[29] Solvent isotope effects on enzyme systems

[29]

SOLVENT ISOTOPE EFFECTS ON ENZYME SYSTEMS

[29] S o l v e n t I s o t o p e E f f e c t s o n E n z y m e

551

Systems

By K. BARBARA SCHOWEN and RICHARD L. SCHOWEN Introduction The introduction of deuterium in place of protium in the hydrogenic sites of water, and its consequent exchange into some positions of enzymes and substrates, produces solvent isotope effects on the kinetic and equilibrium constants associated with the enzymic reaction. These effects, usually expressed as ratios of the appropriate constants in the two isotopic solvents HOH and DOD, are useful in the study of reaction mechanism. There is an extensive and valuable review literature on many aspects of this subject. Readers may wish to consult the following: Isotope effects in general: the volumes of Collins and Bowman I and Melander and Saunders. 2 Isotope effects on enzymic reactions: the books of Cleland, O'Leary, and Northrop, a and of Gandour and Schowen,4 and review articles by Klinman~ and by Cleland. 6 Solvent isotope effects in general: reviews by Laughton and Robertson, 7 Schowen, 8 and Albery. 9 Solvent isotope effects on enzymic reactions: reviews by Katz and Crespi, 1° R. L. Schowen, 1~ and K. B. Schowen. a2 1 C. J. Collins and N. S. Bowman, eds., "Isotope Effects in Chemical Reactions." Van Nostrand-Reinhoid, Princeton, New Jersey, 1970. 2 L. Melander and W. H. Saunders, Jr., "Reaction Rates of Isotopic Molecules." Wiley (Interscience), New York, 1980. 3 W. W. Cleland, M. H. O'Leary, and D. B. Northrop, eds., "Isotope Effects on Enzyme-Catalyzed Reactions." University Park Press, Baltimore, Maryland, 1977. 4 R. D. Gandour and R. L. Schowen, eds., "Transition States of Biochemical Processes." Plenum, New York, 1978. J. P. Klinman, Adv. Enzymol. 46, 413 (1977). W. W. Cleland, this series, Vol. 64, Part B, p. 104. 7 p. Laughton and R. E. Robertson, in "Solute-Solvent Interactions" (J. F. Coetzee and C. D. Ritchie, eds.), p. 400. Dekker, New York, 1969. s R. L. Schowen, Prog. Phys. Org. Chem. 9, 275 (1972). 9 W. J. Albery, in "Proton Transfer Reactions" (E. Caldrin and V. Gold, eds.), p. 263. Chapman & Hall, London, 1975. 10 j. j. Katz and H. L. Crespi, in "Isotope Effects in Chemical Reactions" (C. D. Collins and N. S. Bowman, eds.), p. 286. Van Nostrand-Reinhold, Princeton, New Jersey, 1970. ~1 R. L. Schowen, in "Isotope Effects on Enzyme-Catalyzed Reactions" (W. W. Cleland,

METHODS IN ENZYMOLOGY, VOL. 87

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

552

ISOTOPES AS MECHANISTIC PROBES

[29]

Solvent isotope effects in mixtures of H O H and DOD: reviews by Kresge, 13 Gold, 14 Albery, 9 R. L. Schowen, 11 and K. B. Schowen. TM Biological isotope effects: the review of Katz and Crespi. TM Reviews of narrower aspects of the topic will be referred to at appropriate points in the text. The purposes of the present article are: (a) to give the physicochemical background for the use of solvent isotope effects in biochemical studies, with fairly complete derivations of the requisite algebraic expressions, an account of underlying assumptions, and a review of pertinent experimental and theoretical information; (b) to outline workable procedures for carrying out experiments in this area; and (c) to present the apparatus for interpretation of the results. We shall not, in this article, review the applications of this method nor give any extensive illustrations from the literature. A number of examples of biochemical solvent-isotope effect investigations may be found in reviews published in 197711 and 1978. TM A forthcoming article, 15 which may be considered complementary to the present article, will dwell critically upon examples of application. Background Reaction R a t e s and Equilibria in Isotopic Waters Solvent isotope effects, since they refer to rate or equilibrium constants, have to do with the effect of the isotopic solvent on both initial (reactant) states and final (transition or product) states. It is nearly always desirable to attempt at least in part to untangle these and to deal with the effects on the individual states. Chart I shows how this can be done for the simplest example of an enzyme-catalyzed reaction. In spite of the simplicity of this formulation, we shall see that very few modifications are required to apply the results reached here to even the most complex enzymic processes. The model of Chart I is that of a single-substrate reaction in which reversible binding is succeeded by a single, catalytic step. The Michaelis-Menten expressions are developed for the reactions in H O H and DOD, and then the "ultrasimple" version of the transition state M. H. O'Leary, and D. B. Northrop, eds.), p. 64. University Park Press, Baltimore, Maryland. 12 K. B. Schowen, in "Transition States of Biochemical Processes" (R. D. Gandour and R. L. Schowen, eds.), p. 225. Plenum, New York, 1978. 13 A. J. Kresge, Pure Appl. Chem. 8, 243 (1964). 1, V. Gold, Adv. Phys. Org. Chem. 7, 259 (1969). 15 R. L. Schowen, CRC Crit. Rev. Biochem. (to be published).

[29]

SOLVENT

ISOTOPE

EFFECTS

ON

ENZYME

SYSTEMS

553

CHART I SINGLE-SUBSTRATE REACTION

Simplified one-substrate reaction in HOH (subscript h) and DOD (subscript d): Eh + Ah ' Ku . E A h Ed + Ad "

Ko

- EAd

k. kD

[ E T h ] ---> Eh + Ph

(Ia)

[ E T a ] --* Ed + Pd

(Ib)

Michaelis-Menten equations in HOH and D O D (e = total enzyme concentration): (t)H/eh) -1 = kH-1 + (kH/KH)-I[Ah] -1

(Ic)

(vD/ed)-' = ko -1 + (kD/KD)-I[Ad] -'

(Id)

Rate constants (kL = kn o r ko; KL = KH o r KD) and solvent isotope effects (kn/kD -~ Dk, (kIJKH)/(kD/KD) =-- D(k/K)) in "ultrasimple" transition-state theory, related to Gibbs free energies G z ~, GA t a n d GZAt (I = h o r d) and "defective" Gibbs free energies G*tr for transition states: kL = (kT/h) e x p {-(G*~- - GEA')/RT} (kL/KL) = (kT/h) e x p { - ( G ~ - GE' - GAt)/RT} Dk = e x p {[(G*~ - G ET! *h~ - (GEA d -- GEAh)]/RT} °(k/K) = e x p {[(G*rrd - G *h~rT,- (GE d -- GE h) -- (GA d -- GAd)]/RT}

(Ie) (If) (Ig) (Ih)

Defining the free energy of transfer AGxt of species Xn from HOH into DOD to yield Xd ( w h e r e vx is the total number of hydrogenic sites associated w i t h Xh that must be exchanged t o y i e l d Xd): Xh + (VX/2) D O D = Xd + 0 ' x / 2 ) H O H A G x t = (Gx d - Gx h) - (Vx/2)(Gw d - Gw h)

(Ii) (Ij)

Noting that the total number of hydrogenic sites in the initial and final states must be equal, w e convert the isotope effects to functions of the AGxt: Ok = e x p [ ( A G *t - AGzAt)/RT] D(k/K) = e x p LxI-¢AG*tET-- AGE t -- AGAt)/RT]

(Ik) (I!)

theory 16 is used to express these rate constants as functions of the Gibbs free energy of the transition state and of the two different initial states, (E + A) and EA. An important feature of the transition-state theory emerges in Eq. (Ie) (Chart I): the factor (kT/h) before the exponential is derived from the multiplication of the frequency of decomposition of the transition state (vc) by the partition function for this vibrational motion (taken to be of very low frequency in the "ultrasimple" formulation so that the partition function is kT/hvc). This partition function has been removed, for this purpose, from the complete partition function for the transition state; consequently the free energy for the transition state lacks the contribution from this decomposition motion, and so is "defective," as 16 E. K. Thornton and E. R. Thornton, in "Transition States of Biochemical Processes" (R. D. Gandour and R. L. Schowen, eds.), p. 3. Plenum, N e w York, 1978.

554

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

[29]

indicated by the asterisk in Eq. (Ie). This is significant for two reasons: 1. The omissiont of this motion from the transition-state free energy is the formal way in which primary isotope effects, present when an isotopic center participates in the decomposition (reaction-coordinate) motion, appear in the formulation; this point will emerge more explicitly below. 2. The expression derived in this way is correct only when the reaction-coordinate motion is effectively of very low frequency, i.e., classical in character. If quantum-mechanical tunneling is important, another fomulation is necessary. Equations (Ig) and (Ih) establish the point that the solvent isotope effects on the two kinetic parameters (written in the notation of Clelanda.6), Dk and D(k/K), result from isotopic free-energy differences for a final state (the transition state ET in both cases, with the reaction-coordinate contribution having been cancelled) and an initial state [EA for Ok; (E + A) for D(k/K)]. For more complex kinetic models, this will remain true, except that more than a single molecule may contribute to the effective transition state--which moreover may be different for k and (k/K)--and more than a single reactant species may contribute to each of the effective initial states. The concept of the free energy of transfer of a species from HOH to DOD is introduced by Eqs. (Ii) and (Ij). Most of the species of interest in enzyme mechanisms have exchangeable hydrogenic sites, so that we can usefully think of a typical transfer process as involving two parts: (a) the removal of the solute from its environment in HOH into its new environment in DOD; and (b) the removal of hydrogens from the exchangeable sites and their replacement by deuteriums. We shall always consider only solutions so dilute that the latter process does not sensibly disturb the isotopic purity of the deuterated solvent. A part of the free energy of transfer can in principle be associated with each of these component processes: that deriving from the simple removal of the solute from one environment to the other is often* called the "medium effect T M and is as a rule vastly less important than the second effect, usually called the "exchange effect." Both contributions are combined in Eq. (Ij), although the exchange effect is responsible for the inclusion of isotopic free-energy differences for a sufficient number of solvent molecules to effect the exchange process. The solvent isotope effect is now shown in Eqs. (Ik) and (I1) to arise solely from the difference in free energies of transfer for the ap* The medium effect has itself also been k n o w n as the " t r a n s f e r effect" on occasion, and the reader must be alert to this potential source of confusion.

[29]

SOLVENT ISOTOPE EFFECTS ON ENZYME SYSTEMS

555

propriate initial and final states. The requirements for there to be any solvent isotope effect at all are therefore fairly demanding: at least one of the initial or final states must have a nonzero free energy of transfer; the free energies of transfer for the initial and final states must not be equal. Thus even if some reactant species has a relatively large free energy of transfer, no solvent isotope effect will result unless a change in the free energy of transfer is produced as the transition state is formed. Enzymes, for example, might be expected to have a quite large free energy of transfer because they have so many exchangeable hydrogenic sites, and because they are frequently thought to interact strongly with nearby water molecules. Nearly all of these interactions are likely to be carried over into the transition state unchanged, however. Only the differences upon conversion to the transition state in the contributions of the various exchangeable sites will help to produce a solvent isotope effect. Free

Energies

of Transfer

Free energies of transfer for both electrolytes and nonelectrolytes have been reviewed by Arnett and McKelvey lr and by Jancso and Van Hook, is while questions connected with the "hydrophobic effect" and related matters have also been treated by Ben-Naim, Wilf, and Yaacobi 19 (see also Ben-Naim's book 2°) and by Jolicoeur and Lacroix. 2~ The free energy of transfer of a species reflects its relative stability in HOH and DOD; as defined here, a negative value indicates greater stability in DOD. If solubility from some common reference state, such as the gaseous state or an unhydrated solid phase, were measured in HOH and DOD, the free energy of transfer could be calculated as the difference in free energies of solution. Indeed the relative solubilities* are given by:

sd/s h = e x p ( - AGt/RT), * Standard states can be an important matter; since AGt is often small, its sign can be reversed by a change in standard states. 17 The common scales are molarity, molality, mole fraction, and "aquamolality" = mol/55.51 mol water. Free energies of transfer are nearly equal on molarity, mole fraction and aquamolality scales and are more negative at 298 K by 63 cal mo1-1 than those on the molality scale. The molarity/mole-fraction solubility ratio sd/s h is greater at 298 K than the molality ratio by 1.1117. 1T E. M. Arnett and D. R. McKelvey, in "Solute Solvent Interactions" (J. F. Coetzee and C. D. Ritchie, eds.), p. 344. Dekker, New York, 1969. 18 G. Jancso and W. A. Van Hook, Chem. Rev. 74, 689 (1974). 19 A. Ben-Naim, J. Wilf, and M. Yaacobi, J. Phys. Chem. 77, 95 (1973). 2o A. Ben-Naim, "Hydrophobic Interactions." Plenum, New York, 1980. 21 C. Jolicoeur and G. Lacroix, Can. J. Chem. 51, 3051 (1975).

556

ISOTOPES AS MECHANISTIC PROBES

[29]

and free energies of transfer are often determined from solubility measurements. Thus the following terms are equivalent in significance: negative free energy of transfer; greater solubility in DOD; greater stability in DOD. Table I gives some values for sa/s h for small, neutral molecules. These cases should be easier to understand than those of larger, ionic solutes. First, starting with the two-stage decomposition of the free energy of transfer (medium effect and exchange effect), we can further decompose the medium effect by using the language of scaled-particle theory. 2° Thus transfer exclusive of exchange would be expected to reTABLE I RELATIVE STABILITIES OF SOME SMALL, NEUTRAL MOLECULES IN ISOTOPIC WATERS AT 25 °

Species, X1 Nonpolar molecules Toluene Biphenyl Naphthalene Polar, nonexchanging molecules Carbon dioxide Cyclohexanone Dimethyl sulfate Methyl p-toluenesuffonate 1,3,5-Trinitrobenzene Quinuclidine 1,4-Diazabicyclooctane (DABCO) Polar, exchanging molecules Iodoacetic acid Iodoacetamide Piperidine 1,2-Diaminoethane

Piperazine Zwitterionic amino acids Glycine Alanine Valine Leucine Phenylalanine

sd /s h, molarity or equiv, scale 0.959 1.040 0.833

Reference

a a

0.994 0.915 0.803 0.81 0.82 0.75 0.82

b a a

c

0.79 (undissociated) 0.93 0.85

c c c

c

1.03

c

0.81

c

1.048, 0.995 0.963 0.970 0.964 0.807

a,e e e a

a From the compilation of G. Jancso and W. A. Van Hook, Chem. Rev. 74, 689 (1974). P. Salomaa, A. Vesala, and S. Vesala, Acta Chem. Scand. 23, 2107 (1969). c G. W. Spiegel, Ph.D. Thesis in Chemistry, Washington University, St. Louis, Missouri (1980) (obtained through the courtesy of Professor J. L. Kurz). a G. C. Kresheck, H. Schneider, and H. A. Scheraga, J. Phys. Chem. 69, 3132 (1965). e A. Klimov and V. I. Deshcherevskii, Biofizika 16, 556 (1971).

[29]

SOLVENT ISOTOPE EFFECTS ON ENZYME SYSTEMS

557

quire free-energy changes associated with: 1. Creation of a cavity in DOD, to accommodate the solute, and collapse of a cavity in HOH after removal of the solute. 2. Effects on solute-solvent interactions, including the structures of nearby solvent aggregates. 3. Differences in solute-solvent interactions, such as hydrogen bonding. 4. The differences in electrostatic interactions between solutes and the two isotopic waters. Factors (1) and (2), the cavity-formation effect and the solventstructure effect, may have some importance in solvent isotope effects, although it may be difficult or impossible to separate the two kinds of contribution empirically. The importance of these factors may be greater, in fact, for enzymic than for nonenzymic reactions. The separation of effects as given above aspires to allow the estimation of the magnitudes of cavity-formation free energies on a hard-sphere model [factor (1)], and then to include effects on the association and structure of the waters in factor (2). Accordingly, Jolicoeur and Lacroix~1 have estimated the free-energy changes associated with the formation and collapse of cavities in DOD versus HOH, witti radii up to 9/~. The molar volume of DOD is 18.134 cm 3 at 25°, while that of HOH is 18.069 cm3; the density of particles in HOH is thus greater, so that exclusion of particles from the cavity in DOD reduces the randomness less than in HOH. As a result, cavity formation favors transfer to DOD (AG t contributions are negative). The magnitudes are roughly linear in the square of the cavity radius, with AGt = - - 200 cal mo1-1 for a radius of 9/~. Enthalpy and entropy contributions were also estimated for a cavity that would accommodate a decane molecule: A n t = --900 cal mo1-1, AS t = - 2.4 cal mo1-1 K - 1 , A G t = - 180 cal mo1-1 at 298 K. All of these quantities are not far from experimental values for molecules of this size and probably account for a major part of the free energy of transfer for solutes that do not interact strongly with water. The following points should be recognized: 1. A value of -200 cal mo1-1 for AG t (9 A radius) corresponds to sd/s h = 1.40, a quite small difference in solubility even for a relatively

large molecule. This is true for experimental values of A G t, e v e n for molecules that interact with water (cf. Table I). In a more extensive compilation, Jancso and Van Hook TM list 38 nonelectrolytes with AGt from - 4 8 cal mo1-1 (sd/s h = 1.08) to +200 cal mol -a (Sd/S h = 0.71); the rms value is 67 cal mo1-1 (sd/s h = 0.89). 2. The enthalpic and entropic components of A G t are opposed, with

558

ISOTOPES AS MECHANISTIC PROBES

[29]

a compensation temperature of 375 K for the various calculated values above, so that AGt 0 at this temperature. This is entirely typical of experimental thermodynamic quantities of transfer, a large cancellation of enthalpy and entropy effects to give a small free energy change being almost universal in the temperature range of biochemical interest. This suggests that (a) free energies of transfer for noninteracting solutes and noninteracting parts of other solutes should commonly be small, and (b) the sign of AGt may be hard to predict and may invert as the temperature is changed. 3. Total free energies of transfer for macromolecules such as enzymes may well be large simply because of the cavity-formation contribution. If the linear relation of the free-energy contribution of cavity formation with the square of cavity radius found by Jolicoeur and Lacroix 21 were to hold for larger radii, then the contribution for a globular enzyme of molecular weight about 25,000 would be about -1500 cal mo1-1 (sd/s h = 13.5). In rate or equilibrium processes, however, the very largest part of this effect will be cancelled between initial and final states, so that even here, the net effect is not necessarily expected to make much of a contribution to an experimental solvent isotope effect. =

Factors (3) and (4) above, the hydrogen-bonding and electrostatic effects, should both be of larger magnitude in the case of ions than of neutral molecules; ions will be considered separately below. Even with ions, however, electrostatic interactions are likely to be the same with the two isotopic waters. The vapor-phase dipole moments of HOH and DOD are identical (1.84 D, 100-200°) TM and the dielectric constants of the two liquids are very similar (for HOH: 87.91 at 0°; 78.39 at 25°; 73.19 at 40°; and for IX)D: 87.65 at 0°; 78.06 at 25°; 72.84 at 40° as determined by Vidulich, Evans, and KayZ~). For these reasons, and because the net free energies of transfer for neutral molecules are so small, we shall assume that electrostatic interactions contribute nothing to the free energy of transfer. Hydrogen bonding ought to be different for HOH and DOD (the general question of isotope effects on hydrogen bonding is considered in more detail in another section), yet even for molecules such as cyclohexanone, dimethyl sulfate, and quinuclidine, the solubility ratios (Table I) are not very great. This is probably because there are opposing factors at work in isotope effects on hydrogen bonding, so that ordinary, asymmetrical hydrogen bonds (in which the proton sits much closer to one of the basic atoms than to the other) tend not to show a large isotope effect (see below). Certainly in the vaporization of water, 22 G. Vidulich, D. F. Evans, and R. L. Kay, J. Phys. Chem. 71, 656 (1967).

[29]

S O L V E N T I S O T O P E E F F E C T S ON E N Z Y M E

SYSTEMS

559

hydrogen bonds are completely broken; yet the ratio of vapor pressures 17 at 20° is only PnoM/PDoD= 1.17. This is because 23 the vaporization process leads to a tightening of the potential along the OL bond itself (the familiar shift to higher vibration frequency as the hydrogen bond is broken), which favors vaporization of DOD, but at the same time to a greater freedom of motion external to the molecule, particularly as the hindered molecular rotations, or librations, produced in the liquid by the hydrogen bonding, are converted to the unhindered molecular rotations of the gaseous species; this effect favors vaporization of HOH. The librational effect is larger and the vapor pressure of HOH is thus slightly greater. For hydrogen-bonding effects on the free energy of transfer of solute molecules, the magnitudes should be even smaller, because a ratio of two hydrogen-bonding isotope effects, a small quantity of uncertain direction, is involved. In summary, one can conclude that: (1) Cavity formation will favor a greater solubility for any solute in DOD, but not by more than a factor of about 1.4 for ordinary molecules; for all molecules, a large part of this contribution will cancel out of isotope effects on rates and equilibria. (2) Electrostatic factors should produce negligible effects. (3) Hydrogen-bonding effects should be small and of uncertain directions. (4) Structural effects in the aqueous environment must also be rather small, since all the net effects shown in Table I are so close to unity. The effect of isotopic exchange is the only problem still unconsidered. The molecules in Table I were deliberately chosen so that the stabilities of protium and deuterium in their exchangeable sites are the same as in a bulk water molecule so that this factor makes no direct contribution. Contributions at exchangeable sites can be large, but are better treated by use of isotopic fractionation factors than free energies of transfer. Isotopic Fractionation Factors In the detailed development of reaction mechanisms, we commonly want to identify isotope effects for individual hydrogenic positions, rather than the aggregate effects described by free energies of transfer. As an aid in this endeavor, isotopic fractionation factors can be used. Chart II presents the relationships of isotopic fractionation factors to free energies of transfer and to isotope effects on rate constants, with use of the kinetic model from Chart I. Extension to the case of equilibrium constants is simple and will not be given explicitly. 23 R. A. More O'Ferrall, G. W. Koeppl, and A. J. K r e s g e , J. Am. Chem. Soc. 93, 1

(1971).

560

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

[29]

PROBES

C H A R T II ISOTOPIC FRACTIONATION

FACTORS

Additivity of free energies for (a) nonexchanging parts of Xt (G[, assumed same in HOH, DOD), (b) exchangeable hydrogenic sites internal to X (internal S-sites) and water sites in primary solvation domain (external S-sites), with the total n u m b e r of both kinds of 6-sites, /Xx (contribution o f i t h site, gx~t), and (c) water sites in secondary solvation domain (Z-sites: total n u m b e r of Z-sites, O'x ; average contribution ZxZ): /,tx

Gx I = G~ + ~,, gxl ~ + O-xZxt

(IIa)

t

Free energy of transfer of X (noting that the total n u m b e r of sites Vx = /Xx + O'x): AGx t = ~

(gXl d - g x t ia) + OrX(Zx d -- ZXia) --

(Vx/2)(Gwa -

Gw h)

(lib)

t

Isotopic fractionation factor Sxi for the ith @site in Xl defined relative to a bulk water site (free energy gw t = Gwt/2) as standard:

-RT

In ~bxi =

(gxl d -

g x l h) -

(gw d -

(IIc)

g w h)

Free energy of transfer in terms of (h's: ~x

(-RT

AGx t = ~

in thxl) + O'x[(Zxa - Zxn) - (gw d - gwh)]

(IId)

t

Defining the

medium effect Mx for

the secondary solvation domain:

Mx = (rx[(Zxa - Zxh) - gw d - gwh)] AGx t =

-RT ~

(IIe)

(ln Sxi) + Mx

(IIf)

t

Solvent isotope effect in terms of ~b's [cf. Eqs. (Ik) and (Il)]: Dk = exp / ~

(In ~bEAJ) -- ~T (In **Tt) + [(MET-J

Ok = { ~ j A

MEA)/RT]~ J

t

~bEAJ/~ dp~TI}exp{(MET--MEA)/RT}

"(k/K) = exp { f i

*Ejfi

1

J

~bAj/h --

i

,~TI}exp{(MET--ME--MA)/RT}

(IIg) (IIh)

(IIi,

A useful classification of hydrogenic sites can now be introduced; it corresponds to the various contributions to free energies of transfer. We shall accordingly distinguish:

Internal @sites: exchangeable hydrogenic sites in the structural framework of the solute; where the binding potential differs substantially from the potential at an average bulk-water site so that the contribution to the free energy of transfer should be substantial. External @sites: sites in water molecules that are strongly inter-

[29]

S O L V E N T ISOTOPE E F F E C T S ON E N Z Y M E SYSTEMS

561

acting with the solute, so much so that the binding potential is altered to produce a considerable contribution to the free energy of transfer. Z-Sites: hydrogenic sites in weakly interacting water molecules, or in the structural framework of the solute (e.g., exchangeable sites in prOteins), where the binding potential is so close to that in bulk water that only small isotope effects are produced; such sites will be important if their aggregate isotope effect becomes significant. No distinction of the internal and external classes of Z-sites seems useful. The distinction between internal and external ~b-sites is a mechanistic distinction, and need appear only when mechanistic models are constructed from solvent isotope-effect data. The distinction of the ~b-sites from Z-sites is valuable operationally because the individual isotope effects at the Z-sites are so small and so "uninteresting" that they can be dealt with merely in the aggregate, as an average. Thus Eq. (IIa) (Chart II) shows the exchange-sensitive free energy of a species X expressed as an additive sum of two terms: (1) free-energy contributions of the ~bsites (combined in a single summation), and (2) an aggregate contribution from the Z-sites. Equation (lib) converts to free energies of transfer. Equation (IIc) introduces the idea of an isotopic fractionation factor for a particular site; the isotopic free-energy difference at that site is compared in the fractionation factor to the free-energy difference for an average site in bulk water, which is thus adopted as the standard site. When the free energies of transfer are entered into the equations for Dk and D(k/K), then one obtains the exceedingly simple relationships of Eqs. (IIe) and (IIf): the isotope effect is the ratio of a product over all initial-state fractionation factors to a product over all final-state fractionation factors (thus accounting for all ~b-sites), multiplied by an exponential term in which all the Z-sites of both initial and final states are combined in an aggregate. The free-energy sums in this latter exponential term are denoted by M to emphasize that this is the so-called "medium effect," the individual contributions of water molecules in the Zclass being indistinguishable, and the effect being a highly generalized one. One very great advantage of the fractionation-factor formulation is that a large number of data exist on the magnitudes of fractionation factors, largely obtained by nonkinetic and thus independent experiments. These factors have a systematic relationship to molecular structure, and at least to a fair approximation, isotopic fractionation factors are transferable at the functional-group level (i.e., their magnitudes de-

562

ISOTOPES

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

[29]

PROBES © I

o

% o

<

l,fl

o

o

O

~1~

O '=

•-=

o

ID

e~

o

~.~

m

~

t.q

~q O

Z <

L

f,/,1

O

z

o

m

g <

<

O

N

"~

Z

M~

~D

~o~. ~

o

I

eq ¢,,I

b,

0 rl,

0

I:Z

©

< Z

/

G ©

O

O

z

.<

z <

Z

e

o

r.

6

0

.=

I

O\

/O +

8 .0~ o

.S Z I

,.d Z

..q

,.d

tm

L)

[29]

563

S O L V E N T ISOTOPE E F F E C T S ON E N Z Y M E S Y S T E M S

o

~O

~

e2

e.

e~

I=

.,.,~.,.,.,.,.,.,.,., 0

¢q r,-

,,:,

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II

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%

~o~

6

.4~_ ~

~.d

~_

0

N 0

& •

0 e~

~,-,

000

°uo z

~,

~,~

.

~ ~ - ~ ~ ~ ~ ~ ~.~__. ~

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~

r~

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.-~

a:

L)

¢.2

¢~ i ~ O~

oa e2

564

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

PROBES

[29]

CHART III SOLUTE SPECIES IN ISOTOPIC WATERS

Distribution of isotopic species for a one-site entity X L in mixed isotopic water with atom fraction of deuterium n: [XL] = [ X H ] + [ X D ] = [XH]{1 + [ X D ] / [ X H ] }

(Ilia)

From the definition of an isotopic fractionation factor [Eq. (IIc)]: qbx = e x p [ - ( g x d - g x n ) / R T ] / e x p [ - ( g w a - gwh)/RT] dpx = ( [ x n ] / [ X H ] ) / ( n / [ 1 - n]) [ X D ] / [ X H ] = 4pxn/(1 - n) [ X L ] = [ X H ] I + ndpx/(1 - n) = [XH](1 - n + n~bx)/(1 - n)

(IIIb) (IIIc) (IIId) (IIIe)

[XH] = [XL](1 - n)/(l - n + nSx)

(IIIf)

[ X D ] = [ X L ] n S x / ( 1 - n + n~bx)

(fraction protiated) (fraction deuterated)

(IIIg)

Distribution of isotopic species for two-site model XLaLb : (IIIh)

[XLaLb] = (XHaHb] + [XDaHb] + [ X n a D b ] + [XDaDb] [XLaLb] = [XH~Hb]

[XDaHb] [XHaDb] [XDaDb] 1 + [XH~Hb~ + [XHaHb----~+ ~ J

[XLaL.] = [XHaHb]

1 + ~

[XLaLb] = [XHaHb]

[XLaLb] = [XHaHb]

n~x~

+ ~

n4'xb

n~xa#'xb~

+ (l-~L-~S

(1 - n) ~ + (1 - n ) n ( ~ a + qbxb) + (1 - n) 2

(1 - n + n6x~)(l - n + n6xb) (1 - n) 2

nZC~XadPXb

(IIIi)

(IIIj)

(IIIk)

(IIIl)

(I - n)2 [XH~Hb] = [XLaLb]

(I - n + n~bx~)(l - n + nq~xb)

(Illm)

pend only on the few nearest atoms). Tabulations of isotopic fractionation factors appear in several places, 6,8,11,12 with that of Cleland 6 being the most recent and precise, incorporating many factors determined by isotopic equilibration with enzyme catalysis. Transferable fractionation factors ought not to be employed with more than two significant figures, most particularly in enzymic solvent isotope effects. Table II exhibits some fractionation factors appropriate for use in studies of enzymic mechanisms. The values for some of the entries are developed and discussed elsewhere in this article. Mixed Isotopic Waters

Mixed isotopic waters have great utility in the disentanglement of contributions to the overall solvent isotope effect, permitting in a favorable case the specification of the individual isotopic fractionation factors involved. This is equivalent to separating the overall solindividual

[29]

SOLVENT

ISOTOPE

EFFECTS

CHART

III

ON

ENZYME

SYSTEMS

(Continued)

(1 -- n ) n ~ x a [ X D . H b ] = [ X L a L b ] (1 - n + nt~xa)(l - n + nt~xb) [XHaDb] = [XLaLb]

565

(IIIn)

(1 - n)nt~x b

(IIIo)

(1 - n + n~xa)(1 - n + n~xb)

n'6xa6xb [ X D a D b ] = [ X L a L b ] (1 - n + ntbxa)(1 - n + n~xb)

(IIIp)

I n g e n e r a l , f o r a v-site m o d e l XL1L~ • • • L~ : [XH1H2.

• - H~]=

(1 -

[XL~L~ • • • Lv] ~

n) ~

(IIIq)

I I (1 - n + n + x i ) ~ i

and formulas analogous to Eqs. (IIIm)-(IIIp)for

i n d i v i d u a l s p e c i e s . T h u s , f o r a v-site

model: (1 -

n) ~

Fraction whollyprotonated = v I-I (! - n + n4,xi)

(IIIr)

t

n(1 - n)V-l@xi

F r a c t i o n w i t h D in i t h p o s i t i o n a n d H e l s e w h e r e

I~(1

-

(IIIs)

n + n~xO

t if'

Fraction wholly deuterated =

(IIIt)

v

l-I (1 - n + n6x0 t

vent isotope effect into the contributions that are produced at individual structural sites. Such a separation has obvious value in generating and testing mechanistic hypotheses. The use of isotopic waters to dissect isotope-effect contributions derives from the systematic character of isotopic distribution among solutes. In a mixture of isotopic waters, any solute will exist as a mixture of isotopically substituted subspecies, if the solute has exchangeable hydrogenic sites, and as mixture of isotopically different solvates even if it does not possess exchangeable sites. With appropriate assumptions, simple expressions can be developed to relate the populations of the various subspecies to (a) the composition of the isotopic water, and (b) the isotopic fractionation factors of the solute. This is done in Chart III. The procedure of Chart III is recognizable by enzyme kineticists, being entirely equivalent to that used in developing the distribution of enzyme species in the presence of various ligands, the tb values here

566

ISOTOPES AS MECHANISTIC PROBES

[29]

taking the role of binding constants. The chief assumption involved in Chart III is the rule o f the g e o m e t r i c m e a n . This holds that the value of ~bi for the ith site is independent of the isotopic composition at any other site (j, k, . . . etc.) in the molecule. Alternatively stated, the Rule holds that the ~b values do not change with the degree of deuteration in the isotopic water (i.e., with the atom fraction of deuterium in the mixture, n). The validity of this assumption and effects of its failure have been critically discussed by, Gold, 14 Chiang, Kresge, and More O'Ferral124 and by More O'Ferrall and Kresge, 25 among others. The rule is known to be inexact. If it held precisely, the disproportionation reaction, HOH + DOD = 2HOD would have an equilibrium constant of 4.00. In fact, that constant is 3.78 at 0° and 3.81 at 75° in the liquid phase. TM Nevertheless, violations of the rule at this level, or at levels expected for most cases, produce effects that are well within the errors encountered in most enzyme-mechanism studies. We shall therefore assume that the rule is valid. A c i d - B a s e Reactions in Isotopic Waters The chemistry of acids and bases is, of course, important in studies of enzyme mechanisms because one or another protomer of the enzyme is generally more reactive than others, in a way that is informative of mechanism. In a practical sense, this necessitates the careful control of pH in solvent isotope-effect work. Further, questions of acid-base catalysis are frequently crucial in both enzymic reactions and in the nonenzymic reactions that provide baseline mechanistic information. Thus we consider at this point solvent isotope effects on acid-base reactions. How should the pKa of an acid vary as deuterium is introduced into the aqueous medium? Chart IV derives the expression for the observed K a in mixed isotopic waters [Eq. (IVg)], which suggests that the picture can be quite complicated if there are many 4~-sites and Z-sites in both the acid HA and its conjugate base A-. Treatments as complex as this have been attempted only in a few cases. 14 Generally, simplifying assumptions are introduced as in Eqs. (IVh)-(IVj). The simplest possibility is represented by Eq. (IVh), where only the 24y. Chiang, A. J. Kresge, and R. A. More O'Ferrall, J. Chem. Soc., Perkin Trans. 2 p. 1832 (1980). 25R. A. More O'Ferrall and A. J. Kresge,J. Chem. Soe., Perkin Trans. 2 p. 1840 (1980).

[29]

S O L V E N T I S O T O P E E F F E C T S ON E N Z Y M E

SYSTEMS

567

C H A R T IV K a IN MIXED ISOTOPIC WATERS Ionization o f a n acid in a mixture o f H O H and D O D (L = H or D, n = a t o m fraction of deuterium): LAn + L O L = I-~O + + A nKa n = [An-][L30+]/[LAn]

(IVa) (IVb)

F r o m Eq. (IIIr), modified according to Chart VI below for a m e d i u m effect, the fractions o f e a c h species containing o n l y p r o t i u m are: [Ah-]/[An-] = (1 - n)~A

(1

-

n + n4~AOe - u ^ ' m r

]

(IVc)

(where the 4~Aiare for external 4~-sites); [HaO+]/[I..~O +] = (1 - n)3/(1 - n + nl) 3

(IVd)

(recalling that I = fractionation factor o f l y o n i u m ion, and arbitrarily a s s u m i n g no m e d i u m effect for LaO+); [HAh]/[LAn] = (1 - n)~"

(1

-

n + ndpuAa)e - u . A " m T

•

fiVe)

Combining Eqs. (IVc)-(IVe) with Eq. (IVb), recognizing that Eq. (IVa) m u s t implicitly balance in waters (all t e r m s in [1 - n] ~ cancel),

Ka n =

[Ah-][H30+] [HAh]

fivf) (1 -- n + / ) 8 1 ~

(1 -

n +

n~bal)

t

w h e r e Z = exp[(MA - MHA)/RT]. A s s u m i n g that g a h for the purely protiated species is independent o f n, [~ Ka n = Ka h

(l-n+

II*HAI)] Z n (IVg)

VA (I -

n + n/)~ H

(I -

n

+

n4,~O

t

Simplified forms: N o Z-sites, all

( ~ H A I , (~Ai :

1: ga n

=

gab/(1 -- n + n / ) 3

(IVh)

N o Z-sites, all ~b = 1 for H A , A - e x c e p t for one internal ~b-site, the ionizable proton o f HA: Ka n = Kah(l -- n + n&HA)/(1 -- n + n/)3

(IVi)

N o Z-sites, &'s as in Eq. (IVi) except for VA equivalent external 4~-sites in A - : Ka" = Kah(1 -- n + n~bHA)/(1 -- n + n/)S(1 -- n + n~bA)~^

(IVj)

568

ISOTOPES AS MECHANISTIC PROBES

[29]

TABLE III SOLVENT ISOTOPE EFFECTS ON THE IONIZATION OF ACIDS Acid

pKa h

ApKa

=

PKa d -

pKa h

Some "well-behaved" acids with ApK~ between 0.45 and 0.52 (molarity scale) 2,6-Dihydroxybenzoic acid a 1.23 0.47 Citric acid b,c 2.95 0.49 Salicylic acid a (Kt) 3.01 0.52 Formic acid e 3.67 0.46 Benzoic acid s 4.20 0.49 Succinic acid ° (K0 4.21 0.50 Acetic acid h 4.76 0.51 Succinic acid g (K~) 5.64 0.49 Carbonic acid (CO~)~z (K~) 6.38 0.52 Cacodylic acid b 6.40 0.52 4-Nitrophenol4r 6.98 0.50 Phosphoric acid ~ (K~) 7.20 (6.85) 0.53 (0.52) Boric acid k 9.23 0.52 Some acids with solvent isotope effects outside this range Sulfuric acid k (K~) 1.85 Sulfurous acid (SO~)~z 1.90 Maleic acid t (KI) 1.91 Phosphoric acid m (K0 2.15 Arsenic acid m (/(1) 2.30 Glycine e (KI) 2.85 Fumaric acid t (KI) 3.10 Fumaric acid t (K2) 4.61 Maleic acid t (K2) 6.34 Sulfurous acid i (K~) 7.44 Ammonium ion 9.26 Glycine e (K2) 9.71 Carbonic acid ~ (K2) 10.35 Proline e (K~) 10.58 Phosphoric acid k (/(3) 12.15

0.35 0.66 0.58 0.21 0.30 0.38 0.42 0.38 0.33 0.62 0.61 0.59 0.62 0.65 0.67

Sulfhydryl acids e Pentafluorothiophenol

0.30

2.68

internal tk-sites of the lyonium ion are considered i s o t o p e e f f e c t . S i n c e te = 0 . 6 9 - 0 . 0 2 , t h i s s h o u l d t o p e e f f e c t s Kan/Ka d = 2 . 8 - 3 . 3 w i t h a n a v e r a g e w e r e a n a d e q u a t e m o d e l f o r all a c i d s , all a c i d s D O D t h a n i n H O H b y a f a c t o r o f 3, a n d a p l o t o f

to produce the entire generate overall isoa r o u n d 3.0. I f t h i s would be weaker in (Kan/Kan) 1/3 v e r s u s n

i n m i x e d w a t e r s w o u l d a l w a y s b e l i n e a r w i t h a s l o p e o f ( 0 . 6 9 - 1) = - 0 . 3 1 . T h i s is c e r t a i n l y n o t t h e c a s e i n g e n e r a l . 14 H o w e v e r , a c e r t a i n n u m b e r o f a c i d s c a n b e f o u n d w i t h Kah/Ka o = 2 . 8 - 3 . 3 (i.e., with ApKa = pK~ a - pK~ h = 0.45-0.52.), as shown in Table III. These are

[29]

SOLVENT ISOTOPE EFFECTS ON ENZYME SYSTEMS TABLE III Acid or ion

Thioacetic acid (CHsCOSH) 4-Nitrothiophenol Thiophenol CHsOOCCH2SH Mercaptoethanol Mercaptoacetate ion (-O2CCH2SH) Hydrated metal ions Fe3+ n (NHa)sCo3+ o Cu z+p Gd 3+q La 3+ q

569

(Continued) pKah

ApK~ = pK~d - PKah

3.20 4.50 6.43 7.91 9.61 10.25

0.30 0.31 0.34 0.35 0.35 0.40

2.54 7.16 7.71 8.20 10.04

0.29 0.47 0.49 0.14 0.31

a B. M. Lowe and D. G. Smith, J. Chem. Soc., Faraday Trans. I 69, 1934 (1973). b N. C. Li, P. Tang, and R. Mathur, J. Phys. Chem. 65, 1074 (1961). e R. A. Robinson, M. Paabo, and R. G. Bates [J. Res. Natl. Bur. Stand., Sect. A 73A, 299 (1969)] found ApKa --- 0.53 for (PK1 + pK~)/2. a B. M. Lowe and D. G. Smith, J. Chem. Socl, Faraday Trans. 1 71, 1379 (1975). e W. P. Jencks and K. Salvesen, J. Am. Chem. Soc. 93, 4433 (1971). B. M. Lowe and D. G. Smith, J. Chem. Soc., Faraday Trans. 1 71, 389 (1975). g Robinson, Paabo, and Bates, Ref. c. h Discussed in detail by Gold. 14 P. Salomaa, A. Vesala, and S. Vesala, Acta Chem. Scand. 23, 2107 (1969). Value of pKah from F. A. Cotton and G. Wilkinson, "Advanced Inorganic Chemistry," p. 170. Wiley (Interscience), New York, 1972. P. Salomaa, R. Hakala, S. Vesala, and T. Alto, Acta Chem. Scand. 23, 2116 (1969). t G. Dahlgren, Jr. and F. A. Long, J. Am. Chem. Soc. 82, 1303 (1960). '~ P. Salomaa, L. L. Schaleger, and F. A. Long, J. Am. Chem. Soc. 86 1 (1964). n R. J. Knight and R. N. Sylva, J. Inorg. Nucl. Chem. 37, 779 (1975). o y . Pocker and D. W. Bjorkquist, J. Am. Chem. Soc. 99, 6537 (1977). H. Kakihana, T. Amaya, and M. Maeda, Bull. Chem. Soc. Jpn. 43, 3150 (1970). q T. Amaya, H. Kakihana, and M. Maeda, Bull. Chem. Soc. Jpn. 46, 2889 (1973).

"well-behaved" acids. Even so, these values in some cases almost surely result from cancellation of factors in more complex situations rather than true adherence to the model of Eq. (IVh). Table III also shows examples of overall isotope effects on K a that do not fit Eq. (IVh). Some of these are clearly expected, as in the case of thiol ionization, where it is known (Table II) that ~bna = 0.40-0.46. Introduction of ~bnA = 0.43 into Eq. (IVi) then predicts that ApKa 0.12, still not in good agreement with experiment. The remaining discrepancy probably signals nonunit fractionation factors for waters solvating thiolate anions, ze Others among the observations suggest that 26 W. P. Jencks and K. Salvesen, J. Am. Chem. Soc. 93, 4433 (1971).

570

ISOTOPES AS MECHANISTIC PROBES

[29]

even for oxyacids and ammonium ions, there are either ~b-sites or Zsites that are generating isotope effects. This perhaps ought not to have been unexpected. It can be argued that (1) strong acids should have ~)HA smaller than weak acids, because the ionizable proton should be bound less tightly in strong acids; (2) weak acids should have ~bg smaller than strong acids, if it is imagined that these sites are for solvent molecules hydrogen-bonded to lone pairs of electrons on the atom from which the ionizing proton departed; ~bA might be expected to approach the value for these sites in HO- as the acid becomes very weak (0.70; Table II); the number of such sites/~g would presumably be 3 for oxyacids and unity for ammonium acids. These effects reinforce each other to suggest that ApKa should not be constant at about 0.5, but should increase with PKah. Indeed, it has been pointed out repeatedly that linear relationships of the form ApKa = m(pga h) + b hold for limited groups of acids. Bell27 has an impressive and well-known plot, with m = 0.017, b = 0.44. Mesmer and Hefting2s combined data for HCO3- and HSO4- ionization at 25°, HaPO4 ionization at 0-50 °, and the ionizations of H2PO42- and HOH at 0-300 ° (saturation pressure) to obtain m -- 0.052, b = 0.16 (Kah corrected for symmetry). Salomaa, Hakala, Vesala, and Aalto29 obtained m = 0.034, b = 0.30 for monobasic inorganic oxyacids, and related the result to the model discussed above; they pointed out, however, that no broad general linear relationship for various acids is found. Gordon and Lowe z° reported that m = 0.040, b -- 0.33 for the carboxyl ionization of a number of amino acids. For seven thiols, Jencks and Salvesen 2n found m = 0.012, b -- 0.26. On the other hand, the critical survey of Robinson, Paabo, and Bates al led to the view that, although such a relationship might hold for acids of pKa above about 7, for most stronger acids ApKa is constant at about 0.50 [thus conforming to Eq. (IVh); the value of 0.55 cited 31 is on the molality scale]. A tour-de-force by Ohtaki and Maeda 32 produced the following re27 R. P. Bell, "'The Proton in Chemistry," 1st ed., p. 189. Cornell Univ. Press. Ithaca, New York, 1959 [numerical values from R. P. Bell and A. T. Kuhn, Trans. Faraday Soc. 59, 1789 (1963)]. 2s R. E. Mesmer and D. L. Herting, J. Solution Chem. 7, 901 (1978). 29 p. Salomaa, R. Hakala, S. Vesala, and T. Aalto, Acta Chem. Scand. 23, 2116 (1969). 30 I. N. Gordon and B. M. Lowe, J. Chem. Soc. D p. 803 (1970). 31 R. A. Robinson, M. Paabo, and R. G. Bates. J. Res. Natl. Bur. Stand., Sect. A 73A, 299 (1968). 32 H. Ohtaki and M. Maeda, Z. Naturforsch., B: Anorg. Chem., Org. Chem., Biochem., Biophys., Biol. 27B, 571 (1972).

[29]

SOLVENT ISOTOPE EFFECTS ON ENZYME SYSTEMS

571

suits, where the limits are standard deviations: Six anilinium ions, with pKa - 1.5 to 5, showed ApKa 0.56 +- 0.03. Eleven derivatives of malonic acid, with a range of first and second pKa from -1.5 to 9, gave ApK~ 0.48 +__0.06. Eleven compounds related to malic, succinic, and acetic acids, having p g a 2 to 6, gave ApKa 0.49 +-- 0.06. Eleven phenols with p g a 3 to 10.5 had A p g a 0.55 --I- 0.06. It is not certain whether all these data were converted to a common standard state; if not, an even narrower range could have resulted. In any case, it seems clear that no absolutely general, reliable relationship of ApKa with acid strength holds across all structural classes. The reason for the lack of such a general relationship is probably not that the arguments (1) and (2) cited above are incorrect. Rather it is likely that the electronic effects on the force constants are rather weak, that a number of different vibrations are involved, particularly in the solvating water molecules, and that the summed effects are somewhat irregular. From a practical point of view, it is rather fortunate that large deviations from the simplest possible result--namely, that ApK a = 0.5--do not seem extremely common. One acid-base reaction of great importance in aqueous solution is the ionization of the solvent itself. A knowledge of the autoprotolysis constant for the solvent under particular experimental conditions is often necessary or convenient, and the behavior of water itself can provide a base line for mechanistic judgements. In Table IV are collected the relevant quantities to permit good estimation of the desired constants at various temperatures and ionic strengths. These values are tabulated on the molality scale. Appropriate density data (from readily performed measurements or from the formulas given by Jancso and Van Hook TM)can be used to convert to the more familiar molarity scale under any particular set of experimental conditions if that is wished.

Ionic Stabilities and Solvation in Mixed Isotopic Waters Enzymes are in general ionic, and many substrate molecules are ionic in the free state, in combination with the enzyme, or in both states. Isotope effects on ionic solvation are thus a necessary consideration in biochemical mechanistic uses of solvent isotope effects. The paradigmatic ions in isotopic waters are the lyonium ion I~O + and the lyoxide ion LO-, and because of their internal #b-sites they are subject to larger isotope effects than most other ions. From a combination of

572

ISOTOPES AS MECHANISTIC PROBES

[29]

TABLE IV AUTOPROTOLYSIS CONSTANTS (MOLALITY SCALE), ISOTOPE EFFECTS, AND THERMODYNAMIC FUNCTIONS OF IONIZATION FOR HOH AND DOD AT T w o IONIC STRENGTHS FROM 0 TO 100°a

T (°C)

log KH~°N

log Kl~°wD

I=0 0 25 50 75 100

-14.941 -13.993 -13.272 -12.709 -12.264

-15.972 -14.951 -14.176 -13.574 -13.099

1.031 0.958 0.904 0.865 0.835

-+ 0.023 --- 0.012 -+ 0.013 -+ 0.016 + 0.022

-

-

0.866 0.859 0.846 0.827 0.801

-+ 0.068 -+ 0.034 -+ 0.024 -+ 0.026 -+ 0.029

1.0 0 25 50 75 100

log (Kaw°H/K~ n)

I=

14.725 13.753 13.003 12.404 11.916

AH(cal mo1-1) T(°C)

15.591 14.612 13.849 13.231 12.717

AS(cal mo1-1 K -1)

ACp(cal mo1-1 K -1)

HOH

DOD

HOH

DOD

HOH

DOD

0 25 50 75 100

14954 13340 12111 13063 10045

16130 14350 12980 11820 10720

-13.62 -19.28 -23.25 -26.37 -29.20

-14.00 -20.29 -24.71 -28.17 -31.22

-75.7 -55.3 -44.4 -40.4 -41.7

-83.0 -61.5 -49.5 -44.3 -44.4

I=l.0 0 25 50 75 100

15261 13766 12725 11939 11259

14110 12440 11210 10220 9320

-11.51 -16.76 -20.12 -22.47 -24.35

-19.70 -25.13 -28.70 -31.17 -33.20

-72.3 -49.2 -35.4 -28.5 -26.6

-78.9 -56.3 -43.2 -36.9 -35.8

I = 0

a The data for HOH are from F. H. Sweeton, R. E. Mesmer, and C. F. Baes, Jr. [J. Solution Chem. 3, 191 (1974)], and those for DOD from R. E. Mesmer and D. L. Herring [J. Solution Chem. 7, 901 (1978)]. The ionic strength was adjusted with potassium chloride. The Kw values are apparent ion-concentration products.

studies on the autoprotolysis of the isotopic waters, acidities and other equilibria in isotopic waters, N MR measurements of fractionation factors, analogies with related species such as the methoxide ion in methanol, and theoretical calculations, a satisfying operational picture of these species has emerged. The two structures and their fractionation factors are:

[29]

S O L V E N T ISOTOPE E F F E C T S ON E N Z Y M E SYSTEMS H

573

..Hb--OHe

I

Ha--O-~'" Hb--OHe H

H

Hb--OHe

6~---1 = 0.69

(ha = 1.25 ~bb = 0.70 6,. - 1.oo

Aside from their intrinsic interest, these two structures can be regarded as providing a starting point for the consideration of other ion solvation, and its resultant effects on ion stabilities in isotopic waters. Both lyonium and lyoxide ions are more stable in H O H than in DOD --lyonium ion by a factor of 3.04 and lyoxide ion by a factor of 2.33--giving rise to the large isotope effect of 7.1 on the autoprotolysis constant, s, 23-25 The excess stability of lyonium ion in H O H derives largely from the decreased stretching frequencies of the L - O bonds, compared to those in water, presumably induced by the positive charge on oxygen. 2a The excess stability in H O H of lyoxide is more complex, as the fractionation factors show. aa The effect comes from the loosening of the potential about the hydrogens in the three solvating waters, an effect that seems to occur only in the strongest hydrogen bonds. This contribution is diminished by about 25% from an opposing effect, the tightening of the internal stretching motion, presumably induced by the negative charge on oxygen. Similar considerations, mutatis mutandis, are expected to determine the relative stabilities in isotopic waters of other ions. The general structures, M,,+

I

/O\ (H

X ~' ( H - - O H ) q H)rn

suggest that both cations and anions should tend to show excess stability in HOH, if no other factors come into play. Thus the transfer of positive charge from a cation onto water should induce a fractionation factor that could be as small as 0.69 at each hydrogenic site, if an entire charge is transferred as (presumably) in lyonium ion. Since only a smaller charge should usually be transferred, a fractionation factor closer to unity is expected, and thus a smaller excess stability in HOH. For anions, hydrogen bonding as shown above would also be expected to produce excess stability in HOH. Indeed for anions with no exchangeable hydrogens, no cancellation from internal sites is expected. However, there is evidence that the isotope effect for this kind of hydrogen bonding 33 V. Gold and S. Grist, J. Chem. Soc., Perkin Trans. 2 p. 89 (1972).

574

ISOTOPES AS MECHANISTIC PROBES

[29]

TABLE V RELATIVE EXCESS STASILITIES IN HOH [= exp (AGt/RT)] FOR CATIONS AND ANIONSa

Ion Cations (values relative to Na ÷) Li + Na ÷ K+ NI-I, + Mg~+ Ca 2+ Anions (values relative to Cl-) FCIBrIHCOOCH3CO0(Absolute excess stabilities) COa2SOa~RS-

exp (AGt/RT)

1/1.15 ( 1.00) 1.12 1.10 1.11 1.14 1/1.33 (1.00) 1.03 1.13 1/1.24 1/1.28 1.47~ 2.59 ~ 1.4-2.1 c

a From the data tabulated by G. Jancso and W. A. Van Hook [Chem. Rev. 74, 689 (1974)] unless otherwise indicated. The values are given on the "aquamolality" scale, and are indistinguishable from those for the molarity scale. n Obtained by P. Salomaa, A. Vesala, and S. Vesala [Acta Chem. Scand. 23, 2107 (1969)] from measurements of PKa values and absolute fugacity ratios for COs and SO2. c Estimated from pKa (Table III) and fractionation factor (Table II) data for RSH species.

is very sensitive to basicity, possibly in a quite nonlinear way, and perhaps to other factors, as yet little understood, a4 Thus while both hydroxide and alkoxide ions display strong fractionation with ~b ~ 0.7, anions of lower basicity such as 2-nitrophenoxide and acetate ion show only much smaller effects. 35"36This is generally consistent with the somewhat complicated circumstances attending the isotope effects on hydrogen-bond formation, discussed below. Table V shows some data relevant to these 34 M. M. Kreevoy and T. M. Liang, J. Am. Chem. Soc. 102, 3315 (1980). a5 L. Pentz and E. R. Thornton, J. Am. Chem. Soc. 89, 6931 (1967). 36 V. Gold and B. M. Lowe, J. Chem. Soc. A p. 1923 (1968).

[29]

S O L V E N T ISOTOPE E F F E C T S ON E N Z Y M E SYSTEMS

575

points. Since absolute values of the excess stability of ions in H O H are commonly unavailable, several cations are shown relative to sodium ion, and several anions relative to chloride ion. The greatest utility of these values for biochemical applications is in estimating the solvent isotope effects to be expected from the binding of ions to protein or other macromolecular sites. Usually such binding will result in displacement of another ion of the same charge type into aqueous solution. If the displaced cation were sodium, or a displaced anion were chloride, then the values in the table give a direct estimate of the expected solvent isotope effect. Appropriate ratios may be formed from the values in the table to yield predictions for displacement of any listed ion by any other. Clearly the solvent isotope effects for such reactions will be small for small, singly charged anions, and for all cations. Table V also gives some absolute excess stabilities for a few other species. The doubly charged anions carbonate and sulfite appear to interact reasonably strongly with solvating water molecules, as do the thiolate ions. Perhaps the mechanism of these interactions involves hydrogenbond formation of solvating waters to the ions, or alternatively it may involve some induced change in the hydrogen-bonding pattern or librational freedom of the surrounding water molecules. In any case, in reactions that involve such ions, or that involve protein functional groups analogous to them, one must be alert to the possibility of such isotope effects.

Isotopic Waters in Metal-Ion Environments Ions of metals such as iron, copper, and zinc are special cases of cations that are of great importance in the function of enzymes. Solvent isotope effects are potentially useful in this field, although at the present time a great deal of baseline information from nonenzymic studies is still missing. Table III gave examples of solvent isotope effects on the ionizaton of waters coordinated to such metal ions. The simplest of these cases would appear to be ionization of the pentamminecobalt(III) ion: (NLa)sCo(OI~)3+ + i.,zO = (NI~)sCoOL2+ + [.~O+ w h i c h has Kah/Ka a = 2.95 (PKa d - p K a h = 0.47). This ought to be given by:

K an/ Ka d = ~W32~A315/~bB2~A21513,

where ¢bw3is the fractionation factor for water bound to the + 3 ion, ~b~2is for hydroxide bound to the + 2 ion, ~bA3and ~A2 are for the ammonia ligands of the two ions, and l, as usual, is for 1.30 + (0.69). Since 1-3 = 3 . 0 4 , obviously the entire remaining product of factors contributes essentially

576

[29]

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

nothing. Whether this is because all are unity or nearly unity, or because of a cancellation of effects, one cannot judge. A similar remark would apply to Cu 2÷ ionization, which also has K a h / K a ~ -~ I -z. Fe 3+, in contrast, exhibits a large deviation. For its ionization, (I_~O)sFe(O~)3+ + L~O = (L~O)sFe(OL)~+ + LzO +

Kah/Ka d = 1.95 = ~bwsl~/~bw~l°~bJ~, where the symbols are analogous to

those used above. From the value of l, we have ~bws12/~bw~l°~bs = 0.64. This value of less than unity suggests that conversion of the + 3 ion to the + 2 ion leads to an overall tighter binding at hydrogenic sites in the ligands. How much of this arises from the decreased transfer of positive charge to water ligands, and how much from conversion of a water ligand to a hydroxide ligand, cannot be deduced with certainty. Indeed, some of the effect might in principle come from waters beyond the primary solvation shell. However, a recent study, summarized in Table VI, suggests that only inner-sphere effects are likely to be of much significance. Thus the upper set of ions, with ligands intervening between the ion and the water, show very little effect of solvent deuteration on redox potential. In contrast, extraordinary effects appear when the aquo ions are studied, and may give some indication of the magnitude of isotope effects associated with charge alteration at the metal center. The potential differences given in the table correspond to effects on the redox equilibrium constant of K d / K h = 3.6 (V), 5.4 (Fe), and 9.2 (Cr) for conversion of the + 3 to the + 2 species. These effects probably arise equally from each of the 12 hydrogens of the T A B L E VI SOLVENT ISOTOPE EFFECTS ON THE ELECTRODE POTENTIALS OF METAL IONSa Couple Metal ions protected from the aqueous environment Fe(bpy)a(III)/(II) Ferricinium/ferrocene Co(bpy)a(III)/(II) Co(en)s(III)/(II)b Co(sepulchrate)(III)/(II)c Metal ions exposed to the aqueous environment Fe(III)/(II) Cr(III)/(II) V(III)/(II)

Ellz(HOH) - EIn(DOD) (mY)

0 0 - 5 +8 - 1 -43 -57 -33

a M. J. Weaver and S. M. Nettles, lnorg. Chem. 19, 1641 (1980). A value ofAE1n = - 5 9 mV corresponds to Kd/Kh = 10. b Note that the ligand contains exchangeable sites. c Sepulchrate: 1,3,6,8,10,13,16,19-octaazabicyclo[6.6.6]eicosane.

[29]

SOLVENTISOTOPE EFFECTS ON ENZYME SYSTEMS

577

six inner-sphere water molecules of these ions. The effects are thus about 1.1 to 1.2 per hydrogen, and this should be equal to (,hw~/~bw3). It is, of course, not certain that ~bw2/&w3 for electrochemical reduction of the metal ion will be the same as for decreasing the charge through ionization of a ligand; indeed, it seems extremely likely that the range of 1.1-1.2 should be an upper limit for ligand-ionization cases. In summary, while at least a mild perturbation by large metal ions of water-derived ligands surely occurs, any detailed understanding is lacking. This is further emphasized by the ionization isotope effects for Gd 3+ and La 3+ given in Table III. Isotope Effects on H y d r o g e n Bonding This is a question of obvious significance for enzyme mechanisms, and one that has been plagued by complications. As described above, the expected effects o f hydrogen-bond formation on the bending and stretching motions of the hydrogen are such as to induce opposite isotope effects, the stretching motion becoming freer in the hydrogen bridge, while the bending motion becomes restricted. Experimentally, at least in aqueous solution, there is often a quite large isotope effect on the enthalpy of hydrogen-bond formation, approximately cancelled by an entropic effect in the opposing direction. TM For many cases, as a result, there is little isotope effect on the formation of hydrogen bonds. However, certain hydrogen bonds have recently been discovered to produce very large isotope effects 34 These points are illustrated by the data of Table VII. In the dimerization of various amines, alcohols, and thiols in dilute hexane solution, only the smallest isotope effects are observed; all are slightly inverse. The second system cited involves an initial proton transfer to generate the ion pair (as shown by the ultraviolet spectrum), which then exists in a hydrogen-bonded structure. Such effects as exist seem to be normal in direction, but quite small in magnitude. The third system, by contrast, exhibits large, normal isotope effects. Here, the structures bear a net electrical charge (negative in the examples shown, but Kreevoy and Liang give others with a positive charge), and the reactions occur in a dipolar, aprotic environment. Kreevoy 37 suggests that the conditions for formation of such hydrogen bonds include (1) bases bonding to the proton which are not of greatly different basicity, so that the hydrogen-bond potential is relatively symmetrical, favoring free motion of the proton between the two potential minima, and (2) a nonaqueous solvent, so that the motion of the proton across these minima is 3r M. M. Kreevoy, personal communication.

578

ISOTOPES AS MECHANISTIC PROBES

[29]

TABLE VII SOME ISOTOPE EFFECTS ON HYDROGEN-BOND FORMATION

System Dimerization in dilute hexane solutionsa of: CHaNL~ at 20°C C~HsNL2 at 20°C (CHa)2NL at 20°C (C2Hs)~HL at 20°C CHaOL at 40°C C~HsOL at 40°C CH3SL at 20°C Reaction at 25°C of 2,4-dinitrophenol (H or D)b with: (C~Hs)aN in toluene CsHsN in chlorobenzene CsHt0NL in chlorobenzene CHHsN(CHa)~ in chlorobenzene Reaction in acetonitrile at 25°C with its own conjugate basec of: 4-NO~C6H4OL CFaCOOL 3,5-(NO2)~C6HaCOOL CI~CeOL

KH/Ko

0.95 0.91 0.95 0.85 0.92 0.93 0.96 1.02 -+ 0.02 1.40 -+ 0.05 1.00 -+ 0.01 ~ I. 12 3.2 2.4 3.3 2.5

+- 0.3 -+ 0.3 -+ 0.3 -+ 0.3

a H. Wolff, in "The Hydrogen Bond: Recent Developments in Theory and Experiments" (P. Schuster, G. Zundel, and C. Sandorfy, eds.), p. 1227. North-Holland Publ., Amsterdam, 1976. b R. P. Bell and J. E. Crooks, J. Chem. Soc. p. 3513 (1962). c M. M. Kreevoy and T. M. Liang, J. Am. Chem. Soc. 102, 3315 (1980).

n o t r e s t r i c t e d b y t h e e l e c t r i c a l field o f a s t r u c t u r e d , e x t e r n a l s o l v e n t a g g r e gate.

P r o t e i n S t r u c t u r e a n d S t a b i l i t y in I s o t o p i c W a t e r s In u s i n g s o l v e n t i s o t o p e e f f e c t s f o r m e c h a n i s t i c p u r p o s e s , o n e h a s t o b e c o n c e r n e d w i t h w h e t h e r t h e i s o t o p i c s o l v e n t h a s p r o d u c e d all o r p a r t o f its e f f e c t b y c h a n g i n g t h e s t r u c t u r e o f t h e e n z y m e . O n e c a n e n v i s i o n l a r g e changes--for example, on subunit association or on gross conformation - - o r e x t r e m e l y s u b t l e c h a n g e s , h a v i n g to d o w i t h t h e e x a c t p o s i t i o n o f important functional groups. At the present time, no reliable generalizations can be produced about these points and for the moment they must b e t a k e n u p a s a p p r o p r i a t e in e a c h i n d i v i d u a l c a s e . It is i m p o r t a n t to r e a l i z e t h a t a n y c h a n g e s in p r o t e i n s t r u c t u r e , w h e t h e r l a r g e o r s m a l l , t h a t d o o c c u r in i s o t o p i c s o l v e n t s c a n b e t r e a t e d e x a c t l y a s a n y o t h e r i s o t o p e e f f e c t a n d t h a t t h e y t h e r e f o r e o f f e r , in p r i n c i p l e , u s e f u l i n f o r m a t i o n a b o u t t h e s y s t e m . I f a s u b u n i t a s s o c i a t i o n p r o c e s s is m o r e fa-

[29]

S O L V E N T I S O T O P E E F F E C T S ON E N Z Y M E SYSTEMS

579

vorable in DOD than in HOH, for example, this indicates that in the associated form of the enzyme there are hydrogenic sites--either exchangeable sites of the protein or sites in solvating water molecules--that experience a lighter potential than in the dissociated form of the enzyme. The magnitude of the effect is a measure of the degree of tightness, and can aid in elucidating the structural features of the dissociated and associated proteins. The same considerations apply to solvent isotope effects on conformational equilibria, substrate association processes, etc. Table VIII shows some cases that are pertinent. The first three entries refer to systems that exhibit quite large solvent isotope effects. Thus the formation of octamers from the dimer of/3-1actoglobulin A is greatly enhanced in DOD. This is nearly a pure enthalpic isotope effect, and it is notable that the enthalpy of association is negative in both HOH and DOD, so that the subunit association does not behave as a hydrophobic association process. Rather, it is enthalpy-driven in both isotopic waters and is opposed by a large, unfavorable entropy change that exhibits no solvent isotope effect. Since the association is nevertheless strongly favored by DOD, with an isotope effect of over 200 at 25°, it exemplifies the fact that this kind of solvent isotope effect does not constitute a test for the role of hydrophobic forces. The polymerization of flagellin is also enthalpy-driven, and the association is again opposed by a large entropy contribution. However, both enthalpy and entropy changes are increased proportionally in DOD, so that around 37°, there is practically no solvent isotope effect at all. The association of the monomers of formyltetrahydrofolate synthetase, induced by monovalent cations, was studied in detail in mixtures of the isotopic waters, and the isotope effects have been dissected into their component parts by the proton-inventory method (see below). The overall formation of tetramer from monomer, with incorporation of two cations, is thermodynamically favored in DOD over HOH by a factor of around 40. In mixtures of HOH and DOD, the effect increases as the fouth power of the atom fraction of deuterium, suggesting that the factor of 40 should be regarded as the fourth power of 2.5. This in turn is found to be just the isotope effect for binding of a cation to a single monomer molecule. Although the exact origin of this factor of 2.5 is still not known, this study shows that very large effects, such as a factor of 40, may in fact derive from the accumulation of smaller and much more easily rationalized effects. The final two entries in the table show that many proteins do not respond structurally to solvent deuteration at all. Edelstein and Schachman have used parallel sedimentation equilibrium determinations in HOH and DOD to determine the partial specific volumes of proteins, a method that requires a common structure in the isotopic solvents. Although excep-

580

ISOTOPES AS MECHANISTIC PROBES

, TT,~

177

I

[29]

~

o~

Z

! z~

..~

z m

~'

~ ~z~

o ..~

Z 0

~., Z

i

.~

=

"~1~

•~

~

~ .~-

0

0

.,g,

.-~

.~

~t

~

.~

eL

~o

0

Z

~

Z

o e~

0

~

i~.~

[29]

SOLVENT ISOTOPE EFFECTS ON ENZYME SYSTEMS

581

tions exist, their article tabulates many examples where this requirement appears to be met. The recent study of the properties of the valyl-tRNA synthetase system is especially impressive, since an enzyme with a macromolecular substrate might have been expected to show some solvent isotope effect at least upon its association with such a substrate. At least at pH = pD, no effects were observed on enzyme binding of the nucleic acid, or on its affinity for ATP or valine, or on its thermal stability. On the other hand, a large kinetic solvent isotope effect on the catalytic reaction was seen.

T h e o r y of Solvent Isotope Effects P r o to n Inventories Kinetic studies of solvent isotope effects in a series of mixtures of H O H and DOD can, in some cases, allow the dissection of the isotope effect into its component contributions from different sites in the reactant and transition states. In principle, therefore, one can construct a list of the contributing hydrogenic or protonic sites and the magnitude of the isotope effect for each: a " pr ot on inventory." Chart V shows the requisite mathematical development to extend the fractionation factor treatment of Chart II to mixtures of isotopic waters. An important fact is that the fractionation factors for the transition state, here as in Chart II, have had the contribution of the reaction-coordinate motion removed, in accord with the practice of "ultrasimple" transition state theory. This means that no account of tunneling of labeled protons is included; if it actually occurs, the fractionation factors estimated from experimental data may have unusual values. The conduct of a proton inventory consists of determining the kinetic parameters of interest in a number of isotopic water mixtures of deuterium atom fraction n, so that the data set comprises values kn(n). These data are then fit, by some systematic procedure, to the theoretical expression of Eq. (Vk) to obtain values of the 6Ti* and the ~bRj. In practice, of course, it is commonly not possible to use the perfectly general expression and still obtain a fit to t h e d a t a . Instead, a simplified version must be used, often based on a model that arises from other mechanistic information. The simplest such model is when a single site determines the entire solvent isotope effect. If such a site is in the transition state, kn(n) is a linear function [intercept k0, slope (d~T* -- 1)k0]. If the site is in the reactant state, kn-l(n) is linear. In a similar way, if two sites of the

582

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

[29]

CHART V FRACTIONATION FACTOR TREATMENT F u n d a m e n t a l equation o f transition-state t h e o r y (re = d e c o m p o s i t i o n frequency o f transition state T; X is the reactant molecule): Velocity = k[X] = vc[T] k = vcrT]/[X]

(Va) (Vb)

In a mixture o f H O H and DOD, both T and X are s u m s o f variously deuterated species:

C o n s i d e r two sites in both T and X: kn =

k° =

(VHH[THH] "t" Vnv[Tnv] + VDH[TDH] + VDD[TDD] ([XHH] q- [XHD] "~- XDH]q- [XDD])

(Vd)

(VHH[THH]) [ | + VHD [THn] YaH [TDH] VVV [TDD] - VHH [THH~] + VHH - - [THn] + VHH -[THH] -

(Ve)

/

IX..]

[XDH]+ IX..] [XDD____]] I + ~[XHD] - ~ + [-2~..]

F r o m Eq. (IIId), and noting that k0 = VnH[THH]/[Xnn], n 1 + vHHVHDtk'rl ~ kn = ko

kn = k0

VDH + --VHHd>r2 ~

n

n 1 + t~xa (I - n--------~ + ~x~ ~

n

VDD n2 + --VHH~'rlt~r~ (1 -- n) 2 n2

(Vf)

+ ~bxlthx~ (1 - n) 2

(1 - n) 2 + n(l - H)[(VHD/VHH)~bT1-I- (VDH/VHH)¢bT2] "I- n2[(VDD/VHH)~TI~bT2] (1 - n) 2 + n(l - n)(&xl + ~bxz) + n2t~Xlt~X2 (Vg)

In Eqs. (Vf) a n d (Vg), the 6 t e r m s are all complete, i.e., the transition-state partition functions include the contributions o f the d e c o m p o s i t i o n motion; t h e s e c a n be factored out to give ~b*'s or defective fractionation factors: (VHD/VHH)t~r 1 = (VHI~qHD/VHHqHH)q~*I

(Vh)

and similarly for o t h e r terms. In " u l t r a s i m p l e " transition-state theory, all vcqc = kT/h, so: (1 - n ) 2 + n ( l - n ) [ 6 " 1 + °tDT2J . t , ~ + n 2 W' ~ . ,Tll/~T .~, 2 kn = k0 (1 - n) ~ + n(1 - n)[rxl + 6x~] + n26xl~bx2

(Vi)

kn = k0 (1 - n + n~b*0(1 - n + n~b*~) (1 - n + n&x0(1 - n + n~bx~)

(Vj)

Generalizing to VT and VR sites: kn =/co

n

,/n

(1 - n + n~b*l

t

transition state determine the cubic; etc, A section below is the data during the search for fitting of model equations to ished proton inventories.

(1 - n + n~bm)

(Vk)

J

entire effect, kn(n) is quadratic; if three, devoted to the qualitative examination of adequate models, and to the quantitative the experimental results to produce fin-

[29]

SOLVENT

ISOTOPE

EFFECTS

ON ENZYME

SYSTEMS

583

Enzyme Kinetics in Isotopic Waters Equation (Vk) was developed for a general rate process in which the effective reactant state is R and the effective transition state is T. In Chart VI, the appropriate modifications are introduced to apply the concept to the simple enzyme kinetics of Chart I (see also Chart II), including a distinction of $-sites and Z-sites. For more complex enzyme-kinetic situations, exactly similar expressions apply, except that a careful definition of the effective reactant and transition states is required. In the simple situation of Chart I, the CHART

VI

Starting w i t h Eq. (Vk), and dividing the VT sites into/Zr S-sites and ¢rr Z-sites (all the latter w i t h 65z near one and taken as approximately equal), and similarly for va : 1 ~ ( 1 - n + n~b*l)(1 - n + n~b{,z)~T (kn/ko) =

t 1 ~ ( 1 - n + N6m)(1 - n + n~bRz)~.

(Via)

J

N o t i n g t h a t (1 - n + n~bz) = (1 + n[tbz - 1]), w h i c h is approximately exp [n(6z - 1)] for d~z n e a r one,

I ] ( 1 - n + nd~*0 (kn/ko)

~ {exp [Crx(t~*z - I) - trR(tbRz -- 1)]}" l ~ ( 1 - n + N$m)

(VIb)

J

IF[ (1 - n + n ~ i )

(kn/ko)

Ja {exp [(MT -- MR)/RT]} n 1-[ (1 - n + N~bm)

(VIc)

J

I n v e r t i n g E q s . (IIIh) and (IIIi) and expanding them by a n a l o g y w i t h Eq. (VIc), ~tET

n

(1 - n + n ~ T i )

(k./ko) = ,,~^

{exp ](MET -- MEA)/RT]}"

(VId)

I'I (1 - n + NCJ~Aj) ,J gET

(k/K). (k/K)o

F I (1 - n + nq~Tl ) ~^ [exp [(M~T -- Mr - MA)/RT]}" lei (I - n + N ~ j ) H (1 - n + n 6 m ) J

(Vie)

J

O r m o r e briefly (Phi-TSC, transition-state 6 - c o n t r i b u t i o n ; P h i - R S C , r e a c t a n t - s t a t e (b-contribution; Z, total Z-contribution):

(k./ko) = {[Phi-TSC(n)]k/[Phi-RSC(n)]k}(Zk)"

(VIf)

( k / K ) . = {[Phi.TSC(n)]~m/[Phi_RSC(n)]ktK}(Z~m). (k/K)o

(VIg)

584

[29]

ISOTOPES AS MECHANISTIC PROBES

effective reactant state for the parameter k is the complex EA, and the effective transition state is simply that for the single catalytic step postulated. The effective reactant state for the parameter ( k / K ) is the combination E + A, free enzyme and free substrate, while the effective transition state is still that for the single catalytic step. For any enzyme-kinetic circumstance, the kinetic parameters can also be cast into forms k (referring to the passage of some complexed form of the enzyme to a transition state), and ( k / K ) , referring to the reaction of some free substrate with some form of the enzyme to generate a transition state (in general different from that of the k term). The equations of Chart VI may then be applied to each of the kinetic parameters, with the use of the proper definitions of the reactant and transition states. Consider as an example a simple form of Ping Pong Bi Bi mechanism: A E

P EA

FP

B F

Q FB

FQ

E

The phenomenological forward rate law for this mechanism is always as follows, written in reciprocal form, with e the total enzyme concentration: v-le = ( k / K a ) - l A -1 + ( k / K b ) - l B -1 + (k) -1

Thus if complete kinetic analyses are carried out in a series of isotopic waters, values of three kinetic parameters--k, (k/Ka), and ( k / K b ) - can be obtained in all of the isotopic waters. Each of the parameters can be examined and fitted to its own model independently of the others. The appropriate model depends on the actual mechanistic situation. For example, suppose the rate-determining step were the k3 step (k7 > > ks) and that both substrate binding steps were reversible (k2 > > k3 ; ke > > kr). Then the following situation prevails:

Kinetic parameter k/Ka k/Kb k

Definition EffectiveTS klka/k~ ksk,/ka ks

ETa ET7 ETa

EffectiveRS E+A F+B EA

where ETa and ETr refer to the transition states for the k3 and kr steps. Expressions like those of Eqs. (VIf) and (VIg) can then be used directly for modeling the experimental data for each of the parameters.

[29]

SOLVENT ISOTOPE EFFECTS ON ENZYME SYSTEMS

585

Notice that a c o m m o n transition state is postulated for k and (k/Ka). In the case in which there was no solvent isotope effect on the binding o f A to E, the proton inventories for these two p a r a m e t e r s would then be identical, and otherwise not. A m o r e important circumstance arises if these simplifications cannot be made. In that case, m o r e than one transition state or reactant state m a y be significant for a particular kinetic parameter. Consider the general rate law for the m e c h a n i s m as written above; then the situation is

Kinetic parameter

Definition

Effective TS

k/K~ k/Kb k

[k~-' + (k~k3/k~)-~]-~ [ks-1 + (kskT/ke)-~]-~ (ks-l + k7-1)

w~ET~+ waETa wsET5+ wTET~ w~ETa; w~ET7

Effective RS E +A F+B WEAEA;WFBFB

In this situation, all the kinetic p a r a m e t e r s are m o r e complicated. F o r the k / K terms, a single reactant state set continues to contribute, but two different transition states h a v e to be considered as partially ratedetermining. In the k term, both effective reactant and transition states are combinations (not necessarily linear) of species. The w t e r m s a b o v e are weighting factors and are described further below. The simplest resolution to a circumstance of this kind is w h e n a method exists to determine the individual rate constants separately. F o r example, if both k3 and k6 could be determined in various isotopic waters, then a proton inventory could be carried out for each by itself. The effective reactant and transition states for k3 would be E A and ET3, respectively, and those for k7 would be FB and ETT. This is rarely a realistic possibility, however. The c o m m o n circumstances is that one determines aggregate p a r a m e ters, and that one actually has no clear idea w h e t h e r more than a single state is contributing to the effective transition state. N o t infrequently, the picture is just as dark for the effective reactant state. T w o different app r o a c h e s for treating such cases have been emerging. One has b e e n w o r k e d out particularly by O ' L e a r y , Cleland and N o r t h r o p , and used by others, and m a k e s use of the c o m m i t m e n t - f a c t o r concept. The second, described by Schowen 3a and applied with practical effect by Stein, 39 employs the c o n c e p t o f the virtual transition state. The a p p r o a c h of c o m m i t m e n t factors describes the relative rates of 3s R. L. Schowen, in "Transition States of Biochemical Processes" (R. D. Gandour and R. L. Schowen, eds.t, p. 77, Plenum, New York, 1978. 39R. L. Stein, J. Org. Chem. 46, 3328 (1981).

586

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

[29]

contributing steps in terms of a fraction known variously as a "partitioning factor, ''4° a "commitment to catalysis ''41 or merely a "commitment,"4~ or a "ratio of catalysis."41 Appropriate fractions can be defined for the forward and the reverse directions of reaction. Considering the forward direction for the Ping Pong mechanism shown above, we have for the parameter k, k = k ~ , / ( k 3 + kT) = ks~(1 + C), where C = (ks/kT) is known by one of the names above. We shall join Cook and Cleland42 in calling it simply a commitment factor. Commitment factors are most useful when the entire isotope effect falls in a single step, say in ks. Then we write,

kn = ksnk, I(ksn + kr)

(kn Iko) = (1 + C)l[(kso Iksn) + C],

where the C always refers to the reaction in HOH and is thus a constant in the various isotopic waters. One can choose an appropriate model for (ksn/kso) and then by fitting procedures find both (a) the th values that enter the proton-inventory expression for ks, as well as (b) the best-fit value of C. Clearly similar expressions can be derived for k / K terms, when a single step therein is responsible for the solvent isotope effect. Note that when C is very small (kr > > ks, so that ks clearly determines the rate), kn/ko becomes equal to kzn/k~o and the situation is simple. Similarly, when C is very large, kn/ko is unity for all n because kr is rate-limiting and exhibits no isotope effect. For intermediate values, k,/ko is between unity and the "intrinsic isotope effect" ksn/kso. The virtual transition state approach is algebraically equivalent, but assumes that more than a single step will in general contribute to an isotope effect. The observed isotope effect is then a weighted average, with the form of the weighting factors, their magnitudes and the prescription for generating the observed average being functions of the mechanism. When a particular isotope effect is determined, it is difficult as a general matter to know with certainty how many steps contribute to it. The apparent structure of the transition state that one derives from a measured isotope effect is thus the structure of an imaginary species, which is in fact the weighted average of one or more real transition states. This construct is called a virtual transition state, s8 The virtual transition state is itself of some interest in a practical sense; its properties may allow the more or less accurate prediction of the dynamic behavior of the system, at least under sufficiently limited conditions that the weighting factors in the observed average do not alter drastically. 40 M. H. O ' L e a r y , this series, Vol. 64, Article [4], and previous work. 41 D. B. Northrop, Annu. Rev. Biochem. 50, 103 (1981), and previous work. 42 p. F. Cook and W. W. Cleland, Biochemistry 20, 1790 (1981).

[29]

SOLVENT ISOTOPE E F F E C T S ON E N Z Y M E

SYSTEMS

587

In the example treated above, assume (1) an isotope effect in both ka and k7 ; (2) that both EA and FB have all ~b values equal to unity; and (3) that only one ~b-site (and no Z-sites) in each of ET3 and ET7 contributes. Then the expected form of kn(n) is ( k n l k o ) - ' = [(kolkao)l(1

-

-

n + n6J<)] + [(kolk7o)l(l

-

n + ntb*)]

as can be deduced by writing the expressions according to Eq. (VId) for each of the rate constants and then adding their reciprocals. Note that the quantities (ko/kao) and (kolk7o) are weighting factors, called w~ and w; in the tabulation on p. 585, for the relative importance of the transition states ETa and ET7 in HOH as solvent. In such a simple case as this, the experimental data could be fitted to the above expression to obtain best-fit values of the two transition-state fractionation factors and the two weighting factors as well. In the most favorable cases, the solvent isotope effect experiment itself may reveal that an apparent transition state is in fact virtual; for example, in the case just described, the isotopic data are used to calculate the weighting factors (k0/kao) and (ko Ik7o) as well as the two fractionation factors for the contributing transition states. The weighting factors are those for the contributing states in the virtual average in HOH; the actual weights change with deuteration of the solvent (as reflected in the expression for k n / k o ) , and this introduces a detectable change in the apparent isotope effect. In other cases, where the isotope effects are not very different for the two transition states or where the conditions are not varied sufficiently, an apparent structure may be derived without any explicit warning from the data. For example, under maximum-velocity conditions, with many substrates, catechol O-methyltransferase (COMT) catalyzes two parallel reactions, the m e t a - and p a r a - m e t h y l a t i o n of its catecholic substrates. The kinetic expression is k = k. +km and for the proton inventory: ( k n / k o ) = (kpo/ko)(1 - n + n4~* ) + (km/ko)(1 - n + nqb* )

if we assume for the sake of simplicity that there is a single site contributing in the transition state for each of the parallel pathways, ff the simultaneous reactions are ignored or have not been noticed, then the data might be observed as excellently linear in n, and fit to a linear function ("oneproton" expression): ( 1 - n + nt~app) = wp(1 - n + n4)*) + Wm(1 - n + n4)*) (1 - n + nt~*app) = (Wp -{- Win)(1 -- n) + n(wp4~* + Wmt~*m) 6*pp = w,4,* + Wm6*~

588

ISOTOPES AS MECHANISTIC PROBES

[29]

where we have noted that Wp = (kpo/ko) and W m = (kmo/ko) are the weighting factors for the contributions of the two transition states along the parallel pathways (and that they sum to one) and that therefore the apparent transition-state fractionation factor is just the simple arithmethic weighted average of the two contributing fractionation factors with the weights being those for HOH. Isotope Exchange at Protein Sites The simplest approach to enzymic solvent isotope effects requires that isotopic equilibrium exist between "exchangeable" hydrogenic sites of the enzyme and the substrate, on the one hand, and the isotopic water solvent, on the other. However, the term "exchangeable" is inherently ambiguous. Sites tend to be classified as exchangeable ff equilibration occurs rapidly on the time scale of the experiment of interest, and as nonexchangeable otherwise. Thus the classification of a given site can depend on the experiment being conducted. Protein hydrogenic sites are generally considered exchangeable if the hydrogen is bound to oxygen, nitrogen, or sulfur, and nonexchangeable if the hydrogen is bound to carbon. However, O,N,S-bound hydrogens may exchange slowly (over hours or days), particularly if the site is deeply buried within the protein and is only rarely exposed to the solvent. CBound hydrogen may also exchange over short periods (minutes or hours) under specialized circumstances. Since the time required for the kinetic experiments involved in mechanistic solvent isotope-effect measurements can vary from microseconds to days, no general rule can be offered for the likely degree of exchange in a particular protein under particular kinetic circumstances. The problem must be approached on a case-by-case basis. Most generally, a complete study of the exchange rates of the various sites in the target enzyme may be carried out by established methods. Frequently, however, so extensive an examination of the question is not required. Instead, the following measures can be employed: 1. Enzyme can be incubated in HOH, DOD, and 1:1 H O H : D O D stock solutions for varying periods before a control kinetic experiment in the 1 : 1 mixture. When injection of a very small quantity of any of these stock solutions into the 1 : 1 mixture leads to an identical rate, it is demonstrated that the degree of significant exchange during the incubation period is equivalent to that which occurs during the period between injection and initiation of data collection. Significant exchange refers to exchange at sites which generate an isotope effect. (Exchange at other sites is, of course, not of interest.) When the experiment yields different rates, then significant exchange is still in progress at the end of the incubation proce-

[29]

SOLVENTISOTOPE EFFECTS ON ENZYME SYSTEMS

589

dure. If the incubation period is long compared to a kinetic experiment, exchange at these sites can sometimes be employed as an experimental variable. Injection of an HOH stock solution directly into various isotopic waters permits measurement of solvent isotope effects e x c l u s i v e of those arising from the sites in question. Incubation of the enzyme for sufficiently long periods in various isotopic waters and use of these stock solutions for kinetic experiments in HOH yields the isotope effect for the slowly exchanging sites a l o n e . 2. The kinetic behavior can be monitored carefully during kinetic runs in isotopic waters, for which either enzyme/HOH or enzyme/DOD stock solutions have been employed. Simple behavior indicates equilibration at significant sites. Complex behavior, not seen when only HOH is present, signals significant exchange during the run. Exchange at substrate O-, N-, and S-bound sites can be checked by similar measures. For small substrates, carbon-bound hydrogen exchange (which may occur at activated positions or at positions activated by action of the enzyme) can be checked by reisolation and examination of substrate after partial reaction, and by similar examination of product.

Product Isotope Effects and Rate Isotope Effects In cases where an exchangeable hydrogen becomes attached to carbon in the course of the enzymic reaction, the product will contain both protium and deuterium when the reaction is carried out in a mixture of isotopic waters. O'Leary and his collaborators 43,44 have shown the utility of product-ratio isotope effects and their comparison with other data, such as rate measurements in the isotopic waters and heavy-atom isotope effects. Northrop 45 has extended and refined some of these considerations. As one example, consider the simple Ping Pong mechanism discussed above on p. 587, where the rate isotope effect at saturation by substrates is ( k n / k o ) -1 = w 3 / ( 1

- n + n4~*) + w T / ( 1 - n + n4~*)

Assume, as above, that all fractionation factors in EA and FP are unity; further assume that a single monoprotic acid in EA, in continuous equilibrium with solvent, donates a hydrogen in the k3 step to a carbon-bound site of the substrate (and that this then continues unchanged to product). The product isotope ratio Rpn will be given by Rpn

----

(PD/PH)n In/(1 -=

n)](k3D/k~n)

= In/(1 - n)~b~

43H. Yamadaand M. H. O'Leary,J. Am. Chem. Soc. 99, 1660(1977). 44M. H. O'Leary, H. Yamada, and C. J. Yapp, Biochemistry 20, 1476 (1981). 45D. B. Northrop, J. Am. Chem. Soc. 103, 1208(1981).

590

ISOTOPES AS MECHANISTIC PROBES

[29]

In such a case ~b* can be calculated as Rpn[(1 - n)/n] and no longer need be treated as a variable. The proton-inventory data can then be used to determine ws, wr, and ~b*. Suppose that wa = 1, wr = 0, so that only the k8 step limits the rate. Then the observed rate isotope effect, kn/ko, will be equal to (1 n + nRpn). This relationship of the rate and product isotope effect establishes (a) that a single step limits the rate, (b) that the proton transfer to the precursor of the eventual product molecule in this step produces both the rate and product isotope effects, and (c) that, therefore, there are no other, secondary contributions to the solvent isotope effect. A failure to observe this relationship of rate and product isotope effects can, of course, have many explanations. Yamada and O'Leary43 originally noted that in certain circumstances, product isotope effects may be used to infer the nature of the enzymic acid function which protonates the product precursor. Consider a monoprotic acid that initially equilibrates with solvent, is then shielded from exchange with solvent, and then in this circumstance protonates the product precursor. The fraction of enzyme that bears a protiated acid function will generate PH, while that with deuterated acid will generate PD. If the fractionation factor of the enzymic acid is unity (as will frequently be the case for nonsulfhydryl groups--see Table II), then Rp n will simply be [n/(1 - n)]: there will be no isotopic discrimination. Indeed, this is observed in the enzymic decarboxylation of glutamic acid. 43 If a diprotic acid were to perform the same protonation, however, a chance for isotopic discrimination would exist because the product precursor would face some functions that bear both protium and deuterium, and intramolecular discrimination could occur. The amount of discrimination would depend on both the stereochemical constraints on the proton transfer and the kinetic isotope effect, but commonly some discrimination would occur and Rp~ would not equal [n/(1 - n)]. In the case of a triprotic acid, a still greater chance for discrimination would obviously exist. In a realistic situation, of course, the actual amount of discrimination observed would be a sensitive function of the relative rates of isotope exchange, proton transfer, and other features of the system. Northrop 41'45 has examined some of these complexities algebraically and by means of computer simulations. One result is that the limits of isotope discrimination for the diprotic and triprotic acids are smaller than might at first appear. Consider an experiment at n = 0.5, and allow the intramolecular isotope effect to be very large; then only H is transferred when there is a choice. The predictions are: -

Monoprotic acid: Enzymes forms EH and ED with [EH] = lED]; EH--*PH, ED-~PD, and [PHI = [PD].

[29]

SOLVENT ISOTOPE EFFECTS ON ENZYME SYSTEMS

591

Diprotic acid: Enzyme forms EHH, EHD, EDH, EDD, all in equal concentration; all H-forms-.PH, EDD---~PD, and [PH] = 3[PD].

Triprotic acid: Enzyme forms E H H H , E H H D , EHDH, E D H H , EHDD, EDHD, EDDH, EDDD, all in equal concentration; all Hforms---~PH, EDDD--~PD, and [PH] = 7[PD]. Northrop's treatment 45 considers more reasonable intramolecular isotope effects and suggests what further conclusions might be drawn from comparison of rate and product isotope effects. Methods and Procedures Preparation of Mixed Isotopic Waters Ordinary distilled water should be passed through an ion-exchange column or otherwise completely deionized before use and should subsequently be stored so as to prevent reabsorption of carbon dioxide. Heavy water of 99.7-99.8% purity is readily obtained from commercial sources. It is recommended that all manipulations involving heavy water be carried out in a glove box that has been flushed with dry nitrogen, so as to avoid exchange with atmospheric moisture (and thus loss of deuterium content) as well as to prevent absorption of carbon dioxide. Contamination of neutral DOD-containing solutions upon brief ( < 3 - 4 rain) exposure to normal atmospheric moisture has been claimed to be negligible, 46 which agrees with general experience. Frequently, commercial DOD is somewhat alkaline and in that case can be distilled from glass through a short fractionating column while protected from the atmosphere. The deuterium content should be determined after this procedure as discussed in the following section. Mixtures of light and heavy water for proton inventory work are prepared from the H O H and DOD described above. Generally at least five solutions are needed where n, the atom fraction of deuterium (or mole fraction of DOD), is approximately 0.85, 0.65, 0.5, 0.35, 0.15. Often as many as 8-10 solutions are used. These solutions are most accurately prepared gravimetrically whereby known weights of H O H and DOD are combined. They may also be prepared volumetrically with burets using densities of any desired precision. TM The mixed solvents thus obtained can then be used to prepare solutions of the required pH and/or ionic strength (by dilution of weighed buffer pairs or of KCI, for example) for later use, or of the complete reaction solution (by dilution of the appropriate quantities of reagents, buffers, salts, etc.) Such direct use of the mixed solvents is somewhat 46 K. Mikkelsen and S. O. Nielsen, J. Phys. Chem. 64, 632 (1960).

592

ISOTOPES AS MECHANISTIC PROBES

[29]

TABLE IX~ PREPARATION OF MIXED H O H - D O D TRIs-BUFFERED SOLUTIONS FOR PROTON INVENTORY OF ACYLATION OF ot-LYTIC PROTEASE BY p-NITROPHENYL ACETATE

Volume HOH buffer (ml)b'c

Volume DOD buffer (ml)°,c

8.00 7.00 6.00 5.00 4.00 3.00 2.00 1.00

1.00 2.00 3.00 4.00 5.00 6.00 7.00

-

-

Mole % deuterium oxidea 0.00 12.38 24.77 37.15 49.53 61.92 74.30 86.68

a Taken from D. Quinn, Ph.D. Dissertation, p. 63. University of Kansas, Lawrence (1977). b Buffers were prepared by mixing the indicated volumes by pipetting with a 1000-/zlEppendorf pipet. c Total [buffer] of all buffers was 0.05 M, [Tris-HCl]/[Tris] = 1.27, and /z = 0.028 M. Concentrations of the individual components of HOH and DOD buffers were determined by weighing the required amounts of Tris-HCl and Tris on an analytical balance and dissolving them in 50 ml of HOH or DOD in a volumetric flask. Isotopic dilution in the DOD buffer from exchange between the buffer components and solvent has been accounted for. Calculated from the volume composition of each buffer, with corrections applied for the slightly different molar volumes of HOH and DOD. w a s t e f u l o f D O D , a n d a l t e r n a t i v e m e t h o d s w h e r e s t o c k s o l u t i o n s o f rea g e n t s a n d / o r b u f f e r s a n d / o r salt in p u r e H O H o r p u r e D O D are dil u t e d w i t h the a p p r o p r i a t e w e i g h t s o r v o l u m e s o f p u r e D O D a n d p u r e H O H are o f t e n u s e d . T a b l e I X i l l u s t r a t e s s u c h a p r o c e d u r e . I n e s t i m a t ing the final v a l u e o f n for s o l u t i o n s p r e p a r e d b y v o l u m e t r i c o r g r a v i m e tric p r o c e d u r e s , a c c o u n t m u s t b e t a k e n o f a n y e x c h a n g e a b l e H c o m i n g f r o m the r e a g e n t s o r buffers.

D e u t e r i u m C o n t e n t of Isotopic W a t e r s T h e e x p e c t e d o r c a l c u l a t e d d e u t e r i u m c o n t e n t o f the p u r e D O D a n d o f the i s o t o p i c w a t e r m i x t u r e s is b e s t verified e x p e r i m e n t a l l y . It is ess e n t i a l to do this for the s o l v e n t s u s e d to p r e p a r e the r e a c t i o n m e d i a o r the s t o c k s o l u t i o n s , a n d d e s i r a b l e (but m o r e difficult) for the a c t u a l r e a c t i o n s o l u t i o n s b o t h j u s t b e f o r e a n d j u s t after a k i n e t i c r u n . T h e d e u -

[29]

SOLVENT ISOTOPE EFFECTS ON ENZYME SYSTEMS

593

terium content of isotopic water mixtures can easily and conveniently be determined by integration of the PMR spectrum of the desired solution containing an internal "proton counting" standard. The standard can be pure, dry acetonitrile, dioxane, or similar solvent; the relative amounts of standard and of HOH present should be adjusted so that the NMR peak heights (or areas) for the two substances are similar. The experimental procedure as outlined below is best suited for solutions containing less than 5% HOH. The relative amounts of standard and solvent to be analyzed can, of course, be adjusted to account for solutions of greater HOH content or, alternatively, proportionately more weighed DOD can be added to the solution prior to analysis. A typical procedure follows.4r Weigh a clean, dry, capped NMR tube; record the weight, W,1. Add 500/xl of solvent to be analyzed (>95% DOD), recap and weigh; record the weight, W2. The difference in weighings ( W 2 - W1) is the weight of water (HOH and DOD), gw- (If a solution containing substances other than water--e.g., buffers, salts, reagents--is to be analyzed, their combined weight (Ws) from their known concentrations must be accounted for. In this case, gw = W2 - W1 - W~. ) Add 25/zl of pure, dry acetonitrile to the tube, recap and weigh; record the weight, gA~. Measure the NMR spectrum of the sample from 0-6 ppm downfield from TMS. The CH3CN peak will appear at - 2 ppm; the HOD (or HOH) signal will appear at - 4 . 6 ppm. Integrate the spectrum several times and calculate an average integration ratio: R =

area of CH3CN signal 3 (mol AN) = area of water H signal mol H

(1)

The desired quantity is the atom fraction of D, n: n = (mol D)/(mol H + tool D) Now, the weight of the mixed waters, gw, may be treated as the summed weights of the atoms: gw = (mol H)AH + (mol D ) A D + (moles O)Ao where the A values are atomic weights. Since mol O = (1)(mol H + mol D), mol D = (2/Ma){gw - ½Mh (mol H)} mol H = 3 (mol AN)/R where Mh and Md are the molecular weights of HOH and DOD, respectively (18.015 and 20.028). 47 D. Quinn, Ph.D. Dissertation, p. 384. University of Kansas, Lawrence (1977).

594

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

[29]

I n d e p e n d e n t confirmation o f such analyses m a y be obtained by other analytical methods. 4s p H a n d Its E q u i v a l e n t in I s o t o p i c W a t e r s It is generally important in e n z y m e kinetics to maintain a constant k n o w n p H throughout a run. This is a c c o m p l i s h e d by m e a n s o f buffers (or s o m e t i m e s the p H Stat method) and careful m e a s u r e m e n t o f p H by a p H meter. Complications arise and corrections m u s t be applied in solvent isotope effect w o r k b e c a u s e o f (1) a difference in glass electrode r e s p o n s e (and thus m e t e r reading) to a given L + or O L - concentration, as a function o f solvent deuterium content, and (2) a difference in p H - r a t e b e h a v i o r for a given reaction as a function o f solvent deuterium content. An empirical relation b e t w e e n p H - m e t e r reading and k n o w n p D (negative logarithm o f [D+]) for a p u r e D O D solution, p D = (meter reading) + 0.4 was first o b s e r v e d by Glasoe and L o n g 49 and has been confirmed for a variety o f acids and bases, p H ' s , t e m p e r a t u r e s , electrodes and instruments. 5°-52 The m e t e r reading is low by a constant value o f 0.40 _-+ 0.02 p H units; the difference (hpH)n, where n = 1, b e t w e e n the correct numerical value for p D and the o b s e r v e d reading is thus e x p r e s s e d as (ApH)n=I = p D - m e t e r reading = 0.4

(2)

F o r mixed H O H - D O D solutions, this difference has been reported by two different groups to be o f a quadratic form, (ApH)n = 0.076n 2 + 0.3314n

(3)

for 0.01-0.1 M solutions o f LCI or LCIO449 and (ApH)n = 0.173n 2 + 0.221n

(4)

for 0.0007-0.001 M LCI solutions at both 25 ° and 370. 52 The p H - m e t e r reading is thus e x p e c t e d to be low for any given solution o f a t o m fraction deuterium equal to n by the a m o u n t (ApH)n. The m a x i m u m disc r e p a n c y in (ApH)n calculated f r o m these two experimentally derived 48 I. Kirshenbaum, "Physical Properties and Analysis of Heavy Water" (H. C. Urey and G. M. Murphy, eds.). McGraw-Hill, New York, 1951. 49 p. K. Glasoe and F. A. Long, J. Phys. Chem. 64, 188 (1960). 50 p. Salomaa, L. L. Schaleger, and F. A. Long, J. Am. Chem. Soc. 86, 1 (1964). 5~A. K. Covington, M. Paabo, R. A. Robinson, and R. G. Bates, Anal. Chem. 40, 700 (1968). 52 K. B. J. Schowen, J. K. Lee, and R. L. S c h o w e n , to be published.

[29]

SOLVENT ISOTOPE EFFECTS ON ENZYME SYSTEMS

595

equations is 0.03 pH units. More work is needed to determine whether the discrepancy is meaningful. For greatest accuracy it may be desirable to prepare a calibration curve of meter reading versus n for the particular system, pH, instrument, temperature, and electrodes to be used in a given investigation. A method for determining such an experimental curve is summarized in the following paragraph. The pH meter should be calibrated with standard buffer solutions and adjusted to the desired temperature. The electrode (a combination electrode is most convenient) is then immersed in a thermostatted cell containing the desired solvent (pure HOH, pure DOD, or any of several H O H - D O D mixtures of known n) and allowed to come to thermal equilibrium (about 5 min). Longer soaking or preequilibration of the electrode with the particular solvent being studied is not necessary. Use of a fairly tight-fitting cover for the cell through which the electrode can penetrate, plus a minimal but positive pressure of dry nitrogen, suffices to prevent significant loss of deuterium by exchange during the 6-7 min required for each determination. After equilibration, a small known volume of concentrated standard acid or base (HCI, for example) is added, the solution stirred and the pH meter read. A curve of pH-meter reading versus n can thus be obtained. The n value should be corrected for the H introduced via the added acid or base. This correction will normally be quite small. For example, 20/~l of 11.45 N HCI added to 5 ml of DOD where n -- 0.9976 gives a 0.05 M DCI solution where n = 0.9937. Polynomial regression analysis (see, e.g., BMDPR 53 or other statistical program packages) can then be applied to the data to obtain empirical coefficients of a polynomial representation of (ApH)n(n) under the exact conditions of interest. U s e of Equivalent p L in E n z y m e Kinetics 54

Because almost all acid pKa values change with isotopic solvent composition (see above), enzyme p H - r a t e profiles also shift with deuteration of the aqueous environment. Figure 1 shows a typical example. The plateau at high pL is lower in DOD because of a kinetic isotope effect on the pH-independent rate constant, while the inflection point of the curved portion is shifted to a more basic pL in DOD. Obviously, in a range where the rate is pL-dependent, it will give an incorrect result to measure velocities at pH = pD. In the simplest cases, where the pL dependence arises from ionization of enzyme funcW. J. Dixon and M. B. Brown, eds., "BMDP-79, Biomedical Computer Programs, P-Series." Univ. of California Press, Berkeley and Los Angeles, 1979. 54 C. K. Rule and V. K. LaMer, J. Am. Chem. Soc. 60, 1974 (1938).

596

[29]

ISOTOPES AS MECHANISTIC PROBES

o

~4

& t

t

&

i

0

•

i

4

6

8

I0

pt_ FIG. 1. p L - R a t e profiles (L = H,D) for the a-lytic protease-catalyzed hydrolysis ofpnitrophenyl acetate. Pseudo-first-order enzymic rate constants, Vma~/Km, in HOH (circles) and DOD (triangles). Data obtained from Ref. 47, pp. 78-83. The open symbols are total rate constants to which background corrections have been applied to generate the enzymic rate constants.

tional groups, velocities at pH = pD will differ in part because of an isotope effect on the pL-independent parameters and in part because of a change in the fraction of enzyme in the active protomeric form. If solvent isotope effects are to be determined at only a single point on the p L - r a t e profile, it is thus essential that this point be either (a) in a relatively pL-independent region (such as that above pH or pD 8 in Fig. 1); or (b) maintained at a corresponding position on the p L - r a t e profile in every isotopic water, i.e., at an equivalent pL. The concept of equivalent pL can be illustrated with a system like that of Fig. 1. If Yn is the pL-independent velocity in an isotopic water with fractional deuteration n, then the velocity Vn is given by Vn = Y n K E / ( K E + aL)

where K~. is the value of lyonium-ion activity aL at Vn = Yn/2. KE might be the acidity constant of an enzyme functional group in a simple case, but we allow for more complex potentialities (e.g., that the pL-dependence arises from a change in rate-determining step). Generally, aL is controlled by a buffer, for which we let R be the acid-base ratio and KB the acidity constant, so that

Vn = YnKE/(KE + KsR) Now we define a ratio p n = ( V n -- V n ) / V ~

[29]

SOLVENT ISOTOPE EFFECTS ON ENZYME SYSTEMS

597

This is, in general, the ratio o f deficit velocity (i.e., amount by which the actual velocity is below the maximum at optimal pL) to achieved velocity. Since Pn is also equal to aLIKE, then in a simple case, Pn = [inactive e n z y m e protomer]/[active e n z y m e protomer] It is the quantity Pn that must be maintained constant in the isotopic waters; an equivalent pL is therefore one for which Pn is a constant for all n. The simplest situation is that for which the buffer acidity constant K s and the enzymic inflection point KE show equal solvent isotope effects at all values of n. This is because

On = KnnR/KE n so that if KBn/KE n is a constant with n, then an equivalent p L is obtained simply by maintaining the buffer ratio R constant in all isotopic waters. Of course, in such a series of experiments, the p L changes with isotopic solvent. In the commonest case, where pKs ~ - pKBh = 0.5, then pD = p H + 0.5. In the mixed isotopic waters, p L will be more basic than pH by an amount less than 0.5 units, increasing with n. The change in p L will cause the experiment to " f o l l o w " the p L - r a t e profile as it also shifts, always keeping the velocity at an equivalent point on the profile. The best way to make use of this technique is to determine the p L rate profile in both H O H and DOD. Then a buffer can be selected with a pKs a - p K B h closely approximating the enzymic value, p K ~ f - p K E h. Such a buffer can be used at constant ratio with reasonable safety in isotopic water mixtures. Only in very unusual cases should K~./KB be the same in H O H and DOD but different in mixtures o f the two. Often experiments are carried out at equivalent p L on the assumption that both e n z y m e and buffer will show equal solvent isotope effects; indeed, acids with similar pK values do often exhibit very similar isotope effects (see above). The procedure is thus not without a basis, but it can involve some risk. The p L - r a t e profile in Fig. 1 has pKE d - pKe h = 0.6, although most acids that buffer in this range have pKs d - pK~ h = 0.5. The effect o f this kind of mismatch in an equivalent-pL experiment can be estimated from

(Vn/Vo)/(Yn/Yo) = (1

+ /90)/(1 + P n )

If the buffer and e n z y m e solvent isotope effects match, Pn = P0 and there is no error. If they fail to match, pn :/: P0; instead, Pn = poG(n) where log G(n) = (pKEn - pKE h) -- (pKBn -- pKBh). A mismatch of 0.1

598

ISOTOPES AS MECHANISTIC PROBES

[29]

units would therefore produce (1 + p0)/(1 + pl) = 0.89 for P0 = 1. The overall solvent isotope effect would thus be in error by about 13%; partial solvent isotope effects in mixed isotopic waters would be in error by smaller amounts. Mechanistic

Interpretation

General The most immediate characteristic of an overall solvent isotope effect, VHoH/VDoD,is its direction: whether the rate is greater in HOH or DOD. Table X shows the customary nomenclature and most straightforward interpretations of such observations. Thus a normal isotope effect signifies a loosening at hydrogenic sites upon formation of the transition state, and an inverse isotope effect signifies a tightening. To draw any further conclusion, however, one must be more definite about the nature of the velocity ratio under consideration. The best circumstance is when one is dealing with resolved parameters of the rate law. Then the effective reactant states and effective transition states are fixed by the mechanistic characteristics of the system under study. For example, in the hydrolysis of ester substrates like N-acetylL-tryptophan ethyl ester, catalyzed by a-chymotrypsin, the kinetic parameters kea t and k c a t / g m have these effective reactant and transition states55:

keat: (keat//gm)"

Effective reactant state is acyl-enzyme; effective transition state is that for the deacylation process. Effective reactant state is free substrate and free enzyme; effective transition state is that for the acylation process.

The observed overall isotope effects56 (pL-independent) are keat,HO a/kcat,DO D = ( k e a t / g m ) H O H / ( k e n t / g m )DOD :

2.7 1.8

Thus the conclusion can be drawn that some loosening of bonds at hydrogenic sites occurs in both acylation and deacylation processes. 55 M. L. Bender, "Mechanisms of Homogeneous Catalysis from Protons to Proteins." Wiley (Interscience), New York, 1970. 56 M. L. Bender, G. E. Clement, F. J. K6zdy, and H. d'A. Heck, J. Am. Chem. Soc. 86, 3680 (1964). The value for k c a t / K m is as calculated by M. F. Hegazi, D. M. Quinn, and R. L. Schowen, in "Transition States of Biochemical Processes" (R. D. Gandour and R. L. Schowen, eds.), p. 355. Plenum, New York, 1978.

[29]

S O L V E N TISOTOPE EFFECTS ON ENZYME SYSTEMS

599

TABLE X GENERAL INTERPRETATION OF OVERALL ENZYME SOLVENT ISOTOPE EFFECTS

Observation

Significance

VHoH/VDoD > 1

"Normal" isotope effect: net binding at contributing hydrogenic sites is looser in effective transition state than in effective reactant state "Inverse" isotope effect: net binding at contributing hydrogenic sites is tighter (stiffer) in effective transition state than in effective reactant state

VHoH/VDo D < 1

Semiquantitative formulationa I-I 6rt < 6r~ i j ]~

l~

~ ck*l > ~ 4'm

Cf. Chart V. Among the cautions that must be observed are: (a) each isotope effect may arise from a single site or from multiple sites; and (b) each reactant and transition state may be either a single species or a mixture of species (virtual state, with weights for individual contributing states, as described above). Means for determining the degree o f " p u r i t y " of reactant and transition states are few and controversial. This is an area of current research in which most questions are still unresolved. A greater extent o f progress has been possible with respect to the n u m b e r o f contributing sites; here the proton-inventory approach is helpful and is described in later sections. As noted above, the interpretation of solvent isotope effects proceeds best from data on individual kinetic parameters. The interpretation o f simple velocity ratios is a far riskier undertaking. For example: (a) if the velocities are obtained in a p L - d e p e n d e n t range, they must be at equivalent p L in H O H and DOD or else the apparent effect will be confounded by shifts in the p L - r a t e profile. (b) If the velocities are obtained in a range where substrate-concentration dependence exists, the apparent effect may represent a mixture of effects on V and V / K terms. C u r v a t u r e of P r o t o n - I n v e n t o r y Plots The proton-inventory method allows one to obtain information on the number o f hydrogenic sites contributing to the overall solvent isotope effect and on the magnitudes of the individual isotope effects or fractionation factors. One effective procedure for using the method involves two stages:

600

[29]

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

1. Inspection of the observed plots of Vn(r/) versus n, and generation of hypothetical models--i.e., postulation of equations of the form of Eqs. (VId,e) with particular values of PET, /~E, and /~g" Model generation is based on consideration of (i) whether Vn(r/) is linear; (ii) if nonlinear, the character of its nonlinearity. 2. Testing of the various postulated models against the data, with rejection of improbable and retention of probable models. Part (1) of this procedure will be discussed in this section, and part (2) in the next section. Equations (VId) and (Vie) are of some complexity, and a proper description of each model generation involving them would probably be both elegant and difficult. At the current stage of their use in mechanistic biochemistry, the data are never of sufficient abundance or precision to justify such a discussion. We shall adopt a greatly simplified version which is appropriate for most applications. Figure 2 shows the customary way of representing proton-inventory data, in which Vn is plotted versus n. Here V, is plotted in units of V0 for convenience. Another useful plot is of Vn IV1 so that the intercept at n = 0 displays the overall isotope effect Vo/VI. Each of the eight

2 /___..I-~

~~~e

I

~

J

,

~

o.5

d"---.........~(l-?+n/B) 2r~]

,

1

f

0.5 °

FI/(l-n+4n)] n ~

FIG. 2. Some typical shapes of proton-inventory curves. The error bars s h o w n on curves (b) and (g) are for an error of -+ 5%.

[29]

SOLVENT ISOTOPE E F F E C T S ON E N Z Y M E SYSTEMS

601

curves shown in Fig. 2 is labeled with the function Vn(n)/Vo that was used to generate it. By comparing these with Eqs. (VId) and (Vie) and with the actual curve shapes, we can develop a few simple rules for visual inspection of the curves and deduction of possible models. All the curves are drawn for an overall isotope effect of 4. Those in the upper section [curves (a)-(c)] are for an inverse effect (V1/Vo = 4), while those in the lower section [curves (d)-(h)] are for a normal effect (Vo/VI = 4). Note that the scales of the ordinate differ for the two sections of the figure. The appearance of the curves is dominated by a threefold distinction in the character of the curvature: 1. Some curves are linear [(a) and (e)]; these are for isotope effects generated by a single site, located in the transition state. 2. Some of the curves are bowl-shaped ("bulging down"12;"bowed downward ''u) [(b), (c), (f), (g) and (h)]. These correspond to one of the following models: (i) the isotope effect is generated by a very large number of sites in either reactant or transition state [Co) and (g)]; (ii) the isotope effect is generated by multiple sites in the transition state [(f)]; (iii) the isotope effect is generated by a single site in the reactant state [(c) and (h)]. 3. One of the curves [(d)] is dome-shaped ("bulging up"~2; "bowed upward"H); the corresponding model is for an inverse contribution, overriden by a larger normal contribution. These curves are examples of general types that are encountered in practice. Other examples and further discussion of procedures for visual inspection and model generation are given in earlier articles. 9,n,12 Following are some useful principles. Linear curves. The simplest model is always a one-site model for the transition state. It is not difficult, however, to generate a linear Vn(n) from far more complex models? 7"~ Such models may make good mechanistic sense if a single-site model seems chemically unreasonable. Bowl-shaped curves. The greatest amount of curvature is generated by a model in which the entire isotope effect is attributed to a single reactant-state site [(c) and (h) in Fig. 2], just as the smallest amount of curvature (linear curve) is generated by a single transition-state site [(a) and (e)]. If multiple sites are introduced in the reactant-state model, the curvature is decreased; if multiple transition-state sites are introduced, the curvature is increased. Both tendencies continue as the number of postulated sites is increased. When the number of sites becomes very 57 A. J. Kresge, J. Am. Chem. Soc. 95, 3065 (1973). W. J. Albery, Faraday Discuss. Chem. Soc. 10, 160 (1975), and elsewhere in that volume.

602

ISOTOPES AS MECHANISTIC PROBES

[29]

large ("infinite-site model"), Vn(n) becomes exponential (all Z sites, no ~b-sites; see Chart VI). Then it is not possible to ascribe the sites to a particular state, and the two models converge on a single curve [(b) or (g)]. Bowl-shaped curves can also arise from isotopically induced shifts among parallel reaction pathways, and such models can reasonably be considered if (a) there is independent evidence in their favor; (b) a simple multiple-site model gives unreasonable isotope-effect contributions; or (c) if the observed curvature in Vn(n) cannot be matched by a simple multiple-site model. Dome-shaped curves. Such a curve can easily arise if there are opposing contributions tending to generate inverse and normal isotope effects. Thus curve (d) of Fig. 2 is produced by a model in which a single ~-site of the transition state gives a normal isotope effect of 8 (such as might come from a primary isotope effect in proton transfer), with this opposed by a Z-site effect of 2 in the inverse direction. Another possible source of dome-shaped curves is a shift among serial transition states in determining the rate, induced by deuteration of the solvent. Also, the presence of reactant-state sites tending to generate an inverse isotope effect together with dominating transition-state sites that give normal effects can produce the dome shape. Curve (d) provides an example of this if the Z-sites that generate the inverse contribution of 2 are imagined as reactant-state sites where binding stiffens upon formation of the transition state. It is often useful to estimate the amount of curvature expected for various models for the overall solvent isotope effects and then to compare it with proton-inventory data in a preliminary way. A handy technique is to use midpoint solvent isotope effects. These can be estimated from the formulas in Table XI and compared with experimental data at or near n = 0.5 to obtain an idea of which models are worthy of pursuit. Albery a has developed a particularly complete apparatus for using midpoint solvent isotope effects to survey the range of possible reactant-state and transition-state models consistent with mixed isotopicwater data. Readers may want to consult his writings 9"5s for further information.

Testing of Mechanistic Models and Estimation of Fractionation Factors 9,11,12 Once a general idea has been obtained of a good fit with various types of models--for example, by the means just described--a more

[29]

SOLVENT ISOTOPE EFFECTS ON ENZYME SYSTEMS

603

TABLE Xl FORMULAS FOR MIDPOINT ISOTOPE EFFECTS

VII~/Vi

Model One transition-state ~b-site Two transition-state ~b-sites All Z-sites (generalized solvation model) One reactant-state 6-site Two reactant-state ~b-sites

½[1 + (VolVO] ~[1 + (VolV,),~,] ~ (VolVO '1~ 1/(½[1 + (V, lVo)]} 1/{¼[1 +

(V,/Vo)V2]2}

quantitative interpretation is often desired. In the case of a one-site isotope effect, Vn(n) is a linear function, and the determination of its intercept, V0, and its slope, V0(~b* - 1), provide a complete characterization. Least-squares fits are the most common. With bowl-shaped curves, the simplest models (as already noted) are multisite ones. The difficulty of distinguishing one multisite model from another (say, a two-site from a three-site model) is great for small overall isotope effect (Vo/V1 < 2) but decreases sharply as the isotope effect becomes larger. Table XII shows this in terms of midpoint isotope effects. For an isotope effect of 2.0, the two-site and "infinitesite" effects differ at midpoint by only 3%, as do the one-site and twosite predictions. Thus a "multisite" model may be distinguished from a "one-site" model with very good data, but a further resolution would be most problematical. On the other hand, with an overall effect of 4, the dispersions are on the order of 11-12%, and reach 25-35% with an overall effect of 10. One means of deciding the maximum number of sites permitted by a TABLE XII MIDPOINT ISOTOPE EFFECTS FOR VARIOUS MODELS AS A FUNCTION OF OVERALL ISOTOPE EFFECT

Vll2/VI for Vo/Vx

One-site model

Two-site model

Infinite-site model

1.20 1.50 2.00 3.00 4.00 10.0 50.0

1.100 1.250 1.500 2.000 2.500 5.500 25.5

1.098 1.237 1.457 1.866 2.250 4.331 16.3

1.095 1.225 1.414 1.732 2.000 3.162 7.071

604

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

[29]

particular data set, on a simple multisite model, is to note that VT

Vn(n) = I-I (1 -- n + n~b*i) i

is a polynomial in n of order vT. The data can be fitted by polynomial regression procedures, ~ and the statistical significance of each term evaluated by an F-test. The last statistically significant term then defines the maximum value of VT. It must always be recognized, however, that further terms may be concealed in the experimental noise. In principle, one can relate the coefficients of the best fit polynomial to the ~b* values. This rarely is satisfactory in practice. Instead, the best procedure is to choose one or more models for fitting and to use standard curve-fitting procedures 53 to estimate the values of each of the fractionation factors, as well as V0, which should be made a parameter of the fit. These procedures commonly allow for proper weighting of the individual measurements, and yield proper measures of dispersion for each estimate, such as standard deviations. Furthermore, they usually provide the covariance matrix; this is very informative, since the values of the ~b~ and ~ba terms obtained are often highly correlated with each other. Models obtained for such data are, of course, not unique. They must be considered in the light of other mechanistic data for the same and related systems and interpreted with appropriate caution. Deductions from the Magnitudes of Fractionation Factors Once a satisfactory model or, more commonly, models have been obtained by testing against solvent isotope-effect data, and the magnitudes of the fractionation factors or medium-effect parameters have been determined, one ordinarily wishes to interpret these in terms of reaction mechanism. This is a highly individual matter, but a few principles will be pointed out here. Insofar as medium-effect (Z-site) parameters go, it is not generally possible to decide the exact origin or detailed character of the corresponding sites. They are best considered as generalized solvation effects for proteins or substrate or as arising from a multiplicity of protein-structural sites. Their essential characteristics are that they are (a) large in number and (b) generate only a small isotope effect per site. With both Z-sites and ~b-sites, it should always be borne in mind that the definition of a fractionation factor makes the standard of comparison an average site o f bulk water. A given fractionation factor thi (in either reactant or transition state) can then be thought of as repre-

[29]

SOLVENT

ISOTOPE EFFECTS

ON ENZYME

SYSTEMS

605

senting the isotope effect for the hypothetical process Average site of bulk water --~ ith site

KDOO/KnoH= $i

Thus ~bi > 1 reveals that binding in the ith site is tighter or stiffer than in a bulk-water site, while ~bi < 1 shows binding in the ith site to be looser than in a bulk-water site. Binding can be thought of as reflecting general restriction to motion in all dimensions: tighter means more restricted, looser means less restricted. If a hydrogenic site is involved in the reaction-coordinate motion in a transition state, then its mass will contribute to the reduced mass for the critical decomposition motion vc (see Chart V). This contribution is removed from ~b* and cancelled out in the usual formulation of transition-state theory (thus the asterisk of $*). The result is that ~b*, if the amplitude of the hydrogenic site i is large in the reaction coordinate, may have an exceedingly small value. For instance, if a water molecule were the proton donor in a proton-transfer reaction exhibiting a primary kinetic isotope effect of 8, the fractionation factor for the transferring hydrogen in the transition state would be ~b* = ~ = 0.13. If a different species than w a t e r - - s a y , a sulfhydryl group--functions as proton donor, one can estimate the expected kH/kD by interposing a hypothetical equilibrium for exchange with water; since for SH, ~b - 0.5: --SH

+ B: t

0.5

k~l/kt, = 4

/ /

HOH

, --S---H---B //~

= i/8 = 013

/

Of course, if the functional group that serves as proton donor has ~b = 1, no correction is necessary: --OH

+ B:

~ --O---H---B

HOH

These cases shown, it may be noted, would show quite different proton-inventory curves. For the s u l ~ y d r y l proton donor, Vn = V0[1 - n + n(0.13)]/[1 - n + n(0.5)] For the hydroxyl donor, Vn = V,[1 - n + n(0.13)] This would readily be detected by the midpoint solvent isotope effect (see Table XI). Thus for the sulfhydryl donor, Vl12/V ~ = 3, while the prediction for a single transition-state 4~-site [linear Vn(n)] is 2.5. Thus

606

ISOTOPES AS MECHANISTIC PROBES

[29]

the curve is bowl-shaped. For the hydroxyl case, V m / V 1 = 4.5, in accord with the linear prediction from one transition-state ~b-site. If transition-state ~b-sites do not have amplitude in the reaction coordinate, their contributions are secondary isotope effects. Most straightforwardly, one can estimate expected ~b* values by analogy with the examples o f Table II. For example, 0

--S

II

--SH

+/C\

~

6a = 0.5

\

/ OH

/C\ qr~ = 1.2

This kind o f addition is e x p e c t e d to exhibit a substantial overall solvent isotope effect, V1/Vo = 1.2/0.5 = 2.4 and a characteristic proton inventory, Vn = V0(1 - n + n[1.2])/(1 - n + n[0.5]) In some cases, reliable fractionation factors are not available. Often they can be placed within reasonable limits. F o r example, a metalbound water site H M+~ _ 3 + 0 / \ H

ought to have ~b ~ (0.69) 8, with 8 measuring the degree o f positive charge transfer from the metal. The value o f 0.69 is just I for LaO +. If limits can be placed on 8, a range o f possible ~b values follows (but see Table II, note o). A similar scheme has been used to estimate ~b values for intermediate kinds o f structures such as water in the course o f nucleophilic attack: H

\e+ /O---E

H

Here again, if8 or a range o f 8 values can be estimated, ~b should be about (0.69) 8. As already noted, the a priori estimation and a posteriori interpretation o f ~b values is a matter o f individual approach, in the context o f a particular mechanistic study. One o f the messages o f this article is that theory and experience set moderately stringent limits on the a priori enterprise, while experimental methods exist to provide a degree o f definition to the a posteriori effect.