to the three terminal subunits of the filament that form the The helical actin filament is illustrated schematically in Fig.

Transcription

1 Proc. Natl. Acad. Sci. USA Vol. 82, pp , November 1985 Biochemistry A model for actin polymeriation and the kinetic effects of ATP hydrolysis (ATP cap/steady state/cap-bdy interface) DOMINIQUE PANTALONI*t, TERRE L. HIS, MARIE-FRANCE CARLIER*t, AND EDWARD D. KORN* *Laboratory of Cell Biology, National Heart, Lung, and Blood Institute, and tlaboratory of Molecular Biology, National Institute of Arthritis, Diabetes, and Digestive and Kidney Diseases, National Institutes of Health, Bethesda, MD 225 Contributed by Terrell L. Hill, July 1, 1985 ABSTRACT A model for actin polymeriation is proposed in which the rate of elongation of actin rdaments depends on whether adenosine 5'-triphosphate or.adenosine 5'-diphosphate is bound to the two terminal subunits of the filament. This model accounts quantitatively for the experimental data on the kinetics of filament elongation and explains the effect of ATP hydrolysis on actin polymeriation. Actin, one of the two major proteins of muscle, also occurs in all eukaryotic nonmuscle cells, where it is a dominant component of the cytoskeleton and has an essential role in numerous and diverse motile activities. Much of the actin in nonmuscle cells is in an unpolymeried state but only filamentous actin is known to be biologically active. Therefore, understanding the mechanism of actin polymeriation and its regulation is of obvious biological importance. The G-actin monomer (Mr 42,) assembles into a singlestart, left-handed helical F-actin polymer containing at least a thousand subunits. An important feature of the polymeriation process is the accompanying hydrolysis of G-actinbound ATP to F-actin-bound ADP. Oosawa and co-workers (1, 2) developed the first theory for the kinetics of ATP-actin polymeriation; modified analyses of the spontaneous polymeriation process have since been proposed (3-7). In all the models, ATP hydrolysis has been assumed, at least implicitly, to be tightly coupled to the polymeriation reaction. Therefore, no deviations from simple reversible polymeriation kinetics were introduced, except by Wegner (8), to allow for the possibility of different critical concentrations at the two filament ends. However, as predicted by Cooke (9), recent studies (1-12) have shown that ATP hydrolysis is not necessarily coupled to polymeriation; this results in the formation of an ATP "cap" at the ends of actin filaments. The existence of the ATP cap explains the nonlinear dependence ofthe rate of elongation of actin filaments on the concentration of G-actin (12), a situation similar to that previously observed with microtubules in the presence of GTP (13-17). More recently, it was found (18) that the ATP-F-actin equilibrium polymer that forms transiently during actin polymeriation in ATP has a critical concentration 1 times higher than that of F-actin at steady state. This greater stability of F-actin at steady state may be caused by the particularly strong interaction between ATP-actin and ADPactin subunits, which maintains a very stable interface between the ADP filament core and the ATP cap (19). These results led to a reconsideration of the kinetics of actin polymeriation and the effects ofatp hydrolysis. In the model proposed in this paper, the rate of actin polymeriation depends on the nature of the nucleotide (ATP or ADP) bound The publication costs of this 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. 727 to the three terminal subunits of the filament that form the elongating site. MODEL The helical actin filament is illustrated schematically in Fig. 1. All subunits at positions n 2 3 interact strongly with their two adjacent neighbors at positions n - 1 and n + 1. The fact that a nucleation step occurs in the polymeriation process suggests that they also interact with subunits at positions n - 2 and n + 2 (see ref. 2). The two terminal subunits, however, differ from subunits n 2 3 in their number of contacts with neighbors. A new ATP-containing subunit, therefore, is incorporated into the filament in three steps, in the course of which its four interaction areas are sequentially converted into polymer bonds. The environment of the actin protomer is modified during these three steps, causing an increase in its stability and a possible modification of its properties, in particular its ability to hydrolye ATP. Both the a and b ends, then, can exist in four different conformations, F, F1, F2, and F3, differing in number of terminal ATP-containing subunits (, 1, 2, 3). These conformations interconvert by association and dissociation of ATP-actin and by ATP hydrolysis, according to Scheme 1: DEPOLYMERIZATION -D D -D -D F POLYMERIZATION COUPLED UNCOUPLED -TI-D T -T -D-T -T --\/ L D T -T -T -D-T -T -D -T -T -T-T F. F2 F3 ADP ATP Scheme 1 T and D represent ATP-actin and ADP-actin subunits, respectively, not ATP and ADP. Filaments Fj are defined as having i terminal ATP subunits, all subunits at position n > i being ADP-containing subunits. For simplicity, and since only the three terminal subunits differ on a structural basis, only FO, F1, F2, and F3 species are represented in Scheme 1. Filaments Fj (i > 3) have the same parameters as F3 for the association-dissociation reactions of ATP-actin. ATP is hydrolyed at position n = i of filaments Fj (i 2 3), that is, on the subunit at the interface between the ATP cap and the ADP core. Depending on the ATP-G-actin concentration, F, filaments exist in different fractions, fi, and, therefore, different steady-state rates of growth can be observed. tpermanent address: Laboratoire d'enymologie, Centre National de la Recherche Scientifique, 9119 Gif-sur-Yvette, France. +

2 728 Biochemistry: Pantaloni et al. a end 1',d b d b ~~~~2' %d' +n+2 b > b C n-1 1 \ i 3 "a n n-ik2 \do b end FIG. 1. Structure of an actin filament. This scheme illustrates the four different areas a, b, c, and d with which subunits can interact with neighboring protomers. A subunit at position n inside the filament (solid line) has four areas in contact with four neighbors at positions n - 2, n - 1, n + 1, and n + 2 (dashed lines). At the a end, areas a and c are exposed on subunit 1', and area a on subunit 2'; conversely, at the b end, the complementary 2 areas (b and d on subunit 1, b on subunit 2) are exposed. At positions 3 and 3', all areas of interaction are in contact with neighbors. The kinetics of elongation within Scheme 1 can be described by a series of differential equations for the diferent fractions, fi, of filaments Fj and the concentrations of ATPand ADP-containing subunits. The rate of elongation of an average filament, J. varies with actin concentration according to the following equation: A~c) = (kocd - k-o)fo + (kifoc - k-_f 1) +(k2f1c - k-2f2)+ E(k3fb-c kbj), - i = 3 where CD and c are the concentrations of ADP-G-actin and ATP-G-actin, respectively; ki (i = 1, 2, 3...) is the rate of association of ATP-G-actin to an Fj_j filament and k-i is the rate of dissociation of ATP-actin from an Fj filament; and ko and k-o are the on and off rate constants of ADP-G-actin with Fo filaments. We will assume that elongation takes place at steady state; that is, the rate of interconversion among all the Fi conformations is fast relative to the change ic canint con-- centration. This means that the proportion of the different filaments is kept constant with time; i.e., dfiadt =. The rate of ADP-G-actin binding to Fh filaments is negligible as long as the concentration of ADP-G-actin is lower than the critical concentration. Fo Therefore, filaments essentially depolymerietdpcnain at subunits. the other hand, the rate of whrcidatinn cfatphe-tintoncf and F i exactin D G balanced by the rates of dissociation; i.e., fi = foklc/k-l = foklc [21 f2 = fik2c/k_2 = foklk2c2. [31 In the subsequent steps involving F2, F3,... filaments, the rate of association of ATP-G-actin is balanced by the sum of the rate of dissociation and of the rate of ATP hydrolysis. Therefore, the net on flux of actin subunits is equal to the rate of ATP hydrolysis: k3fi-1c - k-3fi= khfi, [4] where kh is the rate of ATP hydrolysis at position n = i on F, (i. 3). By iteration, the following equation can be derived: fi = f2[k3c/(kl3 + kh)]'-2 = foklk2c2(k'c)-2, [5] where K' = k3/(k3 + kh). Note that, unlike K1 and K2, K' is not an equilibrium constant. The steady-state rate of elongation, J(c), then can be written: J(c) = -klofo + kh (1 - fo - fl - f2). [61 The first term in Eq. 6 represents the off flux of ADPcontaining subunits from FO filaments and the second term represents the net on flux of ATP-containing subunits, which is equal, at steady state, to the rate of ATP hydrolysis, JH. The normaliation relation for thef provides an equation for fo: 1 =fo +fi +f2 + *fi Proc. Natl. Acad. Sci. USA 82 (1985) = fo [1 + K1c + KjK2c2(1 + K'c + (K'c)2 + (K'c)3 +.)] = fo {1 + Klc + [KlK2c2/(1 - K'c)]} (K'c < 1). [7] Then the steady rate of elongation becomes J(c) -k- (1 - K'c) + khklk2k'c3 (1 - K'c) (1 + K1c) + K1K2c2 and the steady rate of ATP hydrolysis is khklk2k'c3 JH = K+ (1- K')1 + Kic) + KjK2c 2 When c = 1/K' = c' the series in Eq. 7 diverges and J = JH = kh. At concentrations above c', Eq. 6 is no longer valid because ATP-actin subunits are incorporated at a faster rate than ATP is hydrolyed. Then, the sie of the ATP cap is no longer defined and increases indefinitely with time. At concentrations above c', the Fo, F1, and F2 species become negligible and only reversible polymeriation of ATP-actin onto ATP ends of the filaments (12) has to be considered, according to the rate equation J(c) = k3c - k-3 = kh + k3(c - c'). [11 In this region, the ATP cap grows at the rate k3(c - c') while the ADP part of the polymer grows at a constant rate JH = kh, independent of c. Therefore, the concentration c' = 1/K' represents a transition point between steady-state polymeriation with coupled ATP hydrolysis at the interface of the ATP cap and polymeriation uncoupled from ATP hydrolysis. The validity of this model can be assessed by comparing

3 Biochemistry: Pantaloni et al. Proc. Natl. Acad. Sci. USA 82 (1985) 729 (1) :5 U- I- 4cat U. C,) w :u C.) cc U- O -J I-. 4u -J J wl ACTIN CONCENTRATION, AM ACTIN CONCENTRATION, sm FIG. 2. G-actin concentration-dependence of the initial rate of elongation of actin filaments, J(c), and of the fractions of different filament species. (C and D) Symbols correspond to the experimental J(c) plots published in refs. 12 and 18. The heavy line is the theoretical J(c) curve according to Eq. 8 and fitted to the data using the parameters given in Table 1. The light line is the extrapolation of the upper part of J(c) plot showing C3 = 3,uM for the big ATP cap and k 3 = 5 s-l. (A and B) Evolution, in the same range of actin concentration, of the fractions fo, fi, andf2 of different filament species FO, F1, and F2 that have, 1, and 2 terminal ATP-subunits; 1 - fo - fi - f2 represents the sum of all filaments having at least three terminal ATP-subunits. theoretical and experimental polymeriation curves and the theoretical and experimental rates of filament growth as a function of actin concentration. Rate of Growth as a Function of Actin Concentration [(c) Plot]. Eqs. 8 and 1 can be adjusted to experimental J(c) plots (12, 18) obtained in the presence of.2 mm ATP and 1 mm Mg2. Under these conditions, as well as under physiological ionic conditions (data not shown), the experimental plot shows a downward curvature at and below the critical concentration and deviates upward linearly with a slope k3, at concentrations above 11,&M [Fig. 2C; this upward deviation has independent verification (18)]. For the theoretical curves, the experimentally determined values of some parameters were used (Table 1) and the remaining unknowns, that is K1, K2, and kh, were calculated or adjusted to give the best fit to the data as follows. Given 1/K' = 11 IiM, kh could be calculated using the values determined for k3 and k3 and the relation kh = (k3/k') - kl3; at the critical concentration, J =, and the value of K1K2 could be derived from Eq. 8. Then, an estimation must be made of the ratio K1/K2, which must be less than 1 to achieve a good fit. Fig. 2 shows the fit obtained by using Eqs. 8 and 1 and the Table 1. values of the parameters given in Table 1. On the same figureare shown the corresponding evolutions of the fractions of Fj filaments and of the rate of ATP hydrolysis in the same range of actin concentrations. It is striking that filaments F2 represent more than 5% of the population of filaments over a large range of actin concentrations (.3-7 KM). At very low actin concentrations, F filaments are predominant and contribute to the rapid depolymeriation. Above the critical concentration, the rate of polymeriation J(c) becomes parallel to the evolution of 1 - fo fi f2 (see Eq. 6). - - Time Course of Nucleated Polymeriation. Eqs. 8 and 1 can be integrated to obtain the time course of polymeriation at a given concentration of filaments and a series of initial ATP-G-actin concentrations. Theoretical polymeriation curves were generated by computer using the Runge-Kutta numerical integration method. Fig. 3 shows the fit to data obtained under the same conditions as for the J(c) plot. The same parameters were used as in Fig. 2. It is important to note that the model accounts for the fact that the elongation curves obtained at different actin concentrations are not superimposable and are not exponential, in contrast to what is expected for a reversible polymeriation model. The devia- Thermodynamic and kinetic parameters for actin filament elongation cc, c3, ct, kl, k3, AM,uM IsM soli 1.tyM 1 k-3, s1 kh, s1 K1, UM-1 K2, M Direct measurement.55 3* ± O.lt f Best fit cc, is the critical concentration-i.e., the concentration at which J = ; c3 is the critical concentration for reversible polymeriation of ATP-G-actin into an ATP-F-actin polymer; c' is the concentration at the transition point-i.e., the concentration above which polymeriation uncoupled from ATP hydrolysis can be observed. *From ref. 18. tfrom ref. 2. tcalculated from kh = (c' - c3)k3.

4 721 Biochemistry: Pantaloni et al. uj 2 xu x Lu 11 1 TIME, s FIG. 3. Time course of polymeriation of nucleated actin at different G-actin concentrations. The ordinate is in percent of polymer formed (1%o polymer is formed at the plateau). Symbols are experimental data obtained by fluorescence monitoring of the polymeriation using N-pyrenyliodoacetamide-modified actin (21) (1%o of labeled actin). Polymeriation buffer consisted of 1 mm Tris chloride, ph 7.8/.1 mm CaCl2/.2 mm dithiothreitol/.2 mm ATP/1 mm MgCI2. Elongation was started by addition of 1.25 /AM F-actin, previously polymeried to steady state under the same conditions, to solutions of G-actin at the following concentrations: A, 1 ILM; *, 3 ILM; o, 6,uM. The light lines represent theoretical curves computer generated by integration of Eq. 8, in which the same values of the parameters as in Fig. 2 were used. The thick curve is the classical exponential time course of elongation obtained by integrating the equation -d(c - cc)/dt = k+[f](c - cc) and using the value of 1.7 AM-1 s-1 for k+ (refs. 12 and 2). tion from the exponential increases as the critical concentration is approached. Role of ATP Hydrolysis in Actin Polymeriation. The model proposed here for actin polymeriation accounts for the observed nonlinear dependence of the rate of polymeriation on actin concentration that was previously attributed to the involvement of ATP hydrolysis in the polymeriation process (12, 18). In addition, a satisfactory fit to the polymeriation curves is achieved over a large range of actin concentrations, including the region of low actin concentrations where the largest deviation from the classical exponential process is observed. The model thus eliminates the need to include fragmentation or reannealing steps that were previously introduced (6, 7, 22) to fit the kinetic data. The main characteristics of the new model can be summaried as follows: the elongation site of the filament consists of the two terminal subunits that can have either bound ATP or bound ADP, leading to three different types of interaction with ATP-G-actin. Hence, different species of filaments, Fo, F1, F2, and F, (i 2 3), according to the number of terminal ATP-containing subunits, need to be considered in the expression of the rate of polymeriation. The nonlinear behavior of the J(c) plot is due to the change in the proportions of the different filament species with ATP-Gactin concentration and to the fact that ADP-containing subunits dissociate about 1 times faster than ATP-containing subunits from F2 filaments. The model implicitly assumes that the rate of nucleotide exchange on the three terminal subunits is slow compared to the rates of association-dissociation of actin subunits. If this were not the case, the proportion of the different Fj filaments would not vary with actin concentration and the experimental J(c) plot would be linear. Filaments Fo, F1, F2, and Fj (i 2 3) are linked together through the association-dissociation reactions of ATP-Gactin and ATP hydrolysis. Polymeriation takes place at Proc. Natl. Acad Sci. USA 82 (1985) steady state at actin concentrations below c' = 1/K'. In this range ATP hydrolysis establishes the steady state of all Fi filaments through the relationshipfi/fi-, = K'c. The steadystate growth hypothesis used to fit the model to the data is justified because in the in vitro assay the concentration of filaments is 3-4 orders of magnitude lower than the concentration of monomeric actin, so that at least a hundred association-dissociation events can take place at the ends without affecting the concentration of monomeric actin by more than 1 to 1%. At concentrations above c' = 1/K', filaments incorporate long stretches of ATP-containing subunits of undefined length, in a scheme of reversible polymeriation uncoupled from ATP hydrolysis. The fast rate of ATP hydrolysis at the interface between the ATP cap and the ADP body of the filament acts like a barrier to the establishment of the regime of reversible polymeriation of ATP-G-actin. This is why the uncoupling between ATP hydrolysis and polymeriation cannot be experimentally observed below a concentration of c' = 1/K'. In other words, the concentration c' can be understood as the critical concentration for the nucleation of the ATP-F-actin polymer, while the concentration c3 = kl3/k3 is the reciprocal of the elongation equilibrium constant K3 for the ATP-F-actin polymer. In previous studies (11, 12), the rate constant for ATP hydrolysis on F-actin was obtained assuming random ATP hydrolysis in the ATP cap at all positions, giving an exponential process that fit the experimental data. The new model assumes that ATP hydrolysis occurs at the cap-body interface and progresses linearly toward the tip; thus, the same data give a value for the rate constant for ATP hydrolysis that is 3 orders of magnitude higher than in the previous model. The apparent exponential process actually observed for ATP hydrolysis after fast polymeriation (when it should be linear according to the new model) can be explained by the length distribution of the ATP cap. This explanation is the same as that which gives rise to an apparent exponential process for the linear depolymeriation of actin monomers from filament ends. While this new model may be a simplification of the real processes, it provides a reasonable fit to the data and offers the advantage of a simple analytical solution for the involvement of ATP hydrolysis in the kinetics of actin polymeriation. The best way to choose between these two models (i.e., random hydrolysis in the cap or site-specific hydrolysis of ATP at the cap-body interface) is to measure directly the rate of ATP hydrolysis by actin filaments as a function of ATP-G-actin concentration (a JH plot). This JH plot should allow the direct determination of JH = kh above 11,uM actin if the present model is correct. A similar treatment (GTP hydrolysis restricted to the interface) has been developed for microtubules (13), which exhibit similar kinetic features for the involvement of GTP hydrolysis in tubulin polymeriation. A more realistic model would incorporate the slight possibility of ATP hydrolysis inside the ATP cap and would include different rates of ATP hydrolysis according to the nature (ATP or ADP) of the neighboring subunits. Such a refinement requires more elaborate Monte Carlo calculations, as in the case of microtubules (13, 14, 17, 23-25). At the critical concentration (cc =.55 um in the chosen example) the Fi filaments coexist in the following proportions (Fig. 2):fo =. 19;fi =.91;f2 =.76;hi>3 =.4. The net rate of growth is ero because the depolymeriation of Fo filaments is counterbalanced by the growth of ATP-capped filaments. This situation is similar to the case of microtubules for which large length fluctuations owing to the phase transition between GTP-capped and uncapped conformations have actually been experimentally observed (15, 16) and analyed within the same model (17). Actin has three, rather than two, regimes: uncapped (c < cc); small cap (cc < c < c'); and large cap (c > c'). A phase transition is not to be expected

5 Biochemistry: Pantaloni et al. between the first two regimes because a small fluctuation suffices to create or abolish the small caps (near c = cc). Although this model does not question the concept of head-to-tail polymeriation (treadmilling) proposed by Wegner (8), it reveals an additional consequence of ATP hydrolysis at the two ends of actin filaments that complicates the analysis of the overall monomer-polymer exchange kinetics at steady state. While treadmilling considers that net depolymeriation at one end (say a, in Fig. 1) compensates for net polymeriation at the other end (b) at steady state, in the present model the transitions between the ATP-capped and the uncapped (Fo) conformations at the same end will result in the depolymeriation of F and lengthening of the Fj (i > ) filaments. In contrast to treadmilling, it is not necessary to have both a and b ends free to observe the preceding reaction. This consideration, then, provides an easy means to determine the relative importance of the two mechanisms of monomer-polymer exchange at steady state. The role of ATP ih actin polymeriation can also be viewed in the following way. The two pairs of ab and cd interactions between ADP-containing subunits in the ADP core of F-actin are strong enough to prevent the dissociation of a subunit from the inside of the filament but the single pair of bonds at the tip of the filament is not strong enough to maintain the polymer in the presence of a low monomer concentration; this is why a large critical concentration (8,4M) is found for ADP-actin. The same is true for the equilibrium homopolymer of ATPactin that has a dissociation constant of 3 ILM. ATP hydrolysis could be associated with the formation of the bond between the a-interaction area on the third subunit from the tip (3' in Fig. 1) and the b-interaction area of the first subunit (1') of the filament. ATP hydrolysis would thus facilitate the first two polymeriation steps of ATP-G-actin and a steady-state ATP cap would maintain dynamically the stability (critical concentration,.55 4M) of an otherwise unstable polymer. CONCLUDING REMARKS In summary, the model for ATP-actin polymeriation developed in this paper accounts for several recent observations that are incompatible with previous models: (i) ATP hydrolysis occurs on filaments subsequent to the elongation step and an ATP-cap is thus maintained at the filament ends; (ii) the rate of filament elongation is not directly proportional to the concentration of ATP-actin monomer; (iii) the time course of elongation is not a true exponential; (iv) the critical concentration for the reversible polymeriation of ATP-actin is larger than the critical concentration of ATP-actin at steady state. The present model proposes the existence of four distinct species of filaments having ATP bound to ero, one, two, or three terminal subunits. The fact that ATP is hydrolyed rapidly at the interface between the ATP cap and the ADP core, but not on the two terminal subunits, promotes the growth of the ADP core, while keeping two ATP subunits at the tip of the filament, over a large range of actin concentration. This mechanism also provides a remarkable Proc. Natl. Acad. Sci. USA 82 (1985) 7211 stability to the filament ends-i.e., a low critical concentration-while the homopolymers of both ATP-actin and of ADP-actin have high critical concentrations. We believe that this model goes a long way toward explaining the role of ATP hydrolysis in the polymeriation of actin. One must recognie, however, that cells contain numerous actin-binding proteins (26-28) that modulate the intrinsic polymeriation properties of the actin molecule. For example, actin-binding proteins may significantly affect ATP hydrolysis by actin, thus altering the distribution of ATP cap lengths, and the length of the ATP cap may, in turn, significantly influence the interaction of actin-binding proteins with filament ends. D.P. was supported in part by the Centre National de la Recherche Scientifique and the Ligue Nationale Francaise Contre le Cancer. 1. Oosawa, F. & Kasai, M. (1962) J. Mol. Biol. 4, Oosawa, F. & Asakura, S. (1975) Thermodynamics of the Polymeriation of Protein (Academic, New York). 3. Wegner, A. & Engel, J. (1975) Biophys. Chem. 3, Tobacman, L. S. & Korn, E. D. (1983) J. Biol. Chem. 258, Frieden, C. (1983) Proc. NatI. Acad. Sci. USA 8, Cooper, J. A., Buhle, E. L., Walker, S. B., Tsong, T. Y. & Pollard, T. D. (1983) Biochemistry 22, Frieden, C. & Goddette, D. (1983) Biochemistry 22, Wegner, A. (1976) J. Mol. Biol. 18, Cooke, R. (1975) Biochemistry 14, Pardee, J. D. & Spudich, J. A. (1982) J. Cell Biol. 93, Pollard, T. 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