the protein may exist in one conformation in the absence of ligand the ligand that binds. On the contrary, the binding of different

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1 Proc. NatL Acad. Sci. USA Vol. 80, pp , January 1983 Biochemistry Alternate model for the cooperative equilibrium binding of myosin subfragment-l-nucleotide complex to actin-troponin-tropomyosin (muscle contraction/regulation/calcium/ising model/matrix method) TERRELL L. HILL*, EVAN EISENBERGt, AND LOIS E. GREENEt *Laboratory of Molecular Biology, National Institute of Arthritis, Diabetes, and Digestive and Kidney Diseases, and tlaboratory of Cell Biology, National Heart, Lung, and Blood Institute, National Institutes of Health, Bethesda, Maryland Contributed by Terrell L. Hill, October 4, 1982 ABSTRACT In this paper we introduce an alternate model for the equilibrium binding of S-i-N (S-1, subfragment 1 of myosin; N. nucleotide) on the troponintropomyosin--actin complex, including the influence of Ca2+ on this binding. In our previous model [Hill, T. L., Eisenberg, E. & Greene, L. E. (1980) Proc. NatL Acai Sci USA 77, ], we assumed that each tropomyosin unit, including one troponin-tropomyosin molecule and seven actin sites on the actin filament, could exist in two conformational states which presumably differed in the position of the tropomyosin on the actin. The binding of S-i-N or Ca2+ to the tropomyosin unit shifted the equilibrium between the two states but did not affect the intrinsic conformation of each state. In contrast, in the present model, we assume that tropomyosin can in principle occupy a continuum of positions on the actin filament. However, in any particular circumstance (N, Ca2+, salt, temperature), the tropomyosin occupies only a single position rather than existing in a dynamic equilibrium between two positions as in our earlier model. The binding of S-i-N or Ca2+ changes the position of tropomyosin on the actin filament and the exact position that the tropomyosin occupies depends on the nucleotide bound to S-1. In an earlier paper (1) we introduced a model for the cooperative equilibrium binding of myosin subfragment 1 (S-1) to the troponin-tropomyosin-actin complex (regulated actin). In this model, the tropomyosin units, consisting of seven actin sites and one troponin-tropomyosin molecule, exist in two conformational states: state 1, in which actin binds S-1 wealdy, and state 2, in which actin binds S-1 strongly. Presumably, the tropomyosin molecule occupies a different position on the F-actin filament in each of these states; intermediate positions (or states) are assumed not to occur. One of the key properties of this model is that, under all conditions, there is an equilibrium between the two structural states of the tropomyosin units; the binding of Ca2O or S-1 to the tropomyosin units shifts the equilibrium between the two states but does not affect the intrinsic conformation of each state. This model can account for the interaction of regulated actin with S-1, S-1PADP, S-l adenosine 5'-[,Sy-imido]triphosphate (S-1'AMP-P[NH]P), and S-1FATP (2, 3), although it is necessary to assume that the ratio of S-1 binding affinity to the strong and weak states (i.e., K2/K1) of the tropomyosin units depends on the nucleotide bound to S-1. This ratio is quite high with S-1 and S-1 ADP, whereas it is almost 1 with S-1 ATP. As we have discussed elsewhere (4), the interaction of a protein and a ligand can occur in a somewhat different manner. Rather than the protein existing in two intrinsically stable conformations, independent of the presence or absence of ligand, The publication costs ofthis article were defrayed in part by page charge payment. This article must therefore be hereby marked "advertisement" in accordance with 18 U. S. C solely to indicate this fact. 60 the protein may exist in one conformation in the absence of ligand and then, when ligand binds, be induced to enter a quite different conformation. In this well-known type of "induced fit" model, the conformation of the protein is not independent of the ligand that binds. On the contrary, the binding of different ligands-e.g., S-1PADP and S-1 AMP-P[NH]P--could induce somewhat different conformations. In the present paper we apply this approach to the cooperative equilibrium binding of S-1 to regulated actin by allowing the tropomyosin to occur in a continuum of positions on the actin filament. In a way, this new induced fit model is as simple as our "two-conformation" model. Although the tropomyosin can exist in a continuum of positions, this model has the simplicity that under any particular circumstance (N, Ca2+, salt, temperature), the tropomyosin can occur in only one position rather than in two positions, as in our earlier model. The new induced fit model is able to fit our experimental data as well as our earlier two-conformation model, although to fit our data it is necessary to assume that the position into which the tropomyosin is "pushed" depends on the nucleotide bound to S-1. This is analogous to the requirement in our earlier model that the ratio of S-1 binding strength to the two states of the tropomyosin units depends on the nucleotide bound to S-1. THE ASSUMED MODEL In the absence of Ca2', we assume that the tropomyosin-troponin complex can exist in two different states, labeled 10 and 2 in Fig. 1. Each tropomyosin-troponin complex covers seven actin sites for the binding of S-1-N (N, nucleotide). If no S-1- N is bound to a seven-site tropomyosin-troponin-actin unit, then the tropomyosin-troponin is in state 1 The tropomyosintroponin complex in this state blocks the access of S-i-N to the actin sites. In order for an S-1-N to be able to bind to an actin site, the S-1-N must, as part of the binding process, push the entire tropomyosin-troponin complex to a new position or state, relative to actin, labeled 2 in Fig. 1. The figure is, of course, schematic. The positive free energy required to move the tropomyosin-troponin is denoted AG. It is assumed that the location of state 2 relative to state 10 and the value of AG could be different when different nucleotides (or no nucleotide) are attached to S-1, because the bound nucleotide may affect the angle at which S-1 binds to actin. Thus, state 2 is not an intrinsic second location for tropomyosin-troponin but rather a location forced on tropomyosin-troponin by the binding of S-1-N. Fig. 1 represents, schematically, the position of tropomyosin in states 10 and 2, as well as in states 1I and III. As discussed below, state 1 is divided into the three states 10, I1, and 1II, depending on the number of Ca2' bound to troponin C. Fig. 1 Abbreviations: S-1, myosin subfragment 1; S-1iAMP-P[NH]P, S- ladenosine 5'-[13,y-imido]triphosphate.

2 Biochemistry: BProc. Hill et al Natl. Acad. Sci. USA 80 (1983) 61 Tm Tn Tn+Ca2+ Tn+2Ca2+ OLD State 1o 11 II i 2 Actin Sits S-1- FIG. 1. Schematic representation of four. states of tropomyosintroponin in model. Troponin undergoes a conformational change (circle,square, rectangle) when Ca2" is bound (heavypoint), causingtropomyosin-troponin (Tm-Tn) to change position in relation to the actin sites when tropomyosin-troponin is in state 1 (substates 10, 1i, III, according to number of Ca2+ bound). When S-1-N is bound, tropomyosin-troponin is in state 2, with a position relative to actin sites that is independent of troponin conformation. can alternatively represent the relative free energies of the four states in these positions. After the first S-1-N is bound to a unit, subsequent S-1-Ns can be bound without any further movement of or free energy expenditure on the tropomyosin-troponin complex. If the subsequent S-1-Ns bind with binding constant K2, then the binding constant of-the initial S-1-N to a unit is e-g/ktk2, or K2/L, if we define L as eag'it. k Thus L 1. When, as usual, L > 1, the binding of the first S-1-N to a unit is inhibited by the extra work that has to be done to move the tropomyosin-troponin. In general, both K2 and L-will depend on N. In our previous model (1), an independent parameter K1 (not needed here) was the "early" binding constant (here it is K2/L). In our earlier model (1) we postulated that, with no S-1-N bound, a dynamic equilibrium exists between tropomyosin-troponin in two different states 1 and 2, with equilibrium constant L. This L is necessarily independent of N; it is an intrinsic property of the actin filament. Here there is no such dynamic equilibrium. Instead, with no S-1-N bound, tropomyosin-troponin is in state 10; with one or more S-1-N bound, tropomyosin-troponin is in state 2. Thus, the state change is part of the binding process. L in the present model depends on N because the state 2 position of tropomyosin-troponin, established by the binding of S-1-N, is generally different for different N. Fig. 2 shows very schematically the physical origin of the difference between the binding constants K2/L and K2. In the former case, work has to be done separating tropomyosin-troponin from the row of actin sites, but this is not so in the latter case. Incidentally, if K' (Fig. 2) is the binding constant for S-1- N on actin in the absence of tropomyosin-troponin, we would expect K2 > KS if there is an attractive interaction between tropomyosin-troponin and S-1-N (Fig. 2). A small effect of this sort is observed (2, 3): k/k2 3 when N = ADP or AMP-P[NH]P. In this model, as previously (1), we assume that there are no direct interactions between neighboring bound S-1-N molecules. This assumption is based on the observed independent binding of S-1 ADP on actin in the absence of tropomyosintroponin (2). The assumption, above, that a single S-1-N bound on any of the seven sites of a unit will move the entire tropomyosin-troponin into a new state, state 2, has been relaxed in ref. 5. If we call the present case R = 7, where R is the number of actin sites undergoing the 10 -> 2 state change when an S-1-N is bound to one of the two end sites of a tropomyosin-troponin unit, then it can be shown (5) that R = 6 and R = 5 are also possible choices for a model, but not R 4. In the absence of structural or other - information to the contrary, we are adopting the R = 7 model here for mathematical simplicity. Perhaps tropomyosin, a twochain a-helical coiled -coil (6), is sufficiently rigid to indicate R = 7 i any case. We have found experimentally (2, 3) that, if tropomyosintroponin is saturated with Ca2", the value of L decreases while. K2/L: e'i + K2: (s) + K0: (s) + Tm Tmfi - SXi Afi lb S- Tm' FIG. 2. Schematic relation between binding constants of S-1-N. K2/L: The first S-1-N bound to a tropomyosin-troponin-actin unit must move tropomyosin-troponin relative to actin, as part of the binding process. K2: Subsequent S-1-N molecules bind to the unit without havingto move tropomyosin-troponin. K02: Bindingof 5-1-N to unregulated actin occurs without interaction between S-1-N and tropomyosintroponin. See text. Tm, tropomyosin. K2 is not significantly affected. These facts are reflected in our model by assuming (Fig. 1) that when no S-i-N is bound (state 1), binding of Ca2+ on troponin also moves tropomyosin-troponin relative to the actin sites, possibly through a conformational change in troponin; but when S-1-N is bound (state 2), the binding of Ca2" has no effect on the state 2 position of tropomyosin-troponin relative to the actin sites. Thus, if we include the possibility of Ca2" binding, state 1 is subdivided into three states, 10, 1I, 1i (Fig. 1), the second index (0, I, or II) referring to the number of Ca2" bound-on the two regulatory sites of troponin. Fig. 1 is very schematic; it is meant to indicate that the free energy AG necessary to move tropomyosin-troponin from state 1 into state 2, to accommodate S-1-N binding, is decreased when two Ca2" ions are bound on troponin (state III) as compared to the case (discussed above) in which Ca2" is absent (state 10). So to speak, state III has a head start over state 10 in the direction of state 2 (state 11 is intermediate). Consequently, L is smaller at high Ca2" than at low Ca2+ concentration. We assume that the more extensive movement of tropomyosin-troponin relative to actin, induced by S-1-N binding, supersedes any Ca2` effect on the position of tropomyosin-troponin. Thus, state 2 is not altered in position and K2 is not affected by Ca2+ binding. Strong cooperative binding of S-1-N on regulated actin is observed in some cases (2, 3). To account for this, we have assumed previously (1), and continue to do so here, that there are nearest-neighbor interactions between the ends of units-that is, between the ends of tropomyosin molecules (7). This is referred to as the tropomyosin-tropomyosin interaction below. These tropomyosin ends are, in fact, known to be in contact one way or another (6). We assume here that the optimal interaction between two ends occurs when both tropomyosin molecules are at the same position (i.e., state) relative to the actin sites and that the greater the difference in position (Fig. 1) of the two molecules (or states) the more the optimal interaction between the ends is decreased. For the four possible states in Fig. 1, there are 10 possible kinds of nearest-neighbor pairs, which we denote by (using the second state 1 index): 00, 01, OII, 02, I I, III, 12, II II, 112,.and 22. Some examples are shown in Fig. 3. This is schematic. Contact between ends is not shown in the figure but it is presumably always present, though with some

3 62 Biochemistry: Hill et al, Proc. Natl. Acad. Sci. USA 80 (1983) Tm-Tn Pair 00 OIl State , 2 2 Actin S-1-N FIG. 3. Notation used to designate the four states and the 10 kinds of nearest-neighbor pairs in the model. See text. distortion when the two molecules are in different states. If, for any one of these pairs, wi1 is the free energy of interaction of the two ends relative to their infinite separation, then we define X - ewkt. We can simplify notation for all of the like pairs, with optimal interaction (see above), by using a single x: X = Xoo = XI I = XII II=X22. [1] In view of the comments above, xi, = xji and we have, for example, the inequalities: X22 = X x112 2 Xx12 X02, [2] Xoo = X X- x01 X01 X02- In both lines, the separation. (Fig. 1) between the two states increases as we move to thetight. An equality would occur only if the levels (Fig. 1) of the two states were the same, in a special case. As will be seen below, these inequalities describe qualitatively the dependence of the tropomyosin-tropomyosin interaction parameter Y (1) on nucleotide and on Ca + that is predicted by the present model. FORMULATION AND PROPERTIES OF THE MODEL We can use the matrix method, as before (1, 5), to deduce the exact properties of the nearest-neighbor Ising model outlined above. Because there are four states (Fig. 1), we have to start with a 4 x 4 matrix. The grand partition functions for the four states are: 10: 1 11: 2Kap 111: aka:p2 2: L-'[(1 + K2c)7-1](1 +-2Kbp + fk2bp2)-2&. the concentration of Ca2", c is the concentration of Here p is S-1-N, Ka is the binding constant of the first Ca2+ on state 1 (the factor of 2 for 11 implies the assumption of two equivalent binding sites), aka is the-binding constant of the second Ca2+ on state 1, and Kb and /3 are similar for the state 2 binding of Ca2+. In 2- L-1 [ ] (Eq. 3), unity is subtracted within because state 2, by definition, never has all actin sites empty (state 1 takes care of this case). The factor L' multiplies each of the remaining terms in [ ] because L is involved (in K2/L) only for the first S-1-N bound, but not in any subsequent binding. The binding constant Ka (in 1I) includes any contributions from a troponin conformational change and from forced movement of tropomyosin-troponin from position 10 to 1I (Fig. 1). Thus, Ka for Ca2+ is analogous to K2/L-for S-1-N. The factor a takes care of any possible difference in free energy required, for a troponin conformational change and for tropomyosin-troponin movement, between 10 -o 1I and I11; a also takes care of Ca2+-Ca2+ repulsion (in HI), if any. The factors f2 and (Eq. 3) are independent (can be simply multiplied) because Ca2+ binding in state 2 is assumed not to perturb the position of tropomyosin-troponin (as it does in state 1). Therefore, Kb does not have a contribution from tropomyosin-troponin movement (as Ka does), and 8 is related only to the troponin conformational change (as in a) and to possible Ca2+-Ca'- repulsion. If the second Ca2+ is bound with more difficulty than the first (negative cooperativity), a, ( < 1. We would expect from the above discussion that Ka < Kb, but whether a < 83 or (3 < a depends on the free energy differences between states 10, 11, and 111 for tropomyosin-troponin. The 4 X 4 matrix (1, 5), including nearest-neighbor interactions, is x 2KapxoI akap2x0j1 i XoI 2Kapx ak~p~xju ~AXKp 12 XO{I 2K'aPXIII akap x (24XI12 X02 2KaPXI2 aknp XI12 24X v 24X02 Even with simplifying assumptions about regularity in the x., the largest eigenvalue y)m (m means maximum) of this matrix has to be found by solving numerically (5) a quartic equation in y. The binding isotherms for S-1-N and Ca2+ then can be calculated, numerically, from (1, 5) 70 = aln ymalnc and o- = aln ym/alnp, [5] respectively, where 0 0 S 1 (for S-1-N) and o- 2 (for - Ca2+). Actually, experiments at intermediate p (Ca2+ concentration) are not available yet. The important special cases, for comparison with experiment, are p- 0 and pa- oo. In the former case, only states 10 and 2 (with 4X, = 1) are important. Thus, we have a 2 X 2 matrix: / x X02 X [6]- VL-1[ ]0 L-'[,JI o) where [ ] appears in Eq. 3. In the latter case (p-k oo), only states OII and 2 (with 4, = 3Kbp2) are significant. The 2 X 2 matrix is 1 akap x akpz X112 [7] 3KJp%4.XII2 OKbO x If, to normalize, we divide each element of this matrix by ak'p2 (this will not affect Eq. 5 for 0), we have x X112 [3] (P [8] ), where Thus L(oo), defined here, has the same significance for p -> 00 that L does for p -a Because we expect Ka < Kb (free energy is required to move tropomyosin-troponin when Ca2" is bound in state 1), presumably L > L(oo), as is observed experimentally (1, 3). As a numerical example, on fitting the two Ca2+ binding curves in Fig. 2 ofref. 8 using f (S-1 present) and the equivalent (a (S-I absent), we find Ka = 3.6 X 106 M-l, Kb = 9.5 X 106 M-l, a 15, = = 4 [10] These values predict, from Eq. 9 that L = 18.6 L(oo) for S-i binding (note, however, that the experimental curves refer to all four Ca2" sites on troponin, not just the two regulatory sites). Kb, a,. and A will not generally be available to allow application of Eq. 9. The usual procedure will be to find L and L(oo) independently by curve fitting of S-1-N binding data, in the absence and in the presence of Ca2+. But we can anticipate, from Eq. 9, that L 3 L(oo). Reliable values of K&, L(oo)-'[ IX,,2 L(oo)-'[ ]x = L(oo) L(aKa 2/pK2b). Eqs. 8 and.6 have the same form except that L(oo) replaces L and x112 replaces x02. We define nearest-neighbor interaction [4] [9]

4 Biochemistry: parameters (1, 5) Hill et al Y(oo) = (X/X112)2, Y = (XIX02), [11] where Y(oo) applies when p -+00 and Y applies when p -+ Because state 11I is nearer to state 2 (Fig. 1) than is state 10 (on account of Ca2+ binding in the former case), x112 > x02 (Eq. 2) and Y > Y(oo). It is not possible to determine experimentally whether Y > Y(oo) because the slight cooperativity observed in the presence of Ca2+ makes the data very insensitive to Y(oo) (3) Ȧccording to our model and assumptions, the greater the separation in position and free energy between states 10 and 2 (Fig. 1), the larger the value of L and the smaller the value of x02 (Eq. 2). If the nucleotide N is changed, in S-1-N, state 10 is unaffected but the position of state 2 may be higher or lower (e.g., because of a change in binding angle of S-1-N). For example, if state 2 is higher, L increases and xo decreases. The value of x is unaffected (it is the optimal interaction parameter for neighboring tropomyosin ends when the two tropomyosin molecules are at the same level-that is, in the same state). Thus, in this example, in view of Eq. 11, Y increases as well as L. In general, when either N or Ca2+ (present or absent) is changed, the values of L and Y are expected to go up or down together. The following analysis is based on Eq. 6, using L and Y (p -+ 0). But, in the presence of Ca2' (p-+ oo), the-same equations are applicable if we merely replace L and Y by L(oo) and Y(oo). The other parameter, K2, is not affected by Ca2'. The larger eigenvalue Ym of Eq. 6 is given by 2ym = a, + a2 + [(a, - a2)2 + 4al42y-l]"l2, [12] where a, = x and a2 = xl-'[ ]. The fraction 6 of actin sites occupied by S-1-N molecules then is found from Eq. 5 to be 6 = P202, [13] where 62 = K2c(1 + K2c)6/[(1 + K2c)7-1] [14] P2 = 2aY-/'f(l - a + V) [15] a-a2/a, = f2 = L'[(1 + K2c)7-1] [16] [(1 - a)2 + 4aY-']1/2. Here, P2 is the fraction of units in state 2, 62 is the fraction of state 2 actin sites that are occupied by S-1-N, and a is the equilibrium constant, per unit, for the transition 10 -± 2 between a filament with all units in state 10 and a filament with all units in state 2 (the interaction parameter is x in both cases and cancels). The transition (as c increases) is half-completed (P2 = 1/2) when a = 1. Note that a does not depend on Y; but the steepness of the transition at a = I (P2 = 1/2) increases with Y. The above results resemble those in ref. 1, but there are significant differences. Here 61 = 0 (no binding on state 10 units) and 62 is not a simple Langmuir binding isotherm [except when K2c>> 1, in which case K2c/(1 + K2c)]. In fact, when K2c - 0, 62-> 1/7 (a state 2 unit necessarily has at least one S-1- N bound). The equilibrium constant a has a correspondingly different form than in ref. 1. Also, Y and L have more explicit physical interpretations here. In particular, L is related to the alteration of the tropomyosin-troponin position on binding the first S-1-N to a unit; L is not an isomeric equilibrium constant as in ref. 1. Incidentally, adsorption with perturbation of the absorbent by the adsorbate, as in the present model, is a common-phenomenon (9). Proc. Natl. Acad. Sci. USA 80 (1983) 63 When K2c -> 0, /7, a-- 7K2c/L, P2 a/y, 6-- K2c/LY. [17] Thus, the initial slope of O(c) is K2/LY. This is the effective binding constant in the initial binding of S-1-N. Y appears in this last expression because each new S-i-N bound on a chain of units practically devoid of bound S-i-N will convert a state 10 unit into a state 2 unit (binding constant K2/L) and, at the same time, transform two-00 neighbor interactions (parameter x) into two less favorable 02 interactions (parameter x02). When Y = 1 (no interaction effects), P2 = a/(l + a). The binding isotherm 6(e) still shows positive cooperativity in this case, if L >> 1, because of the groups of seven sites switching state (10 -+2) as a unit. When L>> 1, there is strong inhibition of early binding (K2/L) followed by relatively easy binding (K2) on state 2 units. When L = 1 as~well as Y = 1, Eq. 13 reduces to simple Langmuir binding: 6 = K2c/(l + kc), with no cooperativity at all. Evaluation of Parameters. The usual procedure in evaluating parameters, given a set of experimental O(c) data, is the following: (i) From each of the high c points (usually 6> 5), K2 is calculated by using K2 = 6/(1-6)c (or, if necessary, Eq. 14) and then is averaged. (ii) By using this (averaged) K2, the curve 02(c) (Eq. 14) is calculated and drawn. This curve should be a good representation of the high c points. (iii) A smooth curve O(c) is drawn through the experimental data in the (inflection) region that includes the point where 6 (experimental) = 02/ 2. The values of 6 and c where 6 = 62/2 are denoted 6' and c' (see Fig. 4). This is the point at which P2 = 1/2 (Eq. 13) and a = 1. (iv) The value of L may now be calculated from Eq. 16: L= (1 + K2c')7-1. [18] There is a corresponding, but not independent, connection between 6' and L: 6' = [(L + 1)/7 -_1](L + 1)6'7/2L. [19] (v) The full theoretical curve. 6(c) is calculated now from Eq. 13 (by using K2 and L found above), adjusting Y for best fit [especially to match the observed slope in 6(c) in the neighborhood of c = c']. (vi) Because there is some flexibility in drawing the smooth curve in step iii, -above, the overall agreement between the theoretical and experimental 6(c) can sometimes be improvedbyadjusting 6' and c' slightly, consistentwith 6' = 02(c')/ 2. Fig. 4, which examines the binding of both S-1ADP and S- 1AMP-P[NH]P to regulated actin -under identical conditions (the absence of Ca2+,,- = 18 M, 25 C), illustrates the above procedure. These data have been analyzed previously by using our original model (3). For the AMP binding data (Fig. 4A), the parameters that give the solid theoretical line with our new model are K2 =-1.46 X 10J M-1, c' 52,uM, 6' = 22, L = 51.1, and Y = 2 Stnilar parameters were obtained when we fitted these data with our old model (3). In applying our new model to the AMP-P[NH]P data, we could not get a.very good fit to the data employing the same values for L and Y- that we used above for the ADP data (dashed line in Fig. 4B). However, by decreasing the values of both L and Y, we were able to obtain a much better fit. -The solid theoretical line for the AMP-P[NH]P data (Fig. 4B) was obtained with the parameters- K2 = 1.27 X 104 M-1, c' = 50,uM, O' - 20, L-= 32, and Y-= 4. Note that these same values of L and Y give a poor fit-to the ADP data (dashed line in Fig. 4A). Thus, to account for our data with the new model, it is necessary to assume that S-1.AMP-P[NH]P does not push the tropomyosin to quite the same position on the actin as does S-1 ADP;

5 n A Kn Biochemistry: Hill et'al.,,.6 0 0~~~~ 45 - o / / 0 / C, /M FIG. 4. Fitting of the model to data obtained for the binding of S- 1'ADP and S-1 AMP-P[NH]P to regulated actin in the absence of calcium (,u = 18 M, 250C). (A) For the ADP data, the solid theoretical line was obtained by using K2 = 1.46 x 106 M-1, L = 51.1, and Y = 2 The dashed line was obtained with the same value of K2 and L = 32 and Y = 4. (B) For the AMP-P[NH]P data, the solid theoretical line was obtained by using K2 = 1.27 x 104 M-1, L = 32, and Y = 4. The dashed line was obtained with the same value of K2 and L = 51.1andY= 2 the binding of S-1-ADP requires a larger change in the position of the tropomyosin than does the binding of S-1.AMP-P[NH]P. This might occur if S-1-AMP-P[NH]P binds to actin at a somewhat different angle than does S-1-ADP (10-12). DISCUSSION In this paper we have presented a new cooperative model that accounts for our binding data as well as does the old model. The major difference between the two models lies in the nature of J Proc. Natl' Acad.- Sci. USA 8a (1983) the states of the tropomyosin. units of regulated actin. In the old model, no matter what the conditions, there are only two intrinsic states for each of the units, states 1 (weak binding) and 2 (strong binding). The binding of Ca2' or S-i affects the equilibrium between-the two states. In addition, the S-i binding constants K1 and. K2, to units in states 1 and 2, are different for different.s-l-nucleotide complexes, which also affects the 1 a± 2 equilibrium.. Thus, the lack of a cooperative effect with S-1FATP is attributed, in this model, to nearly identical binding of S-FATP to states 1 and 2 (K1 -K2). In contrast, in the new model, a tropomyosin molecule is, in principle, able to occupy a continuum of positions (relative to the groove) on the. F-actin filament. The binding of Ca2+ shifts the position of the tropomyosin molecule as does the binding of S-i. In the latter case, the position to which the tropomyosin is shifted depends on the nucleotide bound to S-i. Thus, with the new model, the lack of a cooperative effect with S- 1ATP would be attributed to the ability of an S-1ATP to bind. to actin without causing any significant change in the position of the corresponding tropomyosin molecule on the F-actin filament. It may be possible to distinguish between the two models experimentally. A fluorescence change occurs when S-i or Ca2' binds to regulated actin. If a tropomyosin molecule can occupy a continuum of positions on the F-actin filament, then the magnitude of this fluorescence change might depend on.the position of the tropomyosin on the F-actin filament, which, in turn, could be affected by the nucleotide bound to the S-i. IfS-1ADP and S-1AMP-P[NH]P push the tropomyosin to different positions, the cooperativity observed in the regulated actin S- 1ATPase activity also might be different, depending on whether ADP or AMP-P[NH]P is present in addition to ATP. These experimental approaches must be. explored to distinguish between our original model and the model presented in this paper. 1. Hill, T. L., Eisenberg, E. & Greene, L. E. (1980) Proc. NatL Acad. Sci. USA 77, Greene, L. E. & Eisenberg, E. (1980) Proc. Nati Acad. Sci. USA 77, Greene, L. E. (1982) J. Biol Chem., in press. 4. Hill, T. L. & Eisenberg, E. (1981) Q. Rev. Biophys. 14, Hill, T. L. (1981) Biophys. Chem. 14, Phillips, G. N., Fillers, J. P. & Cohen, C. (1980) Biophys. J. 32, Wegner, A. (1979)J. MoL BioL 131, Bremel, R. D. & Weber, A. (1972) Nature (London) New BioL 238, Hill, T. L. (1950)J. Chem. Phys. 18, Marston, S. B., Rodger, C. D. & Tregear, R. T. (1976) 1. Mol. BioL 104, Chalovich, J. M., Greene, L. E. & Eisenberg, E. (1982) Biophys. J. 37, 263a (abstr.). 12. Eisenberg, E. & Greene, L. E. (1980) Annu. Rev. PhysioL 42,