kinetic isotope effects as probes for hydrogen

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Progress in Reaction Kinetics and Mechanism. Vol. 28, pp119–156. 2003 1468-6783 © 2003 Science Reviews

KINETIC ISOTOPE EFFECTS AS PROBES FOR HYDROGEN TUNNELING, COUPLED MOTION AND DYNAMICS CONTRIBUTIONS TO ENZYME CATALYSIS Amnon Kohen Department of Chemistry, University of Iowa, Iowa City, IA 52242, USA E-mail: [email protected]

Contents

2.

ABSTRACT

120

1. INTRODUCTION 1·1 Enzyme catalysis 1·2 Contribution of various physical phenomena to enzyme catalysis 1·3 Kinetic complexity and the chemical step 1·4 Kinetic isotope effects as probes of the chemical step

121 121 122 124 125

THE SWAIN–SCHAAD RELATIONSHIP 2·1 Semiclassical relationship of reaction rates of H, D and T 128 2·2 1 Swain-Schaad relationship and studies of intrinsic KIE 128 2·3 2 Swain-Schaad relationship 130 2·3·1 Mixed labeling experiments as probes of tunneling and 1°–2° coupled motion 131 2·3·2 Upper semiclassical-limit for 2′ Swain-Schaad relationship 133 2·3·2·1 Zero point energy and reduced mass considerations 134 2·3·2·2 Vibrational analysis calculations 135 2·3·2·3 Effect of kinetic complexity 138 2·3·2·4 The new effective upper limit 141

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3.

4.

TEMPERATURE DEPENDENCY OF KINETIC ISOTOPE EFFECTS 3·1 Temperature dependency of reaction rate and of KIEs 3·2 Tunneling investigation using KIE on Arrhenius preexponential factors 3·3 Tolmanís interpretation of curved Arrhenius plots

142 142 142 145

PROTEIN DYNAMICS ROLE IN CATALYSIS 4·1 Protein motionís role in enzyme catalysis 4·2 Effects of enzyme dynamics on the chemical step and environmentally coupled (vibrationally enhanced) H-tunneling 4·3 MM/QM models and simulations 4·4 Comparison to non-enzymatic reactions

148 148

148 149 150

5.

CONCLUSIONS

152

6.

ACKNOWLEDGEMENTS

152

7.

REFERENCES

152

ABSTRACT Since the early days of enzymology attempts have been made to deconvolute the various contributions of physical phenomena to enzyme catalysis. Here we present experimental and theoretical studies that examine the possible role of hydrogen tunneling, coupled motion, and enzyme dynamics in catalysis. In this review, we first introduce basic concepts of enzyme catalysis from a physical chemistry point of view. Then, we present several recent developments in the application of experimental tools that can probe tunneling, coupled motion, dynamic effects and other possible physical phenomena that may contribute to catalysis. These tools include kinetic isotope effects (KIEs), their temperature dependency and H/D/T mutual relations (the Swain–Schaad relationship). Several theories and models that assist in understanding those phenomena are also described. The possibility that these models invoke a direct role for the enzyme’s dynamics (environmental fluctuations and rearrangements) is discussed. Finally, the need to compare the enzymatic reaction to the uncatalyzed one while investigating contributions to catalysis is emphasised. Prog React Kinet Mech 28:119–156 © 2003 Science Reviews

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Kinetic isotope effects as probes for hydrogen tunneling

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KEYWORDS: enzyme catalysis, isotope effect, tunneling, enzyme dynamics, coupled motion, reaction rate, temperature dependence 1. INTRODUCTION 1·1 Enzyme catalysis Enzymes are the biological catalysts that direct, control and enhance the chemistry in biological systems. Their catalytic power is many orders of magnitude at physiological temperature and pressure. These catalysts are also stereo- and regio-specific, a quality that is crucial to their function. In many cases the limitation in assessing their catalytic power lies in the capability of measuring the rate of a relevant uncatalyzed reaction [1,2], a fact that imposes major limitation on the assessment of the enzyme’s “catalytic power”. From the physical chemistry point of view, biology is faced with two, almost contradicting, tasks that are closely related to enzyme catalysis: one is catalyzing a reaction at the rate most suitable for organism function and the other is imposing the order needed to prevent alternative side-processes. In other words, an enzyme not only catalyzes the reaction of interest, it also inhibits the alternative reactions between the reactants, and between reactants and the solvent, that would otherwise occur in solution. Both rate-effects (catalysis and inhibition) are many orders of magnitude greater than the reaction in solution, and of substantial biological importance. Commonly, the first is denoted catalysis and the second is denoted specificity. This review focuses on issues important to enzyme catalysis, while the specificity is only mentioned as a limitation in assessing the rate and nature of the uncatalyzed reaction. From the kinetic point of view, “Catalytic Power” is the ratio between the reaction rate of the catalyzed reaction and the uncatalyzed reaction. Catalytic power = kcat/kun

(1)

where kcat is the rate of catalyzed reaction and kun is the rate of the uncatalyzed reaction. By definition, catalysis should be unit-less (a ratio of rate constants) and care must be practiced while determining the ‘catalytic power’ of an enzyme. Many enzymatic systems seem to catalyze reactions by factors close to 1017. Several factors complicate this assessment. First, most relevant uncatalyzed reactions are too slow to be measured under the same conditions as the catalyzed reaction. Uncatalyzed reactions are usually measured under extreme conditions of temperawww.scilet.com

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ture and pressure that are extrapolated to physiological conditions for comparison with the enzymatic reaction [1,2]. Recent studies used computational approaches to elucidate the shape of free energy surfaces of various reference reactions. For example, a reference solution reaction for serine proteases was evaluated in [3]. Second, the rates measured for the uncatalyzed reactions are proportional to the reactant(s) concentration and at lim[A]→∞ = ∞ where v is rate and [A] is a reactant concentration. Enzymatic reactions, on the other hand, reach a limiting rate (denoted Vmax) when the catalyst is saturated by substrate. When comparing catalyzed vs. uncatalyzed rates, special attention must be paid to the units and to the nature of the rates. This last point is explained and deliberated below in Section 1·3.

1·2 Contribution of various physical phenomena to enzyme catalysis Mechanistic studies of enzymes can be conducted at different levels. Biochemists may investigate regulatory issues (from expression to allosteric regulations). Organic chemists have recognized important features such as general-acid and general-base catalysis, and inorganic chemists have described the essential role of metals in enzyme catalysis [4]. From the point of view of physical chemistry, important questions include: the relations between the reactive complex’s structure and function; the location and structure of the transition state (TS) along the reaction coordinate; TS stabilization; ground state destabilization; the significance of quantum mechanical (QM) phenomena; the dynamics of the system; energy distribution through the normal modes of an enzyme; and contributions of entropy and enthalpy. Some of these phenomena are defined and their contribution to enzyme catalysis is discussed in more detail below. Many experimental and theoretical studies have attempted to isolate specific contributions of phenomena such as QM tunneling or protein dynamics from the whole catalytic cascade. In some of the following sections we attempt to make similar distinctions, which is always a somewhat artificial process. The quantitative degree of each contribution is inherently model dependent. Nevertheless, once an aspect is determined it is of interest to assess its importance to catalysis. In Sections 2–4 we present some recent examples of how one model can be distinguished from another and how experimental and theoretical tools can identify specific contributions. Two of the terms mentioned above need to be clearly defined, namely, QM tunneling and protein dynamics: www.scilet.com

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Quantum mechanical tunneling: Tunneling is the phenomenon by which a particle transfers through a reaction barrier due to its wave-like properties [5,6]. Figure 1 graphically illustrates this for a symmetric double well system such as the C–H–C hydrogen transfer. It is important to note that the tunneling probability is affected by both the distance of R and P wells and their symmetry. A lighter isotope would have a higher tunneling probability than the heavier one, since a heavy isotope will have lower zero point energy and its probability function would be more localized in its well. Consequently, kinetic isotope effects (KIEs) are effective tools for studying tunneling. Two practical applications are described below: the Swain-Schaad exponential relationship (Section 2) and temperature dependency of KIEs (Section 3). Dynamics: The definition of dynamics, and how to distinguish dynamic motion from entropy effects, is controversial [7–10]. Does a single molecule fluctuating in time represents the same phenomenon as the distribution of conformations found for a large number of molecules in a given time? Is that distribution

Figure 1 An example of ground-state nuclear tunneling. The reactant well (R) is on the left and the product well (P) is on the right. The fine line is the probability (nuclear 2) of finding the nuclei in the reactant or the product wells. The greater the overlap of the R and P probability functions, the higher the tunneling probability.

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of conformations equivalent to the conformational space covered by a single fluctuating molecule? Furthermore, how can we conceptualize the coupling between the enzyme’s dynamics and the reaction coordinate? Several approaches that offer formalisms which couple the reaction coordinate to the fluctuations of its environment are discussed below. For example, a useful approach is to think of a system progressing in time and apply a Fourier transformation to examine the assembly of frequencies in the system (e.g., molecular normal modes). One can then ask how these oscillators may affect the reaction coordinate. In this review, we will consider any nuclear motion as a dynamic phenomenon regardless of it being in thermal equilibrium with the environment (Boltzman distribution) or not.

1·3 Kinetic complexity and the chemical step An inherent limitation in studying the chemical transformation catalyzed by an enzyme is that enzyme kinetics are rather complex. Enzymes catalyze reactions by forming a reactive complex with the reactant(s) prior to the chemical transformation. As illustrated in Figure 2, the formation of that complex and the decomposition of the product complex are part of the overall kinetic cascade which must be considered while investigating the chemical transformation.

Figure 2 A reaction energy profile for enzyme catalyzed A + B → Q + P reaction. The chemical step is marked and the rate-limiting step is product Q release.

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At reactant (substrate) concentration much higher than the dissociation constant of the reactive complex (known as the Michaelis complex) [4,11] the reaction rate is independent of the substrate concentration because the catalyst is saturated by substrate. Since all the enzyme active sites are occupied by substrate the rate is determined by the concentration of the enzyme (all captured as a reactive complex). At low substrate concentration, on the other hand, the reaction rate is proportional to both substrate and enzyme concentrations (as for any bi-reactant reaction) [11]. Consequently, the propagation stage of an enzyme catalyzed reaction (defined as the steady-state) is characterized by two rate constants, first order rate constant kcat at high substrate concentration and second order rate constant kcat/KM (V/K) at low substrate concentration. For a single substrate, this simple model is described by the Michaelis–Menten equation: =

(kcat ) ⋅[E]⋅[S] (K M )obs + [S]

(2)

where [E] and [S] are the total enzyme and substrate concentrations, respectively, and the subscript obs stands for “observed” constant. For complex reactions that involve multiple-substrates, multiple-products and sometimes several subunits of the catalyst, this picture becomes very complex and is characterized by the reactants binding and products release patterns. Nevertheless, even for relatively simple kinetic patterns, the observed rate constants rarely represent the rate and nature of the chemical transformation catalyzed by the enzyme (depicted below as the chemical step). That step is, in most systems, the process of covalent bond breaking and forming and its investigation is at the heart of exploring enzyme catalysis at the molecular level. In an enzyme catalytic cascade, the chemical step(s) is always accompanied by substrate binding, product release and protein rearrangement steps, which are rate determining for many enzymes. The resulting, multi-barrier reaction path is sometimes very different from the non-enzymatic or uncatalyzed one. This complex kinetic cascade is a major obstacle for experimental studies of the chemical step. 1·4 Kinetic isotope effects (KIEs) as probes of the chemical step KIE is the ratio of rates of two reactants that only differ by their isotopic composition. This ratio of rates between the light and heavy isotopes is characteristic of the reaction coordinate and the nature of the transition-state (TS). The KIE www.scilet.com

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measured for a bond cleavage or formation between isotopically labeled atoms is depicted as primary (1°) KIE. This KIE results from the energy of activation differences of the bonds being cleaved and much of its magnitude is due to the differences of zero point energy (ZPE) between the ground state and the TS of the reaction (Figure 3). For example, for C–H bond cleavage a vibrational stretch (3000 cm–1) ground state (GS) is transferred into a translational mode at the TS. The KIE measured with isotopically labeled atoms that are not on a bond that is being cleaved is depicted as secondary (2°) KIE. The hydrogen on the same carbon center as the bond that is being cleaved (for 2° KIE) or other centers (for KIE, KIE, etc.) change their bonding force constants and vibrational frequencies during the reaction. This change in bond order results in a 2° KIE if those uncleaved atoms are isotopically labeled. Those 2° KIEs are normally smaller than 1° KIEs as the change in bond order between GS and TS is much

Figure 3a, Different energies of activation (Ea) for H, D, and T resulting from their different zero-point energies (ZPE) at the ground state (GS) and transition state (TS). The GS-ZPE is constituted by all degrees of freedom but mostly the ZPE stretching frequency, and the TS-ZPE is constituted by all degrees of freedom orthogonal to the reaction coordinate. This type of consideration is depicted as “semiclassical”. Such a model predicts KIE for any two isotopes (e.g., 1 and 2) following ln(k1/ k2) = (Ea 1 - Ea 2)/ RT, where R is the gas constant and T the absolute temperature. b, Tunneling correction for a transition state theory (TST) type of model. In addition to its higher ZPE, the lighter isotope tunnels at a lower energy under the top of the barrier, resulting in smaller Ea (relative to that of the heavier isotope) than in the semiclassical model.

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smaller. Additionally, 2° KIEs are normally smaller or equal to the relevant equilibrium isotope effect (EIE). The EIE, is determined by the full change in bond order from reactants to products, while the KIE is only affected by the change from GS to the TS. Two issues must be addressed while studying KIEs in enzymatic reactions: (1) isotope effects on binding and, (2) kinetic steps that are not isotopically sensitive but still mask the isotope effect, hence impairing the interpretation of the data. The first is commonly addressed by careful design of the isotopic labeling. For example, the ubiquitous cofactor adenine-nicotinamide dinucleotide (NADH), is bound to the enzyme’s active site through many interactions. In studies of its oxidation, only the hydrogen which is to be abstracted as hydride (on the C4 position) should be isotopically labelled, so that isotope effects on binding are negligible. The second issue, denoted ‘kinetic complexity’, is that multiple kinetic steps may mask the KIE. Its mathematical treatment is rigorously described in several published reviews [12,13]. If only one kinetic step is isotopically sensitive, then the general relationship between the intrinsic KIE (KIEint – which is the value of mechanistic interest) and the observed KIE (KIEobs) is given by [12]: KIEobs =

KIEint + C f + Cr ⋅ EIE

(3)

1 + C f + Cr

where EIE is the equilibrium isotope effect and Cf and Cr are the forward and reverse commitments to catalysis, respectively. The term commitment to catalysis is the ratio between the isotopically sensitive rate constant to isotopically insensitive steps affecting the observed KIE. The important implication for this discussion is that if the reaction is practically irreversible the KIEobs can vary from KIEint and unity (no KIE). On the other hand, if the reaction is reversible and Cf is small, the KIEobs may vary between KIEint and EIE. Techniques that allow one to calculate the intrinsic effect, given the observed one, are discussed below in Section 2·2·1. Overall, once an intrinsic KIE is extracted from the experimental data, it imposes a strict constraint on any mechanism, theoretical model, analysis, or simulation addressing the system under study. As described in the following section, intrinsic KIEs are unique as they are a direct measurement of a reaction’s TS and can be directly applied to the reaction potential surface and other physical features. www.scilet.com

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2. THE SWAIN–SCHAAD RELATIONSHIP 2·1 Semiclassical relationship of reaction rates of H, D and T The kinetic relationship among the three isotopes of hydrogen has wide uses as a mechanistic tool in organic and physical chemistry. The Swain–Schaad exponential relationship (EXP, as defined in eqn (4) below) is the semiclassical (no tunneling) correlations among the rates of the three isotopes of hydrogen, and was first defined by Swain et al. in 1958 [14]. The relationship can be predicted from the masses of the isotopes under examination [15]. This EXP results primarily from zero point energy (ZPE) differences between the ground state (GS) and the transition state (TS). Since H has a higher GS ZPE relative to D and T, it should react faster than D and T (Figure 3a.). Several investigators examined this relationship under extreme temperature (20–1000°K) and as a probe for tunneling [16–18]. Experimental and theoretical studies also used this isotopic relationship to suggest a coupled motion between primary and secondary hydrogens in hydride transfer as well as elimination reactions in the gas phase and in organic solvents [19–21]. Two main uses of the Swain–Schaad relationship in enzymology are described in the following sections. 2·2 1° Swain–Schaad relationship and studies of intrinsic KIE The Swain–Schaad exponential relationship was originally defined for primary (1°) KIEs [14]: kH  kH  =  kT  kD 

EXP

or EXP =

ln(kH / kT ) ln(kH / kD )

(4)

where ki is the reaction rate constant for isotope i. If tunneling does not contribute to the H-transfer, EXP can be calculated from [15]: EXP =

 ln(kH / kT )  1 = – 1 H T  ln(kH / kD ) 

 1  – 1  T  D 

(5)

where i is the reduced mass for isotope i. The original EXP was calculated for H/T vs H/D KIEs and yielded a value of 1·44 (using atomic masses). If tunneling contributes to the reaction rate, EXP would be smaller than 1·44. In the investigation of H-tunneling it is more common to use T as a frame of reference and to compare H and D in the following way: www.scilet.com

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EXP =

 ln(kH / kT )  1 = – 1 H T  ln(kD / kT ) 

 1  – 1  T  D 

129

(6)

Equation (6) defines the relationship of H/T to D/T KIEs, for which the semiclassical EXP is 3·26 (for atomic masses: i = mi). Nevertheless, either measurements (using Eqn (5) or Eqn 6) establish the relationship among the three isotopes for a one step reaction (single barrier). If the chemical step is masked by kinetic complexity (Eqn (3)) the observed KIE (KIEobs) will be smaller than the intrinsic one. In the case of Eqn (6), this will affect H/T KIE more than D/T KIE and the observed EXP in Eqn (6) will be smaller than the intrinsic one. For the experiment pertinent to Eqn (5), kinetic complexity will have an opposite effect and the observed EXP will be larger than the intrinsic one. Northrop [12,22] developed a simple method that allows calculating the commitment to catalysis and the intrinsic KIE from the observed KIEs. This method assumes no significant deviation of the intrinsic 1˚ KIE from their semiclassically predicted values.a In analogy to the method described by Northrop [12,22], Eqn (3) can be written again while subtracting 1 from both sides of the equation: (kH / kT )int + C f + Cr ⋅ EIE  kH  –1   –1= 1 + C f + Cr  kT  obs = =

(kH / kT )int + C f + Cr ⋅ EIE – 1 – C f – Cr 1 + C f + Cr

(7)

(kH / kT )int – 1 + Cr ⋅ (EIE – 1) 1 + C f + Cr

At the limit of the H/T EIE goes to 1, which is not a bad assumption for 1° EIE, Eqn (7) becomes:

aIt

is important to note that for 1° KIEs and the resulting 1° Swain–Schaad exponents, no EXP values larger than 3.6 were found in available literature. Even in cases for which tunneling was evident from mixed labeling experiments (Section 1.3.1) or from the temperature dependency of KIEs (Section 3.2), the 1° EXP did not significantly differ from the semiclassically predicted value of 3.3 for ln(kH/kT)/ln(kD/kT) or 1.4 for ln(kH/kT)/ln(kH/kD).

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 kH  (kH / kT )int – 1   –1= 1 + C f + Cr  kT  obs

(8)

Now, Eqn (8) for H/T KIE is divided by the same equation for H/D KIE. Since no isotopic rate constant appears in the denominator, if the expression for H/T KIE-1 is divided by the expression for H/D KIE-1, the denominator will be canceled leaving the ratio of KIEints –1 on the right side: (kH / kT )obs – 1 (kH / kT )int – 1 = (kH / kD )obs – 1 (kH / kD )int – 1

(9)

From Eqn 5: (kH/kD)int = ((kH/kT)int)1/1·44

(10)

And Eqn (9) can be written as: (kH / kT )obs – 1 (kH / kT )int – 1 = (kH / kD )obs – 1 ((kH / kT )int )1/1.44 – 1

(11)

Even though Eqn (11) (and equivalent equations for other KIE experiments) has only one unknown (KIEint), it cannot be solved analytically (due to transcendental functions). After dividing the observed KIEs minus one by each other, a numeric solution can be preformed.b,c 2·3 2° Swain–Schaad relationship Most experimental evidence for a tunneling contribution has come from the breakdown of this relationship for a secondary hydrogen, i.e., not the hydrogen bIn the original works of Northrop [12,22], tables for various KIE experiments offer solutions for a wide range of KIEs. Today this can be calculated with most calculators or any computer. cIn

cases where the chemical step is reversible and the assumption of small 1° EIE is not valid a solution is not possible without measuring the reverse commitment (Cr). Yet, Cleland [23] identified the KIEint values range between the observed KIE (KIEobs) and the product of EIE and KIEobs for the reverse reaction (KIEobs–rev*EIE).

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whose bond is being cleaved but its geminal neighbor. Furthermore, many of the reported experiments have been mixed labeling experiments, in which a secondary H/T kinetic isotope effect was measured for C–H cleavage while the D/T secondary effect accompanied C–D cleavage [6,24]. In experiments of this type, the breakdown of the Swain-Schaad relationship indicates both tunneling and a coupled motion between the primary and secondary hydrogens. 2·3·1 Mixed labeling experiments as probes of tunneling and 1°–2° coupled motion Mixed labeling experiments consist of a pattern of isotopic labeling that is more complex than the one considered in the original Swain–Schaad relationship. Several theoretical studies in the 1980s have suggested that mixed labeling experiments would be the most sensitive indicator of H-tunneling [19,20]. This section describes the mixed labeling experiment and demonstrates how it serves as a probe for tunneling. For the general reaction illustrated in Figure 4, the C–H bond is being cleaved and the primary (1°) hydrogen is transferred while the secondary (2°) hydrogen changes its bonding from (s–sp3) to (s–sp2). In a mixed labeling experiment, the 1° H/T KIE (kH/kT) is measured with H in the 2° position and is denoted by kHH/kTH, where kij is the rate constant for H-transfer with isotope i in the 1° position and isotope j in the 2° position. The 2° H/T KIE is measured with H at the R position and is denoted by kHH/kHT. The 1° and 2° D/T KIE measurements, on the other hand, are conducted with D in the geminal position, and are denoted by kDD/kTD and kDD/kDT, respectively (Figure 5 and Eqn (12) below). 2° M EXP =

ln(kHH / kHT ) ln(kDD / kDT )

(12)

Figure 4 The general H-transfer reaction discussed in the text.

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Figure 5 The isotopic labeling pattern for a mixed-labeling experiment.

The exponential relationship from such mixed labeling experiments is denoted as MEXP. If the 1° and 2° hydrogens are independent of each other, the isotopic labeling of one should note affect the isotope effect on the other. This is denoted the rule of geometrical mean (RGM [25]): r=

ln(kHi / kHT ) =1 ln(kDi / kDT )

(13)

where i is H or D. The RGM predicts that the that the isotopic label at the geminal position should not affect the MEXP: 2° EXP =

ln(kHH / kHT ) ln(kHH / kHT ) = = 2° M EXP ln(kHD / kHT ) ln(kDD / kDT )

(14)

If the motions of the 1° and 2° hydrogens are coupled along the reaction coordinate a breakdown of the RGM will result in an inflated 2° MEXP. The 1° KIE will have a secondary component, and will be deflated, but since the 2° H/D KIE is very small (~1·2), the expected deflation of the 1° MEXP is very small. The 2° KIE on the other hand, will have a primary component and will be significantly inflated. Tunneling of the 1° H will induce a large 2° H/T KIE (kHH/kHT) relative to the more semiclassical 2° D/T KIE (kDD/kDT), due to the reduced effect of tunneling from D in the primary position. In the mixed labeling experiment, when there is coupled motion between the 1° and 2° hydrogens, tunneling along the reaction coordinate results in inflation of the 2° MEXP because H tunneling is more significant than tunneling of D. It is easy to show that the MEXP is a product of the original Swain–Schaad EXP and RGM (r). www.scilet.com

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r • EXP =

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ln(kHH / kHT ) ln(kDH / kDT ) ln(kHH / kHT ) M • = = EXP ln(kDH / kDT ) ln(kDD / kDT ) ln(kDD / kDT )

(15)

A mathematically rigorous explanation of the high sensitivity of the mixed labeling experiment to H-tunneling can be found in [26] and [27]. Both Huskey [26] and Saunders [28,29] have independently shown that exceptionally large values for MEXP are computed only for 2˚ KIEs resulting from coupled motion and tunneling. They concluded that the extra isotopic substitution is an essential feature of the experimental design. The first enzyme that was studied using mixed labeling experiments was yeast alcohol dehydrogenase (ADH) [30]. The 1° hydrogen in that system is the pro-R hydrogen of benzyl alcohol which is transferred from the alcoholate carbon to the re face of C4 of the nicotinamide cofactor (NAD+). Two 2° hydrogens are changing their bond orders as the reaction progresses from reactants to products (Figure 6). 2·3·2 Upper semiclassical-limit for 2° Swain–Schaad relationship Only experimental EXP values outside the semiclassical range may be used as evidence for tunneling. For EXP as defined in Eqn (6) {EXP = ln(kH/kT)/ln(kD/kT)} any value under its semiclassical lower limit can be explained by kinetic complexity (see above) but values larger than its upper limit could serve as an evidence for tunneling. Thus, the determination of the upper limit for this EXP with no tunneling contribution is critical. Until recently, the upper semiclassical limit used was 3·34 as calculated following Streitwieser’s suggestion to

Figure 6 The alcohol dehydrogenase catalyzed reaction. The arrows represent atomic motion along the reaction coordinate.

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Table 1 Experimental examples for inflated EXPs from mixed labeling studies of enzymesa Enzyme

1° EXP

2° EXP

Ref.

Yeast ADH

3·58 (±0.08)

10·2 (±2·4)

[30]

L57V liver ADH

3·14 (±0.05)

4·55 (±0·75)

[31]

F93W liver ADH

3·31(±0.02)

6·13 (±0·50)

[31]

L57 F liver ADH

3·30 (±0.03)

8·50 (±0·99)

[31]

3·50 (±0.05)

13·9 (±4·8)

[32]

NA

4·4 (±1·3)

[33]

b

ADH-hT (65 ˚C) c

TIM aAll

the data (excluding ADH-hT) are reported for experiments at 25 ˚C. ADH from Bacillus stearothermophilus. c Triosephosphate isomerase. b

use the reduced masses of 12C–H, 12C–D, and 12C–T [20,34]. We have recently calculated a new upper limit that is more realistic and relevant to the commonly used mixed labeling experiment [35]. The results of this study are summarized below. 2·3·2·1 Zero point energy and reduced mass considerations A simple reduced mass consideration leads to the commonly used upper limit of ln(kH/ kT)/ln(kD / kT) = 3·34 calculated from Eqn (6) and the reduced masses of C–H, C–D and C–T [6,20,24,34]. This limit was calculated from the reduced masses of only the two atoms whose covalent bond is being cleaved in the reaction. Thus, the stretching mode between these two atoms was considered as a pure, isolated vibrational mode. There is no straightforward reason why this should hold for a system with 1°–2° coupled motion. If the motion of atoms other than the hydrogen and the carbon whose bond is being cleaved are part of the reaction coordinate, a different reduced mass has to be considered. We calculated MEXPs from Eqns (10) and (11) using reduced masses of a variety of coupled modes for the general reaction illustrated in Figure 4. In the case of three nuclei with equal contribution to the reactive mode, a simple analytical solution may be applied. The reduced masses for nuclei moving in opposite directions can be calculated from = mi/mi where mi is the mass of an atom. It is possible to consider the coupling of another coordinate to the cleaved C–H bond analytically through the secular equation [36]. |FG – I| = 0

(16) www.scilet.com

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Here F is the force constant matrix, G defines the reduced mass and coupling for each internal coordinate, and I is the identity matrix. The unknown ( ) is related to a vibrational frequency ( = 4 22; if F is of dimension N N, there will be N values of that satisfy Eqn 16, i.e. N vibrational frequencies) of the system and can be used to calculate the zero point energy and, hence, MEXP. Of course, for the one-coordinate case one obtains Streitwieser’s value of 3·34. However, Eqn (16) can also be used to consider a slightly more complex case, such as the coupling of two bonds in a CH2 group. In this case the frequencies are a function not only of the three masses but also the H–C–H angle (), and it is easily shown that both the 1° and 2° MEXP()’s reach a maximum at MEXP(180°) = 3·34. A similar analysis of the coupling between the H-C-H bend and the C-H stretch yield 1° and 2° MEXPs that are functions of the C-H bond length, , and the ratio of the force constants for the stretch and bend. However, an exhaustive search of the parameter space yielded only a few MEXP > 3·34. The maximum MEXP found in these studies was 4·25 [35]. 2·3·2·2 Vibrational analysis calculations Since there is no analytical solution for a system in which more than two coordinates are coupled, a numerical simulation was employed. The goal, again, was to find whether the semiclassical component of a KIE measured by a mixed labeling experiment can lead to an inflated 2° MEXP and to find a new maximum for a semiclassical MEXP (SC MEXP). Such a maximum, if greater than 3·34, will serve as a new upper limit above which tunneling must be invoked. Yeast ADH was chosen as a model for the general reaction in Figure 4, since it has been studied intensively and kinetic data, including data from mixed labeling experiments, are available. The experimental findings of Cha et al. [30] for yeast ADH were 7·13 ± 0·07 and 1·73 ± 0·02 for 1° H/T and D/T KIEs, respectively, and 1·35 ± 0·015 and 1·030 ± 0·006 for 2° H/T and D/T, respectively. These KIEs lead to a 1° MEXP of 3·58 ± 0·08 and a 2° MEXP of 10·2 ± 2·4 (the error propagation was described elsewhere [32]). We found that those results are reproducible and reliable [37] and thus used them in constructing a model system based on the ADH reaction. Those data were used to parameterize the coupling constants of a truncated model (Figure 7) of the alcoholate substrate and the cofactor (NAD+). With this the contribution of semiclassical KIEs to inflated MEXPs was studied. www.scilet.com

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Amnon Kohen

Figure 7 Truncated model for the ADH-catalyzed reaction. is the C-H-C angle and is the dihedral angle between the two secondary hydrogens, namely, the relative orientation of the two reactants close to the TS. (Reproduced with permission from ©2002 Am. Chem. Soc. [35]).

Rucker and Klinman [38] have used vibrational analysis and the Bigeleisen-Mayer equation to calculate isotope effects for the ADH-catalyzed oxidation of benzyl alcohol by NAD+ (Figure 6). In the context of transition state theory and normal statistical analysis, KIEs can be calculated from the Bigeleisen–Mayer equation [39,40]: KIE = MMI•EXC•ZPE

(17)

where MMI is a mass-moment of inertia term, EXC is a vibrational excitation term and ZPE is a zero-point energy term. A truncated system was used with empirical force constants and geometric parameters for the cutoff model of reactants and proposed transition states. The Bell tunneling correction [41] was applied to calculate KIEs and the experimental results of Cha et al. [30] were used to parameterize the potential surface and coupling constants. Rucker and Klinman found that only a model with both substantial H-tunneling and coupling between the reaction coordinate and vibrational modes could fit the experimental results. In accordance with previous studies, they also found that the 2° D/T KIE is expected to be the most sensitive parameter to changes in reaction coordinate properties. Rucker and Klinman studied a linear ( = 180°) C-H-C transfer and a dihedral angle between the two 2° hydrogens () equal to 0° (i.e., 2° hydrogens and the C–H–C system were in one plane). www.scilet.com

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In [35] we used the Rucker–Klinman model and methodology to examine the semiclassical (no tunneling) values of the 2° MEXP in the cross-labeling experiment. We repeated their calculation for a large conformational space (240°>>120° every 20° and 360° for in 60° increments). and define the relative conformation of the reactants and were constrained while calculating the vibrational frequencies and KIEs at each conformation (Figure 7). Other geometric parameters and the force constants, excluding the off-diagonal coupling constants, were set as described in [38]. All calculations were done on a 500 MHz Macintosh G4 using the program BEBOVIB IV [42]. This program solves the vibrational secular equations for the molecules of interest using user-defined force fields. The program then computes isotope effects using the calculated vibrational frequencies and the Bigeleisen equation with, or without, tunneling correction. At each conformation, the coupling constants were adjusted while fitting the calculated KIE to the experimental value reported in [30]. The parameter space was fully swept through (-180