[30] Deuterium and tritium kinetic isotope effects on initial rates

[30]

DEUTERIUM

[30] Deuterium

AND TRITIUM KINETIC ISOTOPE EFFECTS

607

and Tritium Kinetic Isotope Effects on Initial Rates By

DEXTER

B.

NORTHROP

Introduction An enzyme-catalyzed reaction may display a very broad range of values for kinetic isotope effects, depending on reaction conditions. Because our methods of determination and ways of thinking about isotope effects originate with chemical reactions (which do not behave this way), the variation of values has led to considerable conceptual difficulty, poorly designed experiments, and faulty interpretation of results. The purpose of this chapter is to define the variety of hydrogen isotope effects within a logical and practical framework, and to outline appropriate experimental designs for the determination of each. Nomenclature, Definitions, and T h e o r y In an effort to reduce the complexity of kinetic expressions, the nomenclature of Northrop 1 will be used, with some recent additions. The conventional expression for an isotope effect, kH/kD, will be written Dk; similarly, the deuterium isotope effect on Vmax/Km [i.e., (VH/KH)/(VD/KD)]will be written DV/K, and so forth. The family of deuterium isotope effects on a hydride transfer reaction following an ordered Bi Bi reaction mechanism is illustrated in Fig. 1. What one measures directly is an "observed" isotope effect, DV, depicting a single data point for one set of reaction conditions, and as such it is dependent upon the entire steady-state distribution of enzyme forms. DV/Kappand DV~pp are "apparent" isotope effects extrapolated from observed effects to very low and very high substratc concentrations, respectively, and termed apparent because the cosubstrate (or activator, in this case substrate B), is fixed at an arbitrary level. DV/K and DV are "limiting" isotope effects at low and high cosubstrate concentrations, respectively, and represent the true kinetic constants describing the steady-state expression of the isotope effect. The limiting isotope effects share a portion of the reaction sequence in common. The "internal" isotope effect, Dk', is the effect on this common portion, representing all steps of the catalytic sequence followD. B. Northrop, in " I s o t o p e Effects on E n z y m e Catalyzed R e a c t i o n s " (W. W. Cleland, M. H. O ' L e a r y , and D. B. Northrop, eds.), p. 122. University Park Press, Baltimore, Maryland, 1977. METHODS IN ENZYMOLOGY, VOL. 87

Copyright © 1982by AcademicPress, Inc. All rights of reproduction in any form reserved. ISBN 0-12-181987-6

608

ISOTOPES AS MECHANISTIC PROBES

[30]

DV

f

D,V~pp

I f

I

- -

~

~

DVapp

°VIK

I

~

E +

I

f

I

I

r

f

I

I

I A ~kl' E A

.~

Dk'

I I

I I I

I

I

I

I

- - " ~

°k

I ks

I

~

DV

I

I

I

kS~

k7 i

k9

kll

I

k13

+ B~k4EAB ~--~EAB ~ E P Q ~IoEPQ - - P + E O - - " Q + E

FIG. 1. The steady-state dependency of a family of deuterium isotope effects.

ing substrate binding and leading up to and including the first irreversible step (in this case, the release of the first product). Finally, Dk is the "intrinsic" isotope effect and expresses the full effect imposed on hydride transfer, 2 which is analogous to isotope effects in chemical reactions. Intrinsic isotope effects may be different in the forward and reverse reactions, thus giving rise to an isotope effect on the equilibrium constant, O g e q = Dkf/Dkr, which is often easier to calculate than to measure. 3"4 Comparisons between isotope effects in this family (or between deuterium and tritium, see below) requires a zero point of reference. By the convention of expressing isotope effects as a ratio, the absence of an effect has a value of 1. Correction to a zero point of reference is therefore accomplished by subtracting 1 from the value of isotope effect; an isotope effect minus one is hereafter referred to as a "diminished" isotope effect? The kinetic expression for observed isotope effects on an enzyme-catalyzed reaction is the ratio of both entire rate equations describing the disappearace of hydrogen and deuterium substrates. The isotopically sensitive step appears in multiple terms and cannot be factored out. As a result, observed effects cannot be interpreted quantitatively, but may have some qualitative value. 2 To achieve factoring and subsequent simplification to useful values, it is necessary to examine the limits of rate equations 2 D. B. Northrop, Biochemistry 14, 2644 (1975). 3 W. E. Buddenbaum and V. J. Shiner, Jr., in "Isotope Effects on Enzyme Catalyzed Reactions" (W. W. Cleland, M. H. O'Leary, and D. B. Northrop, eds.), p. 1. University Park Press, Baltimore, Maryland, 1977. 4 W. W. Cleland, this series, Vol. 64, Article [5]. 5 The term is borrowed from musical nomenclature, in which the interval in a diminished chord is one note less than its parent.

[30]

DEUTERIUM AND TRITIUM KINETIC ISOTOPE EFFECTS

609

at low and high substrate concentration, where enzyme reactions approach apparent first-order and zero-order kinetics, respectively, as illustrated in Fig. 2. At low substrate concentrations, measured isotope effects approach DV/K, governed by equations which can be arranged into the following form 1:

°V/K = (Dk + C~ + Cr DKeq)/(1 + Cf + Cr)

(1)

Cf is the "forward commitment to catalysis" and represents the tendency of the enzyme complex poised for hydride transfer to continue forward through catalysis, as opposed to its tendency to partition back to free enzyme and unbound substrate. Similarly, Cr is the "reverse commitment to catalysis" and represents the tendency for the first enzyme complex following hydride transfer to undergo reverse catalysis, as opposed to partitioning forward through the first irreversible step leading to unbound product. As the reverse commitment to catalysis becomes large, the reaction step containing hydride transfer nears a chemical equilibrium, and the isotope effect expressed on V/K approaches the value of the equilibrium isotope effect. Obviously, the lower the commitments to catalysis, the more fully the kinetic isotope effect will be expressed on V/K. The (a) .......................

v (

~

/~1

I

...................Vo

L

KH

I

h

I KH

I

1

I IOK H

5K H

tOK H

[$I

i

I

5K H

(b)

o

VH

J

IS]

t

FIG. 2. The relationship of isotopic and nonisotopic velocities as a function of substrate concentration. (a) Michaelis-Menten-type plots, with the asymptotes identified as V and V/K. (b) Isotope effects from the ratios of curves in A with the limits of DVand DV/Kidentified. The broadened portion of the curves indicate the usual range of velocities and substrate concentrations employed in steady-state kinetics.

610

[30]

ISOTOPES AS MECHANISTIC PROBES

algebraic expression for the commitment factors of a linear sequence obeys the series°: C =

1 + ki_3 \ 1

• • •

(2)

where the/th step is hydride transfer. For the ordered mechanism in Fig. 1, the apparent isotope effect at low concentrations of substrate A and fixed levels of substrate B (i.e., DV/Ka) is goverened by the commitment factors k3 [B])]

,10(

(3)

kl,

C, = ku \ 1 + "~'~3)

(4)

Substitution of Cfa for Cf in Eq. (1) reveals the apparent isotope effect on V/Ka to be dependent on the concentration of B. At very high [B], the commitment of A becomes large and the isotope effect is abolished. At very low [B], the expression approaches the limiting isotope effect on V/K (i.e. DV/Kb) where the forward commitment factor reduces to

(5)

cfb = (k,/kr) (1 + ks~k,)

Hence, Eq. (1) describing the limiting isotope effect on V/K is independent of the concentration of A. (Note that the concentration of B necessary to halve the value of the diminished V/K isotope effect is equal to K~aKJKa). 1In equilibrium ordered reaction mechanisms, kz is very large; thus for practical purposes, Eq. (3) approximates Eq. (5), and the limiting V/K isotope effect is obtained at either low [A] or low [B]. For random mechanism, expressions for the commitment factors are more complex due to the branch points: the expression forward through the common sequence is divided by the net rate constant ~ away through the separate branches. For example, in the random segment

''%EAB

ka

*EAB

kll

,

(6)

EB

6 M. H. O'Leary, in "Isotope Effects on Enzyme-catalyzed Reactions" (W. W. Cleland, M. H. O'Leary, and D. B. Northrop, eds.), p. 233. University Park Press, Baltimore, Maryland, 1977. 7 W. W. Cleland, Biochemistry 14, 3220 (1975).

[30]

DEUTERIUM AND TRITIUM KINETIC ISOTOPE EFFECTS

611

the net rate constant for the release of A is koff = ks + k~ = ks + k2 k4/(ks + ks[B])

(7)

The forward commitment to catalysis at low [A] is therefore Cfa

=

kll[ ka ] kl----~ 1 + ks + k2k4/(k2 + ks[B])

(8)

Similarly, the forward commitment of catalysis at low [B] is

Cfb

=k,,k,o I-[ 1 + k, + ksks/(ks + kr[A])]

(9)

Examination of Eqs. (8) and (9) reveals that the isotope effects at low concentrations of one substrate should be determined at low concentrations of the other, in which case both equations reduce to k,x(

k0 ) 1 + !,, + k----C

(10)

Equation (10) also obtains when A and B are in rapid equilibrium from binary complexes EA and EB (i.e., both ks and ks are much greater than V/Et) when either substrate is at low concentration; hence limiting °V/K equals apparent DV/Kb and DV/Ka. Alternatively, if either substrate is rapidly released from the ternary complex (i.e., either k4 or ks is large) the forward commitment factor simplifies to Cif =

kn/klo

(11)

which is an "internal" commitment to catalysis for Eq. (6). At high substrate concentrations, measured isotope effects approach DV, governed by equations that can be arranged into the form8: DV =

Dk + Rf ~El A- CrDKeq 1 + R f / E f + Cr

(12)

Cr is the reverse commitment to catalysis defined above, in common with the kinetic expressions for isotope effects on V/K. Rf is the "ratio of catalysis," and is responsible for the familiar usage of isotope effects to identify the rate determining step. Rf consists of the arithmetic sum of the ratios of the rate constant for hydride transfer to the net rate constants (indicated by primes) for each of the other forward steps. The net rate constants preceding hydride transfer, however, do not include the hydride transfer step. Rf has the general form

s D. B. Northrop,

Biochemistry 20, 4056 (1981).

612

ISOTOPES AS MECHANISTIC PROBES

.,__[.

ki

• "~

k,

~'----~ + k;+----~+ --7-- " " •

[30]

]

ki+4

(13)

where the ith step is hydride transfer. The other new term, JEt, is the "equilibration preceding catalysis." If this equilibration is unfavorable, the hydride transfer step will act as a "gate" to enzyme turnover, with the result that an isotope effect may be expressed despite subsequent steps that are slower than the isotopically sensitive step. The term Ef consists of a geometric series of the reciprocal of the equilibrium constants for each of the steps preceding hydride transfer but following substrate binding. It has the general form [ 1 + K__~qH 1 (1+" 1_ ( 1 + Keq,~_

El=

" "))]

(14)

where the ith step is hydride transfer. For the mechanism in Fig. 1, the expression for the ratio of catalysis divided by the equilibration preceding catalysis contributing to the apparent ° V at saturating A is written: Rf

ka/~[B] + ka/~ + kglk, + kg/k~ + k~/kh + ka/k,5

Eff =

(15)

1 + (ks/kO[1 + (kJks)(1 + k4/ka[B])]

where net rate constants are k'~ = k5kT/(k8 + kO, ~ = k3k'g/(k4 + ~ ) , and k~l = knk13/(k12 + k13). Internal isotope effects cannot be obtained by manipulation of substrate concentrations, but may be approached by studying limiting isotope effects as a function of some other variable, such as pH dependence. For example, in the mechanism EH .

k4H

k~A

k3

EAH

ksH

k~

k~A

E

EA

kz

k'

~EP

ko

~E+P

(16)

where k' represents an unknown number of steps, lowering the pH draws enzyme away from EA and the catalytic sequence, thus lowering the external commitment to catalysis: k' Cef

:

k2 + keka[H]/(k6 + kO

(17)

Thus, at low pH, DV/K approaches the limit DV/K = °k' + Cef=

1 + col

IDk'l"
(18)

[30]

DEUTERIUM

AND TRITIUM KINETIC ISOTOPE EFFECTS

613

Also, at low pH, the conversion of EAH to EA attains an unfavorable equilibrium, affecting the maximal velocity. At first thought, one expects the isotope effect to be abolished rather than enhanced by a shift in the rate-limiting step to something other than catalysis; but the velocity is limited not because a forward rate constant is reduced, but rather because the preceding equilibrium has become less favorable, elevating the hydride transfer step to the status of a "gate." The external ratio of catalysis divided by the pH-dependent equilibration preceding catalysis for the reaction in Eq. 06) is [see Eqs. (13) and 04)] Ref k' / k~ + k' / k9 Ef = 1 + ks[H]/k,

(19)

Thus, at low pH, DV approaches the limit: DV =

Dk' + R e f / E f 1 + Ref/Ef

(20)

In addition to determining the internal isotope effect, variation of pH may distinguish between reaction mechanisms .9.10 Variants of the mechanism of Eq. (16) are illustrated in Fig. 3, together with the form of expected plots of diminished limiting isotope effects as a function of pH. While only the first variant may be readily distinguished from the other three by normal pH kinetics,ll the additional use of the isotope effect may distinguish among all four, provided the limiting isotope effects at neutral pH do not already equal the internal effect. The determination of the intrinsic isotope effect is based on the fixed relationship between deuterium and tritium isotope effects, defined by Swain et al., 12 Tk = Dkl.~2

(21)

This relationship does not appear to hold for apparent, limiting, or internal isotope effects, which reflects the fact that none of these have been corrected to a common zero point. For diminished isotope effects, T k - 1 = Dk1"442-- 1

(22)

a fixed relationship between deuterium and tritium is maintained. In the absence of an equilibrium isotope effect, the general expression for diminished V / K isotope effects becomes a simpler function of the diminished

9 p. F. C o o k and 10 p. F. Cook and 11 W. W. Cleland, 12 C. G. Swain, E. 5885 (1958).

W. W. Cleland, Biochemistry 20, 1797 (1981). W. W. Cleland, Biochemistry 20, 1805 (1981). Adv. Enzymol. 45, 273 (1977). C. Stivers, D. F. Reuwer, Jr., and L. J. Schaod, J. Am. Chem. Soc.

80,

614

ISOTOPES AS MECHANISTIC PROBES

MECHANISM

LOG (Dv-1) P L O T S

[30]

LOG(DV/K-1) PLOTS

l

I

pH

pH

FIG. 3. The pH dependence of diminished isotope effects on V and V/K. The symbols for mechanisms represent variants of the mechanism in Eq. (16), signifying the presence and absence of steps governed by ks, ks, ~ , and ks.

intrinsic effect [see Eq. (1) at aKeq = 1]: Dk-- 1 DV/K - 1 = 1 + Cf + Cr

(23)

The commitment factors do not contain isotopic steps; hence the parallel tritium expression for Eq. (23) has the same denominator, leading to a cancellation of commitment factors in the ratio

DV/K- 1 TV/K-

Ok- 1

1 = Tk- 1

(24)

Substitution of the Swain relationship yields la DV/K- 1 vV/K - 1

-

Ok- 1 Dk1'442-- 1

(25)

Consequently, bk is calculable from values of DV/K and TV/K. In the presence of an equilibrium isotope effect, DKeq = Dkf/Dkr, the expression

[30]

DEUTERIUM

AND TRITIUM KINETIC ISOTOPE EFFECTS

615

for diminished isotope effects is not cleared of commitment factors as in Eq. (23), but takes the form H

° k f - 1 + Cr(OKeq - 1) °V/K - 1 =

1 + Cf + Cr

(26)

and the ratio of diminished effects becomes DV/K

TV/K-

-

1 1

Ok -

1

+ Cr(DKeq -

1)

(27)

Dkl'44z -- 1 + Cr(DKelh442 -- l )

With two unknowns (i.e., Dk and Cr), an exact solution is not possible, but limits can be set for the intrinsic isotope effect because at DK,eq > 1 DKeq t>

(bkf/Dkr)CALC

(28)

Therefore, it follows that the intrinsic isotope effect lies between the calculated limits 1~

°Keq (Dkr)CALC/> Dkf t> (Dkf)cALc

(29)

TO obtain the upper limit, (Dkr)CALCcan be calculated from data in the forward reaction by first multiplying the limiting isotope effects by the equilibrium isotope effect, because (30)

D ( W K ) f / D ( V / K ) r -- DKeq

Therefore

°(V/K)f/OKeq- 1 O(v/IOr- 1 °(kr)CALC- 1 T(VIIOflDIOeh442 - 1 -T(VIK)r- 1 --D[/.~l.442\~rICALC 1 -

(31)

-

The isotope effects of Fig. 1 generally get larger as one moves down the figure. Comparison between the diminished form of these isotope effects can be employed to determine the relative steady-state contribution of different components to the limiting values of kinetic parameters. Comparing diminished intrinsic to diminished V/K isotope effects provides a measure of the sum of commitment factors, as shown by the following rearrangement of Eq. (23): Dk-- 1 Cf d- C r -- D V / K _ 1

1

(32)

Similarly, comparing diminished intrinsic to diminished internal isotope effects provides a measure of the isotopic suppression within the catalytic sequence, and comparing diminished internal to diminished limiting isotope effects provides a measure of the "stickiness" of substrates [cf. Eqs. (10) and (ll)].

616

ISOTOPES AS MECHANISTIC PROBES

[30]

Practical Aspects

Noncompetitive versus Competitive Measurements. Noncompetitive measurements employ total or near-total isotopic substitution of substrates and are usually associated with deuterium. 13 At less than total substitution, observed isotope effects are easily corrected by the relationship DV =

VH(obs)F

(33)

VD~obs)-- Vmobs)(1 -- F) where F is the fraction of deuterium substitution. The term noncompetitive derives from the fact that labeled and unlabeled substrates are reacted with enzyme in separate experiments; hence they do not "compete" with each other for the enzyme. Competitive measurements usually employ trace labeling of substrates and are usually associated with tritium (and heavy atom isotope effects6). The large disparity between the concentrations of labeled and unlabeled substrates makes it impossible to measure their separate velocities. Instead, normal isotopic discrimination is measured, as expressed in either isotopic enrichment of the substrate or isotopic depletion of a product. The kinetics of initial velocities of competing substrates obey equations of the form 13 DH (V/K)H[SH] V-"T-= (V/K)T[ST]

(34)

which reveals that isotopic discrimination measures only V/K isotope effects. Because of the isotopic enrichment of substrate, the conditions of competitive experiments constantly change as a function of the progress of the reaction. Integration and rearrangement of Eq. (34) yield log(1 - f )

TV/K = log[1 - f(SA)p/(SA)o

(35)

where f is the fractional conversion of substrate to product, (SA)o is the initial specific radioactivity of labeled substrate, and (SA)p is the specific radioactivity of product at a fractional conversion f. To determine TV/K experimentally, one incubates an enzyme with tritiated substrate of accurately known specific activity, stops the reaction after a short period of time, determines the extent of the reaction, isolates the tritiated product, and determines its specific activity. Each of these steps may present experimental problems, depending on each enzyme system. For example, with some enzymic reactions, it may be more convenient or accurate to determine (SA)o from the specific activity of products after driving the 13 H. Simon and D. Palm, Angew. Chem., Int. Ed. Engl. $, 920 (1966).

[30]

DEUTERIUM AND TRITIUM KINETIC ISOTOPE EFFECTS

617

reaction to completion, because identical procedures can then be used for determining (SA)p. Equations (34) and (35) also assume reactions are irreversible, and they do not take into account the additional isotope effect of a back reaction. Errors of this origin are normally avoided by stopping reactions after only low levels of substrates have been converted to product (i.e., f < 5%). Under these conditions, the isotope effect is obtained directly from the ratio of specific activity, independent o f f , because Eq. (35) approaches the limit TV/K = I(SA)0/(SA)pLr_,o

(36)

Highly precise determinations of isotope effects by the competitive method have recently shown that the specific activity ratio quickly falls off as a function off, even below the arbitrary 5% level, ~4"15indicating that T(V/K) should be obtained from several measurements at different fractional conversions and extrapolated back to zero f. Competitive experiments have the disadvantage of yielding less information (i.e., TV/K only) than noncompetitive. However, they do have several advantages. First, one less extrapolation step is necessary for computing a limiting isotope effect; hence, fewer experiments are required. Second, impurities in substrates are less of a problem. Inhibitors that only interfere with the binding of substrates will not alter a competitive isotope effect, because isotopic and nonisotopic rates are equally affected. In the noncompetitive mode, both DVapp and °V/gapp a r e affected bY unequal contamination of labeled and unlabeled substrates, and DVapp may be shifted towards DV/Kappby equal contamination. (The assumption that valid isotope effects could be obtained if labeled and unlabeled substrates were equally free of inhibitors has been disproven). 16 Third, competitive experiments are independent of the concentration of the labeled substrate, whereas noncompetitive measurements require accurate knowledge of substrate concentrations for computation of DV/K. Fourth, stereochemical purity is not required in competitive experiments employing a trace label, but is necessary to avoid possible secondary isotope effects in the noncompetitive mode; hence the preparation of labeled substrates is easier. Fifth, competitive experiments yield greater precision. By definition, conditions such as temperature, pH, enzyme concentration, changes in enzyme activity, etc., are exactly the same for reactions of competing labeled and unlabeled substrates, and incorporating a ~4C

~4 R. K. Goitein, D. Chelsky, and S. M. Parson, J. Biol. Chem. 253, 2963 (1978). ~5 H. G. Bull, J. P. Ferroy, E. H. Cordes, A. Ribbi, and R. Apitz-Castro,J. Biol. Chem. 253, 5186 (1978). ~s C. E. Grimshaw and W. W. Cleland, Biochemistry 19, 3153 (1980).

618

I S O T O P E S AS M E C H A N I S T I C

PROBES

[30]

label into the experimental design to yield a doubly labeled product offers the added high precision of T/14C ratio counting (i.e., ---0.5%) 14,~s as a means for estimating (SA)0 and (SA)p. Computation of Apparent and Limiting Isotope Effects. The isotope effects obtained by a competitive measurement may be either apparent or limiting V/K effects, depending upon the reaction mechanism and which substrate is labeled. For the mechanism of Fig. 1, if B carries the tritium label, then the limiting TV/K is obtained directly; if A carries the label, the isotope effect is dependent upon the concentration of B, which must be extrapolated to zero to obtain the limiting TV/K. A linear extrapolation is possible on a plot of the reciprocal of the apparent diminished isotope effect versus the concentration of B, described by the equation [cf. Eqs. (1), (3), (5), and (26)]:

1 TV/Kap p -

1 1

T V / K b -- 1

+

Cro[B]ka/k2 Ok -- 1

+ Cr(DKeq -

1)

(37)

For random mechanism, a similar extrapolation is necessary regardless of whether A or B carries the label. However, the equation analogous to Eq. (37) is nonlinear due to the complexity of a branch-point commitment factor, and some alternative extrapolation is needed (see Miller and Kilnman, this volume, Article [34]). The isotope effects obtained by noncompetitive measurements must always be extrapolated as a function of substrate and cosubstrate concentration, regardless of which substrate carries the deuterium label. When data are plotted on a double-reciprocal plot, they look like a noncompetitive inhibition pattern, and should be analyzed in much the same way, in that one looks for slope and intercept effects, DV/KaDpand VVapp , respectively. There are important differences, however. The analysis of inhibition patterns has two distinct objectives: model discrimination and model fitting. ~r'is The emphasis is usually on the former; hence many concentrations of varied substrates are chosen, with a focus near the Michaelis constant. In contrast, the analysis of isotope effect data is concerned only with model fitting at high and low substrate concentrations. Model discrimination is not an immediate objective because, in most instances, the presence or absence of slope and intercept changes will depend on the relative size of different rate constants and not on their presence or absence. It is also not the purpose of these experiments to establish that double-reciprocal plots are linear. (That will have been done in earlier kinetic studies establishing the steady-state mechanism, which must be known for proper design of the isotopic experiments.) Model fitting is truncated in lr p. j. F. H e n d e r s o n , Tech. Life. Sci. [Sect.]: Biochern. Vol. B1/II, p. 1 (1978). is S. E. D a m g a a r d , Biochemistry 20, 5662 (1981).

[30]

DEUTERIUM

AND TRITIUM

KINETIC ISOTOPE EFFECTS

619

comparison, because the usual parameter estimation is not an objective; instead, a ratio of parameters is sought. The choice of substrate concentrations expresses these differences as illustrated in Fig. 2B: at Michaelis concentrations, observed isotope effects are midway b~tween DVapp and DV/Kapp, reflecting a minimal discrimination between parameters; at 10 K and 0.1 K, the observed isotope effects are within 90~ of DVapp and oV/Kapp, respe,ctively. Therefore, to determine apparent isotope effects on V and V/K by the noncompetitive method, a four-point pattern is recommended, reflecting high and low substrate concentrations in the presence and absence of deuterium. Minimizing the number of data points provides the opportunity to collect enough replicates of each to obtain a measure of their variance and perform the extrapolation by linear regression analysis, yet avoid the statistical pitfalls of fitting to the reciprocal plot. TM The computation of limiting isotope effects from apparent values requires extrapolations as a function of the cosubstrate concentration. The problem is, the apparent isotope affects are not simple functions of the cosubstrate [see Eqs. (8) and (15)]. Instead, the extrapolations must be made prior to computing the isotope effects, using two four-point patterns representing high and low cosubstrate concentrations and computing the limiting isotope effects from the equivalent of slope and intercept replots. In practice, however, the errors introduced by using apparent values in place of limiting ones may not be significant, because the error is dependent upon the difference between ~V and DV/K. For example, if DV equals DV/K, then observed isotope effects are independent of the cosubstrate concentration. Determination of Intrinsic Isotope Effects. An estimation of the intrinsic isotope effect is dependent upon a comparison of diminished deuterium and tritium isotope effects [Eq. (27)]. With few exceptions (see Klinman, this volume, and Ref. 17), these have been obtained by a combination of noncompetitive and competitive methods. This combination approach suffers not only the questionable accuracy of obtaining values by differing experimental design (in which the former is subject to errors from impurities) but also from the labor and time differences of collecting enough data to extrapolate both sets of data to limiting V/K isotope effects needed for the comparison. As a result, multiple determinations sufficient to establish true confidence limits have been absent. In addition, Albery and Knowles TM examined the basis of Eq. (27) and calculated that the experimental error on kinetic data.must be less than 3% for determining the relative free energies of two transition states that are both partially rate-limiting (which is a somewhat different question, but the 19 W. J. Albery and J. R. K n o w l e s ,

J. Am. Chem. Soc. 99, 637 (1977).

620

ISOTOPES AS MECHANISTIC PROBES

[30]

analysis is valid.) Consequently, a new experimental design is necessary, because a new question is being asked. Deuterium isotope effects are normally determined by the noncompetitive method, but need not be. Raftery and co-workers introduced an elegant double-label technique for competitive determination of deuterium isotope effects and achieved sufficient precision to apply it to the detection of small secondary isotope effects. 2° In this design, the reaction progress of the deuterated substrate is followed with a carbon-14 trace label and compared to the reaction progress of the nondeuterated substrate, followed with a tritium trace label, to take advantage of the precision available by T/~4C channels ratio scintillation counting. Improvements in the precision of the method have subsequently appeared.~4,15 The advantages of the competitive method, together with this new design, now invite the elaboration of a "triple-competitive" method for determining primary intrinsic isotope effects, z~ To illustrate the method, consider the following sequence of reactions of lactate dehydrogenase: [14C]NADD + [14C]pyruvate -* [14C/D]lactate + [14C]NAD

(38)

[T]NADH + p4C]pyruvate~ [14C/H]lactate + [T]NAD

(39)

NADT + [~4C]pyruvate~ [~4C/T]iactate + NAD

(40)

Reaction (38) represents deuteride transfer accompanied by the production of NAD labeled in the adenine ring with 14C. Reaction (39) represents hydride transfer accompanied by the production of tritiated NAD, also labeled in the adenine ring. Consequently, the deuterium isotope effect if expressed in the T/14C ratio of NAD. Reaction (40) represents tritiide transfer, which produces tritiated lactate. All three reactions are accompanied by the production of 14C-labeled lactate, which will reflect hydride transfer providing N A D H is trace-labeled in the transferable position (see below). Consequently, the tritium isotope effect is expressed in the T/14C ratio of lactate. In order to convert counting ratios to isotope effects, reference determinations in the absence of isotope effects are needed. Several approaches are possible and await experimental verification, but one example is to drive the reaction to completion by the addition of NADase, followed by T / 1 4 C ratio determination in isolated lactate and ADP-ribose. This latter step follows the removal of several samples from the initial reaction, in order to extrapolate to a zero level of conversion [Eq. (36)]. The net result is a determination of the intrinsic isotope effect from a single incubation, which should provide increased accuracy and precision; identical reaction conditions are assured for both isotope effects, errors of 20 F. W. Dahlquist, T. Rand-Meier, and M. A. Raftery, Biochemistry 8, 4214 (1969). 21 C. J. Newton and D. B. Northrop, unpublished results.

[30]

DEUTERIUMAND TRITIUM KINETIC ISOTOPE EFFECTS

621

the noncompetitive method are avoided, one extrapolation step is eliminated, and advantage is taken of the precision of double-label counting methods, which should yield an estimate of the ratio of limiting isotope effects to within +- 2%. In addition, the reduction of labor makes it practical to perform replicate determinations for the purpose of establishing confidence limits and to determine intrinsic isotope effects as a function of other variables such as alternate substrates, temperature, pH, and isozymes. Three precautions are unique to the triple-competitive method and must be exercised. First, the deuterated substrate must be a negligible species, not only to avoid decreasing the measure of hydride transfer in the acceptor product (i.e., [~4C]lactate) but also, in the event of less than total isotopic substitution, to avoid decreasing the measure of hydride transfer in the donor product (i.e., [T]NAD). Second, as 'mentioned above, competitive experiments with a trace label do not normally require stereospecifically labeled substrates, but here they do because of the confusion with a second nontransferable label. NADT must be stereospecific to avoid production of NAD(T). Third, impurities that normally do not alter the measurement of a competitive isotope effect may have to be removed. For example, NAD is a common impurity in NADH, and any labeled NAD would have to be removed in the above design. T h e final computation of the intrinsic isotope effect from the ratio of diminished deuterium and tritium isotope effects requires that one first prepare a series of approximations because of the exponential character of the Swain relationship. This may be accomplished by constructing a graph, 2 a table, 22 or an iterative computer program such as given in the appendix. E n z y m e - M e d i a t e d Solvent Isotope Effects Solvent isotope effects are usually obtained by measuring enzymic reaction rates with an excess of substrate in H20 and in D20 as a function of pH(D). A proton inventory in the form of a variation of the atom fraction of deuterium determines whether the effect observed is of single or multiple origins. Rarely, if ever, is the concentration of substrate varied to obtain the V / K isotope effect. Consequently, with only the single value of DV available, interpretations of the kinetic functions governing the expression of solvent isotope effects are severely limited. This is unfortunate, because the influence of substrate concentration on solvent isotope ef22Appendix D, in "Isotope Effectsin Enzyme-CatalyzedReactions" (W. W. Cleland, M. H. O'Leary, and D. B. Northrop, eds.), p. 280. UniversityPark Press, Baltimore, Maryland, 1977.

622

ISOTOPES AS MECHANISTIC PROBES

[30]

fects may distinguish mechanisms o f direct transfer between solvent and substrate from mechanisms of enzyme-mediated transfer) 3 Only the latter will show substrate dependence of isotopic discrimination (obtained by performing competitive experiments in 50: 50 1-120: I)20) and will do so only if the mediating group is partially shielded by the substrate from exchange with the solvent. If fully shielded, a monoprotic mediating group will show no isotopic discrimination at high substrate concentration, 24 but di- and triprotic mediating groups may still display isotopic discrimination because of intramolecular isotope effects. This discrimination must be less than intrinsic isotope effects because alternate reaction pathways of di- and triprotic mechanisms compensate for the isotope effect and, in effect, dilute the expression of the intramolecular component. The upper limits of isotope discrimination are:

(PH/PD)diprotie (PH/PD)tr~p~otic= =

(3°k + 1)/(Dk + 3)

(41)

[7(Dk)2 + 10°k + 1]/[(Dk)2 + 10°k + 7]

(42)

Maximal values of shielded isotopic discrimination are calculated from Eqs. (41) and (42) after first determining the intrinsic isotope effect from Eq. (27). Because the commitment factors governing the V/K isotope effect were shown to be at least as great as those governing intramolecular isotopic discrimination, minimal values are calculated by substituting the V/K effect for the intrinsic in Eqs. (41) and (42). If isotopic discrimination is less than both minima, the group is monoprotic; if less than the triprotic minimum, the group is either a shielded diprotic or a partially shielded monoprotic; if less than the triprotic maximum, all three groups are candidates at various levels of shielding; and if greater than the triprotic maximum, then proton transfer must be mediated by an unknown group which cannot be fully shielded from exchange with solvem. Limitations The primary limitation encountered in the use of isotope effects in cnzymology is the need for extreme precision. Unlike steady-state kinetics, which are concerned with parameter estimations, isotope effects are defined as ratios o f parameters that, in addition, have ~ t a t i v e signifi= cance only as diminished ratios to yet another isotope effeeL such as illustrated by Eqs. (24) and (32). The limitation compounds because percentage errors are additive when ratios are calculated. A c o r o l l a ' o f this limitation is that the amount of labor necessary to achieve and docu, ~3 D. B. Northrop, J. Am. Chem. Soe: 103, 120~(t981). 34 H. Y ~ atld M. H. O'Leary, J. A m . Chem.. Soc. ~P), 1660"(197:'/).

[30]

DEUTERIUM AND TRITIUM KINETIC ISOTOPE EFFECTS

623

ment the requisite precision while striving for accuracy can easily become prohibitive. For example, in order to fully document isotopic differences between a natural and an alternate substrate, four complete sets of initial velocity patterns are needed, preferably determined on the same day with identical reagents. To overcome this limitation, alternative experimental designs are needed to replace those that have evolved from steady-state kinetics, because different kinetic questions are being asked with an isotopic substitution of a substrate than are being asked by varying the concentration of that substrate. The triple-competitive isotopic experiment and the four-point pattern have been expressly devised in an attempt to overcome this limitation. As a result of the prohibitive labor involved, apparent isotope effects and even observed isotope effects have been accepted as limiting or intrinsic isotope effects, simply on the basis that they are "large." But how large is "large?" From transition-state theory, a value of 7 is predicted for a full deuterium isotope effect on C - H bond cleavage, but smaller observed values have been accepted as intrinsic and larger values have been reported, z5 A full complement of limiting, internal, and intrinsic isotope effects is needed to relieve these uncertainties. Concluding Remarks A discussion of applications and interpretations of isotope effects data was deliberately avoided. When full cognizance of the general equations is taken, many of the experimental designs and mechanistic conclusions in the literature become quite disturbing. Often the wrong questions are being asked of the isotope, yet it is not altogether clear what the right questions are. We are caught in a moment of transition as isotope effects emerge from being a minor curiosity, used occasionally to attempt identification of a rate-determining step to become perhaps the most powerful tool we have for distinguishing between alternative mechanisms of enzymic catalysis. This latter prophecy derives from the unique property of a primary isotope effect to perturb a single catalytic step in an enzymatic reaction, yielding a point of reference unavailable by any other means. It was stated earlier that model discrimination was not an immediate objective of isotope effect studies. Model discrimination retums, however, when isotopic effects are used as a point of reference in examining enzymic catalysis as a function of another reaction variable. 26The effect of pH

25 j. p. Klinman, in "Transition States of Biochemical Processes" (R. D. Gandour and R. L. Schowen, eds.), p. 165. Plenum, New York, 1978. 2e D. B. Northrop, Annu. Rev. Biochem. 50, 103 (1981).

624

ISOTOPES

AS M E C H A N I S T I C

PROBES

[30]

variation, illustrated in Fig. 3, is but one exampie of a promising future in enzymology for the isotope effect. Appendix The following computer program was written by Cleland (personal communication) for the purpose of calculating intrinsic isotope effects. By successive approximations, the program calculates the limits on both the forward and reverse intrinsic isotope effects, by fitting to Eqs. (27) and (31). The input data are placed on a single card and consist of the V/K deuterium isotope effect, the V/K tritium isotope effect, and the rcciproCALCULATION-OF INTRINSIC |IOTORE EFFECTS DATA INPUT * DEUTERIUM ISOTOPE EFFECTw TRZTIU~ EFFECTe BETA IN THREE FtO,q PIELDSo FDLLOWE~ BY ANY DESIRED TITLE, AS MANY SETS OF DATA MAY BE PROCESSED AS ~ESIREO. THE PROGRAM IS sTOPR[D ~ITM A BLANK CARD. g REAO le DXw Tie BETA IFtDI) It,It,tO | FORHATt3F|O.SeoOH TITLE 10 PRTNT lw DIw Tie BETA VF • ( O Z . t . ) / ( T I = l , ) YR • ( O I * B E T A - I , ) t ( T I * O E T A t * I , O ~ E ' I * ) V ~ YF Ms1

, s 11~

S DO E • leo VC • C A . t , ) / t A * * t , 4 a 2 - t , 1 Dy • ( l , 4 ~ 2 * A t * o ~ 4 2 " 1 , ' e O U 2 * A * t I o ~ 4 ~ ) / t A * t S o ~'10)e*2 A • I + (yeyC)/OY GO To ( 3 w ~ ) ~ M 3AFsA BAF • AF*BETA T&F • A l * | e ~ a 2 PTAF • BETA**I~4U2 *TAF H • • V 8 YR GO TO S UARsA ARR • AR/BETA TAR • A * * I . ~ U ~ TARB • TAR/BETA**I,~aE PRINT be ARBe IF b FORMAT{48H L|HIT$ ON FORHARD DEUTERIUH IsoTOPE EFFECT ARE FI0,Se I ?H AND F|O,~) PRINT T# TARBe TAF 7 FOR~AT(a3H CORRESPONDIN~ TRITIUM ISOTOPE EFFECTS ARE F10oSe | 7H ANO F l 0 o ~ ] PRINT Pe ARe BAF_ S FORNIT(QSH LIMIT8 O~ REVERSE nEIJTERIUM IsDTNPE EFFECT ARE FIO,Se t TH AND FIOD~) PRINT 7e TARe ~TAF PRIr~T tE 12 FURMATftH / / / 1 GO To 9 11P~INT 13 13 rOR~IATtI~H PROGRAM COMPLETED/) STOP END

[31]

DETERMINING TRANSITION-STATE STRUCTURE

625

cal of the equilibrium isotope effect (BETA) in three successive F10.5 fields. Columns 31-78 of the card are for a title; anything here is printed out verbatum to identify the output. Acknowledgments This article draws on work supported by grants from NSF (PCM 75-16704) and NIH (GM00294).

[31] The Use of Isotope Effects to Determine Transition-State Structure for Enzymic Reactions By W. WALLACE CLELAND

Isotope effects have been used for a long time to determine the structures of transition states for chemical reactions. 1 Such analysis is made easier by the fact that the chemical reaction being looked at is often the sole rate-limiting step. [However, in reactions such as hydroxide-mediated hydrolysis of esters there are at least two steps, with hydroxide adding to form a tetrahedral intermediate, which then breaks down to release the alcohol, and the partition ratio of the tetrahedral intermediate for release of alcohol and hydroxide (that is, reverse reaction) affects the isotope effects actually observed. 2] For enzymic reactions, on the other hand, it is rare for the chemical bond-breaking step to be totally rate-limiting, and other steps, such as product release, or conformation changes that precede or follow the bond-breaking step, are usually partly rate-limiting. Further, there are different isotope effects on V and on V/K for each reactant, and it is not always easy to determine the intrinsic isotope effect on the bond-breaking step, which is the value that gives information on the structure of the transition state. Hence in this chapter we will review the current status of techniques for determining intrinsic isotope effects, and then discuss the interpretation of these values in terms of transitionstate structure. i For recent books summarizing the field, see L. Melander and W. H. Saunders, Jr., "Reaction Rates of Isotopic Molecules." Wiley, New York, 1980; "Transition States for Biochemical Processes" (R. D. Gandour and R. L. Schowen, eds.). Plenum Press, New York, 1978; "Isotopic Effects in Chemical Reactions" ((2. J. Collins and N. S. Bowman, eds.), Van Nostrand Reinhold, New York, 1970. There are also some useful chapters in "Isotope Effects on Enzyme-Catalyzed Reactions" (W. W. Cleland, M. H. O'Leary, and D. B. Northrop, eds.), University Park Press, Baltimore, Maryland, 1977, and in particular a very complete list of bibliographies and reviews on isotope effects through 1976. 2 M. H. O'Leary and J. F. Marlier, J. Am. Chem. Soc. 101, 3300 (1979). METHODS IN ENZYMOLOGY, VOL. 87

Copyright © 1982 by Academic Press, Inc. All rights of reproduction in any form reserved. ISBN 0-12-181987-6