Dec. 5, 1062
KINETICSOF DEHYDROGENASE-CATALYZED REACTIONS
[CONTRIBUTION FROM
THE
4375
DEPARTMENT OF CHEMISTRY, UNIVERSITY OB WISCONSIN, MADISON6, WIS.]
Multiple Intermediates in Steady-state Enzyme Kinetics. 111. Analysis of the Kinetics of Some Reactions Catalyzed by Dehydrogenases BY VICTORBLOOM FIELD,^^ LEONARD PELLER'~ AND ROBERT A. ALBERTY RECEIVEDJUNE 7, 1962 Steady-state kinetic data for several dehydrogenase systems are analyzed in the fashion described in reference 2. Various kinetic parameters and lower limits for the rate constants are calculated. Evidence requiring the consideration of isomers of the binary complexes in some of these systems is cited. A consideration of the dual role of hydrogen ion in affecting the experimentally determined kinetic parameters and calculated lower limits is presented. I t is suggested that diffusioncontrolled reactions in the combination of substrates with the enzyme may be indicated. Finally, an examination is made of the feasibility of determining certain kinetic parameters by direct experimental measurement.
Introduction The only difference between the rate laws for the In the preceding paper2 (designated as 11), we two mechanisms is in the presence of terms involvhave obtained some formal results for the steady ing (A)(B)(Q) and (B)(Q)(R) in the denominator state kinetics of enzyme-catalyzed reactions in- of the former. It may be shown that the expresvolving two reactants and two products. In sions 9a and 9b of reference 2, which are used to particular the form of the rate law, the expressions calculate KABQand K B Q Rin terms of kinetic pafor the kinetic parameters and their interrelation- rameters for the forward and reverse reactions alone, ships, and the lower limits for the various rate will give rise to infinite values for these quantities constants presented there are valid irrespective if ternary complexes are non-existent or kinetically of the number of intermediates in the reaction. We insignificant. Such a situation indeed appears t o have, however, assumed an ordered, or more pre- be the case for the reactions catalyzed by liver alcohol dehydrogenase and malic dehydrogenase concisely a preferred, path of reaction. I n the present paper we apply these results to sidered below. the data for a number of reactions catalyzed by the Lactate Dehydrogenase dehydrogenases, which comprise a relatively well Schwert and c o - w ~ r k e r s ~have ~ . ~ extensively studied class of such systems. It was found profitable in paper 11,in the interests studied this system and have provided sufficient of writing the over-all rate law in a succinct and data for determination of the relevant kinetic symmetrical form, to depart somewhat from the parameters a t four pH values.3b In determining the entries in Table I1 we have customary nomenclature used by other workers in this field. To facilitate transposition of results assumed an order of combination with the enzyme by the reader, we present in Table I a summary of in the forward reaction of D P N + before lactate, the two other widely used schemes and the equiva- and in the reverse direction of DPNH before pyrulents in our symbolism. It is to be noted that con- vate. In the absence of kinetic studies of the type sideration of the complete rate law in paper I1 for described in the previous paper2 and earlier4 to such reactions has necessitated the introduction of elucidate the order of reaction, we may refer to further parameters not considered by these authors. binding studies of the type cited below. The redundancy relations 8a-9b of reference 2 have been used to calculate KAQ,KBR,KABQ and KBQR. TABLE I Isomers of the binary complexes derived from the EQUIVALENT EXPRESSIONS FOR SONEOF THE KINETIC combination of the enzyme with D P N + are strongly PARAMETERS IN THE RATELAWFOR suggested by the fact that the values of KABTHE FORWARD REACTIOX VQR~VABKAKB are greater than unity a t pH 6.15 1" IIb IIIC and 6.98. This is a sufficient condition for reVAR VAB* ( E )O/@ quiring the consideration of binary isomers.2 A t KA KAB*/KB* 412/4? pH 6.15 a bimolecular rate constant for the combiKB KAB*/KA* 412/@1 nation of DPNH with the enzyme of a t least the KAB KAB* +Q/~z order of lopM-l set.-' is apparent. Reference 2 , paper 11. *See for example, ref. 15. Somewhat disconcertingly, the values of KABQ " K . Dalziel, Acta Chew. S a n d . , 11, 1706 (1957). The a t pH 6.98 and 8.02 are negative. The equivalent quantities for the reverse reaction are designated and KBQR by superscript primes in this reference. calculation of these parameters involves a subtraction of quantities of considerable numerical unIn the succeeding analysis we shall calculate certainty. The negative values may simply arise values of the parameters on the reasonable assump- from the accumulation of experimental errors. tion that account must be taken of ternary comThe value of the dissociation constant of the plexes in the mechanism. As may be seen from lactate dehydrogenase-DPN+ complex of 3.9 f paper 11, the expressions for the general lower 0.7 X M from an ultracentrifugal determinalimits on the rate constants are identical whether tion2 in 0.1 A4 phosphate buffer a t pH 6.80 a t 25' ternary or only binary complexes are considered. agrees only nioderately well with K A a t pH 6.98 in Table 11. The difference in experimental ( 1 ) (a) National Science Foundation Predoctoral Fellow 1959-1962; (b) National Institutes of Health, Bethesda 14, Md. (3) (a) Y.Takenaga and G. W. Schwert, J . Biol. Chcm., 223, 157 (2) V. Bloomfield, L. Peller and R. A. Alberty, J . A m . Chcm. SOL., (19.56); (b) A. D.Winer and G. W. Schwert, ibid., 231, 106.5 (19.58). 84,4367 (1962). (4) R. A. Alberty, J . A m . Chem. Soc., SO, 1777 (1958).
4376
VICTORBLOOMFIELD, LEONARD PELLER AND R. A. ALBERTY
K I A B T I C PARAMETERS FOR
TABLE I1 DEHYDROGEXASES A S D CALCULATED LOWER LIMITS (a= ADPX+, R = DPNH)
Yeast alcohol dehydrogenases B = C~HIOH. Q = CHaCHO
Lactate d e h y d r ~ g e n a s e ~ ~ B = lactate, Q = pyruvate
6.15
6.98
8.02
9.01
7.15
6.0
0.75 2.6 3.7 6.6
1.25 2.5 3.3 3.3
2.9 1.8 1.7 1.1
4.7 1.3 1.1 1.0
0.62 2.6 8.4 4.2
0.38 1.7 12.3 16
7.0 9.2 1 . 1 3.8 2.6 11 :3 2.2 6.0 100 1.2 1.3 0.53 10.8 130 5.5 0.44 1.2 1.2 -0.035 -0.57 5.0 -1.9 -0.41 48 1.25 4.5 5.1 4.8 6.1 0.95
5 0 7 1 13 5.5 33 6.0 3.0 31 1.2 0.38
3 8 4.2 5.15 1.3 8.8 6.1 -3.0 -5.8 0 21 0.041
10.3 3.9 2.8 0.37 i.3 1.4 1.3 7.2 0.42 0.30
--
RATEC O L S I A S I S Malate dehydrogenase
Ri bit ol dehydrogenases B = D-ribitol, Q
-
Liver alcohol dehvdrosenase'l B = CzHaOH, Q = CHaCHO
-
D-
ribulose
-
oxal
8.0
6.38 1.41 3.19
0.031 1.3 0.55 0.033
0.049 .16 ,092 ,0089
1.67 7.0 0.28 0.56
0.74 0.28 1.0 0.067 1.3 o.ni5
,082 .86 7.9 0.113 1.3 o . oo79
5.7 5.0 1.1
71 .,5 133 34.7 850 10.1 4.0 27.8
'I
I,
'I
m"
,
4.7
P h y g a a r d and I1 lheurell, i l r t r i Cheirz ScumJ , 9 , 1800
=
acetate
9.0
2.9
Yeast Alcohol Dehydrogenase The alcohol dehydrogenase from yeast has been subjected to fairly detailed kinetic s t ~ d y .In ~ Table I1 are found values a t two pH's for the kinetic parameters appearing in eq. 3 of reference 2. It should be emphasized that as in the previously discussed case of lactate dehydrogenase a mechanism invoking ternary complexes has been presumed coupled with the assumption that the coenzyme combines with the enzyme before the other substrate. Accordingly, use has been made of the redundancy relations, ey. Sa-9b of reference 2 , to calculate several of the parameters in Table I1 from those supplied by direct experimental measurement.j We note the existence of lower limits for the bimolecular rate constants of 10b-107 121- 'see: I . The negative values of KABQ and K B Q Ra t pH 6.0 are again probably the result of experimental uncertainties. However, the fact that KABVQK' L r ~ ~ k 7is & greater -~ than unity a t both pH values a t which data are available argues for isomers
malate, Q
7.0
1.25
conditions for these two measurements niay be sufficient to account for this. Agreement between these differently determined dissociation constants can be construed as support for the presumed order of combination of substrates with the enzyme.
B = L-
8.0
0.75
10.9 k - a , k-8, k - , X lo-'', sec. 10.3 KQRVAA/VQRKQKR 0.25 K.IBVQR~VAEKAKB 9 . 4
(3) A (lgjj)
OF
VOl. 84
(1
3,
16 5.9 1.0 1.0
of the binary complexes arising from association of D P N + with the enzyme. A comparison of K A and KR in Table I1 a t pH 7.15 with the values of the dissociation constants, 2.6 X lop4 JI and 1.3 x 10-5M, respectively, determined by an ultracentrifugal technique a t p H 7.8 (at 0-5') is reasonably g 0 0 d . ~ ' ~ Ribitol Dehydrogenase Recently, Nordlie and Fromm have presented the results of a steady state study of the kinetics of the reaction of D-ribitol with DPN+ to produce Dribulose and DPNH.8 The molecular weight of the enzyme, ribitol dehydrogenase, is unknown, so calculations of VAB/(E),and VQR/(E)O cannot be made. From the reported kinetic parameters,* it is, however, possible to determine certain others by use of relations previously given.2 The values of these constants appear in Table 11. In determining the additional parameters use has been made of the value VAB/VQK= 0.ti6 from rneasurenients of the maximum velocities in the forward and reverse directions a t the same concentration of (6) J . E. Hayes, J r . , and S. l p . Vrlick, .I. H i d . Chem., 207, 226 (1 s x ) .
( 7 ) A . P. Sygaard and H. 'l'heorell, A d a Chem. Scaizd., 9, 1.551 (19jZ). 18) I K A , i t follows that the lower limit for k, must be greater than for kf (cf. relations 12a and 13a of reference 2). Similarly as K Q > K R , the lower limit for k - ~ , + ~ must , be greater than that for L g ( c f . relations 12b and 13b of reference 2).
4377
combination of ethanol and acetaldehyde with the enzyme-coenzyme complex if ternary complexes are kinetically insignificant. The values of KAa t pH 7 and 9 compare reasonably well with the values of the equilibrium constant for the dissociation of the enzyme-DPN+ complex determined by a spectrophotofluorimetric techniquelj: 1.6 X 10-4Llf and 0.12 X 10-4M, respectively. Similarly, K R a t the two pH values is in essential agreement with the equilibrium values of 0.31 X loF6 ,If ($H 7) and 0.65 X X (pH 9). It should be remarked that K A and K R as derived from kinetic studies represent specifically defined dissociation constants. For them to be coniparable to the quantities determined by spectrofluorimetry, the latter technique must specifically distinguish free enzyme from the bound coenzyme, however many isomers of the latter may be present. Departures from agreement may be attributed in part to the failure of the physical measurement used to make this distinction.
Liver Alcohol Dehydrogenase This enzyme obtained from horse liver was chronologically the first dehydrogenase to be studied extensively. 9-15 We can now consider Malate Dehydrogenase the results of 10 years of study of the kinetics of this reaction carried out by Theorell and coQuite recently Raval and Wolfel8 have published workers. The most recent data, obtained by data on the steady state kinetics of malate dehydroTheorell and XcKeelj and presented in Table 11, genase isolated from pig hearts.lg Their results for show some interesting features not encountered the four kinetic parameters a t PH S.0 for the with lactate dehydrogenase, yeast alcohol dehydro- forward reaction (reduction of DPN+) and the genase and ribitol dehydrogenase. symmetrically related parameters for the reverse In terms of the Dalziel notation, Theorell and reaction (oxidation of DPNH) were reported in NcKee’j found that 41q52/412 = do’and 41’&’/~12’Dalziel’s symbolism described in column I11 of = 40 a t both pH 7.0 and 9.0. In our symbolism Table I. these relations are equivalent to the kinetic criteria We have expressed their results in our notation for isomers of binary complexes being equal to and calculated the four additional kinetic paramunity. As a further consequence, the calculated eters from equations 8a-9b of reference 2 and values of K A B Qand K B Q R are effectively infinite. the lower limits for the various rate constants. More precisely, these latter parameters are so large This enzyme system exhibits the same adherence that the steady state concentration of the ternary to the Dalziel criteria for the kinetic insignificance complexes is essentially negligible compared to that of ternary complexes as liver alcohol dehydroof the binary The mechanism without genase, discussed above. However, the lower limits kinetically significant ternary complexes was orig- for the bimolecular rate constants kf and k - , are inally proposed for this reaction by Theorell and not so low as in that case. Chance. lo Glutamic Dehydrogenase Lye see from Table I1 that the lower limits for The reaction catalyzed by this enzyme involves the bimolecular rate constants k , and k-(n+l) for the combination of DPN and DPNH, respectively, two reactants and three products, viz. T P N + NH4+. The with the enzyme are of the usual order of magni- glutamate a T P N H f glutarate tude, l U f to lo7 M-l sec.-l. However, kf and k-, stoichiometry is like that of all the dehydrogenase(the latter would be k-f if there were in fact no catalyzed reactions, but NH4 is produced rather ternary complexes) are significantly smaller than the than Hf. This latter distinction nullifies the sinicorresponding rate constants for the other systems plification employed above of regarding one of the considered here. TT-hile part of this effect may be products as held a t a fixed concentration, ie., clue to the influence of pH (see below), this result (H+) being determined by the added buffer. The finds rationalization in the idea that there must be steady state kinetics is more complicated than for rather stringent orientational requirements, and the previously discussed cases. Lye summarize hence a large negative entropy of activation,” in the here some of the results which are of principal utility in an analysis of kinetic data for such a ( 0 ) H. T h w r e l l :ind R . Hunnichsen, Aclu Chem. S r u n d . , 6 , 1103 system. Reference 20 contains a fuller account (1951). (10) H. ‘Theordl a i d €3 Chaiice, ibid., 6 , 1127 (1Y:l). of the kinetic formulation. (11) H. Theorell, 4 . P. S y g a a r d a n d R. Bonnichsen, ibid., 9, 1148 For an enzyme-catalyzed reaction A B = (1955). P Q R with 3 mechanism of the type (12) H. ‘I‘heorrll diicl A . I ) . \Viner. .4rLh. Biockein. Biuphrs., 83, 291
+
+
+ +
(1 !l.>9),
( 1 3 ) H. ’llieorell, Federuliopi P r o c . , 20, 9ti7 ( 1 Y t i l ) . (14) H . l h e o r e l l and J. RIcKee, S u t u r e , 192, 47 (1961). (13) H. Theorell and J. RIcKee, Arlri Ciiev?. Srujid., 16, 1707 ( 1 N l ) . (11;) H. Thevrell a n d J. McKee, 15, 1811, 1831 ( I O i i l ) . (17) Sec, e.g., S. Glasstone, K. J. Laidler a n d H. Eyriiig, “ l l i c
+
T h e t r y uf K a t c l ’ r < x c ~ ~ e sllcGraw-Ilill ,” Bwjk Co.,Iiic , Nrw Yurk, .U. Y . , 1941. (18) D. N. R a v a l and R. G . Wolfe, Biochrwz., 1, 203 (1962). (19) R . G. R’olfe and J. B. Keilands, J . Biol. Chern., 221, D l (1961i). (20) 1‘. Bluomfield, Doctoral Dissertation, 1)eyt. o f Chemistry, University of Wisconsin, 19ti2.
VICTORBLOOMFIELD, LEONARD PELLER AND R. A. RLBERTY
4378
Vol. 84
simplifications are, however, not of general validity.20 From the steady state kinetics of the oxidation of TPNH, Frieden inferred that NH4+ combined with the enzyme before glutarateaZ4With the customary assumption that T P N H reacts first with the enzyme, the order is completely specified. In Table I11 we present Frieden's data23for the reaction in the symbolism of column I of Table I. The minimum values for the various rate constants according to relations 3a-4b have also been calculated. It is to be noted that inasmuch as VPQRKAB/VABKAKBequals 6.2 under the condition of study more than one binary complex must be considered in the combination of T P N + with the enzyme.
It may be shown that lower limits on the rate constants are given by the expressions ki 3 VABKBIKAB (E)o k- (n + 1) VPQRKPQIKPQR (E)o k-q 3 VIQRKPKRIKPQR (E)o kb > VABKA/KAB (E)o k - , > VPQRKQRIKPQK (E)o k a + I ) , k(Xg+ I ) , hY+I ) , k ( s + I ) > VAB/(EO) k - a , k - p , k - y , k-8 > VPQR/(E)O
(3c) (3d) (3e)
VPQRKABIVABKAKB >1
(Sa)
>
(3a) (3b)
(.ia)
(4b)
If more than one binary intermediate of type X, must be considered while if
+
VABKPQR(KPKQI< KPQKQ)/ VPQRKPKQRKPQKR > 1 (5b)
there is more than one complex Xa and/or more than one X,. Frieden has made a fairly detailed study of this enzyme s y ~ t e m . ~ His ~ - ~analysis ~ of data for the oxidation of T P N H was based on a rate law derived for a scheme with a specific number of intermediates whose formz3 is confirmed by eq. 2 on setting (A) = (B) = 0. Frieden's results were presented using the symbolism of column I1 of Table I and with the experimentally justijed simplification^^^ in terms of our nomenclature (column I) that (1) KPKR = KPR, (2) KPQR= K ~ K Q Rand (3) KPQR = K R K ~ Q .The above (21) C. Frieden, J . B i d . Chem., 234, 809 (1959). (22) C. Frieden, ibid., 234, 815 (1959). (23) C. Frieden, ibid., 834, 2891 (1959). (24) Such a n inference can be conceivably made for t h i s system by examining t h e contributory terms t o t h e reciprocal rate law when (A) = (B) = 0. This possibility arises because of t h e presence of terms in (R) ( Q ) a n d (P) ( Q ) but not (P)(R) in t h e expression.23
TABLE I11 KIKETIC DATAFOR GLUTAMIC DEHYDROGENASE, AS STUDIED B Y FRIED EN,^^ AND LOWERLIMITSOF VARIOUSRATECosSTANTS. 0.01 M TRISACETATE BUFFER. p H 8 0 , 25' -4,T P N ; B, glutamate; P, glutarate; Q , NH4+; R, TPXH VAB/(E)O, sec.-l 33 KA, M 2 . 3 x 10-4 KR, M 8.9 x 10-3 KAB,M 2 4.2 x 10-7 1.0x 103 VPQR/(E)o,sec. -l KP, Af T x 10-4 RR,M 2.6 X 10-5 K Q R ,Jd2 8.3 x 10-8 KPK,&I2 i.8 X KPQ,M 2 2 . 2 x 10-6 5.8 x 10-11 KPQR,M 3 7 o x 105 k l , M-I SCC.-' kb. 4f-l SeC.-' 1.s x 104 33 ka+ I ) , k ( g + I ) , kY+ I ) , k ( ? + I ) , set.-' 3 . 8 x 107 k + + 1 ) , M-l set.-' 3 . 1 X lo6 k-q, M-1 sec.-l 1 . 4 X 10' k+, 4f-l set? 1 . 0 x 103 k - a , k-Xg, k--,, k-6, set.-' VPQRKABI V.kBKAI8); G. G. IIammcs and I
