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*Laboratory of Biological Modeling, National Institute of Diabetes and Digestive and Kidney Diseases, National Institutes of Health, Bethesda, Maryland 20892-5621; and Department of Biochemistry and Molecular Biology, Brody School of Medicine at East Carolina University, Greenville, North Carolina 27858-4354
Correspondence: Address reprint requests to Joseph M. Chalovich, Dept. of Biochemistry and Molecular Biology, Brody School of Medicine at East Carolina University, 600 Moye Blvd., Greenville, North Carolina, 27834. Tel.: 252-744-2973; Fax: 252-744-3383; E-mail: chalovichj{at}mail.ecu.edu.
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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). A further eightfold increase in ATPase rate occurs with the addition of high concentrations of NEM-S1 (Williams et al., 1988), a modified myosin subfragment that binds tightly to actin even in the presence of ATP. In solution, the binding of NEM-S1 or other "activating" forms of myosin can elevate the ATPase activity to about the same level regardless of the presence of Ca2+ (Williams et al., 1988). Furthermore, the affinity of S1-ADP and rigor S1 for actin is enhanced by both Ca2+ and binding of S1 to actin (Greene and Eisenberg, 1980). The rate of S1 and S1-ADP binding are also increased by both Ca2+ and the occupancy of actin with bound S1 (Trybus and Taylor, 1980; Geeves and Halsall, 1987). In contrast, the affinity of S1 to actin during the hydrolysis of ATP is not greatly affected by Ca2+ or by S1 binding (Chalovich et al., 1981; Chalovich, 1992). These observations have been interpreted in terms of different models of regulation. Identifying reasonable models of regulation is important because of the role of defects in muscle regulatory proteins in cardiac failure (Thierfelder et al., 1994) and other disorders.
Tropomyosin-troponin undergoes a rapid structural change upon activation (Kress et al., 1986) but the number of discrete functional states and the role of each of these states is uncertain. This change was postulated to be a movement of tropomyosin into the actin groove (Parry and Squire, 1973; Haselgrove and Huxley, 1973). Recent observations suggest that there may be three positions of tropomyosin on actin (Vibert et al., 1997; Poole et al., 1995) although it is uncertain if there are three discrete functional states (Perry, 2003). The possibility has been raised that troponin contributes to the observed movement (Hai et al., 2002; Chalovich, 2002). In the relaxed state tropomyosin overlaps the putative site of S1-ADP binding but not the site of S1-ATP binding (Craig and Lehman, 2001). Thus there is good agreement between structural and biochemical data.
We favor the view that regulation of striated muscle contraction involves transitions among two or more parallel pathways of ATP hydrolysis. Ca2+ and ATP-free myosin binding are activators that stabilize states of the actin filament that have enhanced activities toward actin-activated ATP hydrolysis (Chalovich et al., 1981; Hill et al., 1981; Chalovich and Eisenberg, 1982). The parallel pathway model is based upon the idea that the cooperative activation of ATPase activity (Eisenberg and Weihing, 1970; Murray et al., 1982) results from multiple states of actin with distributions that are determined by Ca2+ and S1-ADP or rigor S1 binding (Greene and Eisenberg, 1980, 1988). Fig. 1 A shows this model having parallel pathways of ATP hydrolysis for free S1, and for S1 bound to actin in state 1 (inactive) and state 2 (active). A model based on two functional actin states of actin was sufficient to simulate equilibrium binding of S1 to actin (Hill et al., 1980) and actin-activated ATPase activity (Hill et al., 1981).
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). We recently used Monte Carlo methods to analyze binding kinetics by the parallel pathway Hill model (Chen et al., 2001). The two-state Hill model predicted the same binding kinetics as a model that included the blocked state. However, that former analysis was too restricted to adequately test either model's ability to simulate real data under a variety of experimental conditions. A rigorous test of models of regulation requires an extensive data set that includes equilibrium binding, binding kinetics, and ATPase activities under the same conditions.
We have now compiled a data set for the binding of S1-ADP to regulated actin for testing models of regulation. Equilibrium binding was measured at six free Ca2+ concentrations at each of six ionic strength conditions. Binding kinetics were measured for many of these same conditions. Both the Hill model and a model containing a blocked state were fitted to the equilibrium binding curves to obtain the parameters describing equilibrium binding as a function of ionic strength and free Ca2+. Inclusion of a blocked state was unnecessary for simulating equilibrium binding and binding kinetics. Thus two states are sufficient for explaining regulatory function. The model is also consistent with a third functional state should that be required. The results are consistent with the idea that tropomyosin-troponin is an allosteric switch (Hill et al., 1981; Resetar et al., 2002; Heeley et al., 2002).
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MATERIALS AND METHODS |
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; Eisenberg and Kielley, 1972), myosin (Kielley and Harrington, 1960), and troponin and tropomyosin (Eisenberg and Kielley, 1974) were isolated from rabbit back muscle. Myosin-S1 was made by digestion of myosin with chymotrypsin (Weeds and Taylor, 1975). Protein concentrations were determined by light absorbance at 280 nm, corrected for scattering at 340 nm, using the following extinction coefficients (
0.1%) for 280 nm: actin, 1.15; myosin-S1, 0.75; tropomyosin, 0.33; troponin, 0.37.
Actin was labeled with N-(1-pyrenyl)iodoacetamide (Kouyama and Mihashi, 1981) and stored as a 40-µM stock in 4 mM imidazole (pH 7.0), 1 mM dithiothreitol, 2 mM MgCl2, and 40 µM phalloidin. In each experiment actin was mixed with troponin and tropomyosin in a 1:1:1 molar ratio to insure saturation of actin at the low concentrations used for binding studies.
Equilibrium binding measurements
The equilibrium binding of myosin-S1 to actin-tropomyosin-troponin was determined by the decrease in phalloidin-stabilized pyrene-labeled actin fluorescence that occurs with S1 binding (Dancker et al., 1975; Kouyama and Mihashi, 1981; Criddle et al., 1985). The procedure used was essentially that described by others (Tobacman and Butters, 2000). Fluorescence measurements were made in a Spex Fluorolog fluorescence spectrophotometer (Spex Industries, Edison, NJ) with a cell maintained at 25°C with a circulating water bath. Pyrene fluorescence was measured at 384 nm with excitation at 364 nm. To reduce scattered light the 0.5-mm slits were used during the measurements. Each data point was the average of a 5-s data collection interval.
Binding experiments were done in a solution containing 20 mM MOPS pH 7.0, 110 mM KCl, 5 mM MgCl2, 1 mM dithiothreitol, 2 mM ADP, 0.2 mg/ml bovine serum albumin, sufficient KCl to reach the target ionic strength, and Ca-EGTA to reach the desired pCa. The actin concentration in binding experiments was either 75 nM (at 120 mM ionic strength) or 150 nM. The solution also contained 14 units/ml hexokinase and 1 mM glucose to scavenge ATP and 20 µM Ap5A to inhibit ATP formation through the myokinase reaction. Ca-EGTA buffers were used to fix the free Ca2+ concentration (Fabiato and Fabiato, 1979). Ca2+-EGTA was mixed with EGTA in ratios as follows (in mM) 0:2, 0.8:1.2, 1:1, 1.4:0.6, 1.8:0.2, 2:0, and the highest free Ca2+ concentration was obtained by using 0.1 mM CaCl2 without EGTA. The free Ca2+ content in the solution was verified by a Corning Ca2+-sensitive electrode (Corning, NY) with a Fisher Scientific AccuMet pH meter (Hampton, NH), which gave 
0, 0.6, 3, 10.3, 16.75, 20, and 70 µM free Ca2+, respectively.
Titrations were done by first measuring the fluorescence of a 2-ml volume of pyrene-labeled actin-tropomyosin-troponin. A small volume of a concentrated S1 stock was added to the reaction mixture in a stirred cuvette; the sample was allowed to equilibrate for at least 5 min before measuring the fluorescence. It was important to make each measurement only after the signal had stabilized so that the constants determined were equilibrium values. The sample was exposed to incident light only during the period of observation. The fluorescence intensity and the concentration of all proteins were corrected for the volume change caused by adding S1. The total volume change for an entire binding isotherm was <10%.
Values of 
(S1 bound to actin total ratio) and free S1 concentration were calculated using the equations:
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The Hill model of cooperative equilibrium binding was fitted to the binding curves (Hill et al., 1980). A long actin filament saturated with troponin and tropomyosin can be in either state 1 with low affinity for S1-ADP given by K1 or in state 2 with a higher affinity given by K2. Each unit of seven actin sites and one tropomyosin-troponin changes state as a group. This together with interactions between tandem groups gives rise to cooperativity. The saturation of actin with S1 () as a function of free S1 concentration can be described as follows (Hill et al., 1980):
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i, fraction of actin units in the i state occupied by S1; K1,2,constants for S1-actin binding; C, free S1 concentration; n, number of actin monomers in one actin-tropomyosin-troponin unit (assumed to be 7); L, is the equilibrium constant for transition of an isolated actin-tropomyosin-troponin unit with no neighbors, no bound Ca2+, and no bound S1 from state 2 to state 1; Y, the overall cooperativity parameter; Yij, individual cooperativities between units in states i and j (Yij = Yji was assumed for symmetrical case), xij, yij, and zij represent the free energy of nearest-neighbor tropomyosin interactions (Wij) in exponential form e–Wij/kT (Hill et al., 1980), ka,b, affinity of Ca2+ for troponin in states 1 and 2 with values of 105 and 106 M–1, respectively (Zot and Potter, 1987), and 
is the free Ca2+ concentration. We assumed that the values of ka,b did not change over the ionic strength range in this study. The simulated curves were not very sensitive to the value of K1, so simulations were normally carried out with the assumption that K1 = K2/8 (Greene and Eisenberg, 1988). However, values of K1 less than this gave similar results.
We used the MLAB modeling system (Civilized Software, Bethesda, MD) to produce a global fit of the Hill model to all data collected at a single ionic strength condition but at varied Ca2+ concentrations. Because the pyrene fluorescence change monitors only the second step of the two-step binding of S1-ADP to actin the term (2*p2) of the Hill model was fitted to the fluorescence data. The constants xij, yij, zij, L, and K2 were allowed to float to reach a best global fit. Y() and L'() were then calculated from the results of the fit using formulas for Y() and L'() from the set of equations (Eq. 1). Parameters of the fit, i.e., L, L'() = LY11()/LY22(), Y(), and K2 (where L' is equilibrium constant for transition between states for the entire actin filament with Ca2+, but without S1) at different ionic strengths and free Ca2+ concentrations were used for further mathematical simulations.
In doing a global fit it is possible to limit the predictions of the model by placing constraints on various parameters. It is common to limit the value of the cooperativity parameter Y to values >1 thus giving positive cooperativity. In this special case we constrained Y to be >1 by constraining xij, yij, zij, to values >0 and by constraining xii > xij, yii > yij, zii > zij. We also examined the more general case where nearest-neighbor interactions could be both favorable and unfavorable by allowing Y to be any value >0. The two fitting procedures gave identical results at physiological ionic strength or lower although a slightly better fit was obtained at higher ionic strength with negative cooperativity.
The McKillop and Geeves model (McKillop and Geeves, 1993) was also fitted to our experimental equilibrium binding curves. In this model each unit of seven actins and one tropomyosin-troponin can exist in three states. In the blocked state myosin-S1 cannot bind to actin. Binding can occur in the closed state with an equilibrium constant 
but the isomerization resulting in force production cannot occur. Note that a superscript G is used to distinguish the Geeves parameter from the Hill parameter. Binding of S1 to actin in the open state occurs in two steps with equilibrium constants and 
The transition described by 
is thought to be associated with the force producing conformational change. The transitions from blocked to closed and closed to open occur with equilibrium constants Kb (or [closed]/[blocked]) and Kt (or [open]/[closed]). Binding in this case is described by the equations as follows (McKillop and Geeves, 1993):
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n is the size of the cooperative unit, and c is the free [S1].
This equation was fitted to our experimental data to evaluate the constants 
Kb,and Kt.
Fitting with the McKillop and Geeves model was more complex because many combinations of parameter values could describe the equilibrium and kinetic data. Curve fitting was possible only after the values of some parameters were restricted as described earlier (McKillop and Geeves, 1991). We restricted Kb to be <10 and set 
so this was excluded from the fit. Our fitting procedure was different from that described earlier in that we varied several parameters simultaneously. We used the kinetics of binding to determine the value of Kb + Kb x Kt in the absence of Ca2+. Kb and Kt at other Ca2+ concentrations were allowed to float.
Binding kinetics
Rates of binding of S1 to pyrene-labeled actin were measured under the conditions used for equilibrium binding using a DX17.MV/2 sequential stopped flow spectrofluorometer (Applied Photophysics, Leatherhead, UK). The excitation wavelength was set at 364 with a monochrometer with 0.5-mm slits. Emission was measured through a 384-nm long-pass filter. All measurements were made at 25°C. Experiments were performed with either actin or S1 in excess of actin and S1, i.e., 0.5 µM actin and 2.5 µM S1, or 2.5 µM actin and 0.5 µM S1. Unless otherwise stated, the values of concentrations given are those at the instant after mixing in the stopped flow. For further analysis and comparison with computer simulations the maximum fluorescence changes were normalized to unity.
Theoretical binding curves were calculated from the equilibrium parameters using the procedure described earlier (Chen et al., 2001). K1 and K2 (see above) are the equilibrium binding constants of S1-ADP to the inactive and active states of actin, respectively. The corresponding rate constants describing the two-step S1 binding and dissociation are k1, 
k2, 
where the apostrophe indicates the reverse rate constant. L is the equilibrium constant for transition of an isolated actin-tropomyosin-troponin from the active state to the inactive state. The rates of transition of the actin filament are 
m and ßm, where m is the number of S1 bound to the actin-tropomyosin-troponin unit (from 0 to 7). For one unit we have:
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It can be assumed that
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represents how (K1/K2)m is partitioned between and ß.
When cooperativity is present, the values of and ß can be expressed with respect to parameters from equilibrium binding (Chen et al., 2001):
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is a constant between 0 and 1, Y, L', K1, K2 describe the equilibrium binding. 0 can be calculated from L, because L = ß0/0, and the limits for ß0 have been chosen from earlier simulations (Brenner and Chalovich, 1999). ß0 was set near 3 at all ionic strengths and 0 was changed in proportion to L. The other unknown parameters, 
, and were taken from our earlier simulations (Chen et al., 2001) with some modifications. The code for the Monte Carlo simulations was written in C++ by Dr. Bo Yan of the Mathematical Research Branch in the National Institute of Diabetes, Digestive and Kidney Diseases, National Institutes of Health. |
RESULTS |
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Fig. 2 shows , the fraction of actin sites bound to S1, as a function of the free S1 concentration at 120, 180, 225, 260, 300, and 375 mM ionic strength and at several free Ca2+ concentrations. Binding was measured in the presence of ADP to produce an activating state of S1 that would stabilize the active state of regulated actin. The high S1 concentration region of the curves was generally omitted from the figure so that the cooperativity is visible. Examples of full binding curves are shown as insets. At low free Ca2+ concentrations the curves exhibit the characteristic S shape signifying positive cooperativity. At high free Ca2+, the curves appear hyperbolic. Increasing ionic strength weakens the binding so that saturation occurs at progressively higher free S1 concentrations. A more quantitative evaluation of the effects of varying the conditions was obtained by fitting the Hill model to the data.
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Values of L'() and Y() were calculated from Eq. 1 assuming that the affinity of troponin for Ca2+ was independent of ionic strength. The results are shown in Fig. 4. Fig. 4, A and B, show the results that were obtained if both positive and negative cooperativity were allowed. Fig. 4, C and D, are the corresponding results when Y was restricted to values giving positive cooperativity (Y > 1). The cooperativity parameter, Y, increased with increasing Ca2+ concentrations at ionic strengths from 120 to 260 mM. At higher ionic strength the cooperativity decreased with increasing Ca2+ to give negative cooperativity (Fig. 4 A) unless constrained (Fig. 4 C). At the intermediate ionic strength values used, the value of Y was approximately independent of Ca2+. When Y was unconstrained the value of L' decreased with increasing Ca2+ concentrations and increased with increasing ionic strength (Fig. 4 B). That is, whereas Ca2+ stabilized the active state (state 2), increases in ionic strength at any Ca2+ concentration tended to stabilize the inactive state (state 1). When the value of Y was limited to values >1 the values of L' at the highest ionic strengths were decreased at all values of Ca2+. This resulted in a more narrow distribution of values of L' as shown in Fig. 4 D.
Little information is available about the salt dependence of the binding of Ca2+ to actin-tropomyosin-troponin in both the inactive and active states. We assumed that the affinity for Ca2+ did not depend on the ionic strength. In the case of pure troponin C the affinity at 300 mM ionic strength is 0.5 that at 120 mM (Ogawa, 1985; Iida, 1988). We repeated the fitting of the 300-mM ionic strength binding curve assuming that the affinity decreased by an even larger fraction to 0.2 of the value at 120 mM ionic strength. The values of the binding parameters at 300 mM ionic strength changed slightly when the Ca2+ affinity was assumed to be lower. In the absence of Ca2+ the values of Y and L' were 10% smaller than with the higher Ca2+ affinity. At saturating Ca2+ the values of Y and L' were 10% higher than with the higher Ca2+ affinity. That is, with the assumption of a decreased Ca2+ affinity at high ionic strength the relationship of Y and L' was slightly flattened out. Because the data of Fig. 4 B are on a log scale the change resulting from this assumption was small.
Binding kinetics
We recently demonstrated the feasibility of simulating the kinetics of S1 binding to regulated actin based on the parameters obtained from equilibrium binding using the Hill model (Chen et al., 2001). The kinetics of binding are ionic strength dependent with a nonlinear relationship between the duration of the lag in binding and the ionic strength (Resetar et al., 2002). Simulation of binding kinetics from equilibrium binding parameters (L, L', Y, K2) at several different ionic strength conditions provides a robust test of the model.
Several examples of the observed kinetic traces of S1-ADP binding to regulated actin and the theoretical curves are shown in Fig. 5. Fig. 5, A and B, show changes in pyrene actin fluorescence as a function of time with S1 in excess over actin. Binding in the presence of Ca2+ (Fig. 5 B) occurred with 10 times the rate as that in the absence of Ca2+ (Fig. 5 A). Binding in the absence of Ca2+ is initially slow but the rate increases with time. This gives rise to a lag in the binding in the absence of Ca2+ that is often explained by a large fraction of regulated actin that is blocked by tropomyosin so that S1-ADP cannot bind. In this blocked state model S1-ADP binding to available sites results in a cooperative decrease in the population of blocked regulated actin so that the rate of binding increases. The theoretical curves calculated from our equilibrium binding parameters show the same trends that were found experimentally without the inclusion of a blocked state.
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The kinetics of binding could be simulated using equilibrium binding values from an unconstrained fit as well as with the constraint of positive cooperativity (Y > 1). In the unconstrained case negative cooperativity occurred in EGTA when the ionic strength was at 225 mM or greater and in Ca2+ when the ionic strength was 260 mM or greater. Examples of a simulation at high ionic strength (300 mM) with the constraint of positive cooperativity are shown in Fig. 5, A and B (dashed-dotted line near curve d). The fits are reasonable in this case but the value of ß, the rate constant from the inactive state to the active state had to be reduced three times to obtain a good fit. It was impossible to decide between the two cases (Y > 0 and Y > 1) on the basis of fitting either the equilibrium binding or kinetic data.
Table 1 summarizes the values of the parameters used to simulate both equilibrium binding and binding kinetics from 120 to 375 mM ionic strength at both saturating Ca2+ and in the virtual absence of Ca2+.
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We showed previously that the duration of the lag in binding increased with increasing ionic strength even beyond physiological values of ionic strength (Resetar et al., 2002). However, we did not know the reason for this increased lag. Fig. 6 compares the lag duration from the experimental time courses with those obtained from simulations in Fig. 5. As in our earlier data the lag duration increased with ionic strength. It is also apparent from Fig. 6 that the simulations accurately predicted the lag duration. That means that the increased lag may be attributed largely to the increased stability of the inactive state of regulated actin, state 1.
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. Our analysis produced a steeper increase in both Kb and Kt with increasing free calcium than shown in their Fig. 7 B. Furthermore the value of n decreased with increasing free Ca2+ in our analysis whereas n increased in their Fig. 7 B. The parameters for the McKillop and Geeves model are given in Table 2.
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DISCUSSION |
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) or by changes in the expression of troponin isoforms (Anderson et al., 1991). Seemingly small changes in the primary structure of troponin I or troponin T result in disorders including myopathies (Braun et al., 1996; Nowak et al., 1999) and cardiomyopathies (Thierfelder et al., 1994). Our goal is to provide a framework for studying these adaptive and pathologic changes. We propose that changes in the binding of tropomyosin-troponin to actin result in the formation of multiple parallel pathways with different rates of ATP hydrolysis. Ca2+ and myosin act as effectors that stabilize the active state of regulated actin (Fig. 1).
This view was based on observations that the binding of myosin to regulated actin during steady-state hydrolysis of ATP is not greatly affected by Ca2+ (Chalovich et al., 1981; Brenner et al., 1982; Resetar et al., 2002), that the increase in ATPase rate in response to Ca2+ results from an increase in the kcat (see Chalovich, 1992), and that the mode of interaction between myosin and regulated actin can be classified into two states depending on the nucleotide bound to myosin (Chalovich, 1992; Brenner and Chalovich, 1999). Thus S1-ADP (but not S1-ATP) binding to regulated actin is cooperative with increasing S1 concentration and the binding is also somewhat enhanced by Ca2+. The parallel pathway model could simulate the equilibrium binding of S1 and S1-ADP to regulated actin (Hill et al., 1980) and the general features of the steady-state ATPase activity (Hill et al., 1981). However the inclusion of an additional state was thought to be required to explain binding kinetics (Trybus and Taylor, 1980; McKillop and Geeves, 1993). The effect of Ca2+ on binding kinetics was used to define the fraction of regulated actin in that additional state, the blocked state (Head et al., 1995). A subsequent analysis showed that the two-state Hill model gave the same predictions of binding kinetics as a model including a blocked state (Chen et al., 2001).
We now show that the Hill model could simulate equilibrium binding of S1-ADP to regulated actin and the kinetics of binding at several ionic strengths and over a range of free Ca2+ concentrations. All equilibrium binding data at a single ionic strength were analyzed globally so that all curves within a series could be predicted just by varying the free Ca2+ concentration. The resulting binding parameters could be used to simulate the rate of binding of S1-ADP to actin-tropomyosin-troponin both with actin in excess over S1 and with S1 in excess over actin. In the latter case the binding rate increased with time as the regulated actin filament became increasingly more active as S1-ADP stabilized the active state. The two-state version of the Hill model can describe binding kinetics in addition to the equilibrium binding and ATPase activities that were analyzed earlier (Hill et al., 1980, 1981).
We also evaluated our 225-mM ionic strength data using the McKillop and Geeves model. Our fitted values differ from published data obtained under similar conditions (Maytum et al., 1999) in several respects as shown in Fig. 7. In both cases Kb and Kt increase with increasing free Ca2+ concentration but our analysis of our data gave lower values of Kt in the absence of Ca2+ and our curve increased more steeply with increasing Ca2+ than does their curve (Maytum et al., 1999). We found that K1 is higher than they reported and our value increased slightly with Ca2+ whereas theirs decreased slightly. Similarly, we find that the value of n, the size of the cooperative unit, decreased with Ca2+ whereas they found higher values of n that increased with increasing Ca2+ (Geeves and Lehrer, 1994). These effects should be considered when rationalizing the physical significance of n.
The reasons for the different values in Fig. 7 are both experimental and analytical. Our equilibrium titrations were done over a much longer time course to ensure that each measurement reported an equilibrium value. In fitting the model to data we reduced the number of fixed parameters and simultaneously minimized the errors of all remaining parameters.
Previous examinations of the blocked state model were done by letting K1 and Kt float with fixed values of Kb and K2 (Maytum et al., 1999). The value of n was assumed to be an integer between 5 and 15. Values of n giving the minimum least-square difference were used in subsequent modeling. That value was different at low and high free Ca2+. The value of Kb was assumed to be >10 in the presence of Ca2+ and was excluded from the fit. In the absence of Ca2+ the value of Kb was estimated from the ratio of binding rates in the absence (k–Ca) and presence (k+Ca) of Ca2+. The assumption was made that the reduction in rate was due to a decrease in available actin concentration as a result of steric blocking. Then Kb was calculated using the formula k–Ca/k+Ca = 1 – 1/(1 + Kb(1 + Kt)) with the assumption that Kt = 0. The fitting algorithm was repeated with this value of Kb to find the best values of K1 and a new value of Kt. The value of n was optimized by exploring integers between 5 and 15. The new value of Kt was usually smaller than that in the presence of Ca2+. The final results are not a universal solution because of the indeterminate uncertainty in Kt.
We attempted to improve this fitting procedure but we could not avoid fixing some parameters. In accord with previous simulations (Maytum et al., 1999) we set K2 = 20 and restricted Kb between 0 and 10. Likewise, we assumed that in the presence of Ca2+ there was no blocked state. In the absence of Ca2+ the value of Kb was determined from the equation k–Ca/k+Ca = 1 – 1/(1 + Kb(1 + Kt)) (Maytum et al., 1999). In this case k–Ca/k+Ca = 1/3, so Kb(1 + Kt) = 1/2. At this point our modeling differed. We did not fix Kb by assuming a value of Kt; rather, we allowed the MLAB fitting routine to obtain the best values of Kt, Kb, n, and K1 taking into account the above-mentioned equation. The constraint that Kb(1 + Kt) = 1/2 was relaxed when the Ca2+ concentration was elevated although Kb(1 + Kt) appeared to be >1/2 at all free Ca2+ concentrations.
The blocked state model has a great deal of flexibility in the parameters and a large sensitivity to the initial assumptions. When fitting multiple parameters simultaneously, as we have done, the fit was not highly sensitive to the value of n. Also, values of Kb and Kt could largely compensate for each other so that it was difficult to determine which parameter was more sensitive to Ca2+ and of greatest regulatory importance. It is possible that Ca2+ may have a larger effect on Kt than previously thought. In that case, like the Hill model, the major regulatory event would be a transition between two bound states rather than a change in binding. It is known experimentally that Ca2+ has a large effect on the kcat of regulated actin-activated ATPase activity (King and Greene, 1985; Wagner, 1984; El-Saleh and Potter, 1985; Chalovich and Eisenberg, 1982) and the large effect on the rate of product release (Heeley et al., 2002). Such large effects require a change in the rate of transition between two bound states.
Earlier studies of the kinetics of binding of S1 to regulated actin reported that the effect of Ca2+ on the rate of binding did not occur at ionic strengths below 50 mM (Head et al., 1995). Because the reduction in the rate of binding at low Ca2+ was assumed to result from a fraction of regulated actin that could not bind to S1 this observation led to the proposal that the blocked state exists only above 50 mM ionic strength. A later study showed that the Ca2+ sensitivity of binding was even more complex than previously thought. The lag in binding with excess S1 and no free Ca2+ increased in a nonlinear manner from 30 mM ionic strength to 300 mM ionic strength (Resetar et al., 2002). We are now able to explain that effect on the basis of an increased stability of state 1 of regulated actin at high ionic strength. Fig. 6 shows that the duration of the lag in binding is well described by the salt-dependent changes in the equilibrium binding parameters.
The stabilization of state 1 with increasing salt concentration was seen as an increase in the value of L with ionic strength. L is the equilibrium constant favoring state 1 for a single regulatory unit in the absence of tropomyosin-tropomyosin interactions and without bound S1 or Ca2+. The value of L' also increased with increasing ionic strength so that the inactive state becomes more stable. L' is equal to L times the ratio of the cooperativity parameters for two tropomyosin units in state 1 compared to state 2 at an arbitrary Ca2+ concentration, . The increase in L contributes to the increase in L' with ionic strength although it is possible that the ratio Y11()/Y22() also increases to give an overall greater salt dependence of L'. A salt effect on the ratio Y11()/Y22() could be due either to stabilization of the interaction between adjacent tropomyosin molecules in state 1 or destabilization of adjacent tropomyosin molecules in state 2. This change is the same at high and low Ca2+ concentrations.
Values of Y tend to decrease as the ionic strength is increased so there is less tendency of adjacent regulatory units to be in the same state. In fact, the best fit of some data sets occurred when the value of Y was between 0 and 1, that is when there was negative cooperativity between adjacent tropomyosin molecules. It was possible to obtain almost as good of a fit if the value of Y was constrained so that negative cooperativity was impossible. In those cases, the value of Y tended to approach 1 (no cooperativity) with increasing ionic strength. The decrease in the value of Y may be related to a decrease in the end-to-end interaction of tropomyosin with increasing ionic strength. Sousa and Farah measured the linear polymerization of tropomyosin as a function of ionic strength (Sousa and Farah, 2002). At 8 µM tropomyosin the fraction polymerized was 90% at 25 mM NaCl, and 20% at 150 mM NaCl. Beyond 200 mM NaCl there was no appreciable polymerization. This lack of coupling between tropomyosin molecules at high ionic strength would be expected to reduce the value of Y as we have observed.
Negative cooperativity could occur if steric repulsion or other repulsive forces were dominant at high ionic strength when the end-to-end interactions among tropomyosin molecules were small. Unfavorable interactions could occur at high ionic strength between adjacent tropomyosin molecules, between tropomyosin and troponin, or between other protein pairs. At this time we have no compelling argument for or against negative cooperativity at high ionic strength. The key points are that positive cooperativity occurs at physiological ionic strength and that the Hill model functions well with both assumptions over a wide range of conditions.
Several of the parameters defining equilibrium binding varried with the free Ca2+ concentration. The value of L' decreased with increasing Ca2+ as expected so that state 2 was more stable in the presence of Ca2+. The Ca2+ dependence of Y is more complex. Below 225 mM ionic strength, the value of Y increased with increasing Ca2+ whereas above 260 mM ionic strength Y decreased with increasing ionic strength. There is some uncertainty in the Ca2+ dependence of the parameters L' and Y because of the assumption that the affinity of troponin C for Ca2+ is ionic strength independent. We have evaluated the binding parameters for the 300-mM ionic strength condition with the assumption that the affinity for Ca2+ decreased to 0.2 times the value at 120 mM ionic strength. This value was chosen because it exceeds the observed change in affinity of pure troponin C for Ca2+ (Ogawa, 1985; Iida, 1988). The effects of introducing a change in Ca2+ affinity on the parameters Y and L' were small and do not alter the conclusions.
We are unaware of models other than the Hill model that have been shown to predict ATPase activities, equilibrium binding and binding kinetics. Smith, Maytum, and Geeves have proposed a new type of regulatory model that is largely based on available structural data (Smith et al., 2003). That model, like the earlier Geeves model, has three states of tropomyosin corresponding to the three observed positions of tropomyosin on actin (Vibert et al., 1997). That model also incorporates the idea of a flexible tropomyosin molecule so that adjacent actin protomers within the influence of a single tropomyosin molecule may exist in states having different activities. The kink size of the flexible tropomyosin was found to track the size of the cooperative unit of their earlier model. Our own evaluation of the cooperative unit (Fig. 7 or Table 2) differed from that reported by Maytum et al. (1999) and would not exhibit the same relationship. The model of Smith et al. (2003) was able to fit curves for the equilibrium binding of S1 to regulated actin in the presence and absence of Ca2+ (Smith and Geeves, 2003). Although the full predictive ability of that model has not been explored this is an interesting new approach.
In the original Hill model the simplifying assumption was made that the ATPase activity increased in going to state 2 but did not change in going from state 1 with no bound Ca2+ to state 1 with bound Ca2+. Although this two-state model is sufficient to explain function in solution it will be necessary to include an additional functional state if the third observed structural state has a unique function. The Hill model can be readily expanded to three states (Fig. 1). The major states 1 and 2 can exist with either 0, 1, or 2 bound Ca2+ (Fig. 1). Because of the cooperativity of Ca2+ binding it is most likely that a regulatory unit will have either 0 or 2 bound Ca2+. Therefore, the most reasonable possibilities are states 10, 12, 20, and 22. The latter two states seem to be similar if not identical in terms of their properties (Greene and Eisenberg, 1988) so that it would be quite reasonable to have three functional states, 10, 12, and 2. Treating state 12 as a third state would mean assigning this state with an ATPase activity intermediate between those of states 10 and 2. However all three states would retain the ability of binding to S1 during steady-state ATP hydrolysis; that is, none of these would be a blocked state. We have not been able to distinguish two-state and three-state models by an analysis of equilibrium binding and binding kinetics. The number of functional states may be apparent from a more detailed analysis of the effect of Ca2+ and ionic strength on the regulation of ATPase activity.
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ACKNOWLEDGEMENTS |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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This work was supported by the National Institutes of Health (grant AR40540 to J.M.C.).
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FOOTNOTES |
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Submitted on March 25, 2004; accepted for publication June 22, 2004.
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REFERENCES |
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); 180 mM (
); 225 mM (
); 260 mM (
); 300 mM (x); and 375 mM (*). Both axes are on a log scale.
).