Proc. Natl. Acad. Sci. USA Vol. 88, pp. 5562-5566, July 1991
Biophysics
Cooperativity in axonemal motion: Analysis of a four-state, two-site kinetic model (enzyme kinetics/positive cooperativity/mathematical modeling/dynein)
CHARLOTTE K. OMOTO*t, JEFFREY S. *Program in Genetics and Cell Biology, and
PALMER0§, AND MICHAEL E. MOODY*:
*Department of Pure and Applied Mathematics, Washington State University, Pullman, WA 99164-4234
Communicated by Howard C. Berg, February 19, 1991 (received for review November 7, 1990)
ABSTRACT A kinetic model for axonemal motion based upon a four-state mechanochemical cycle of dynein with two active sites is described. Our model analysis determines the pseudo-steady-state concentrations of enzyme species for specified rate constants, most of which are experimentally determined, with given substrate and product concentrations. The proportion of enzyme species in which both active sites are detached from the microtubule (denoted as "both detached"), numerically calculated from the model, appears to be proportional to experimental observations of flagellar beat frequency. This correlation between beat frequency and the both-detached enzyme species is maintained over a wide range of substrate concentrations and exhibited an apparent positive cooperativity at low substrate concentrations, which we call "obligate cooperativity." The unusual obligate cooperativity exhibited by flagellar beat frequency parallels that seen in the calculated proportion of the both-detached enzyme species and is interpreted as a requirement for a molecule of substrate to bind to each active site in a multimeric dynein in order to produce oscillatory motion. Furthermore, the proportion of the bothdetached enzyme species correlates with experimentally observed changes in beat frequency with a nucleotide analog and with product inhibition.
40
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Beat Frequency / [ATP] (hz./iMN)
Movement of eukaryotic cilia and flagella is based upon the cyclical interaction of dynein ATPase with microtubules in the axoneme. In reactivation experiments, flagella are demembranated and move in a solution of known composition. Normally, the concentration of axonemes is so much lower than that of the substrate, MgATP, that the experiment is at pseudo-steady-state. In reactivation experiments, the beat frequency varies as a function of [MgATP] and exhibits saturation kinetics (1). The assumption of pseudo-steadystate is supported by the fact that beat frequency remains constant during the course of such experiments. Production of organized movements that result in a beating axoneme unquestionably involves many complex mechanochemical steps. In spite of this, the observation that beat frequency follows saturation kinetics invites a kinetic approach. At very low substrate concentrations, beat frequency deviates from simple Michaelis-Menten behavior (2-5). This deviation is emphasized by plotting the beat frequency as a function of beat frequency/[ATP] (i.e., in a Eadie-Hofstee plot, Fig. 1). An enzyme with simple saturation kinetics yields a linear Eadie-Hofstee plot with the slope equal to -Km and the y intercept equal to Vmax. A deviation from linearity, such as that in Fig. 1, is evidence of positive cooperativity (e.g., refs. 6 and 7). Positive cooperativity can be visualized and quantified using the Hill plot (8). The Hill plot of the beat frequency data is shown in Fig. 1 Inset. The Hill coefficient, a measure ofcooperativity given by the slope
of the Hill plot, exceeds 1 only at the lower substrate concentrations. For a typical enzyme with positive cooperativity, the maximum Hill coefficient is observed when the enzyme is half-saturated with the ligand (9, 10). A Hill coefficient >1 at the lowest concentration of substrate suggests that more than one molecule of substrate must bind to produce a reaction. This is usually not the case with conventional enzymes (9, 10). That the maximum Hill coefficient in beat frequency is found at the lowest substrate concentrations is thus unusual. One possible interpretation of this apparent positive cooperativity in beat frequency is as a requirement for a molecule of substrate to bind to each active site in a multimeric complex of a mechanochemical enzyme, dynein, in order to produce oscillatory motion. We called this apparent cooperativity "obligate cooperativity." Such an interpretation of beat frequency data prompted our examination of a kinetic model of dynein that uses two active sites. Axonemal dyneins are characterized as large multiprotein complexes with two or three very high molecular weight polypeptides (reviewed in ref. 11). These high molecular
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. §1734 solely to indicate this fact.
Abbreviation: 2-Cl-ATP, 2-chloroadenosine 5'-triphosphate. tTo whom reprint requests should be addressed. §Present address: Dakota State University, Madison, SD.
5562
FIG. 1. Plot of beat frequency as a function of beat frequency/ [ATP] [Eadie-Hofstee (6) or Eadie-Scatchard plot (7)]. Data are replotted from ref. 4 with the permission of the author. Arbacia punctulata sperm was reactivated at 220C. Km in this example was 0.29 mM. (Inset) Hill plot ofthe same beat frequency data. A line with a slope of 1.0 is drawn. The slope for the data below the arrow is 1.45.
Biophysics: Omoto et al.
Proc. NatL. Acad. Sci. USA 88 (1991)
weight polypeptides contain the ATP-binding and hydrolysis site. These high molecular weight polypeptides are associated with the large globular structures that contain the ATP-sensitive microtubule-binding site. The dyneins in a typical flagellum are comprised of inner and outer dynein arms. The outer dynein arms have either two or three high molecular weight polypeptides. The inner dynein arms are multifarious with three distinct dynein complexes, each with two high molecular weight polypeptides (11, 12). Since the role of these dyneins in the determination of beat frequency is not clear, for simplicity our model assumes two active sites.
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concentrations, binding of one active site in a multimeric dynein to microtubules is affected by the state of the other sites (14). However, preliminary versions of this model using a higher rate of association of dynein to microtubules of "tethered" dynein did not significantly affect the results. Thus the final model treats each dynein active site as independent. The high molecular weight polypeptides of dyneins are clearly not identical; however, we simplified our treatment and considered them as indistinguishable. (ii) Onlyfour states ofdynein are used in the current model. The product release step as shown in the model occurs only from the dynein attached to a microtubule. Preliminary versions ofthis model incorporated product release from, and substrate binding to, detached active sites, shown with stippled lines in Fig. 2A. Extensive numerical simulations showed that the proportions of detached active sites not bound to substrate or product was insignificant even when the rate of dynein-P rebinding to microtubules (k3) was as low as 30 s-1 (simulations not shown). (iii) All kinetic coefficients are constant. Undoubtedly, the rates of steps 1, 3, and 4 depend upon the shear or orientation of the dynein with respect to the microtubule-binding site. However, the number of dynein molecules over the length of the whole axoneme is very large. We suggest that the position-dependent differences in rate constants might be averaged for the purposes of this model. (iv) Only one product release step is specified. It is assumed that the product release step corresponds to ADP release since phosphate is not a strong inhibitor of beat frequency (15). Biochemical evidence suggests that phosphate release is relatively fast and that ADP release is rate limiting (16).
METHODS In this model, we treat the axoneme as a homogeneous enzyme system. Our model assumes two active sites for dynein, each governed by the four-state mechanochemical cycle for one active site depicted in Fig. 2A. This cycle is analogous to the Lymn-Taylor model for actomyosin (13). Considering all possible transitions between states, our model requires 10 different enzyme species (Fig. 2B). The set of differential equations describing the model is given in Fig. 2C. The following simplifications and assumptions are incor-
porated into this model. (i) There is a single class of dynein with two independent, indistinguishable active sites. Biochemical and genetic studies of dynein have clearly shown a multiplicity of dyneins in the axoneme. For simplicity, this model uses only a single class of dynein with only two active sites. It is possible that in a dynein with two active sites, the state of one site may affect the rate oftransition of the other site. At low nucleotide
C k4AP 2kAoS 2k.4AoP + k...T,
d dt
A
d dt
=
2k4A12p + 2k-4AoP - k4Ap - k1ApS - k-3Ap- k4ApP + k3Tpo + k-.Tp,
dt2p dt
-
k_4APP 2k4A2p 2k-3A2p + k3T2, -
dt
=
kTTp, + k-2Tpo + 2kAoS - k-4TP - kTS - k2T, + 2k-1D2, - k-_T,
dTp,
-
kjAPS + k-4TP + k.2T2p + k3Dp, k4Tp,
dt
dT2p
B Both Attached (A)
Tethered (T)
AO
Both Detached (D) Ts
(PP)
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dt
k_2T
dt
dt
p
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(
T2P
0
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dS dt
+
k2D, k
k4T2p + k3Ap - k-2Tpo- k4TpoP- kTpoS - k3Tpo + k-,Dp
+ k-3T2, - 2k-2D2p - 2k3D2p
-2Dp, + kTS 2k2D2, -2k-D2,
=
2k-2D2p + 2k2D2, + k-3Tp, - k..Dp, + kTpoS - k2Dp
=
-kS(2Ao+A,+T,+Tpo)+k-.(Dp,+2D2,+§p1+T,)
dt
Eh
k2Tp, - k_3Tp, - k-,Tps
2k.3A2p + k2Tp, + 2k3D2p + k-4TpoP - k-2T2p - k-372p - k4T2p - k3T2p
dt
A2p
-
dt
dt,
P
-
-
k-2Dps - k3Dps
~~~~~~dP dt k,(Ap + Tps + 2A2p + T2p) - -4P(2Ao+
FIG. 2. Four-state kinetic model. (A) Diagram of the four-state mechanochemical cycle for one active site. Each square or circle indicates site of dynein. Squares, active sites bound to a microtubule; circles, active sites detached from a microtubule; S, substrate; P. product. The numbers represent steps in the cycle and are used as subscripts to the kinetic rates for that step. Counterclockwise is the forward direction; clockwise the reverse direction is indicated by the minus sign. The stippled steps were included in a preliminary model but are eliminated in the current model after numerical analysis showed them to be insignificant. (B) Ten different enzyme states possible with two independent indistinguishable active sites, each executing the above four-state cycle. Symbols as in A. Labels for each enzyme species are indicated at the right. The three enzyme species in which both active sites are detached from the microtubule are considered together and labeled Both Detached. (C) Set of differential equations specifying this model. Symbols as in A and B. one active
Biophysics: Omoto et al.
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Table 1. Range of rate constants Rate Event constant ATP-induced k, dissociation ATP hydrolysis k2 ATP synthesis k.2 Dynein-P binding to k3 microtubules Dynein-P release k-3 from microtubules with vanadate Product release from k4 dynein-microtubules Product binding to kL4 dyneinmicrotubules The concentration of microtubules
Proc. Natl. Acad. Sci. USA 88 (1991)
Rate 1.6-4.7 ,tM-s1
Ref(s). 14, 17
55-100 s-1 10-30 s-1 1.6-6 jLM-1-s-
16, 18 16, 19 14
0.6 s-1
18
30-60 s-1 0.1-0.5 ,uM- s1
15
= 1.6-5 mM.
With these assumptions, the concentrations of the various species are numerically determined by the system of differential equations (Fig. 2C) (Livermore solver for ordinary differential equations) using the double-precision com-
of k3 in the range of 20-600 s 1 are conceivable. Accordingly, k3 values of 30-600 s-1 are used. There is currently no good estimate of the rate kL3. Experimental determination of the dissociation of dynein-product intermediate from microtubules has utilized the phosphate analog vanadate and obtained a maximum rate of 0.6 s-1 (20). Since the value for k3 is quite high, using k-3 in the range of 0.1-10 s-1 does not have a significant effect on the proportion of enzyme species calculated. Rates of 1 s1 and 0.1 s1 are used for k-3. The maximum beat frequency is used as a guide for the choice of values for k4. The maximum beat frequency depends upon the cells and conditions used. In our simulations, we use k4 in the range of 30-90 s-1. The reverse rate for the binding of ADP, kL4, is calculated from the K, of MgADP for beat frequency (15) and ATPase activity (16) (0.4 mM) and the forward rate, k4. Using the middle value of 45 s-1 for the forward rate, a value of 0.1 ,uM-1's-1 for kL4 is calculated.
enzyme
puter program.
Table 1 shows the experimentally measured and estimated of rates for the steps in the dynein mechanochemical cycle. The actual rate constants employed in our model analyses are shown in Table 2. The second-order rate constant of dynein dissociation from MAP-free microtubules of 1.6 jM-1 s-1 (14) was used as a starting point for k1. The rate k1 describes the combination of ATP binding to dynein and dissociation of dynein-ATP complex from the microtubule. In in vitro assays, ATP binding and the subsequent dissociation of the dynein are very rapid, with the maximal rate >1000 s-1 (14). Since the aforementioned value for the dissociation rate was measured in vitro using microtubules and dynein free in solution, and our modeling is for the microtubules and dynein confined in the axoneme, we use rates down to 0.5 AM-1 s-1. Simulations 1-3 assume that ATP-dependent dissociation, step 1, is irreversible. Simulations 4-6 assume step 1 to be reversible with a range of kL1 from 1 to 150 s-1. range
The rates k2 and k2 of 100 s-1 and 30 s- (19) were used for the final model calculations. A range of rates, down to 50 s-1 and 10 s-1, respectively, for k2 and kL2 gave qualitatively similar results as long as a similar ratio of k2 to k.2 was maintained. The rate k3 is taken as the product of the experimentally determined second-order association rate constant of dynein to microtubules (1.2-6 x 104 M-1's-1) (14) and the estimated effective concentration of tubulin in the axoneme. The concentration of tubulin in the axoneme, with the central microtubules included, is -1.6 mM. The effective concentration of tubulin in the vicinity of microtubule-binding site of dynein may be as high as 5-10 mM. With this range of values, rates
RESULTS We chose a sufficiently low concentration of enzyme (10 pM) in our model to maintain pseudo-steady-state conditions, even at the lowest substrate concentration of 1 ,M, over a time course of 1-1000 s, the range of times studied experimentally. For [ATP] between 1 and 1000lM, the simulated decrease in [ATP] at 1000 s is limited to =1%. The simulated rate of product formation is linear over 1-1000 s for all [ATP] tested. Fig. 3 displays the proportions of the enzyme species at selected values of substrate concentrations, using the rate constants in row 1 of Table 2. The rate of hydrolysis using the same rate constants exhibits simple Michaelis-Menten behavior with a Km of 54 AM and Vm of 64 nmol/100 s. This corresponds to 5 ,uM min-1mg-i or 64 s-1, assuming a molecular weight of 750,000 per active site. This rate is about 10-fold higher than that measured in isolated dynein, which is as expected (14, 21). We numerically calculate the proportion of enzyme species for which both active sites are detached from microtubules (Both Detached in Fig. 2A) at t = 100 s. The change in the proportion of the both-detached enzyme species with ATP concentration exhibits nonlinear behavior at low substrate concentrations (Fig. 4) similar to that observed in the EadieHofstee plot of beat frequency (Fig. 1). This nonlinearity is not evident from the Eadie-Hofstee plot of all detached active sites-that is, the proportion both-detached and half of the tethered species combined (not shown). Therefore, the apparent positive cooperativity at low substrate concentrations is an effect of requiring that both active sites of a dimeric enzyme be detached, an obligate cooperativity, as opposed to the total proportion of detached active sites per se. The qualitative shape of the Eadie-Hofstee plot for the bothdetached enzyme species at low substrate concentrations is
Table 2. Parameters for simulations Sim. no. 1 2 3
Kinetic parameters for "both-detached"
Rate constant*
ki
k2
k3
k4
k-1
k-2
k-3
k-4
Vmax
Kin, /AM
0.6 100 300 60 0 30 1 0.1 21.6 113 2 100 600 60 0 30 1 0.1 17.8 35 2 100 300 90 0 30 1 40 0.1 31.8 4 2 100 300 60 1 30 0.1 0.1 21.6 38 5 2 100 300 60 10 30 0.1 0.1 21.8 44 6 2 100 300 60 150 30 0.1 0.1 22 107 The initial concentrations of enzyme species and product were zero except for Ao = 10-5 /AM. Substrate concentrations of 1-1000 lOM were used to determine kinetic parameters. Sim. no., simulation number. *First-order rate constants in units of s-1; second-order rate constants in units of uM-' s-1.
Biophysics: Omoto et al.
Proc. Natl. Acad. Sci. USA 88 (1991)
relatively insensitive to changes in many of the rate constants. Extrapolating from the linear region at the higher substrate concentrations of the Eadie-Hofstee plot of the both-detached enzyme species, Vmsa (intercept) and Km (slope) are determined. The results are shown in Table 2 for the selected rate constants. The values of Km estimated by this procedure are similar to the Km for beat frequency. To quantify this obligate cooperativity, the Hill plot of the proportion calculated from our model of both-detached enzyme species is shown in Fig. 4 Inset. As with the Hill plot of the actual beat frequency data shown in Fig. 1 Inset, the Hill plot in Fig. 4 Inset shows a slope >1 at the lowest substrate concentrations. As a further test of this model, we simulate the changes in rates presumed to occur with an ATP analog, 2-chloroadenosine 5'-triphosphate (2-Cl-ATP). Previous experimental work showed that the Km for beat frequency was lower with 2-Cl-ATP (5). It was suggested that the anti-conformation of the nucleotide favored by the analog is the preferred conformation of the nucleotide at the active site. Thus the 2-chloro modification will result in increasing the concentration of the nucleotide in the preferred conformation. This can be simulated in our kinetic model by increasing the rate of k1. Estimation of the dissociation rate of the phosphate analog vanadate (22) suggests that the rate of either k3 or k4 is increased with 2-Cl-ATP. The rates given in Table 2, row 1, result in a Km for the both-detached enzyme species similar to the Km for beat frequency with ATP. The increase in the rate of k1 and k3 shown in Table 2, row 2, decreases the Km and Vsax calculated for the both-detached enzyme species. These changes in parameters are comparable to the changes in parameters for beat frequency observed with 2-Cl-ATP. In that experiment, Michaelis constants for beat frequency with ATP and 2-Cl-ATP were 120 and 32 ,uM, respectively (S). The Vmax for beat frequency with 2-Cl-ATP was -30% less than that for ATP. Increasing the k1 significantly decreases Km yet does not affect Vmax (simulations not shown). Increasing k, and k4 (Table 2, row 3) decreases Km but increases Vmax. These observations suggest that the increased rate of vanadate dissociation with 2-Cl-ATP may result from increasing the rate analogous to k3. As a further check on the model we simulate the effect of ADP, a competitive inhibitor of beat frequency (15). The [ADP] can be manipulated to produce similar beat frequencies using different [ATP]. For example, beat frequencies of
5565
1.8-2.2 Hz are observed for 17-20 tLM MgATP or, alternatively, with 2 mM ADP and 70-90 ILM MgATP (E. Pate and C. J. Brokaw, personal communication). If we use these values for substrate and product concentrations and the rate constants in Table 2, row 1, we obtain from the model the relative proportions of enzyme species shown in Fig. 5. Only the proportion of the both-detached enzyme species is similar between the two conditions. In particular, although the enzyme species A2p, Tps, and T2p also exhibit a nonlinear Eadie-Hofstee plot at low substrate concentrations, the relative proportions of these species are very different for the two cases examined in this paper. These observations further support our hypothesis that the proportion of the both-
detached enzyme species best correlates with beat frequency. As expected for a competitive inhibitor, ADP increased Km but had little effect on Vma, for beat frequency (15). A similar effect on Km and Vma,, for both detached enzyme species is observed in our model (not shown).
DISCUSSION Our four-state, two-site kinetic model for axonemal motion is based on and mimics the conditions of reactivation experiments. We use experimentally determined rate constants from the literature and assumptions for homogeneous enzyme systems in solution. We compare the experimental observations of beat frequency to the relative proportions of the various enzyme species calculated using the model. The proportion of the both-detached enzyme species is strongly correlated with beat frequency with respect to changes in substrate and product concentrations and with changes in rate constants that represent a nucleotide analog. We suggest the term obligate cooperativity to describe the underlying mechanism for the cooperative behavior of beat frequency. The standard mechanism of enzyme cooperativity involves a change in the rate or affinity of one subunit by the binding of substrate by another subunit of a multimeric enzyme. However, when the reaction is mechanochemical, and one of the products of the reaction is motion, we suggest that other mechanisms of cooperativity may be operative. This alternative type of cooperativity may be important for motor molecule function. Our modeling suggests that only four steps of the dyneinmicrotubule mechanochemical cycle are needed to describe a
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FIG. 4. Eadie-Hofstee plot of proportion of the both-detached species numerically calculated from the model using the rate constants in Table 2, row 1. (Inset) Hill plot of the same data. enzyme
5566
Proc. Natl. Acad. Sci. USA 88 (1991)
Biophysics: Omoto et al.
E >'20N C
10
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Ap
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Enzyme Species FIG. 5. Change in the percentage of different enzyme species with ADP inhibition. Rate constant as in Table 2, row 1. 11, With 17 AM ATP, no ADP; ED, with 70 jsM ATP and 2 mM ADP. Note that only the both-detached enzyme species have similar percentages under the two conditions.
at least one parameter of axonemal motion, beat frequency. Thus we suggest that in the axoneme, the dissociation of the occurs only from dynein-product complex attached to the microtubule. This interpretation further supports the notion that in the axoneme, dynein ATPase is tightly coupled to interaction with microtubules (23). The observation that the ATPase activity in the model is -10x higher than experimentally observed ATPase activity of isolated dynein ismconsistent with our current understanding of dynein mechanochemical coupling. The rate-limiting step, that of product release, in the ATPase cycle of isolated dynein is significant (6 s'1) and microtubules activate dynein ATPase activity about 6- to 10-fold (14, 21). The model, for simplicity, uses two active sites. The good fit of the both detached enzyme species to beat frequency using only two sites suggests that the dyneins involved in beat frequency determination have two active sites. The experimental data we use were all derived from sea urchin axonemes, which have two high molecular weight polypeptides in the outer dynein arms. Chlamydomonas have three high molecular weight polypeptides in the outer dynein arms. If the obligate cooperativity seen in beat frequency is related to the number of subunits in outer dynein arms, we might expect a higher Hill coefficient with Chlamydomonas axonemes. The slope of the Hill plot of beat frequency (Fig. 1 Inset) becomes noticeably greater than 1 at an ordinate value below -1. The lowest [MgATP] in published data on the ATP dependence of beat frequency of Chlamydomonas have values of log [V / (Vmax - v)] of approximately -0.7 and -0.4 (24). This may not be low enough to observe the obligate cooperativity. Our work prompts an extension ofreactivation studies of wild-type and mutant Chlamydomonas missing the outer dynein arms (25) at lower [MgATP]. The kinetic rates used in the model suggest insights into steps in the axonemal dynein-microtubule mechanochemical cycle. The values for Km using a rate (0.6 ,uM-1 s-) much lower than that observed in solution (1.6-4.7 ,uM- s-1) or by increasing the rate kL1 suggest that in the axoneme step 1 is
product (ADP)
significantly reversible (26) or that the dissociation rate is reduced in the confines of the axoneme. What is the basis for the correlation between the proportion of the both-detached enzyme species and beat frequency? Obviously, dynein whose active sites are detached from the microtubule cannot exert force. Yet beat frequency may not be limited by force. Even at the highest substrate concentrations (1000 AuM, Fig. 3), although the proportions of the both-detached species are high, the proportions of Tp. and A2p, enzyme species likely to be involved in force production, are even higher. In this kinetic model, the changes in the proportion of the both-detached enzyme species are in concert with those of other enzyme species. The significance of the correlation between the both-detached species and beat frequency must be further tested by manipulation of proportions of enzyme species by the use of nucleotide analogs and products and their effect on beat frequency. The model introduced here is our initial attempt to use kinetic modeling to describe axonemal motion. It may be that only beat frequency is amenable to such analysis. It has been noted previously that beat frequency is modulated independently of other parameters of movement (i.e., ref. 27). Conversely, other parameters of motion, such as sliding velocity, wavelength, bend angle, etc., may be correlated with proportions of particular enzyme species. Evaluation of this model for other parameters of movement will require careful quantification of these parameters of motion over a range of substrate and product concentrations. We thank Dr. S. Penningroth for providing the data plotted in Fig. 1 and Dr. E. Pate and Dr. C. J. Brokaw for providing unpublished data and for critically reading the manuscript. C.K.O. thanks Dr. W. W. Cleland for helpful criticism and discussions. M.E.M. was partially supported by National Science Foundation Grant BSR 8700680. This research was supported in part by National Science Foundation Grant DCB-8918108 to C.K.O. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.
17. 18. 19. 20. 21. 22.
23. 24. 25. 26. 27.
Brokaw, C. J. (1%7) Science 156, 76-78. Gibbons, B. H. & Gibbons, I. R.' (1972) J. Cell Biol. 54, 75-97. Brokaw, C. J. (1975) J. Exp. Biol. 62, 701-719. Penningroth, S. M. & Peterson, D. D. (1986) Cell Motil. Cytoskeleton 6, 586-594. Omoto, C. K. & Brokaw, C. J. (1989) Cell Motil. Cytoskeleton 13, 239-244. Fersht, A. (1985) Enzyme Structure and Mechanism (Freeman, New York), 2nd Ed. Segel, I. H. (1975) Enzyme Kinetics (Wiley, New York), pp. 346384. Hill, R. (1925) Proc. R. Soc. London Ser. B 100, 419. Cleland, W. W. (1970) in The Enzymes, ed. Boyer, P. D. (Academic, New York), Vol. 2, pp. 1-65. Neet, K. E. (1980) Methods Enzymol. 64, 139-192. Piperno, G. (1990) Cell Motil. Cytoskeleton 17, 147-149. Piperno, G., Ramanis, E. F., Smith, E. F. & Sale, W. S. (1990) J. Cell Biol. 110, 379-389. Lymn, R. W. & Taylor, E. W. (1971) Biochemistry 10, 4617-4624. Omoto, C. K. & Johnson, K. A. (1986) Biochemistry 25, 419-427. Okuno, M. & Brokaw, C. J. (1979) J. Cell Sci. 38, 105-123. Holzbaur, E. L. F. & Johnson, K. A. (1989) Biochemistry 28, 5577-5585. Porter, M. E. & Johnson, K. A. (1983) J. Biol. Chem. 258, 65826587. Johnson, K. A. (1983) J. Biol. Chem. 258, 13825-13832. Holzbaur, E. L. F. & Johnson, K. A. (1986) Biochemistry 25, 428-434. Shimizu, T. & Johnson, K. A. (1983) J. Biol. Chem. 258, 1383313840. Gibbons, I. R. & Fronk, E. (1979) J. Biol. Chem. 254, 187-1%. Omoto, C. K. & Nakamaye, K. L. (1989) Biochim. Biophys. Acta 999, 221-224. Omoto, C. K. (1989) J. Theor. Biol. 137, 163-169. Kamiya, R. & Okamoto, M. (1985) J. Cell Sci. 74, 181-191. Kamiya, R. (1988) J. Cell Biol. 107, 2253-2258. Pate, E. & Cooke, R. (1989) J. Muscle Res. Cell Motil. 10, 181-1%. Brokaw, C. J. (1977) J. Exp. Biol. 71, 229-240.
Biophysics
Cooperativity in axonemal motion: Analysis of a four-state, two-site kinetic model (enzyme kinetics/positive cooperativity/mathematical modeling/dynein)
CHARLOTTE K. OMOTO*t, JEFFREY S. *Program in Genetics and Cell Biology, and
PALMER0§, AND MICHAEL E. MOODY*:
*Department of Pure and Applied Mathematics, Washington State University, Pullman, WA 99164-4234
Communicated by Howard C. Berg, February 19, 1991 (received for review November 7, 1990)
ABSTRACT A kinetic model for axonemal motion based upon a four-state mechanochemical cycle of dynein with two active sites is described. Our model analysis determines the pseudo-steady-state concentrations of enzyme species for specified rate constants, most of which are experimentally determined, with given substrate and product concentrations. The proportion of enzyme species in which both active sites are detached from the microtubule (denoted as "both detached"), numerically calculated from the model, appears to be proportional to experimental observations of flagellar beat frequency. This correlation between beat frequency and the both-detached enzyme species is maintained over a wide range of substrate concentrations and exhibited an apparent positive cooperativity at low substrate concentrations, which we call "obligate cooperativity." The unusual obligate cooperativity exhibited by flagellar beat frequency parallels that seen in the calculated proportion of the both-detached enzyme species and is interpreted as a requirement for a molecule of substrate to bind to each active site in a multimeric dynein in order to produce oscillatory motion. Furthermore, the proportion of the bothdetached enzyme species correlates with experimentally observed changes in beat frequency with a nucleotide analog and with product inhibition.
40
-
-l-
o
0
o
N
30
-
1 O
a0
-c
-V.
(-)Ca)
~
1110
-d
100
1000
[ATP] (AM)
20 LL.
1
4-
0)
co
10 Cnv004. 0.025
L 0.075 r.
I
r-r o 1E u"
M
I
0.1 25
Beat Frequency / [ATP] (hz./iMN)
Movement of eukaryotic cilia and flagella is based upon the cyclical interaction of dynein ATPase with microtubules in the axoneme. In reactivation experiments, flagella are demembranated and move in a solution of known composition. Normally, the concentration of axonemes is so much lower than that of the substrate, MgATP, that the experiment is at pseudo-steady-state. In reactivation experiments, the beat frequency varies as a function of [MgATP] and exhibits saturation kinetics (1). The assumption of pseudo-steadystate is supported by the fact that beat frequency remains constant during the course of such experiments. Production of organized movements that result in a beating axoneme unquestionably involves many complex mechanochemical steps. In spite of this, the observation that beat frequency follows saturation kinetics invites a kinetic approach. At very low substrate concentrations, beat frequency deviates from simple Michaelis-Menten behavior (2-5). This deviation is emphasized by plotting the beat frequency as a function of beat frequency/[ATP] (i.e., in a Eadie-Hofstee plot, Fig. 1). An enzyme with simple saturation kinetics yields a linear Eadie-Hofstee plot with the slope equal to -Km and the y intercept equal to Vmax. A deviation from linearity, such as that in Fig. 1, is evidence of positive cooperativity (e.g., refs. 6 and 7). Positive cooperativity can be visualized and quantified using the Hill plot (8). The Hill plot of the beat frequency data is shown in Fig. 1 Inset. The Hill coefficient, a measure ofcooperativity given by the slope
of the Hill plot, exceeds 1 only at the lower substrate concentrations. For a typical enzyme with positive cooperativity, the maximum Hill coefficient is observed when the enzyme is half-saturated with the ligand (9, 10). A Hill coefficient >1 at the lowest concentration of substrate suggests that more than one molecule of substrate must bind to produce a reaction. This is usually not the case with conventional enzymes (9, 10). That the maximum Hill coefficient in beat frequency is found at the lowest substrate concentrations is thus unusual. One possible interpretation of this apparent positive cooperativity in beat frequency is as a requirement for a molecule of substrate to bind to each active site in a multimeric complex of a mechanochemical enzyme, dynein, in order to produce oscillatory motion. We called this apparent cooperativity "obligate cooperativity." Such an interpretation of beat frequency data prompted our examination of a kinetic model of dynein that uses two active sites. Axonemal dyneins are characterized as large multiprotein complexes with two or three very high molecular weight polypeptides (reviewed in ref. 11). These high molecular
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. §1734 solely to indicate this fact.
Abbreviation: 2-Cl-ATP, 2-chloroadenosine 5'-triphosphate. tTo whom reprint requests should be addressed. §Present address: Dakota State University, Madison, SD.
5562
FIG. 1. Plot of beat frequency as a function of beat frequency/ [ATP] [Eadie-Hofstee (6) or Eadie-Scatchard plot (7)]. Data are replotted from ref. 4 with the permission of the author. Arbacia punctulata sperm was reactivated at 220C. Km in this example was 0.29 mM. (Inset) Hill plot ofthe same beat frequency data. A line with a slope of 1.0 is drawn. The slope for the data below the arrow is 1.45.
Biophysics: Omoto et al.
Proc. NatL. Acad. Sci. USA 88 (1991)
weight polypeptides contain the ATP-binding and hydrolysis site. These high molecular weight polypeptides are associated with the large globular structures that contain the ATP-sensitive microtubule-binding site. The dyneins in a typical flagellum are comprised of inner and outer dynein arms. The outer dynein arms have either two or three high molecular weight polypeptides. The inner dynein arms are multifarious with three distinct dynein complexes, each with two high molecular weight polypeptides (11, 12). Since the role of these dyneins in the determination of beat frequency is not clear, for simplicity our model assumes two active sites.
5563
concentrations, binding of one active site in a multimeric dynein to microtubules is affected by the state of the other sites (14). However, preliminary versions of this model using a higher rate of association of dynein to microtubules of "tethered" dynein did not significantly affect the results. Thus the final model treats each dynein active site as independent. The high molecular weight polypeptides of dyneins are clearly not identical; however, we simplified our treatment and considered them as indistinguishable. (ii) Onlyfour states ofdynein are used in the current model. The product release step as shown in the model occurs only from the dynein attached to a microtubule. Preliminary versions ofthis model incorporated product release from, and substrate binding to, detached active sites, shown with stippled lines in Fig. 2A. Extensive numerical simulations showed that the proportions of detached active sites not bound to substrate or product was insignificant even when the rate of dynein-P rebinding to microtubules (k3) was as low as 30 s-1 (simulations not shown). (iii) All kinetic coefficients are constant. Undoubtedly, the rates of steps 1, 3, and 4 depend upon the shear or orientation of the dynein with respect to the microtubule-binding site. However, the number of dynein molecules over the length of the whole axoneme is very large. We suggest that the position-dependent differences in rate constants might be averaged for the purposes of this model. (iv) Only one product release step is specified. It is assumed that the product release step corresponds to ADP release since phosphate is not a strong inhibitor of beat frequency (15). Biochemical evidence suggests that phosphate release is relatively fast and that ADP release is rate limiting (16).
METHODS In this model, we treat the axoneme as a homogeneous enzyme system. Our model assumes two active sites for dynein, each governed by the four-state mechanochemical cycle for one active site depicted in Fig. 2A. This cycle is analogous to the Lymn-Taylor model for actomyosin (13). Considering all possible transitions between states, our model requires 10 different enzyme species (Fig. 2B). The set of differential equations describing the model is given in Fig. 2C. The following simplifications and assumptions are incor-
porated into this model. (i) There is a single class of dynein with two independent, indistinguishable active sites. Biochemical and genetic studies of dynein have clearly shown a multiplicity of dyneins in the axoneme. For simplicity, this model uses only a single class of dynein with only two active sites. It is possible that in a dynein with two active sites, the state of one site may affect the rate oftransition of the other site. At low nucleotide
C k4AP 2kAoS 2k.4AoP + k...T,
d dt
A
d dt
=
2k4A12p + 2k-4AoP - k4Ap - k1ApS - k-3Ap- k4ApP + k3Tpo + k-.Tp,
dt2p dt
-
k_4APP 2k4A2p 2k-3A2p + k3T2, -
dt
=
kTTp, + k-2Tpo + 2kAoS - k-4TP - kTS - k2T, + 2k-1D2, - k-_T,
dTp,
-
kjAPS + k-4TP + k.2T2p + k3Dp, k4Tp,
dt
dT2p
B Both Attached (A)
Tethered (T)
AO
Both Detached (D) Ts
(PP)
D2p
dt
k_2T
dt
dt
p
dDps
(
T2P
0
DPS
dS dt
+
k2D, k
k4T2p + k3Ap - k-2Tpo- k4TpoP- kTpoS - k3Tpo + k-,Dp
+ k-3T2, - 2k-2D2p - 2k3D2p
-2Dp, + kTS 2k2D2, -2k-D2,
=
2k-2D2p + 2k2D2, + k-3Tp, - k..Dp, + kTpoS - k2Dp
=
-kS(2Ao+A,+T,+Tpo)+k-.(Dp,+2D2,+§p1+T,)
dt
Eh
k2Tp, - k_3Tp, - k-,Tps
2k.3A2p + k2Tp, + 2k3D2p + k-4TpoP - k-2T2p - k-372p - k4T2p - k3T2p
dt
A2p
-
dt
dt,
P
-
-
k-2Dps - k3Dps
~~~~~~dP dt k,(Ap + Tps + 2A2p + T2p) - -4P(2Ao+
FIG. 2. Four-state kinetic model. (A) Diagram of the four-state mechanochemical cycle for one active site. Each square or circle indicates site of dynein. Squares, active sites bound to a microtubule; circles, active sites detached from a microtubule; S, substrate; P. product. The numbers represent steps in the cycle and are used as subscripts to the kinetic rates for that step. Counterclockwise is the forward direction; clockwise the reverse direction is indicated by the minus sign. The stippled steps were included in a preliminary model but are eliminated in the current model after numerical analysis showed them to be insignificant. (B) Ten different enzyme states possible with two independent indistinguishable active sites, each executing the above four-state cycle. Symbols as in A. Labels for each enzyme species are indicated at the right. The three enzyme species in which both active sites are detached from the microtubule are considered together and labeled Both Detached. (C) Set of differential equations specifying this model. Symbols as in A and B. one active
Biophysics: Omoto et al.
5564
Table 1. Range of rate constants Rate Event constant ATP-induced k, dissociation ATP hydrolysis k2 ATP synthesis k.2 Dynein-P binding to k3 microtubules Dynein-P release k-3 from microtubules with vanadate Product release from k4 dynein-microtubules Product binding to kL4 dyneinmicrotubules The concentration of microtubules
Proc. Natl. Acad. Sci. USA 88 (1991)
Rate 1.6-4.7 ,tM-s1
Ref(s). 14, 17
55-100 s-1 10-30 s-1 1.6-6 jLM-1-s-
16, 18 16, 19 14
0.6 s-1
18
30-60 s-1 0.1-0.5 ,uM- s1
15
= 1.6-5 mM.
With these assumptions, the concentrations of the various species are numerically determined by the system of differential equations (Fig. 2C) (Livermore solver for ordinary differential equations) using the double-precision com-
of k3 in the range of 20-600 s 1 are conceivable. Accordingly, k3 values of 30-600 s-1 are used. There is currently no good estimate of the rate kL3. Experimental determination of the dissociation of dynein-product intermediate from microtubules has utilized the phosphate analog vanadate and obtained a maximum rate of 0.6 s-1 (20). Since the value for k3 is quite high, using k-3 in the range of 0.1-10 s-1 does not have a significant effect on the proportion of enzyme species calculated. Rates of 1 s1 and 0.1 s1 are used for k-3. The maximum beat frequency is used as a guide for the choice of values for k4. The maximum beat frequency depends upon the cells and conditions used. In our simulations, we use k4 in the range of 30-90 s-1. The reverse rate for the binding of ADP, kL4, is calculated from the K, of MgADP for beat frequency (15) and ATPase activity (16) (0.4 mM) and the forward rate, k4. Using the middle value of 45 s-1 for the forward rate, a value of 0.1 ,uM-1's-1 for kL4 is calculated.
enzyme
puter program.
Table 1 shows the experimentally measured and estimated of rates for the steps in the dynein mechanochemical cycle. The actual rate constants employed in our model analyses are shown in Table 2. The second-order rate constant of dynein dissociation from MAP-free microtubules of 1.6 jM-1 s-1 (14) was used as a starting point for k1. The rate k1 describes the combination of ATP binding to dynein and dissociation of dynein-ATP complex from the microtubule. In in vitro assays, ATP binding and the subsequent dissociation of the dynein are very rapid, with the maximal rate >1000 s-1 (14). Since the aforementioned value for the dissociation rate was measured in vitro using microtubules and dynein free in solution, and our modeling is for the microtubules and dynein confined in the axoneme, we use rates down to 0.5 AM-1 s-1. Simulations 1-3 assume that ATP-dependent dissociation, step 1, is irreversible. Simulations 4-6 assume step 1 to be reversible with a range of kL1 from 1 to 150 s-1. range
The rates k2 and k2 of 100 s-1 and 30 s- (19) were used for the final model calculations. A range of rates, down to 50 s-1 and 10 s-1, respectively, for k2 and kL2 gave qualitatively similar results as long as a similar ratio of k2 to k.2 was maintained. The rate k3 is taken as the product of the experimentally determined second-order association rate constant of dynein to microtubules (1.2-6 x 104 M-1's-1) (14) and the estimated effective concentration of tubulin in the axoneme. The concentration of tubulin in the axoneme, with the central microtubules included, is -1.6 mM. The effective concentration of tubulin in the vicinity of microtubule-binding site of dynein may be as high as 5-10 mM. With this range of values, rates
RESULTS We chose a sufficiently low concentration of enzyme (10 pM) in our model to maintain pseudo-steady-state conditions, even at the lowest substrate concentration of 1 ,M, over a time course of 1-1000 s, the range of times studied experimentally. For [ATP] between 1 and 1000lM, the simulated decrease in [ATP] at 1000 s is limited to =1%. The simulated rate of product formation is linear over 1-1000 s for all [ATP] tested. Fig. 3 displays the proportions of the enzyme species at selected values of substrate concentrations, using the rate constants in row 1 of Table 2. The rate of hydrolysis using the same rate constants exhibits simple Michaelis-Menten behavior with a Km of 54 AM and Vm of 64 nmol/100 s. This corresponds to 5 ,uM min-1mg-i or 64 s-1, assuming a molecular weight of 750,000 per active site. This rate is about 10-fold higher than that measured in isolated dynein, which is as expected (14, 21). We numerically calculate the proportion of enzyme species for which both active sites are detached from microtubules (Both Detached in Fig. 2A) at t = 100 s. The change in the proportion of the both-detached enzyme species with ATP concentration exhibits nonlinear behavior at low substrate concentrations (Fig. 4) similar to that observed in the EadieHofstee plot of beat frequency (Fig. 1). This nonlinearity is not evident from the Eadie-Hofstee plot of all detached active sites-that is, the proportion both-detached and half of the tethered species combined (not shown). Therefore, the apparent positive cooperativity at low substrate concentrations is an effect of requiring that both active sites of a dimeric enzyme be detached, an obligate cooperativity, as opposed to the total proportion of detached active sites per se. The qualitative shape of the Eadie-Hofstee plot for the bothdetached enzyme species at low substrate concentrations is
Table 2. Parameters for simulations Sim. no. 1 2 3
Kinetic parameters for "both-detached"
Rate constant*
ki
k2
k3
k4
k-1
k-2
k-3
k-4
Vmax
Kin, /AM
0.6 100 300 60 0 30 1 0.1 21.6 113 2 100 600 60 0 30 1 0.1 17.8 35 2 100 300 90 0 30 1 40 0.1 31.8 4 2 100 300 60 1 30 0.1 0.1 21.6 38 5 2 100 300 60 10 30 0.1 0.1 21.8 44 6 2 100 300 60 150 30 0.1 0.1 22 107 The initial concentrations of enzyme species and product were zero except for Ao = 10-5 /AM. Substrate concentrations of 1-1000 lOM were used to determine kinetic parameters. Sim. no., simulation number. *First-order rate constants in units of s-1; second-order rate constants in units of uM-' s-1.
Biophysics: Omoto et al.
Proc. Natl. Acad. Sci. USA 88 (1991)
relatively insensitive to changes in many of the rate constants. Extrapolating from the linear region at the higher substrate concentrations of the Eadie-Hofstee plot of the both-detached enzyme species, Vmsa (intercept) and Km (slope) are determined. The results are shown in Table 2 for the selected rate constants. The values of Km estimated by this procedure are similar to the Km for beat frequency. To quantify this obligate cooperativity, the Hill plot of the proportion calculated from our model of both-detached enzyme species is shown in Fig. 4 Inset. As with the Hill plot of the actual beat frequency data shown in Fig. 1 Inset, the Hill plot in Fig. 4 Inset shows a slope >1 at the lowest substrate concentrations. As a further test of this model, we simulate the changes in rates presumed to occur with an ATP analog, 2-chloroadenosine 5'-triphosphate (2-Cl-ATP). Previous experimental work showed that the Km for beat frequency was lower with 2-Cl-ATP (5). It was suggested that the anti-conformation of the nucleotide favored by the analog is the preferred conformation of the nucleotide at the active site. Thus the 2-chloro modification will result in increasing the concentration of the nucleotide in the preferred conformation. This can be simulated in our kinetic model by increasing the rate of k1. Estimation of the dissociation rate of the phosphate analog vanadate (22) suggests that the rate of either k3 or k4 is increased with 2-Cl-ATP. The rates given in Table 2, row 1, result in a Km for the both-detached enzyme species similar to the Km for beat frequency with ATP. The increase in the rate of k1 and k3 shown in Table 2, row 2, decreases the Km and Vsax calculated for the both-detached enzyme species. These changes in parameters are comparable to the changes in parameters for beat frequency observed with 2-Cl-ATP. In that experiment, Michaelis constants for beat frequency with ATP and 2-Cl-ATP were 120 and 32 ,uM, respectively (S). The Vmax for beat frequency with 2-Cl-ATP was -30% less than that for ATP. Increasing the k1 significantly decreases Km yet does not affect Vmax (simulations not shown). Increasing k, and k4 (Table 2, row 3) decreases Km but increases Vmax. These observations suggest that the increased rate of vanadate dissociation with 2-Cl-ATP may result from increasing the rate analogous to k3. As a further check on the model we simulate the effect of ADP, a competitive inhibitor of beat frequency (15). The [ADP] can be manipulated to produce similar beat frequencies using different [ATP]. For example, beat frequencies of
5565
1.8-2.2 Hz are observed for 17-20 tLM MgATP or, alternatively, with 2 mM ADP and 70-90 ILM MgATP (E. Pate and C. J. Brokaw, personal communication). If we use these values for substrate and product concentrations and the rate constants in Table 2, row 1, we obtain from the model the relative proportions of enzyme species shown in Fig. 5. Only the proportion of the both-detached enzyme species is similar between the two conditions. In particular, although the enzyme species A2p, Tps, and T2p also exhibit a nonlinear Eadie-Hofstee plot at low substrate concentrations, the relative proportions of these species are very different for the two cases examined in this paper. These observations further support our hypothesis that the proportion of the both-
detached enzyme species best correlates with beat frequency. As expected for a competitive inhibitor, ADP increased Km but had little effect on Vma, for beat frequency (15). A similar effect on Km and Vma,, for both detached enzyme species is observed in our model (not shown).
DISCUSSION Our four-state, two-site kinetic model for axonemal motion is based on and mimics the conditions of reactivation experiments. We use experimentally determined rate constants from the literature and assumptions for homogeneous enzyme systems in solution. We compare the experimental observations of beat frequency to the relative proportions of the various enzyme species calculated using the model. The proportion of the both-detached enzyme species is strongly correlated with beat frequency with respect to changes in substrate and product concentrations and with changes in rate constants that represent a nucleotide analog. We suggest the term obligate cooperativity to describe the underlying mechanism for the cooperative behavior of beat frequency. The standard mechanism of enzyme cooperativity involves a change in the rate or affinity of one subunit by the binding of substrate by another subunit of a multimeric enzyme. However, when the reaction is mechanochemical, and one of the products of the reaction is motion, we suggest that other mechanisms of cooperativity may be operative. This alternative type of cooperativity may be important for motor molecule function. Our modeling suggests that only four steps of the dyneinmicrotubule mechanochemical cycle are needed to describe a
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FIG. 3. Proportion of enzyme species over a range of substrate concentrations using rate constants in Table 2, row 1.
100
[ATPI (AM)
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10
FIG. 4. Eadie-Hofstee plot of proportion of the both-detached species numerically calculated from the model using the rate constants in Table 2, row 1. (Inset) Hill plot of the same data. enzyme
5566
Proc. Natl. Acad. Sci. USA 88 (1991)
Biophysics: Omoto et al.
E >'20N C
10
A',
Ap
"2p
Ts
Tps T2p
'po
~~~~~Both
Detached
Enzyme Species FIG. 5. Change in the percentage of different enzyme species with ADP inhibition. Rate constant as in Table 2, row 1. 11, With 17 AM ATP, no ADP; ED, with 70 jsM ATP and 2 mM ADP. Note that only the both-detached enzyme species have similar percentages under the two conditions.
at least one parameter of axonemal motion, beat frequency. Thus we suggest that in the axoneme, the dissociation of the occurs only from dynein-product complex attached to the microtubule. This interpretation further supports the notion that in the axoneme, dynein ATPase is tightly coupled to interaction with microtubules (23). The observation that the ATPase activity in the model is -10x higher than experimentally observed ATPase activity of isolated dynein ismconsistent with our current understanding of dynein mechanochemical coupling. The rate-limiting step, that of product release, in the ATPase cycle of isolated dynein is significant (6 s'1) and microtubules activate dynein ATPase activity about 6- to 10-fold (14, 21). The model, for simplicity, uses two active sites. The good fit of the both detached enzyme species to beat frequency using only two sites suggests that the dyneins involved in beat frequency determination have two active sites. The experimental data we use were all derived from sea urchin axonemes, which have two high molecular weight polypeptides in the outer dynein arms. Chlamydomonas have three high molecular weight polypeptides in the outer dynein arms. If the obligate cooperativity seen in beat frequency is related to the number of subunits in outer dynein arms, we might expect a higher Hill coefficient with Chlamydomonas axonemes. The slope of the Hill plot of beat frequency (Fig. 1 Inset) becomes noticeably greater than 1 at an ordinate value below -1. The lowest [MgATP] in published data on the ATP dependence of beat frequency of Chlamydomonas have values of log [V / (Vmax - v)] of approximately -0.7 and -0.4 (24). This may not be low enough to observe the obligate cooperativity. Our work prompts an extension ofreactivation studies of wild-type and mutant Chlamydomonas missing the outer dynein arms (25) at lower [MgATP]. The kinetic rates used in the model suggest insights into steps in the axonemal dynein-microtubule mechanochemical cycle. The values for Km using a rate (0.6 ,uM-1 s-) much lower than that observed in solution (1.6-4.7 ,uM- s-1) or by increasing the rate kL1 suggest that in the axoneme step 1 is
product (ADP)
significantly reversible (26) or that the dissociation rate is reduced in the confines of the axoneme. What is the basis for the correlation between the proportion of the both-detached enzyme species and beat frequency? Obviously, dynein whose active sites are detached from the microtubule cannot exert force. Yet beat frequency may not be limited by force. Even at the highest substrate concentrations (1000 AuM, Fig. 3), although the proportions of the both-detached species are high, the proportions of Tp. and A2p, enzyme species likely to be involved in force production, are even higher. In this kinetic model, the changes in the proportion of the both-detached enzyme species are in concert with those of other enzyme species. The significance of the correlation between the both-detached species and beat frequency must be further tested by manipulation of proportions of enzyme species by the use of nucleotide analogs and products and their effect on beat frequency. The model introduced here is our initial attempt to use kinetic modeling to describe axonemal motion. It may be that only beat frequency is amenable to such analysis. It has been noted previously that beat frequency is modulated independently of other parameters of movement (i.e., ref. 27). Conversely, other parameters of motion, such as sliding velocity, wavelength, bend angle, etc., may be correlated with proportions of particular enzyme species. Evaluation of this model for other parameters of movement will require careful quantification of these parameters of motion over a range of substrate and product concentrations. We thank Dr. S. Penningroth for providing the data plotted in Fig. 1 and Dr. E. Pate and Dr. C. J. Brokaw for providing unpublished data and for critically reading the manuscript. C.K.O. thanks Dr. W. W. Cleland for helpful criticism and discussions. M.E.M. was partially supported by National Science Foundation Grant BSR 8700680. This research was supported in part by National Science Foundation Grant DCB-8918108 to C.K.O. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.
17. 18. 19. 20. 21. 22.
23. 24. 25. 26. 27.
Brokaw, C. J. (1%7) Science 156, 76-78. Gibbons, B. H. & Gibbons, I. R.' (1972) J. Cell Biol. 54, 75-97. Brokaw, C. J. (1975) J. Exp. Biol. 62, 701-719. Penningroth, S. M. & Peterson, D. D. (1986) Cell Motil. Cytoskeleton 6, 586-594. Omoto, C. K. & Brokaw, C. J. (1989) Cell Motil. Cytoskeleton 13, 239-244. Fersht, A. (1985) Enzyme Structure and Mechanism (Freeman, New York), 2nd Ed. Segel, I. H. (1975) Enzyme Kinetics (Wiley, New York), pp. 346384. Hill, R. (1925) Proc. R. Soc. London Ser. B 100, 419. Cleland, W. W. (1970) in The Enzymes, ed. Boyer, P. D. (Academic, New York), Vol. 2, pp. 1-65. Neet, K. E. (1980) Methods Enzymol. 64, 139-192. Piperno, G. (1990) Cell Motil. Cytoskeleton 17, 147-149. Piperno, G., Ramanis, E. F., Smith, E. F. & Sale, W. S. (1990) J. Cell Biol. 110, 379-389. Lymn, R. W. & Taylor, E. W. (1971) Biochemistry 10, 4617-4624. Omoto, C. K. & Johnson, K. A. (1986) Biochemistry 25, 419-427. Okuno, M. & Brokaw, C. J. (1979) J. Cell Sci. 38, 105-123. Holzbaur, E. L. F. & Johnson, K. A. (1989) Biochemistry 28, 5577-5585. Porter, M. E. & Johnson, K. A. (1983) J. Biol. Chem. 258, 65826587. Johnson, K. A. (1983) J. Biol. Chem. 258, 13825-13832. Holzbaur, E. L. F. & Johnson, K. A. (1986) Biochemistry 25, 428-434. Shimizu, T. & Johnson, K. A. (1983) J. Biol. Chem. 258, 1383313840. Gibbons, I. R. & Fronk, E. (1979) J. Biol. Chem. 254, 187-1%. Omoto, C. K. & Nakamaye, K. L. (1989) Biochim. Biophys. Acta 999, 221-224. Omoto, C. K. (1989) J. Theor. Biol. 137, 163-169. Kamiya, R. & Okamoto, M. (1985) J. Cell Sci. 74, 181-191. Kamiya, R. (1988) J. Cell Biol. 107, 2253-2258. Pate, E. & Cooke, R. (1989) J. Muscle Res. Cell Motil. 10, 181-1%. Brokaw, C. J. (1977) J. Exp. Biol. 71, 229-240.
