosin binding is masked by explosive tropomyosin polymerization once cable formation is initiated on actin filaments. In contrast, in silico analysis, characterizing molecular dynamics simulations of single wild-type and mutant tropomyosin molecules on F-actin, is not complicated by tropomyosin polymerization at all. In fact, molecular dynamics performed here demonstrates that a midpiece tropomyosin domain is essential for normal actin-tropomyosin interaction and that this interaction is strictly conserved in a number of tropomyosin mutant species. Elsewhere along these mutant molecules, twisting and bending corrupts the tropomyosin superhelices as they “lose their grip” on F-actin. We propose that residual interactions displayed by these mutant tropomyosin structures with actin mimic ones that occur in early stages of thin-filament generation, as if the mutants are recapitulating the assembly process but in reverse. We conclude therefore that an initial binding step in tropomyosin assembly onto actin involves interaction of the essential centrally located domain.     

Tropomodulin binds two tropomyosins: a novel model for actin filament capping

Tropomodulin, a tropomyosin-binding protein, caps the slow-growing (pointed) end of the actin filament regulating its dynamics. Tropomodulin, therefore, is important for determining cell morphology, cell movement, and muscle contraction. For the first time we show that one tropomodulin molecule simultaneously binds two tropomyosin molecules in a cooperative manner. On the basis of the tropomodulin solution structure and predicted secondary structure, we introduced a series of point mutations in regions important for tropomyosin binding and actin capping. Capping activity of these mutants was assayed by measuring actin polymerization using pyrene fluorescence. Using direct methods (circular dichroism and native gel electrophoresis) for detecting tropomodulin/tropomyosin binding, we localized the second tropomyosin-binding site to residues 109-144. Despite previous reports that the second binding site is for erythrocyte tropomyosin only, we found that both short nonmuscle and long muscle alpha-tropomyosins bind there as well, though with different affinities. We propose a model for actin capping where one tropomodulin molecule can bind to two tropomyosin molecules at the pointed end.     

Effect of the state of oxidation of cysteine 190 of tropomyosin on the assembly of the actin-tropomyosin complex

Tropomyosin, cross-linked at cysteine 190, was found to bind more weakly to actin filaments than uncross-linked tropomyosin. Cross-linking of tropomyosin can cause actin filaments nearly completely covered with tropomyosin to be uncovered almost completely. The critical monomer concentration of actin is not significantly changed by binding of cross-linked or uncross-linked tropomyosin to actin filaments. The binding curves were analyzed quantitatively, thereby taking into account the polar end-to-end contact of tropomyosin molecules bound by actin and the overlap of the seven subunit binding sites along the actin filament. Under the conditions of the experiment (80 mM KCl, 1 mM MgCl2, pH 7.5, 38-42 degrees C), the equilibrium constant for isolated binding of tropomyosin to actin filaments is in the range 1 x 10(3)-3 x 10(3) M-1. The equilibrium constants for binding of tropomyosin to binding sites along the actin filament with one or two neighbouring tropomyosin molecules are in the range of 10(6) or 10(8) to 10(9) M-1, respectively. The equilibrium constants for binding of tropomyosin to binding sites along the actin filament with one or two neighbouring tropomyosin molecules are in the range of 10(6) or 10(8) to 10(9) M-1, respectively. The equilibrium constants for cross-linked and uncross-linked tropomyosin differ by a factor of only about two. Owing to the highly cooperative binding, these differences are sufficient so that actin filaments nearly completely covered with uncross-linked tropomyosin are uncovered almost completely by cross-linking tropomyosin at cysteine 190.     

Equilibrium of the actin-tropomyosin interaction

The actin-tropornyosin interaction was studied by means of light-scattering. The experimental data were analysed on the basis of the model of co-operative binding of large ligands to a one-dimensional lattice with overlapping binding sites. The affinity of tropomyosin for actin filaments was dependent on the magnesium concentration. A fivefold increase of the magnesium concentration (from 0·5 mm to 2·5 mm) enhanced the equilibrium constant twofold (from 700 to 1600 m^?1) for the isolated binding of tropomyosin molecules to actin filaments. At low magnesium concentrations (0·5 mm), tropomyosin molecules were bound to isolated binding sites on an actin filament about 600 times more weakly than to contiguous binding sites. At increased magnesium concentrations (2·5 mm), the tendency of tropomyosin to bind contiguously increased twofold. Due to the co-operative nature of the actin-tropomyosin interaction, a small change in the magnesium concentration may cause a great change of the structural organisation of the complex. A small enhancement of the magnesium concentration (from 1 mm to 1·5 mm) caused bare filaments to be covered almost completely with tropomyosin. The length of tropomyosin clusters and the number of gaps on actin filaments depended strongly on the magnesium concentration. From the values of the experimentally determined equilibrium constants, it was concluded that the end-to-end interaction of tropomyosin was not strong enough to bring about all-or-none behaviour, where actin filaments of physiological length (~1000 nm) are either completely covered with or completely free of tropomyosin.     

Tropomyosin as a regulator of the sliding movement of actin filaments

We examined the capacity of tropomyosin molecules regulating the sliding movement of actin filaments on myosin molecules in the presence of ATP molecules to be hydrolyzed. For this objective, we prepared tropomyosin molecules modified to be a little bit stiffer compared to the intact ones by applying a fixed cross-linker between a pair of twisted tropomyosin monomers. The cross-linked tropomyosin molecules, when complexed with actin filaments, were found to inhibit the sliding movement of the filaments on myosin molecules even in the absence of calcium-regulated troponin molecules. It is then suggested that the mechanical flexibility of tropomyosin molecules may be instrumental to actualizing the proper functional regulation of the sliding movement of actin filaments.     

Direct Measurements of Local Coupling between Myosin Molecules Are Consistent with a Model of Muscle Activation

Muscle contracts due to ATP-dependent interactions of myosin motors with thin filaments composed of the proteins actin, troponin, and tropomyosin. Contraction is initiated when calcium binds to troponin, which changes conformation and displaces tropomyosin, a filamentous protein that wraps around the actin filament, thereby exposing myosin binding sites on actin. Myosin motors interact with each other indirectly via tropomyosin, since myosin binding to actin locally displaces tropomyosin and thereby facilitates binding of nearby myosin. Defining and modeling this local coupling between myosin motors is an open problem in muscle modeling and, more broadly, a requirement to understanding the connection between muscle contraction at the molecular and macro scale. It is challenging to directly observe this coupling, and such measurements have only recently been made. Analysis of these data suggests that two myosin heads are required to activate the thin filament. This result contrasts with a theoretical model, which reproduces several indirect measurements of coupling between myosin, that assumes a single myosin head can activate the thin filament. To understand this apparent discrepancy, we incorporated the model into stochastic simulations of the experiments, which generated simulated data that were then analyzed identically to the experimental measurements. By varying a single parameter, good agreement between simulation and experiment was established. The conclusion that two myosin molecules are required to activate the thin filament arises from an assumption, made during data analysis, that the intensity of the fluorescent tags attached to myosin varies depending on experimental condition. We provide an alternative explanation that reconciles theory and experiment without assuming that the intensity of the fluorescent tags varies.     

Caldesmon reduces the apparent rate of binding of myosin S1 to actin-tropomyosin

Equilibrium measurements of the rate of binding of caldesmon and myosin S1 to actin-tropomyosin from different laboratories have yielded different results and have led to different models of caldesmon function. An alternate approach to answering these questions is to study the kinetics of binding of both caldesmon and S1 to actin. We observed that caldesmon decreased the rate of binding of S1 to actin in a concentration-dependent manner. The inhibition of the rate of S1 binding was enhanced by tropomyosin, but the effect of tropomyosin on the binding was small. Premixing actin with S1 reduced the amplitude (extent) of caldesmon binding in proportion to the fraction of actin that contained bound S1, but the rate of binding of caldesmon to free sites was not greatly altered. No evidence for a stable caldesmon-actin-tropomyosin-S1 complex was observed, although S1 did apparently bind to gaps between caldesmon molecules. These results indicate that experiments involving caldesmon, actin, tropomyosin, and myosin are inherently complex. When the concentration of either S1 or caldesmon is varied, the amount of the other component bound to actin-tropomyosin cannot be assumed to remain fixed. The results are not readily explained by a mechanism in which caldesmon acts only by stabilizing an inactive state of actin-tropomyosin. The results support regulatory mechanisms that involve changes in the actin-S1 interaction.     

Movement of smooth muscle tropomyosin by myosin heads.

It has been proposed that during the activation of muscle contraction the initial binding of myosin heads to the actin thin filament contributes to switching on the thin filament and that this might involve the movement of actin-bound tropomyosin. The movement of smooth muscle tropomyosin on actin was investigated in this work by measuring the change in distance between specific residues on tropomyosin and actin by fluorescence resonance energy transfer (FRET) as a function of myosin head binding to actin. An energy transfer acceptor was attached to Cys374 of actin and a donor to the tropomyosin heterodimer at either Cys36 of the beta-chain or Cys190 of the alpha-chain. FRET changed for the donor at both positions of tropomyosin upon addition of skeletal or smooth muscle myosin heads, indicating a movement of the whole tropomyosin molecule. The changes in FRET were hyperbolic and saturated at about one head per seven actin subunits, indicating that each head cooperatively affects several tropomyosin molecules, presumably via tropomyosin's end-to-end interaction. ATP, which dissociates myosin from actin, completely reversed the changes in FRET induced by heads, whereas in the presence of ADP the effect of heads was the same as in its absence. The results indicate that myosin with and without ADP, intermediates in the myosin ATPase hydrolytic pathway, are effective regulators of tropomyosin position, which might play a role in the regulation of smooth muscle contraction.     

Regulation of muscle contraction: bindings of troponin and its components to actin and tropomyosin.

The bindings of troponin components to actin and tropomyosin has been studied by cosedimentation with actin and affinity chromatography. It is shown that troponin binds to actin and tropomyosin in the presence and absence of calcium but the binding to actin is sensitive to ionic strength. Troponin-I + C binds to actin-tropomyosin in the absence of calcium but not to actin or tropomyosin alone. Troponin-I binds to actin and the binding is improved in the presence of tropomyosin even though troponin-I does not bind to tropomyosin alone. Troponin-C does not bind to actin or tropomyosin. The results suggest that the binding of troponin by actin is influenced by tropomyosin. A model of regulation by troponin is proposed.     

Rate and mechanism of the assembly of tropomyosin with actin filaments

The rate of assembly of tropomyosin with actin filaments was measured by stopped-flow experiments. Binding of tropomyosin to actin filaments was followed by the change of the fluorescence intensity of a (dimethylamino)naphthalene label covalently linked to tropomyosin and by synchrotron radiation X-ray solution scattering. Under the experimental conditions (2 mM MgCl2, 100 mM KCl, pH 7.5, 25 degrees C) and at the protein concentrations used (2.5-24 microM actin, 0.2-3.4 microM tropomyosin) the half-life time of assembly of tropomyosin with actin filaments was found to be less than 1 s. The results were analyzed quantitatively by a model in which tropomyosin initially binds to isolated sites. Further tropomyosin molecules bind contiguously to bound tropomyosin along the actin filaments. Good agreement between the experimental and theoretical time course of assembly was obtained by assuming a fast preequilibrium between free and isolatedly bound tropomyosin.     

Tropomodulin binding to tropomyosins. Isoform-specific differences in affinity and stoichiometry.

Tropomodulin is a human erythrocyte membrane cytoskeletal protein that binds to one end of tropomyosin molecules and inhibits tropomyosin binding to actin filaments [Fowler, V. M. (1990) J. Cell Biol. 111, 471-482]. We have characterized the interaction of erythroid and non-erythroid tropomyosins with tropomodulin by non-denaturing gel electrophoresis and by solid-phase binding assays using 125I-tropomyosin. Non-denaturing gel analysis demonstrates that all tropomodulin molecules are able to bind tropomyosin and that tropomodulin forms complexes with tropomyosin isoforms from erythrocyte, brain, platelet and skeletal muscle tissue. Scatchard analysis of binding data using tropomyosin isoforms from these tissues indicate that tropomodulin binds preferentially to erythrocyte tropomyosin. Specificity is manifested by decreases in the apparent affinity or the saturation binding capacity of tropomodulin for non-erythrocyte tropomyosins. Erythrocyte tropomyosin saturates tropomodulin at approximate stoichiometric ratios of 1:2 and 1:4 tropomyosin/tropomodulin (apparent Kd = 14 nM-1 and 5 nM-1, respectively). Brain tropomyosin saturates tropomodulin at a 1:2 ratio of tropomyosin/tropomodulin, but with a threefold lower affinity than erythrocyte tropomyosin. Platelet tropomyosin saturates tropomodulin at a tropomyosin/tropomodulin ratio of 1:4, but with a sevenfold lower affinity than erythrocyte tropomyosin at the 1:4 ratio. These results correlate with oxidative cross-linking data which indicate that tropomodulin can self-associate to form dimers and tetramers in solution. Since tropomodulin interacts with one of the ends of tropomyosin, varying interactions of tropomyosin isoforms with tropomodulin probably reflect the heterogeneity in N-terminal or C-terminal sequences characteristic of the different tropomyosin isoforms. Isoform-specific interactions of tropomodulin with tropomyosins may represent a novel mechanism for selective regulation of tropomyosin/actin interactions.     

Crosslinking of actin filaments and inhibition of actomyosin subfragment-1 ATPase activity by streptococcal M6 protein

M proteins are antiphagocytic molecules on the surface of group A streptococci having physical characteristics similar to those of mammalian tropomyosin. Both are alpha-helical coiled-coil fibrous structures with a similar seven-residue periodicity of nonpolar and charged amino acids. To determine if M protein is functionally similar to tropomyosin we studied the interaction of M protein with F-actin. At low ionic strength, M protein binds to actin weakly with a stoichiometry different from that of tropomyosin. M protein does not compete with tropomyosin for the binding to actin, indicating that it is functionally different from tropomyosin. M protein does compete with myosin subfragment-1 for binding to actin and induces the formation of bundles of actin filaments. The formation of actin aggregates is associated with a sharp reduction in the rate of ATP hydrolysis by subfragment-1. Intact streptococci having M protein on their surface are shown to bind to actin.     

Effects of the amino-terminal regions of tropomyosin and troponin T on thin filament assembly.

Bacterially expressed alpha-tropomyosin lacks the amino-terminal acetylation present in muscle tropomyosin and binds poorly to actin (Hitchcock-DeGregori, S. E., and Heald, R. W. (1987) J. Biol. Chem. 262, 9730-9735). Using a linear lattice model, we determined the affinity (Ko) of unacetylated tropomyosin or troponin-unacetylated tropomyosin for an isolated site on the actin filament and the fold increase in affinity (y) when binding is to an adjacent site. The absence of tropomyosin acetylation decreased Ko 2 orders of magnitude in the absence of troponin. Tropomyosin acetylation also enhanced troponin-tropomyosin binding to actin, not by increasing cooperativity (y), but rather by increasing Ko. These results suggest that the amino-terminal region of tropomyosin is a crucial actin binding site. Troponin promoted unacetylated tropomyosin binding to actin, increasing Ko more than 1,000-fold. Troponin70-259, which lacks the troponin T peptide (1-69) spanning the overlap between adjacent tropomyosins, behaved similarly to intact troponin. Cooperative interactions between adjacent troponin-tropomyosin complexes remained strong despite the use of a nonpolymerizable tropomyosin and a troponin unable to bridge neighboring tropomyosins physically. The Ko for troponin70-259-unacetylated tropomyosin was 500-fold greater than for troponin159-259-unacetylated tropomyosin, indicating that troponin T residues 70-158 are critical for anchoring troponin-tropomyosin to F-actin. The mechanism of cooperative thin filament assembly is discussed.     

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.     

Interaction between Acanthamoeba actin and rabbit skeletal muscle tropomyosin.

The binding of 125I-labeled muscle tropomyosin to Acanthamoeba and muscle actin was studied by ultracentrifugation and by the effect of tropomyosin on the actin-activated muscle heavy meromyosin ATPase activity. Binding of muscle tropomyosin to Acanthamoeba actin was much weaker than its binding to muscle actin. For example, at 5 mM MgCl2, 2 mM ATP, and 5 micronM actin, tropomyosin bound strongly to muscle actin but not detectably to Acanthamoeba actin. When the concentration of actin was raised from 5 micronM to 24 micronM in the presence of 80 mM KCl, the binding of tropomyosin to Acanthamoeba actin approached its binding to muscle actin. As with muscle actin, the addition of muscle heavy meromyosin in the absence of ATP induced binding of tropomyosin in Acanthamoeba actin under conditions were binding would otherwise not have occurred. The most striking difference between the interactions of muscle tropomyosin with the two actins, however, was that under conditions where tropomyosin was found to both actins, its stimulated the Acanthamoeba actin-activated heavy meromyosin ATPase but inhibited the muscle actin-activated heavy meromyosin ATPase.     

Tropomodulin: a cytoskeletal protein that binds to the end of erythrocyte tropomyosin and inhibits tropomyosin binding to actin

Human erythrocytes contain a Mr 43,000 tropomyosin-binding protein that is unrelated to actin and that has been proposed to play a role in modulating the association of tropomyosin with spectrin-actin complexes based on its stoichiometry in the membrane skeleton of one Mr 43,000 monomer per short actin filament (Fowler, V. M. 1987. J. Biol. Chem. 262:12792-12800). Here, we describe an improved procedure to purify milligram quantities to 98% homogeneity and we show that this protein inhibits tropomyosin binding to actin by a novel mechanism. We have named this protein tropomodulin. Unlike other proteins that inhibit tropomyosin-actin interactions, tropomodulin itself does not bind to F-actin. EM of rotary-shadowed tropomodulin-tropomyosin complexes reveal that tropomodulin (14.5 +/- 2.4 nm [SD] in diameter) binds to one of the ends of the rod-like tropomyosin molecules (33 nm long). In agreement with this observation, Dixon plots of inhibition curves demonstrate that tropomodulin is a non-competitive inhibitor of tropomyosin binding to F-actin (Ki = 0.7 microM). Hill plots of the binding of the tropomodulin-tropomyosin complex to actin indicate that binding does not exhibit any positive cooperativity (n = 0.9), in contrast to tropomyosin (n = 1.9), and that the apparent affinity of the complex for actin is reduced 20-fold with respect to that of tropomyosin. These results suggest that binding of tropomodulin to tropomyosin may block the ability of tropomyosin to self-associate in a head-to-tail fashion along the actin filament, thereby weakening its binding to actin. Antibodies to tropomodulin cross-react strongly with striated muscle troponin I (but not with troponin T) as well as with a nontroponin Mr 43,000 polypeptide in muscle and in other nonerythroid cells and tissues, including brain, lens, neutrophils, and endothelial cells. Thus, erythrocyte tropomodulin may be one member of a family of tropomyosin-binding proteins that function to regulate tropomyosin-actin interactions in non-muscle cells and tissues.     

Structural role of tropomyosin in muscle regulation: analysis of the x-ray diffraction patterns from relaxed and contracting muscles

Previous work has shown that there are significant differences in the X-ray diffraction patterns obtained from relaxed and contracting muscles. We show that some of these changes can be explained in terms of a small movement (~ 5 to 15 Å) of the tropomyosin molecules in the groove of the actin helix. The position of the tropomyosin in relaxed skeletal muscle is such that it might physically block or at least structurally alter the cross-bridge attachment site on actin, whereas in contracting skeletal muscle the tropomyosin moves to a position well clear of the attachment site. The movement of the tropomyosin molecules is apparently smaller in molluscan muscles during tonic contraction than in vertebrate skeletal muscle. We suggest a possible relationship between the smaller movement of the tropomyosin and the “catch” response of molluscan muscles.We also show that any increase of intensity on the 59 Å and 51 Å layer-lines is most likely to be associated with some extra mass (HMM S-1) attaching to the actin molecules. Such a change cannot be explained in terms of a change in tropomyosin structure or in the order within the thin filaments. Since changes on these two layer-lines have been observed during contraction, this provides good evidence for cross-bridge attachment to actin in contracting muscles.     

Structure of the rigor actin-tropomyosin-myosin complex

Regulation of myosin and filamentous actin interaction by tropomyosin is a central feature of contractile events in muscle and nonmuscle cells. However, little is known about molecular interactions within the complex and the trajectory of tropomyosin movement between its "open" and "closed" positions on the actin filament. Here, we report the 8 ? resolution structure of the rigor (nucleotide-free) actin-tropomyosin-myosin complex determined by cryo-electron microscopy. The pseudoatomic model of the complex, obtained from fitting crystal structures into the map, defines the large interface involving two adjacent actin monomers and one tropomyosin pseudorepeat per myosin contact. Severe forms of hereditary myopathies are linked to mutations that critically perturb this interface. Myosin binding results in a 23 ? shift of tropomyosin along actin. Complex domain motions occur in myosin, but not in actin. Based on our results, we propose a structural model for the tropomyosin-dependent modulation of myosin binding to actin.     

Co-operative regulation mechanism of muscle contraction: Inter-tropomyosin co-operation model

A new model is presented on the basis of our experimental data and the “tropomyosin-blocking theory” of muscle relaxation to explain the regulation of certain characteristics of muscle contraction, namely that the relation of contraction to pCa is co-operative while calcium-binding is essentially non-cooperative. Our experiments show that end-to-end interactions between adjacent tropomyosin molecules in the groove of the actin helix are essential for the co-operative regulation. The blocking theory says that the tropomyosin molecule in relaxed muscle sterically blocks the myosin attachment site on actin, whereas in contracting muscle it moves to a position away from the attachment site. In this model a concerted movement of tropomyosin molecules, brought about by their end-to-end interactions, is considered to be the essential mechanism of co-operative regulation, and it is assumed that the positional changes of tropomyosin occur primarily when the four calcium binding sites of troponin on the tropomyosin are saturated with calcium. Theoretical analysis of the model, based upon the two-state allosteric model, leads to a Michaelis-Menten equation for the Ca-binding function together with a co-operative equation for the state function, proportional to the contraction or ATPase activity. These two functions fit well the experimental data. With cardiac muscle the slope of the contraction versus pCa curve is slightly less steep than that obtained with skeletal muscle. This difference can be explained by the difference in the number of Ca-binding sites of troponins.