The Kinetics of Chain Exchange in Two-Chain Coiled Coils: CYCY- and @@-Tropomyosin SUM10 OZEKI, TADASHI KATO, MARILYN EMERSON HOLTZER, and ALFRED HOLTZER* Depdrtiwiit ot Chemistry, Washington University, St. Louis, Missouri 631 30
SYNOPSIS
Measurements are presented o n t h e time course of chain exchange among two-chain ahelical coiled coils of rabbit tropomyosin. All experiments are in a regime (temperature, protein concentration) in which coiled-coil dimers are t h e predominant species. Self-exchange in rue-tropomyosin was investigated by mixing aa species with a*a*, t h e asterisk designating a n a-chain whose lone sulfhydryl (C190) has been blocked by carboxyamidomethylation. T h e overall process aa a*a* 2 2aa* is followed by measurement of t h e fraction ( h )of aa* species as a function of time. Similarly, self-exchange in PP-tropomyosin p*P* F? 2pp*, in which p* is examined by measurements of t h e overall process: /7/3 signifies a P-chain blocked a t both sulfhydryls (C36 a n d C190). T h e observed time course , h ( z) for both chains is well fit by t h e first-order equation: h ( t ) = h (a) ( 1 - e - k l t )with Y 0.5. T h i s long-time limit is a s expected for self-exchange, a n d agrees with experiments t h a t attain equilibrium after slow cooling of thermally dissociated a n d unfolded chains. T h e simplest consonant mechanism is chain exchange by rate-limiting dissociation of dimers followed by random reassociation. Kinetic analysis shows k , t o be t h e rate constant for t h e chain dissociation step, a quantity not previously measured for any coiled coil. This rate constant for p/3 species is about a n order of magnitude greater t h a n for aa. In both, t h e activation enthalpy a n d entropy are very large, suggesting t h a t activation t o a n extensively ( >50% ) unfolded species necessarily precedes dissociation. Experiments are also reported for overall processes: aa + @*P* 2aP* a n d a*a* F? 2a*o. Results are independent of which chain is blocked. Again h ( a )Y 0.5, in agreement with equilibrium experiments, a n d t h e time course is first order. T h e rate constants a n d activation parameters are intermediate between those for self-exchange.
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*
INTRODUCTION Coiled-coil proteins are widely distributed in nature and serve a variety of biological functions.'-3 The coiled-coil structure is formed by the parallel, registered, supertwisted association of two amphipathic t u - h e l i ~ e s .Different ~~~ coiled-coil molecules may comprise chains of vastly different lengths. The shortest known coiled coils comprise chains of -30 amino acids, the longest near 1000.193In a given coiled-coil molecule, the helices may or may not have the same amino acid sequence.6-'2 In more than one biological context, it appears that the homodimer or heterodimer composition has been biologically tiiopdymers. Vol. 91, 957-966 (1991) a 1991 J o h n Wiley & Sons, Inc.
CCC 0006-3525/91/080957-lO$O4.00
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~ e l e c t e d . ~ -For " example, rabbit cardiac muscle is entirely aa in composition, the a signifying a 284residue protein chain of particular, and known, sequence.8 Various other rabbit muscles also contain a genetic variant ( p ) chain of the same length, included in two-chain molecules.6-" Each particular muscle, it appears, has its own relative population of a- and P-chains, assembled into its own mixture of homodimeric and heterodimeric molecular spec i e ~ . ' ~T.h' ~e same type of biological selection of species occurs in nonmuscle coiled coils as well." Such selection of molecular species can be brought about by thermodynamic, kinetic, or biological control.'* Each of the three control mechanisms has its advocates and it is possible that different systems employ different mechanisms~12,15.16-18However, the paucity of experimental
* To whom correspondence should be addressed.
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OZEKI ET AL.
data on fundamental processes that may be involved in the sorting of the species that occurs in the biological system makes it difficult to select among possible mechanisms. For example, evaluating the possibility of thermodynamic control requires measurement of equilibrium constants for heterodimerhomodimer eq~ilibria.".'~ Likewise, evaluation of the possibility of kinetic control requires information on the time scale and mechanism by which the constituent processes occur.14~17~1'~20 In the present paper, we address the latter question in the cases of aa- and 00-tropomyosin of the rabbit. In particular, we report here on experimental studies of the kinetics of chain exchange in coiledcoils. We confine our attention to a regime in which the vast majority of molecules are dimers. Thus, the principal net stoichiometric process under investigation is
xx + YY * 2XY
(1)
To simplify the kinetic analysis, it is desirable to begin with cases in which X and Y are very similar chains, because rate constants for all mechanistic steps (e.g., dissociation into chains, recombination, etc.) may be assumed independent of which chain is involved. Thus, we have studied overall reaction (1) in cases where X is a n a rabbit tropomyosin chain and Y is a n a-chain in which the single cysteine ( C190) is blocked by carboxyamidomethylation. In this manner, the kinetics of self-exchange can be investigated for a-chains. Likewise, we report here studies in which X is a 0 rabbit tropomyosin chain and Y is a 0-chain in which both cysteines (C36 and C190) have been blocked by carboxyamidomethylation. Although we emphasize here this more easily analyzed process of self-exchange, we also report data on the cases in which X is a n intact a-chain and Y a sulfhydryl-blocked 0-chain, and in which X is a blocked a-chain and Y a n intact pchain. Blocking is necessary even when distinguishable native chains are involved, because of the assay employed in measuring the heterodimer populat i ~ n . ' ' . ' ~These sulfhydryl-blocked species are referred t o below as a* and 0*, respectively. It is noteworthy that we employ here a n assay that is direct, i.e., that directly measures the fraction of heterodimeric molecules as a function of time, rather than some physical property that must be interpreted in those terms. T h e experiment consists of incubating a one-to-one mixture of homodimeric molecules a t a temperature sufficiently high that exchange takes place on a feasible time scale, yet sufficiently low t h a t the concentration of monomeric
species is negligible so far as quantitation (but not kinetics) is concerned. At a given time, aliquots are quenched a t low temperature (0°C) where exchange of chains is known to be negligibly slow.'' The resulting dimeric molecular population is then subjected to a disulfide cross-linking procedure that is known to cross-link only intramolecularly, i.e., it acts only on coiled-coil molecules having two adjacent sulfhydryls. Thus, since one chain type has only blocked sulfhydryls, neither its homodimers nor any heterodimers can be cross-linked. The resulting pattern of cross-linked and noncross-linked species can be examined via sodium dodecyl sulfate polyacrylamide gel electrophoresis ( NaDodSO, /PAGE) and contrasted with a blank mixture that has been kept a t low temperature, thereby precluding exchange. Quantitation of the gels to measure the relative 66 kd (dimer) and 33 kd (monomer) regional amounts determines the fraction of heterodimeric species in the original mixture a t the time the exchange reaction was quenched.".l9 T h e kinetics of chain exchange can be quantitatively assessed in this manner, as will be seen. Our principal finding is that the simplest mechanism that satisfactorily explains these data is exchange by dissociation into monomers followed by reassociation. In the cases of self-exchange, this picture enables one to interpret the observed rate constant as the rate constant for dissociation of the dimer and to evaluate the activation parameters for that process.
MATERIALS A N D METHODS Protein Preparations
acu-Tropomyosin was prepared from rabbit cardiac muscle and 60-tropomyosin from rabbit skeletal muscle a s previously Rabbit cardiac tropomyosin contains exclusively a-chains and yields pure aa species. Skeletal muscle tropomyosin contains both a- and /3-chains.6-8The 0-chains were separated from a by fractionation in 8 M urea-formate buffer and 00 species regenerated by dialysis vs benign medium as described earlier.6 Cross-linking of aa species a t C190 or of /3p species a t C36 and C190 was accomplished with ferricyanide in the manner described below. The ability of 00 species to cross-link is always virtually 100%. The ability of aa species t o cross-link is usually nearer 90%, but it can be improved by subjecting the cross-linked product to size-exclusion chromatography.24325 This precaution was therefore observed in the present work; the ability of the purified and re-reduced aa species to cross-link was greater than 97%.
K I N E T I C S OF CHAIN E X C H A N G E
Sulfhydryl-blocked chain species were prepared by carboxyamidomethylation of the CYNor /3/3 species that were reduced by dithithreitol ( D T T ) with iodoacetamide in denaturing medium as described in detail earlier.26 Sulfhydryl titration of the blocked product with Ellman's reagent 1H'27.2X revealed that less than 5% of the original sulfhydryl content remained. T h e protein concentration for any and all species was determined using a Beckman DU-50 spectrophotometer. We used an extinction coefficient a t the 275-nm peak of 0.314 cm2mg-I. Originally, this absorbance peak was reported to be a t 277 nm.29More modern spectrophotometers show it a t 275 nm. T h e numerical value of the extinction coefficient was originally based upon micro-Kjeldahl determination of nitrogen content"; it has been confirmed by the Edelhoch method"" for all species involved in this studv. " Chain Exchange (Hybridization) Experiments
A lyophilized sample of a given chain species and of the distinguishable sulfhydryl-blocked species with which it was to be hybridized were separately dissolved to 10 mg/mL in NaCIBooTrisBo (7.4). We designate complex aqueous solvent media by giving each solute species symbol or abbreviation with its millimolarity as subscript, followed by the pH in parentheses. The solutions were passed through a 0.45-pm Millipore filter and the concentration determined spectrophotometrically (see above). The concentration of each was adjusted to the same desired value, in this study in the range of 0.2-10.0 mg/ mL. T o each solution, D T T was added to a final concentration of 20 m M to reduce any existing interchain disulfide linkages. In experiments involving unblocked flfl species, the final D T T concentration used was 30 m M . In either case, the solution containing blocked species was treated the same way, i.e., D T T was added in the interest of uniformity even though no cross-linked species could be present. Each solution was thermostated a t 37°C for 2 h and then refrigerated ( 4 ° C ) overnight. In the earlier stages of this investigation, chain degradation due to bacterial growth was observed in a few solutions after the long-duration incubations a t 32-42°C required in the kinetic experiments. These data were unusable. In later runs, therefore, NaN3 was added to 1 m M along with the DTT. Mixtures for exchange experiments were prepared from equal volumes of the two solutions (of equal concentrations). T h e mixture was divided into vials and placed into a thermostat bath a t the desired
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temperature; in this study in the range was -3242°C. The most convenient sample volumes were 150 pL for 0.2 and 1.0 mg/mL, and pL for 5.010.0 mg/mL solutions. One mixture sample was stored in the refrigerator as a blank. It has previously been shown, and was confirmed in this study, that such mixtures do not detectably exchange chains a t room temperature or below for weeks, if not more." Vials were removed from the thermostat a t definite times, quenched in an ice-water bath to terminate exchange, marked for identification, and stored in the refrigerator. After all the vials were removed, the 15-20 solutions were put into 9-mm DiaCell capsules ( InstruMed, Inc.) and dialyzed 4X vs 1 L of NaCIBooTrisBo( 7.4) a t 4°C to remove D T T and NaN,3.For the 5.0-10.0 mg/mL solutions, 120 pL of dialysis buffer was added to each solution before dialysis; thus each solution was 150 pL in volume regardless of its protein concentration. T h e assay for heterodimers requires disulfide cross-linking. This was carried out a t room temperature by adding a 2.5 m M K3Fe(CN),, solution and a 0.025 m M CuSO, solution to each capsule. Final concentrations of K3Fe( C N ) 6were varied with protein concentration in order to correspond to about threefold excess over free sulf hydryls. Similarly, final concentrations of CuS04 were varied from 0.44 to 2.7 pM. T h e procedure is not sensitive to these concentrations. The capsules were then dialyzed vs buffer of the same composition, including the oxidation reagents. After 1 h, NaC1500TrisnoNEM,,,EDTA5(7.4) was added to each capsule to stop the reaction and prepare for the NaDodSO,/ PAGE assay. Final concentrations of NEM and EDTA were 1.6 and 0.8 m M , respectively. The capsules were then dialyzed vs NaC1BooTris,50NEM1 6EDTAo,8( 7.4) for 30 min. These numerous dialyses of numerous capsules are greatly facilitated by construction of a raft consisting of a piece of light, stiff plastic 92 X 6: in., into which are cut 24 capsulesized holes. The dialyzate can then be placed in a flat dish on a magnetic stirrer, the raft placed on the surface, and the capsules placed into the holes. For NaDodS04/ P A G E analysis, ( NaDodS04)210Tris18R was added to each capsule to final concentrations of 70 and 63 m M , respectively, and the capsules dialyzed vs ( N a D 0 d S 0 ~ ) ~ , , T rfor i s ~4: ~ h. (See below for a variant of this PAGE procedure t h a t is superior, but was only discovered late in this study.) T h e samples were run on 8% acrylamide minigels of 1.5-mm thickness. Gels were stained either with Coomassie Brilliant Blue G-250 or with Fast Stain (Zoion Research, Inc.) as previously described." In the former case (Coomassie) , gels were
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OZEKI ET AL.
immersed in stain I for 15-24 h and stain I1 for 24 h, and destained in 10% acetic acid for 15 h. In the latter case (Fast S t a i n ) , gels were rinsed 3x 20 min in a n aqueous solution containing 45% methanol and 10% acetic acid ( v / v ) , stained for 30 min, destained 2X for 20 min, and stored in 20% acetic acid for 15 h. No difference in results was seen that depended upon which staining procedure was employed. At least four gels were run and averaged for each capsule solution. Scanning was performed using an Isco Model 1312 scanner and integration was as previously described. T h e fraction of heterodimers (hybrids) h was calculated from h = ( g : - g 2 ) / gs,as previously derived; g 2 ( g ! ) is the fraction of the total integrated intensity in the dimer, i.e., crosslinked, band for the experiment (blank) solution.'" This relationship automatically corrects for a small number of sulf hydryl-bearing homodimers that do not get cross-linked and for differences in dye binding of the two chain species. In the present work, wherein all chain species are rather similar, these are both expected to be very small. T h e blank consists of a mixture of equal numbers of sulf hydryl-bearing and sulf hydryl-blocked homodimers. Only the former can be disulfide crosslinked, so we expect g: N 0.5. In the present work, we always found g: = 0.45-0.5, with blanks involving the pp species perhaps tending more toward the lower part of the range. Considering the experimental error in g2(--tO.02), this is satisfactory. This protocol for the entire experiment is effective, but variants are possible and some were employed in the early stages of this work. Indeed, two of the measurements reported here were made several years ago by one of us ( M E H ) using the much cruder methods, including individual dialysis bags and planimetric integration, then available. One major variant employed with some of the present measurements involved removal of D T T as soon as the initial reduction was accomplished. In such experiments, premature cross-linking by ambient oxygen during exchange was prevented by performing manipulations, dialyses, etc., in N2 atmosphere in a glove bag using Nz-bubbled solvents. There was no difference in the results, unlike earlier work, indicating that the presence of DTT does not affect exchange or interfere with the assay with our present protocols; it need not be removed until cross-linking is to be carried out. As will be seen below, the long-time limit reached in the kinetics runs in all three systems is h N 0.5 and agrees with experiments" in which the equilibrium is reached from the other side, i.e., by cooling
thermally unfolded, dissociated chains. In a small fraction of runs, anomalous results were obtained in which values of h leveled off prematurely a t a magnitude appreciably less than 0.5. This anomaly was particularly prone to occur a t the lowest temperatures and concentrations. In such a case, the NaDodS04/PAGE assay was checked by employing a more recently developed C, reversed-phase high performance liquid chromatography ( HPLC ) assay for dimer content."2T h e HPLC assay on the same long-time solutions showed h = 0.5, as expected. We have no explanation a t present for the failure of our NaDodS04/PAGE assay in those cases. The data were not used. However, it was subsequently discovered that this anomaly can be avoided by dialyzing vs water just before addition of ( N a DodS04)210Tris188 and omitting the subsequent dialysis vs detergent in preparing samples for electrophoresis (see above).
RESULTS Self-Exchange in aa-Tropomyosin
First we examine results of exchange experiments [see reaction Eq. ( 1 ) ] in which X = a , i.e., a n intact a-tropomyosin chain, and Y = a*, i.e., a n a-chain with a carboxyamidomethyl-blocking group a t C190. Since the temperature range is such that dimer species dominate,33 the overall reaction occurring is
T h e time course of the reaction a t 38°C is shown in Figure 1 as h ( t ), the measured fraction of aa* molecules, vs time. T h e data of Figure 1are in the range of protein concentration 2.5-10.0 mg/mL, i.e., a fourfold range, and clearly show that the curve of h vs t is independent of protein concentration. Since this strongly suggests a first-order process, the data were fit to the equation: h ( t )= h ( c c ) ( l - e-'lt)
(3)
wherein h ( x) and k l are constants. In the present case, wherein the two chain species CY and a* are so closely matched, we expect the equilibrium state to be characterized by random chain association, hence we expect he, = 0.5. In fact, this has earlier been shown to be true in this system by experiments in which the mixture was heated to 55°C to convert it to unfolded dissociated chains, then recooled either slowly or rapidly to low tem-
961
KINETICS OF CHAIN EXCHANGE
0.0 L
,
10.0
20.0
'I
I
80.0
I
90.0
timelksec
Figure 1 . Self-exchange in N N species. Fraction h of ( Y N * pseudohybrid ( heterodimeric ) two-chain coiled-coil moleciiles vs time at 38°C. Solid line is Eq. ( 3 ) fitted to s-' and h ( s ) all data points, using h, = 1.30 X = 0.480. Loading dimer concentration (mg/mL): 2.5, open dels; :3.5, open squares; 5.0, solid circles; 10.0, solid diamonds. Note break in abscissa.
perat ure.I8 This equilibrium experiment was repeated in the present work and confirmed. In earlier work, the CY(Y* molecules were referred to as "pseudohybrids," because they are heterodimers of chains that are distinguishable but so very similar." As can be seen in Figure 1, the long-time limit reached in that particular data set is h ( K ) = 0.48, well within error of the expectation value of 0.5. This is a common feature of all the experiments reported here, which showed h ( c c ) values in the range of 0.450.55. The computer fit also determines the time constant for the exchange process, in this particular case k , ' = 7.7 ks, or a bit over 2 h.
wherein k is Boltzmann's constant, T the Kelvin temperature, and AGt, AH', and A S t the activation Gibbs energy, enthalpy, and entropy, respectively, per stoichiometric unit. Accordingly, the plot of Figure 3 is as In( k , / T ) vs ( 1/ T ). The accessible temperature range is narrow and the error substantial from run to run. Nevertheless, the data in each case are tolerably well fit by straight lines. The resulting activation parameters are summarized in Table I. Chain Exchange in cycy
+ @@ Mixtures
Although our primary focus in this paper is on selfexchange, wherein the equivalence of physical properties of all chains may be safely assumed and simplifies the kinetic analysis, we report here some experiments on the more complex exchange processes:
and
Data for such a run a t 36°C are shown in Figure 4, wherefrom it is evident that the resulting time course is independent of which species, cy or is blocked. This result supports the assumption made above
a,
Self-Exchange in @P-Tropomyosin
Results for similar experiments are shown in Figure 2, only at, 36°C and involving the reaction
n
c
a*
signifies a P-chain that has been carin which boxyamidomethylated a t both sulfhydryls, C36 and C190. Here, the protein concentration range is 25fold (0.2-5.0 m g / m L ) and, again, the results are independent of concentration. Once again, we expect a long-time result of nearly 0.5; for this particular run. the fit gives h ( cc ) = 0.54, with a time constant of k I = 0.32 ks. Thus, self-exchange in the PP case is considerably more rapid. The temperature dependence for these observed rate constants for each self-exchange system is shown in Figure 3 for all runs. According to the Eyring theory, the rate constant is given by
OO4O
i t
o.20 0.00
P'
I I, I t
1.0
2.0
It
3
I1
11.0
I
21.0
time/ks
Figure 2. Self-exchange in pp species. Fraction h of pp* pseudohybrid ( heterodimeric ) two-chain coiled-coil molecules vs time at 36°C. Solid line is Eq. ( 3 ) fitted to all data points, using kl = 3.08 X s-l and h(co) = 0.540. Loading dimer concentration (mg/mL): 0.2, open triangles; 5.0, solid circles. Note breaks in abscissa.
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OZEKI ET AL.
t/"C 40 -10
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DISCUSSION 36
32
I
I
t-
-14
-16
k@
\, :
0 0
-18 3.1 5
3.20
3.25
3.30 3 .30
T-'/(kK)-'
Figure 3. Eyring plot, i.e., ln(hlT-') vs T I , for selfexchange of two-chain coiled-coil molecules: aa, lower data set; PO, upper data set. Loading dimer concentration (mg/ m L ) : 0.20, open triangles; 1.0, open circles; 1.75, open diamond; 2.5, open del; 3.5, open squares; 5.0, solid circles; 10.0, solid diamond. (Because of overlap, some symbols a t 38°C are partially or completely obscured.) Solid lines are by least squares and yield the Eyring parameters in Table I.
Self-Exchange in aa-Tropomyosin
We next attempt a kinetic analysis of the overall process described by stoichiometric Eq. ( 2 ) . There are three direct, simple mechanisms by which such chain exchange can be envisioned to occur: ( a ) dissociation into monomeric chains, followed by reassociation to form some heterodimers, i.e., dissociation-reassociation; ( b ) dissociation into monomers, followed by attack of a monomeric chain (say, a* ) on a homodimer (say, a a ), resulting in displacement of a n a-chain and formation of a n aa* heterodimer; and ( c ) encounter of two homodimers, with direct partner exchange, i.e., a double displacement (metathesis ) r e a ~ t i o n . " ~ Since it is physically clear that rate-determining mechanisms ( b ) and ( c ) must both yield time constants for the reaction that are dependent upon protein concentration, contra experiment, these will not be further pursued here, except to note that elementary chemical kinetics suffices to show that, in the present regime of self-exchange in a milieu wherein dimers dominate the population, the result for both mechanisms is of the form
that the blocking groups do not alter the exchange properties of either chain. For this run, the value h (03 ) = 0.46 is obtained, also confirming our earlier This is of the same form as Eq. ( 3 ) with h ( c c ) = finding that the equilibrium state of such mixtures 1/2, which agrees with experiment. However, as anof a - and P-chains is also characterized by nearly ticipated, in cases ( b ) and ( c ) the quantity K depends random chain association, a t least a t these temperupon C, the total formal concentration of dimers in atures. the reaction mixture. In particular, for the dissociaThe temperature dependence of the observed rate tion-displacement mechanism, one finds constants for such formation of a*p or ap* heterodimers is shown in Figure 5 as a n Eyring plot along with the corresponding fit lines of the two self-exchange reactions shown earlier (Figure 3 ) . Eyring parameters for the ap heterodimers are also given wherein k, is the rate constant for displacement of in Table I. T h e reader may gain some measure of either chain of the heterodimer by an unlike chain the continuity of our earlier and present work by and K is the equilibrium constant for dissociation examination of the open circle datum and the open of either homodimer into chains. For the doubletriangle datum of Figure 5. These two points were displacement mechanism, on the other hand, one obtained by one of us ( M E H ) in 1984 in a prelimfinds inary investigation of chain exchange, using handdone planimetry and dialysis bag t e c h n ~ l o g y . ' ~ ~ ' ~ Nevertheless, they are not otherwise distinguishable Table I Kinetic Parameters for Chain Exchange from more recent data collected by different hands, ffa PP ffP on different protein preparations, using different methods, and a t different concentrations. In any 4.00 23.8 0.588 Rate constant-' (36°C) (ks) event, the rate constants for formation of a/3 het22.0 23.2 A c t (36°C) (kcal-su-') 24.3 erodimers by chain exchange are roughly midway 131 156 102 AN' (kcal. su-') between the ones for aa and /3/3 self-exchange, as A S t (cal K-' * su-l) 252 434 347 Figure 5 shows.
KINETICS OF CHAIN EXCHANGE
OSO
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Solution of the kinetic equations is also aided by recognition that monomer species concentrations are small, whence chain conservation and the initial condition of one-to-one mixture require that
c = 2 ( a a ) + ( a a * )= 2 ( a * a * ) + ( ( Y ( Y * ) 0.00
963
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10.0
90.0
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Figure 4. Exchange in N N /3/3 mixtures. Fraction h of a*/3 ( solid circles) and a@*(open squares) hybrid ( heterodimeric ) two-chain coiled-coil molecules vs time a t 36°C. Solid line is Eq. ( 3 ) fitted to data points for cu*/3, using k , = 2.50 X s-' and h ( a ) = 0.460. Loading dimer concentration = 5.0 m g . m L - ' . Separate fit to NO* data points ( n o t shown) gives k l = 2.60 X s-' and h ( a ) = 0.460. Note break in abscissa.
wherein k,, is the rate constant for the double displacement. Since Eq. ( 9 ) or ( 10) requires an exponent in Eq. ( 8 ) that depends on protein concentration, they are each clearly contrary to fact, and mechanisms ( b ) and ( c ) can be rejected. We next examine mechanism ( a ) ,dissociationreassociation, for its ability to fit the kinetic data. This mechanism involves the reactions
Equation ( 1 1 ) makes clear that, in self-exchange experiments, the concentration of the two homodimer species must be the same a t all times, not just initially. Moreover, this means that the small concentrations of monomer must also be the same, i.e., ( a ) = ( a * ). Thus, information on only one homodimer is necessary, the corresponding information being redundant. With these constraints, the necessary kinetic equations stem from the first and third of Eqs. ( 11) :
in which Eq. ( 1 2 ) and ( a )= (a* ) have been inserted. The monomer concentration ( a )can be eliminated by equating two expressions for it obtained from Eq. ( 1 4 ) , yielding
t/"C 40
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a
SYNOPSIS
Measurements are presented o n t h e time course of chain exchange among two-chain ahelical coiled coils of rabbit tropomyosin. All experiments are in a regime (temperature, protein concentration) in which coiled-coil dimers are t h e predominant species. Self-exchange in rue-tropomyosin was investigated by mixing aa species with a*a*, t h e asterisk designating a n a-chain whose lone sulfhydryl (C190) has been blocked by carboxyamidomethylation. T h e overall process aa a*a* 2 2aa* is followed by measurement of t h e fraction ( h )of aa* species as a function of time. Similarly, self-exchange in PP-tropomyosin p*P* F? 2pp*, in which p* is examined by measurements of t h e overall process: /7/3 signifies a P-chain blocked a t both sulfhydryls (C36 a n d C190). T h e observed time course , h ( z) for both chains is well fit by t h e first-order equation: h ( t ) = h (a) ( 1 - e - k l t )with Y 0.5. T h i s long-time limit is a s expected for self-exchange, a n d agrees with experiments t h a t attain equilibrium after slow cooling of thermally dissociated a n d unfolded chains. T h e simplest consonant mechanism is chain exchange by rate-limiting dissociation of dimers followed by random reassociation. Kinetic analysis shows k , t o be t h e rate constant for t h e chain dissociation step, a quantity not previously measured for any coiled coil. This rate constant for p/3 species is about a n order of magnitude greater t h a n for aa. In both, t h e activation enthalpy a n d entropy are very large, suggesting t h a t activation t o a n extensively ( >50% ) unfolded species necessarily precedes dissociation. Experiments are also reported for overall processes: aa + @*P* 2aP* a n d a*a* F? 2a*o. Results are independent of which chain is blocked. Again h ( a )Y 0.5, in agreement with equilibrium experiments, a n d t h e time course is first order. T h e rate constants a n d activation parameters are intermediate between those for self-exchange.
+
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*
INTRODUCTION Coiled-coil proteins are widely distributed in nature and serve a variety of biological functions.'-3 The coiled-coil structure is formed by the parallel, registered, supertwisted association of two amphipathic t u - h e l i ~ e s .Different ~~~ coiled-coil molecules may comprise chains of vastly different lengths. The shortest known coiled coils comprise chains of -30 amino acids, the longest near 1000.193In a given coiled-coil molecule, the helices may or may not have the same amino acid sequence.6-'2 In more than one biological context, it appears that the homodimer or heterodimer composition has been biologically tiiopdymers. Vol. 91, 957-966 (1991) a 1991 J o h n Wiley & Sons, Inc.
CCC 0006-3525/91/080957-lO$O4.00
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~ e l e c t e d . ~ -For " example, rabbit cardiac muscle is entirely aa in composition, the a signifying a 284residue protein chain of particular, and known, sequence.8 Various other rabbit muscles also contain a genetic variant ( p ) chain of the same length, included in two-chain molecules.6-" Each particular muscle, it appears, has its own relative population of a- and P-chains, assembled into its own mixture of homodimeric and heterodimeric molecular spec i e ~ . ' ~T.h' ~e same type of biological selection of species occurs in nonmuscle coiled coils as well." Such selection of molecular species can be brought about by thermodynamic, kinetic, or biological control.'* Each of the three control mechanisms has its advocates and it is possible that different systems employ different mechanisms~12,15.16-18However, the paucity of experimental
* To whom correspondence should be addressed.
957
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OZEKI ET AL.
data on fundamental processes that may be involved in the sorting of the species that occurs in the biological system makes it difficult to select among possible mechanisms. For example, evaluating the possibility of thermodynamic control requires measurement of equilibrium constants for heterodimerhomodimer eq~ilibria.".'~ Likewise, evaluation of the possibility of kinetic control requires information on the time scale and mechanism by which the constituent processes occur.14~17~1'~20 In the present paper, we address the latter question in the cases of aa- and 00-tropomyosin of the rabbit. In particular, we report here on experimental studies of the kinetics of chain exchange in coiledcoils. We confine our attention to a regime in which the vast majority of molecules are dimers. Thus, the principal net stoichiometric process under investigation is
xx + YY * 2XY
(1)
To simplify the kinetic analysis, it is desirable to begin with cases in which X and Y are very similar chains, because rate constants for all mechanistic steps (e.g., dissociation into chains, recombination, etc.) may be assumed independent of which chain is involved. Thus, we have studied overall reaction (1) in cases where X is a n a rabbit tropomyosin chain and Y is a n a-chain in which the single cysteine ( C190) is blocked by carboxyamidomethylation. In this manner, the kinetics of self-exchange can be investigated for a-chains. Likewise, we report here studies in which X is a 0 rabbit tropomyosin chain and Y is a 0-chain in which both cysteines (C36 and C190) have been blocked by carboxyamidomethylation. Although we emphasize here this more easily analyzed process of self-exchange, we also report data on the cases in which X is a n intact a-chain and Y a sulfhydryl-blocked 0-chain, and in which X is a blocked a-chain and Y a n intact pchain. Blocking is necessary even when distinguishable native chains are involved, because of the assay employed in measuring the heterodimer populat i ~ n . ' ' . ' ~These sulfhydryl-blocked species are referred t o below as a* and 0*, respectively. It is noteworthy that we employ here a n assay that is direct, i.e., that directly measures the fraction of heterodimeric molecules as a function of time, rather than some physical property that must be interpreted in those terms. T h e experiment consists of incubating a one-to-one mixture of homodimeric molecules a t a temperature sufficiently high that exchange takes place on a feasible time scale, yet sufficiently low t h a t the concentration of monomeric
species is negligible so far as quantitation (but not kinetics) is concerned. At a given time, aliquots are quenched a t low temperature (0°C) where exchange of chains is known to be negligibly slow.'' The resulting dimeric molecular population is then subjected to a disulfide cross-linking procedure that is known to cross-link only intramolecularly, i.e., it acts only on coiled-coil molecules having two adjacent sulfhydryls. Thus, since one chain type has only blocked sulfhydryls, neither its homodimers nor any heterodimers can be cross-linked. The resulting pattern of cross-linked and noncross-linked species can be examined via sodium dodecyl sulfate polyacrylamide gel electrophoresis ( NaDodSO, /PAGE) and contrasted with a blank mixture that has been kept a t low temperature, thereby precluding exchange. Quantitation of the gels to measure the relative 66 kd (dimer) and 33 kd (monomer) regional amounts determines the fraction of heterodimeric species in the original mixture a t the time the exchange reaction was quenched.".l9 T h e kinetics of chain exchange can be quantitatively assessed in this manner, as will be seen. Our principal finding is that the simplest mechanism that satisfactorily explains these data is exchange by dissociation into monomers followed by reassociation. In the cases of self-exchange, this picture enables one to interpret the observed rate constant as the rate constant for dissociation of the dimer and to evaluate the activation parameters for that process.
MATERIALS A N D METHODS Protein Preparations
acu-Tropomyosin was prepared from rabbit cardiac muscle and 60-tropomyosin from rabbit skeletal muscle a s previously Rabbit cardiac tropomyosin contains exclusively a-chains and yields pure aa species. Skeletal muscle tropomyosin contains both a- and /3-chains.6-8The 0-chains were separated from a by fractionation in 8 M urea-formate buffer and 00 species regenerated by dialysis vs benign medium as described earlier.6 Cross-linking of aa species a t C190 or of /3p species a t C36 and C190 was accomplished with ferricyanide in the manner described below. The ability of 00 species to cross-link is always virtually 100%. The ability of aa species t o cross-link is usually nearer 90%, but it can be improved by subjecting the cross-linked product to size-exclusion chromatography.24325 This precaution was therefore observed in the present work; the ability of the purified and re-reduced aa species to cross-link was greater than 97%.
K I N E T I C S OF CHAIN E X C H A N G E
Sulfhydryl-blocked chain species were prepared by carboxyamidomethylation of the CYNor /3/3 species that were reduced by dithithreitol ( D T T ) with iodoacetamide in denaturing medium as described in detail earlier.26 Sulfhydryl titration of the blocked product with Ellman's reagent 1H'27.2X revealed that less than 5% of the original sulfhydryl content remained. T h e protein concentration for any and all species was determined using a Beckman DU-50 spectrophotometer. We used an extinction coefficient a t the 275-nm peak of 0.314 cm2mg-I. Originally, this absorbance peak was reported to be a t 277 nm.29More modern spectrophotometers show it a t 275 nm. T h e numerical value of the extinction coefficient was originally based upon micro-Kjeldahl determination of nitrogen content"; it has been confirmed by the Edelhoch method"" for all species involved in this studv. " Chain Exchange (Hybridization) Experiments
A lyophilized sample of a given chain species and of the distinguishable sulfhydryl-blocked species with which it was to be hybridized were separately dissolved to 10 mg/mL in NaCIBooTrisBo (7.4). We designate complex aqueous solvent media by giving each solute species symbol or abbreviation with its millimolarity as subscript, followed by the pH in parentheses. The solutions were passed through a 0.45-pm Millipore filter and the concentration determined spectrophotometrically (see above). The concentration of each was adjusted to the same desired value, in this study in the range of 0.2-10.0 mg/ mL. T o each solution, D T T was added to a final concentration of 20 m M to reduce any existing interchain disulfide linkages. In experiments involving unblocked flfl species, the final D T T concentration used was 30 m M . In either case, the solution containing blocked species was treated the same way, i.e., D T T was added in the interest of uniformity even though no cross-linked species could be present. Each solution was thermostated a t 37°C for 2 h and then refrigerated ( 4 ° C ) overnight. In the earlier stages of this investigation, chain degradation due to bacterial growth was observed in a few solutions after the long-duration incubations a t 32-42°C required in the kinetic experiments. These data were unusable. In later runs, therefore, NaN3 was added to 1 m M along with the DTT. Mixtures for exchange experiments were prepared from equal volumes of the two solutions (of equal concentrations). T h e mixture was divided into vials and placed into a thermostat bath a t the desired
-
959
temperature; in this study in the range was -3242°C. The most convenient sample volumes were 150 pL for 0.2 and 1.0 mg/mL, and pL for 5.010.0 mg/mL solutions. One mixture sample was stored in the refrigerator as a blank. It has previously been shown, and was confirmed in this study, that such mixtures do not detectably exchange chains a t room temperature or below for weeks, if not more." Vials were removed from the thermostat a t definite times, quenched in an ice-water bath to terminate exchange, marked for identification, and stored in the refrigerator. After all the vials were removed, the 15-20 solutions were put into 9-mm DiaCell capsules ( InstruMed, Inc.) and dialyzed 4X vs 1 L of NaCIBooTrisBo( 7.4) a t 4°C to remove D T T and NaN,3.For the 5.0-10.0 mg/mL solutions, 120 pL of dialysis buffer was added to each solution before dialysis; thus each solution was 150 pL in volume regardless of its protein concentration. T h e assay for heterodimers requires disulfide cross-linking. This was carried out a t room temperature by adding a 2.5 m M K3Fe(CN),, solution and a 0.025 m M CuSO, solution to each capsule. Final concentrations of K3Fe( C N ) 6were varied with protein concentration in order to correspond to about threefold excess over free sulf hydryls. Similarly, final concentrations of CuS04 were varied from 0.44 to 2.7 pM. T h e procedure is not sensitive to these concentrations. The capsules were then dialyzed vs buffer of the same composition, including the oxidation reagents. After 1 h, NaC1500TrisnoNEM,,,EDTA5(7.4) was added to each capsule to stop the reaction and prepare for the NaDodSO,/ PAGE assay. Final concentrations of NEM and EDTA were 1.6 and 0.8 m M , respectively. The capsules were then dialyzed vs NaC1BooTris,50NEM1 6EDTAo,8( 7.4) for 30 min. These numerous dialyses of numerous capsules are greatly facilitated by construction of a raft consisting of a piece of light, stiff plastic 92 X 6: in., into which are cut 24 capsulesized holes. The dialyzate can then be placed in a flat dish on a magnetic stirrer, the raft placed on the surface, and the capsules placed into the holes. For NaDodS04/ P A G E analysis, ( NaDodS04)210Tris18R was added to each capsule to final concentrations of 70 and 63 m M , respectively, and the capsules dialyzed vs ( N a D 0 d S 0 ~ ) ~ , , T rfor i s ~4: ~ h. (See below for a variant of this PAGE procedure t h a t is superior, but was only discovered late in this study.) T h e samples were run on 8% acrylamide minigels of 1.5-mm thickness. Gels were stained either with Coomassie Brilliant Blue G-250 or with Fast Stain (Zoion Research, Inc.) as previously described." In the former case (Coomassie) , gels were
960
OZEKI ET AL.
immersed in stain I for 15-24 h and stain I1 for 24 h, and destained in 10% acetic acid for 15 h. In the latter case (Fast S t a i n ) , gels were rinsed 3x 20 min in a n aqueous solution containing 45% methanol and 10% acetic acid ( v / v ) , stained for 30 min, destained 2X for 20 min, and stored in 20% acetic acid for 15 h. No difference in results was seen that depended upon which staining procedure was employed. At least four gels were run and averaged for each capsule solution. Scanning was performed using an Isco Model 1312 scanner and integration was as previously described. T h e fraction of heterodimers (hybrids) h was calculated from h = ( g : - g 2 ) / gs,as previously derived; g 2 ( g ! ) is the fraction of the total integrated intensity in the dimer, i.e., crosslinked, band for the experiment (blank) solution.'" This relationship automatically corrects for a small number of sulf hydryl-bearing homodimers that do not get cross-linked and for differences in dye binding of the two chain species. In the present work, wherein all chain species are rather similar, these are both expected to be very small. T h e blank consists of a mixture of equal numbers of sulf hydryl-bearing and sulf hydryl-blocked homodimers. Only the former can be disulfide crosslinked, so we expect g: N 0.5. In the present work, we always found g: = 0.45-0.5, with blanks involving the pp species perhaps tending more toward the lower part of the range. Considering the experimental error in g2(--tO.02), this is satisfactory. This protocol for the entire experiment is effective, but variants are possible and some were employed in the early stages of this work. Indeed, two of the measurements reported here were made several years ago by one of us ( M E H ) using the much cruder methods, including individual dialysis bags and planimetric integration, then available. One major variant employed with some of the present measurements involved removal of D T T as soon as the initial reduction was accomplished. In such experiments, premature cross-linking by ambient oxygen during exchange was prevented by performing manipulations, dialyses, etc., in N2 atmosphere in a glove bag using Nz-bubbled solvents. There was no difference in the results, unlike earlier work, indicating that the presence of DTT does not affect exchange or interfere with the assay with our present protocols; it need not be removed until cross-linking is to be carried out. As will be seen below, the long-time limit reached in the kinetics runs in all three systems is h N 0.5 and agrees with experiments" in which the equilibrium is reached from the other side, i.e., by cooling
thermally unfolded, dissociated chains. In a small fraction of runs, anomalous results were obtained in which values of h leveled off prematurely a t a magnitude appreciably less than 0.5. This anomaly was particularly prone to occur a t the lowest temperatures and concentrations. In such a case, the NaDodS04/PAGE assay was checked by employing a more recently developed C, reversed-phase high performance liquid chromatography ( HPLC ) assay for dimer content."2T h e HPLC assay on the same long-time solutions showed h = 0.5, as expected. We have no explanation a t present for the failure of our NaDodS04/PAGE assay in those cases. The data were not used. However, it was subsequently discovered that this anomaly can be avoided by dialyzing vs water just before addition of ( N a DodS04)210Tris188 and omitting the subsequent dialysis vs detergent in preparing samples for electrophoresis (see above).
RESULTS Self-Exchange in aa-Tropomyosin
First we examine results of exchange experiments [see reaction Eq. ( 1 ) ] in which X = a , i.e., a n intact a-tropomyosin chain, and Y = a*, i.e., a n a-chain with a carboxyamidomethyl-blocking group a t C190. Since the temperature range is such that dimer species dominate,33 the overall reaction occurring is
T h e time course of the reaction a t 38°C is shown in Figure 1 as h ( t ), the measured fraction of aa* molecules, vs time. T h e data of Figure 1are in the range of protein concentration 2.5-10.0 mg/mL, i.e., a fourfold range, and clearly show that the curve of h vs t is independent of protein concentration. Since this strongly suggests a first-order process, the data were fit to the equation: h ( t )= h ( c c ) ( l - e-'lt)
(3)
wherein h ( x) and k l are constants. In the present case, wherein the two chain species CY and a* are so closely matched, we expect the equilibrium state to be characterized by random chain association, hence we expect he, = 0.5. In fact, this has earlier been shown to be true in this system by experiments in which the mixture was heated to 55°C to convert it to unfolded dissociated chains, then recooled either slowly or rapidly to low tem-
961
KINETICS OF CHAIN EXCHANGE
0.0 L
,
10.0
20.0
'I
I
80.0
I
90.0
timelksec
Figure 1 . Self-exchange in N N species. Fraction h of ( Y N * pseudohybrid ( heterodimeric ) two-chain coiled-coil moleciiles vs time at 38°C. Solid line is Eq. ( 3 ) fitted to s-' and h ( s ) all data points, using h, = 1.30 X = 0.480. Loading dimer concentration (mg/mL): 2.5, open dels; :3.5, open squares; 5.0, solid circles; 10.0, solid diamonds. Note break in abscissa.
perat ure.I8 This equilibrium experiment was repeated in the present work and confirmed. In earlier work, the CY(Y* molecules were referred to as "pseudohybrids," because they are heterodimers of chains that are distinguishable but so very similar." As can be seen in Figure 1, the long-time limit reached in that particular data set is h ( K ) = 0.48, well within error of the expectation value of 0.5. This is a common feature of all the experiments reported here, which showed h ( c c ) values in the range of 0.450.55. The computer fit also determines the time constant for the exchange process, in this particular case k , ' = 7.7 ks, or a bit over 2 h.
wherein k is Boltzmann's constant, T the Kelvin temperature, and AGt, AH', and A S t the activation Gibbs energy, enthalpy, and entropy, respectively, per stoichiometric unit. Accordingly, the plot of Figure 3 is as In( k , / T ) vs ( 1/ T ). The accessible temperature range is narrow and the error substantial from run to run. Nevertheless, the data in each case are tolerably well fit by straight lines. The resulting activation parameters are summarized in Table I. Chain Exchange in cycy
+ @@ Mixtures
Although our primary focus in this paper is on selfexchange, wherein the equivalence of physical properties of all chains may be safely assumed and simplifies the kinetic analysis, we report here some experiments on the more complex exchange processes:
and
Data for such a run a t 36°C are shown in Figure 4, wherefrom it is evident that the resulting time course is independent of which species, cy or is blocked. This result supports the assumption made above
a,
Self-Exchange in @P-Tropomyosin
Results for similar experiments are shown in Figure 2, only at, 36°C and involving the reaction
n
c
a*
signifies a P-chain that has been carin which boxyamidomethylated a t both sulfhydryls, C36 and C190. Here, the protein concentration range is 25fold (0.2-5.0 m g / m L ) and, again, the results are independent of concentration. Once again, we expect a long-time result of nearly 0.5; for this particular run. the fit gives h ( cc ) = 0.54, with a time constant of k I = 0.32 ks. Thus, self-exchange in the PP case is considerably more rapid. The temperature dependence for these observed rate constants for each self-exchange system is shown in Figure 3 for all runs. According to the Eyring theory, the rate constant is given by
OO4O
i t
o.20 0.00
P'
I I, I t
1.0
2.0
It
3
I1
11.0
I
21.0
time/ks
Figure 2. Self-exchange in pp species. Fraction h of pp* pseudohybrid ( heterodimeric ) two-chain coiled-coil molecules vs time at 36°C. Solid line is Eq. ( 3 ) fitted to all data points, using kl = 3.08 X s-l and h(co) = 0.540. Loading dimer concentration (mg/mL): 0.2, open triangles; 5.0, solid circles. Note breaks in abscissa.
962
OZEKI ET AL.
t/"C 40 -10
-12
DISCUSSION 36
32
I
I
t-
-14
-16
k@
\, :
0 0
-18 3.1 5
3.20
3.25
3.30 3 .30
T-'/(kK)-'
Figure 3. Eyring plot, i.e., ln(hlT-') vs T I , for selfexchange of two-chain coiled-coil molecules: aa, lower data set; PO, upper data set. Loading dimer concentration (mg/ m L ) : 0.20, open triangles; 1.0, open circles; 1.75, open diamond; 2.5, open del; 3.5, open squares; 5.0, solid circles; 10.0, solid diamond. (Because of overlap, some symbols a t 38°C are partially or completely obscured.) Solid lines are by least squares and yield the Eyring parameters in Table I.
Self-Exchange in aa-Tropomyosin
We next attempt a kinetic analysis of the overall process described by stoichiometric Eq. ( 2 ) . There are three direct, simple mechanisms by which such chain exchange can be envisioned to occur: ( a ) dissociation into monomeric chains, followed by reassociation to form some heterodimers, i.e., dissociation-reassociation; ( b ) dissociation into monomers, followed by attack of a monomeric chain (say, a* ) on a homodimer (say, a a ), resulting in displacement of a n a-chain and formation of a n aa* heterodimer; and ( c ) encounter of two homodimers, with direct partner exchange, i.e., a double displacement (metathesis ) r e a ~ t i o n . " ~ Since it is physically clear that rate-determining mechanisms ( b ) and ( c ) must both yield time constants for the reaction that are dependent upon protein concentration, contra experiment, these will not be further pursued here, except to note that elementary chemical kinetics suffices to show that, in the present regime of self-exchange in a milieu wherein dimers dominate the population, the result for both mechanisms is of the form
that the blocking groups do not alter the exchange properties of either chain. For this run, the value h (03 ) = 0.46 is obtained, also confirming our earlier This is of the same form as Eq. ( 3 ) with h ( c c ) = finding that the equilibrium state of such mixtures 1/2, which agrees with experiment. However, as anof a - and P-chains is also characterized by nearly ticipated, in cases ( b ) and ( c ) the quantity K depends random chain association, a t least a t these temperupon C, the total formal concentration of dimers in atures. the reaction mixture. In particular, for the dissociaThe temperature dependence of the observed rate tion-displacement mechanism, one finds constants for such formation of a*p or ap* heterodimers is shown in Figure 5 as a n Eyring plot along with the corresponding fit lines of the two self-exchange reactions shown earlier (Figure 3 ) . Eyring parameters for the ap heterodimers are also given wherein k, is the rate constant for displacement of in Table I. T h e reader may gain some measure of either chain of the heterodimer by an unlike chain the continuity of our earlier and present work by and K is the equilibrium constant for dissociation examination of the open circle datum and the open of either homodimer into chains. For the doubletriangle datum of Figure 5. These two points were displacement mechanism, on the other hand, one obtained by one of us ( M E H ) in 1984 in a prelimfinds inary investigation of chain exchange, using handdone planimetry and dialysis bag t e c h n ~ l o g y . ' ~ ~ ' ~ Nevertheless, they are not otherwise distinguishable Table I Kinetic Parameters for Chain Exchange from more recent data collected by different hands, ffa PP ffP on different protein preparations, using different methods, and a t different concentrations. In any 4.00 23.8 0.588 Rate constant-' (36°C) (ks) event, the rate constants for formation of a/3 het22.0 23.2 A c t (36°C) (kcal-su-') 24.3 erodimers by chain exchange are roughly midway 131 156 102 AN' (kcal. su-') between the ones for aa and /3/3 self-exchange, as A S t (cal K-' * su-l) 252 434 347 Figure 5 shows.
KINETICS OF CHAIN EXCHANGE
OSO
>
Solution of the kinetic equations is also aided by recognition that monomer species concentrations are small, whence chain conservation and the initial condition of one-to-one mixture require that
c = 2 ( a a ) + ( a a * )= 2 ( a * a * ) + ( ( Y ( Y * ) 0.00
963
20.0
10.0
90.0
timelks
+
Figure 4. Exchange in N N /3/3 mixtures. Fraction h of a*/3 ( solid circles) and a@*(open squares) hybrid ( heterodimeric ) two-chain coiled-coil molecules vs time a t 36°C. Solid line is Eq. ( 3 ) fitted to data points for cu*/3, using k , = 2.50 X s-' and h ( a ) = 0.460. Loading dimer concentration = 5.0 m g . m L - ' . Separate fit to NO* data points ( n o t shown) gives k l = 2.60 X s-' and h ( a ) = 0.460. Note break in abscissa.
wherein k,, is the rate constant for the double displacement. Since Eq. ( 9 ) or ( 10) requires an exponent in Eq. ( 8 ) that depends on protein concentration, they are each clearly contrary to fact, and mechanisms ( b ) and ( c ) can be rejected. We next examine mechanism ( a ) ,dissociationreassociation, for its ability to fit the kinetic data. This mechanism involves the reactions
Equation ( 1 1 ) makes clear that, in self-exchange experiments, the concentration of the two homodimer species must be the same a t all times, not just initially. Moreover, this means that the small concentrations of monomer must also be the same, i.e., ( a ) = ( a * ). Thus, information on only one homodimer is necessary, the corresponding information being redundant. With these constraints, the necessary kinetic equations stem from the first and third of Eqs. ( 11) :
in which Eq. ( 1 2 ) and ( a )= (a* ) have been inserted. The monomer concentration ( a )can be eliminated by equating two expressions for it obtained from Eq. ( 1 4 ) , yielding
t/"C 40
-12
a
