How enzymes harness highly unfavorable proton transfer reactions

Abstract

Acid–base reactions that are exceedingly unfavorable under standard conditions can be catalytically important at enzyme active sites. For example, in triose phosphate isomerase, a glutamate side chain (nominal pK _a ≈ 4 in solution) can in fact deprotonate a CH group that is vicinal to a carbonyl (pK _a ≈ 18 in solution). This is true because of three distinct interactions: (a) ground state pK _a shifts due to environment polarity and electrostatics; (b) dramatic increases in effective molarity due to optimization of proximity and orientation; and (c) transition state pK _a shifts due to binding interactions and the formation of strong low barrier hydrogen bonds. In this report, we review the literature showing that the sum of these three effects supplies more than enough free energy to push forward proton transfer reactions that under standard conditions are exceedingly nonspontaneous and slow.

1. INTRODUCTION

Enzymes are remarkably effective at speeding up reactions at low temperatures, commonly accomplishing reaction rate accelerations of 10⁷–10¹⁹ over the uncatalyzed reaction. ¹ , ² , ³ , ⁴ Particularly efficient enzymes ² , ⁵ give accelerations up to 10²³! A number of interactions at the active site contribute to this impressive lowering of activation energy, including transition state stabilization, ¹ , ⁶ , ⁷ , ⁸ desolvation, ¹ , ⁹ substrate orientation/pre‐organization/orbital steering, ¹⁰ and proximity effects of catalytic groups including charges, nucleophiles, electrophiles, acids, and bases. ¹ , ¹¹

Proton transfer is the most common enzyme‐catalyzed reaction, ⁵ , ¹² appearing in well over half of catalytic mechanisms. Jencks ¹¹ was the first to address the following thermodynamic conundrum regarding some enzymatic proton transfer steps: How can weak bases like carboxylate (asp/glu) or imidazole (his) side chains deprotonate exceedingly weak acids like hydroxyl groups or CH groups alpha to a carbonyl? Table 1 lists some of the more well‐known examples of nonspontaneous proton transfers in enzyme mechanisms.

TABLE 1.

Nonspontaneous proton transfers in enzyme mechanisms

Enzyme	E a ‐base	pK a (HBaq)	pK a e (HBeas a )	S a ‐acid	pK a (HSaq)	pK a e (HSeas)	K eq,aq f	K eq,eas	∆G°eas (kcal/mol)
Serine proteases (e.g., chymotrypsin)	his≡N:	6.5	7	HO—ser	15	15	10–8.5	10−8	11
RNAase, xylose isomerase	his≡N:	6.5	7	HO—S	15	15	10–8.5	10−8	11
Carboxypeptidase, 13 thermolysin, histone deacetylase, mutT diPi hydrolase	glu‐COO:−	4.3	7	H2O	14	7.5–9 b	10–9.7	≈10−1	1‐3
Ketosteroid isomerase	asp‐COO:−	3.9	4.6	H—C(R)—C=O	13	5.6 c	10‐9.1	10−1	1
Acyl‐CoA dehydrogenase 14	glu‐COO:−	4.3	9.2	H—C(R')—C=O	18	8	10–13.7	101.2	−2
Citrate synthase (step 1), triose‐Pi isomerase	asp‐COO:− glu‐COO:−	3.9 4.3	6–6.7 6–6.7	H—C(R')—C=O H—C(R')—C=O	18 18	10–14 10–14	10–14.1 10–13.7	≈10−6 ≈10−6	8
Mandelate racemase	lys‐NH2 his≡N:	10.4 6.5	6.4 d 6.4	H—C(R)—COO:− H—C(R)—COO:−	30 30	9–15 b 9–15 b	10–19.6 10–23.5	≈10−6 ≈10−6	4–12 4–12
Aconitase, fumarase, aspartase	ser‐O:−	15	5.7–6.4 d	H—C(R)—COO:−	30	9–13 b	10−15	≈10−5	4–10
Enolase	lys‐NH2	10.4	6.5 d	H—C(R')—COO:−	≥ 32	< 17 b	< 10−21	≈10−10	14

^aE = enzyme; S = substrate; eas = enzyme active site (pK _a values from References 1, 2, 13, 15, and references therein).

^bSubstrate pK _a is lowered considerably by binding to one or more metal cations.

^cSubstrate pK _a is lowered considerably by H‐bond donors asp₉₉‐COOH and tyr₁₄‐OH.

^dpK _a is substantially below the nominal aqueous value, due to nearby positive charge density (or hydrophobic side chain).

^eActive site pK _a values listed here are measured (or calculated) for the E·S complex. Interestingly, Richard and co‐workers ¹⁶ , ¹⁷ , have found that for triose phosphate isomerase, pK _a of the base‐catalytic glutamate at the active site increases from ≈4 in the free enzyme to ≈6 in the E·S complex, and then to ≈10 in the enzyme complexed with the bound enolate intermediate. The implication is that at the proton transfer transition state, pK _a(glu) is somewhere between 6 and 10. ¹⁸ This would change the calculated value of the C—H acid, but not K _eq,eas or ∆G°_eas.

^f K _eq = 10^{‐(pKa(HS)–pKa(HB))}; ∆G° = −RTln(K _eq).

Consider, for example, the initial, rate‐determining step in triose‐phosphate isomerase catalysis, extraction of a proton from a carbon acid by carboxylate: ¹⁹

$$\mathrm{glu}={\text{glutamate}}_{165}\mathrm{at}\text{the}{\text{enzyme}}^{\prime }\mathrm{s}\text{active site};\text{in the triose substrate molecule},\mathrm{R}=\mathrm{OH}\text{and}{\mathrm{R}}^{\prime }={\mathrm{CH}}_2\mathrm{O}{\mathrm{PO}}_3^{2-}.$$ (1)

Using the nominal aqueous pK _as in Table 1, the equilibrium constant for the triose phosphate isomerase initial proton transfer should be 10⁻¹⁴, and ∆G° = −RTlnK _eq = +19 kcal/mol. So, how can such a nonspontaneous reaction be important in an enzyme mechanism? Incidentally, triose phosphate isomerase is a good example to consider here, because Albery and Knowles ²⁰ , ²¹ showed that this enzyme achieves catalytic “perfection,” increasing the enzyme‐catalyzed rate by a factor of 10¹¹ over the reaction rate in buffer; it essentially functions close to the diffusion‐controlled limit. ²²

Let us summarize the extent of the problem faced by triose‐phosphate isomerase: Clearly, at the typical standard state of all 1 M aqueous solutes, the proton transfer in Equation (1) will not occur to any appreciable extent. For example, starting with 1 mM glutamate and triose‐phosphate substrate (Equation [1]), K _eq = 10⁻¹⁴ predicts equilibrium concentrations of 10⁻¹⁰ M products. Thus only one out of 10⁷ reactants (=10⁻³ M/10⁻¹⁰ M) are converted to products; in a 10 μm diameter spherical cell, there would be only about 30 molecules of each product inside the entire cell at equilibrium.

Of course, most enzymes catalyze reactions that are not at equilibrium. ²³ Furthermore, we must account for the difference between thermodynamics and kinetics. In general, except for electron and hydrogen transfer reactions covered by Marcus Theory, ∆G° tells us little about a reaction's activation energy, E _a. On the other hand, for nonspontaneous reactions, E _a must exceed ∆G°. So, for the triose‐phosphate isomerase proton transfer step, E _a ≥ 19 kcal/mol, which makes for a slow reaction indeed: Given a diffusion‐controlled limit of 10⁹ M⁻¹ s⁻¹ in aqueous solution, ²⁴ from Arrhenius's Law we would have k ≤ 10⁻⁵ M⁻¹ s⁻¹ at 25°C! (For a more detailed discussion of the relationship between E _a and ∆G°, see Supporting Information, Section I.)

So how can a “perfect” enzyme, or for that matter any enzyme, work with such a slow, nonspontaneous proton transfer step? The answer, as is so often the case in enzyme catalysis, lies in the choice of standard state, ⁸ , ⁹ , ¹³ as well as in the effects of proximity, orientation, and local environment. ¹ , ¹¹ Recall that the enzyme‐catalyzed reaction occurs not in aqueous solution with all concentrations 1 M, but at the enzyme's active site, which is often hydrophobic, thus helping to desolvate the substrate and nearby catalytic groups. ²⁵ , ²⁶ There are, in fact, three distinct interactions at the enzyme active site that are important in driving these nonspontaneous proton transfer reactions forward: (a) ground state pK _a shifts due to environment polarity and electrostatics; (b) dramatic increases in effective molarity due to optimization of proximity and orientation; and (c) transition state pK _a shifts due to binding interactions and the formation of strong low barrier hydrogen bonds.

2. GROUND STATE EFFECTS: pK _a AND LOCAL ENVIRONMENT

The aqueous solution value of ∆G° = +19 kcal/mol for the triose‐phosphate isomerase proton transfer reaction was calculated using the nominal aqueous pK _a values listed in Table 1 (Columns 3 and 6). However, in their meta‐analysis, Pace et al reported that protein side chain pK _a values can be up to five units lower or higher than the nominal value in solution. ²⁷ , ²⁸ Clearly, local environment can make a huge difference in the ground state pK _a. (See Supporting Information, Section II, for further discussion of the effects of environment on pK _a.)

For example, in carboxypeptidase, thermolysin, and related hydrolases ¹⁴ (Table 1), the pK _a of the nucleophilic water substrate is lowered from its aqueous value of 14 (Reference 29) to 7.5–9 due to binding to Zn²⁺ at the active site. ¹⁴ , ¹⁵ , ¹⁸ , ³⁰ This allows for easier deprotonation of the water by the nearby basic glutamate, whose pK _a is raised from its aqueous value of 4 up to 7, by a nearby aspartate—COO:⁻. In this case, K _eq for proton transfer rises from 10⁻¹⁰ in aqueous solution to about 10⁻¹ at the active site (Table 1).

Similarly, Ghisla and Thorpe have reported ¹⁶ that at the acyl‐CoA dehydrogenase active site, the pK _a of the glutamate base is raised from 4 to 9.2 due to low local polarity, while the pK _a of the substrate H—C—C=O group decreases from 18 to ≈ 8 as a result of an extensive hydrogen bonding network: K _eq for this proton transfer increases by an incredible 15 orders of magnitude, from ≈10⁻¹⁴ to ≈10¹. A similar substrate H—C—C=O pK _a decrease occurs for ketosteroid isomerase (from 13 to 5.6, as a result of strong hydrogen bond donation), ¹⁷ increasing K _eq from 10⁻⁹ to 10⁻¹. Toney has reported that such ground state interactions in substrate binding at the enzyme active site can cause carbon acid pK _a decreases of 5–23 units, corresponding to decreases in reaction free energy of 7–31 kcal/mol. ¹⁷ Similarly, Gerlt and Gassman have shown that C=O protonation (or hydrogen bond donation) could account for pK _a decreases of up to 16 units. ³¹

For a more typical example, let us return to the triose‐phosphate isomerase active site, where pK _a values for the glutamate proton acceptor ³² , ³³ , ³⁴ and the substrate carbon acid ¹⁷ have been measured to be 6–6.7 (but see ³⁵ ) and 10–14, respectively. ²⁴ , ³⁶ These ground state effects increase the proton transfer K _eq (Table 1) from 10⁻¹⁴ to ≈ 10⁻⁶ (still bad, but better; Zhai et al ³⁷ suggest a slightly larger improvement). In general, such an eight order‐of‐magnitude increase in K _eq (active site vs. aqueous solution) is not uncommon.

It is instructive to examine in a bit more detail the interactions at the triose‐phosphate isomerase active site that cause these two dramatic pK _a shifts. For the glu₁₆₅ proton acceptor, a nearby ile₁₇₀ crowds out hydration waters (i.e., favors dehydration), and raises the local hydrophobicity (i.e., decreases the local dielectric). ³⁸ Shorter, stronger hydrogen bonds with the substrate's acidic C—H groups are thus favored by the dehydration, the lower dielectric, and the steric constraints. ³⁸ , ³⁹ These hydrogen bonds in turn raise the pK _a of the glutamate side chain, making it more likely to be deprotonated and serve as a proton acceptor. Further details on the active site interactions that raise pK _a(glu₁₆₅) have been reported recently by Zhai et al. ³⁷

The substrate's acidic C—H group is alpha to a carbonyl, so its conjugate base is an enolate. ³⁹ This anionic product is stabilized electrostatically by three nearby positive and δ + groups: the neutral his₉₅NH and asn‐NH₂ side chains, and the cationic lys₁₂—NH₃ ⁺ side chain. ³⁸ , ³⁹ These hydrogen bonds, which serve to lower the pK _a of the acidic CH group, are all strengthened by proximity and by the low dielectric. The sum total of all of these hydrogen bonds, lower dielectric, and dehydration interactions figures into the pK _a raising of glu₁₆₅ and pK _a lowering of substrate CH at the active site of triose‐phosphate isomerase. These concerted interactions illuminate some of the “complexities involved in clearly dissecting out the role and contribution of each change to the kinetics and thermodynamics” of enzyme catalysis. ⁴⁰

It is interesting to note in Table 1 that for all of the carbon acid substrates, pK _a values are dramatically lowered at the active site (compared to the value in solution), making them more acidic and thus more reactive. Similarly, for most of the active site bases, pK _a values are higher at the active site (compared to the value in solution), making them more basic and thus more reactive. However, for lysine (mandelate racemase, enolase) and serine (aconitase/fumarase/aspartase family), the pK _a values are lower at the active site, making them less basic and thus less reactive. This thermodynamically counter‐intuitive alteration assures that a significant proportion of the lys‐NH₃ ⁺ and ser‐OH side chains are in the active, deprotonated state at pH 7, ready to deprotonate the substrate.

Of the 11 enzyme‐catalyzed proton transfer steps listed in Table 1, K _eq in aqueous solution ranges from 10⁻⁹ to 10⁻²⁴ (∆G° = +12 to +33 kcal/mol), a steep uphill climb indeed! But note that ground state pK _a shifts at the active site bring three of these 10 steps very close to equilibrium (∆G° = −2 to +3 kcal/mol; see also Reference 35). For the remaining proton transfer reactions, active site pK _a shifts supply roughly 2/3 of the required free energy, so the remaining ∆G° ranges from +4 to +14 kcal/mol. For these nonspontaneous reactions, E _a must be ≥ 4–14 kcal/mol. (Please see Supporting Information, Section I, for a more detailed discussion of the relationship between E _a and ∆G°.) Toney ¹⁷ found that for both ketosteroid isomerase and aspartate aminotransferase, E _a exceeds ∆G° by 8–9 kcal/mol; taking this as a typical value, we can estimate activation barriers of 10–25 kcal/mol for the eight nonspontaneous enzyme‐catalyzed proton transfers in Table 1. Using the Arrhenius law with a pre‐exponential factor (A ₀) of 10⁹ s⁻¹ in aqueous solution ²⁴ , this gives rate constants of ≈ 50 s⁻¹ down to 1(10⁻⁸) s⁻¹ (see Section III of the Supporting Information for tabulated rate constants). Considering that enzymes typically have k _cat values ² ranging from 10 to 10⁶ s⁻¹, clearly they must have further energy inputs that allow them to surmount the proton transfer activation barrier. These inputs come via high effective molarity and transition state/substrate binding interactions.

3. EFFECTIVE MOLARITY QUANTIFIES INCREASES IN REACTION RATE AND SPONTANEITY

Effective molarity is an important concept that was introduced back in the 1950s to calculate the magnitude of the “chelate” effect, ⁴¹ , ⁴² that is, the increase in speed and spontaneity of bimolecular reactions that are “converted” to unimolecular reactions. In other words, for proton transfer, effective molarity quantifies the increase in rate and spontaneity if the donor and acceptor are part of a single molecule. ⁴³

Effective molarity can be calculated from kinetic or thermodynamic results. Comparing a bimolecular (or intermolecular) ligation reaction forming an A‐B bond from A + B, to its unimolecular (intramolecular) counterpart, the effective molarity of reactant B (EM _B) is the concentration of B that would be required in the bimolecular reaction to give the reaction rate achieved in the unimolecular reaction. From this relationship, we can derive (see Supporting Information, Section IV):

$${\mathit{EM}}_{\mathrm{B}}=\frac{k_{\mathrm{uni}}}{k_{\mathrm{bi}}}=\frac{K_{\mathrm{eq},\mathrm{uni}}}{K_{\mathrm{eq},\mathrm{bi}}}\left\{\text{units}=\frac{{\mathrm{s}}^{-1}}{{\mathrm{M}}^{-1}{\mathrm{s}}^{-1}}=\frac{1}{{\mathrm{M}}^{-1}}=\mathrm{M}\right\}$$ (2)

where k _uni, k _bi, K _eq,uni, and K _eq,bi are the rate constants and equilibrium constants for the uni‐ and bimolecular reactions, respectively.

In general, due to the entropic advantage of having the motion of A and B restricted by their covalent attachment, EM _B > 1 M. If the proximity of B to A is optimized by a rigid framework within the molecule, EM _B can exceed 10¹² M, so this can be a very large effect indeed. ¹ , ³ , ¹²

We can use the Arrhenius Law to show that effective molarity in the unimolecular reaction decreases the activation energy of the bimolecular reaction by RTln(EM):

$$E_{\mathrm{a},\mathrm{uni}}=E_{\mathrm{a},\mathrm{bi}}-\mathrm{R}T\ln \left({\mathit{EM}}_{\mathrm{B}}\right)$$ (3)

Similarly, effective molarity can be shown to decrease the free energy of the bimolecular reaction by RTln(EM) (see Supporitng Information, Section IV, for the derivation). One can envision this lowering of both the free energy (∆G°_uni) and the activation energy (E _a,uni) of a reaction by RTln(EM) thusly: The reaction spontaneity is increased by raising reactant free energy by RTln(EM), while leaving the transition state unaltered.

Although Koshland, ⁴⁴ Jencks, ⁴⁵ and others initially believed that EM in aqueous solution was limited to a maximum of 55 M, the concentration of water, Page and Jencks showed that EM values up to 10¹¹ M were possible. ⁴⁶ More recently, values up to 10¹³ M have been found for some strained substrates. ¹ , ¹²

Jencks ⁴⁵ was the first to suggest that, similar to the chelate effect found with bifunctional molecules, non‐covalent binding of the substrate at the enzyme's active site partially freezes the substrate's labile bond(s) in optimal position to react with critical enzyme side chains. In this way, the effective molarity of the critical enzyme side chain can exceed 1 M by many orders of magnitude. ⁷ , ⁴⁶ As Page and Jencks explained, in the bimolecular reaction, two reactant molecules freely diffusing in solution lose substantial translational and rotational entropy as they collide to form the transition state; this entropy loss yields a high activation energy. Because substrate binding to the enzyme “pre‐freezes” the substrate, this avoids the entropy penalty in approaching the transition state, and lowers the activation energy. ⁴⁶ This is sometimes referred to as pre‐organization ⁴⁷ or substrate orientation. More recently it was pointed out that enthalpy can be even more important than entropy in this pre‐organization process. ²⁶

Benkovic, Bruice, and others ²⁵ , ²⁶ have argued that pre‐organization and desolvation at the enzyme active site are critically important sources of catalytic power. Furthermore, the pre‐organization conferred by effective molarity at the active site could well be accompanied by at least partial desolvation.

Effective molarity in enzyme catalysis can be illustrated in this electrostatic example: If the enzyme's active site includes a deprotonated basic side chain (e.g., his‐imidazole or lys‐NH₂) located an optimal distance from the substrate's labile proton, say within 1–1.5 Å, then the effective molarity of the basic side chain will be much higher than it would have been in solution. In a now‐classic 1989 paper, Toney and Kirsch measured the effective molarity of the catalytic lys₂₅₈ basic side chain in aspartate aminotransferase. ⁴⁸ By site‐directed mutagenesis (lysine ➔ alanine), they removed the amine side chain, and then added exogenous bases (e.g., various alkyl amines) to the solution. By measuring the rate constants for the exogenous buffer‐catalyzed reactions and comparing them to k _cat for the wild type enzyme (Supporting Information, Section IV), Toney and Kirsch calculated an effective amine concentration of 10⁶ M at the active site of wild type aspartate aminotransferase. ⁴⁸

Table 2 lists the effective molarities and their associated activation energy declines of a number of critical basic side chains in various enzyme‐catalyzed proton transfer reactions. Effective molarities here range from 10³ to 10⁸ M, ⁷ corresponding to activation energy decreases of 4–11 kcal/mol. Averaging all enzymes, the activation energy decrease due to side chain EM is 7.2 ± 1.9 kcal/mol. This compares reasonably well with quantum mechanical free energy calculations showing, for trypsin and catechol O‐methyl transferase, ∆G_“cratic” = 9–13 kcal/mol.

TABLE 2.

Effective molarities of critical enzyme side chains and their effect on decreasing activation energy of proton transfer steps

Enzyme	Side chain	EM (M)	E a decrease a (kcal/mol)
Mandelate racemase 4	glu317	≥ 3 (105)	≥ 7.5
Mandelate racemase 49	his297	≥ 2,800	≥ 5
Mandelate racemase 49	lys166	≥ 620	≥ 4
Aspartate transaminase 48	lys258	9.8 (105)	8.2
Cytidine deaminase 4	glu104	8 (106)	9.4
Orotidine‐Pi decarboxylase 4	lys93	6 (107)	11
Orotidine‐Pi decarboxylase 50	Pi group	1 (109)	12
Chymotrypsin 51	his57	2 (106)	8.6
Chymotrypsin 51	ser195	105 to 106	7 to 8
Trypsin 47	ser195	2.3 (108)	11.4
Papain 52	cys25	7 (107)	11
Luciferase 53	his44	3.5 (105)	7.6
Ribulose bis‐Pi carboxylase 54	lys191	≈ 105	≈ 7
Carboxypeptidase 55	arg127	≤ 1,000	≤4
Glycerol‐3P dehydrogenase 56	lys120	54 b	2.3
Several enzymes 12	H+ transfer	≤ 106	≤8
antibody 12	—COO:−	4 (104)	6
Enzyme‐bound 5	Ligands	≈ 105	≈ 7

^a E _a decrease = RTln(EM); see Equation (3).

^bThe unusually low EM for lys₁₂₀ in GPDH is due to a strong network of interactions at the mutated active site that optimally orients the exogenous catalytic acid (EtNH₃ ⁺). ⁵⁶ Two other differences that distinguish the GPDH measurement from others in the table are that the proton transfer occurs after the catalytic step ⁵⁷ (hydride anion transfer) and is not rate‐limiting, and it involves acid catalysis rather than base catalysis.

4. TRANSITION STATE EFFECTS: SUBSTRATE BINDING

The ability of enzymes to bind and stabilize the transition state has been analyzed from several perspectives (see for example, References 7, 25, 58), but Menger's split‐site model is especially useful in understanding how enzyme‐substrate binding interactions can lower the activation barrier. ¹ , ³ The substrate is split into two parts: a reactive “R” portion containing the labile bond(s), and a “B” portion that binds spontaneously to the enzyme. Similarly, the enzyme's active site can be split into “R” and “B” parts that bind the respective parts of the substrate. The E(B)·S(B) binding interaction is spontaneous (∆G _ES(B) is negative), involving H‐bonds, salt bridges, hydrophobicity, etc. On the other hand, the E(R)·S(R) interaction is nonspontaneous (∆G _ES(R) is positive), because it requires desolvating parts of both the enzyme and substrate, ²⁵ , ²⁶ deforming the substrate toward its transition state structure, and bringing reactive enzyme side chains into correct orientation and proximity to react with the substrate. (Note that this latter effect suggests that the effects of E(R)·S(R) interactions analyzed below may overlap somewhat with effective molarity effects discussed above.) Finally, because the S(B) portion of the substrate does not change in the transition state, spontaneous binding of ground state S(B) shows up as equally spontaneous binding of transition state S(B)^‡ (i.e., ∆G _ES(B) = ∆G _ES(B)‡). This situation is depicted in the free energy diagram (Figure 1).

FIGURE 1.

Menger's “Split‐Site Model” of enzyme catalysis. The uncatalyzed reaction (S ➔ S^‡) is the purple curve. E' (black) is a hypothetical enzyme with K _m = 1 M (i.e., ∆G _b(E'·S) = 0, so ∆G _E'S(R) = −∆G _E'S(B)). E (red) is a typical enzyme with K _m ≈ 1 μM (i.e., ∆G _b(E·S) = −8 kcal/mol); ∆G _d(E·S) = ∆G _ES(B) + ∆G _ES(R) = x _B + x _R

First, let us consider a hypothetical enzyme E' that has a Michaelis constant K _m' = 1 M (black in Figure 1). Recall that for a Michaelis–Menten enzyme, K _m ≈ K _d(E·S), so for E', ∆G _d(E'·S) ≈ ‐RTln(K _m') = 0. Because ∆G for enzyme‐substrate binding is the sum of ∆G _E'S(R) and ∆G _E'S(B),From Equation (4) we know that ∆G _E'S(R) = −∆G _E'S(B). In other words, ∆G _b(E' + S) = 0 because spontaneous S(B) binding exactly balances nonspontaneous S(R) binding.

(4).

Chemical Formula: $\text{∆}{G}_{\mathrm{b}}\left({\mathrm{E}}^{\prime }+\mathrm{S}\right)=‐\text{∆}{G}_{\mathrm{d}}\left({\mathrm{E}}^{\prime }\mathrm{S}\right)=\mathrm{O}=\text{∆}{G}_{{\mathrm{E}}^{\prime }\mathrm{S}\left(\mathrm{R}\right)}+\text{∆}{G}_{{\mathrm{E}}^{\prime }\mathrm{S}\left(\mathrm{B}\right)}$

At the same time, G _E'·S‡ is lower than G _S‡ by ∆G _E'S(B), so the activation energy for the enzyme‐catalyzed reaction is lower than that for the uncatalyzed reaction by ∆G _E'S(B), that is, ∆G ^‡ _cat,E' = ∆G ^‡ _uncat–|∆G _E'S(B)|. Menger's split‐site model shows that for the hypothetical E' enzyme with K _m' = 1 M, the E_B·S_B binding energy is converted directly into transition state stabilization and catalytic power: The more spontaneous the binding of S_B by the enzyme, the larger is ∆G _E'S(R), and the lower is the activation energy.

For real enzymes, K _m ranges from 100 mM down to 100 nM. ⁵⁹ Let us consider a typical enzyme E with K _m = 1 μM (red in Figure 1; ∆G _b(E + S) ≈ RTln(K _m) = −8 kcal/mol). According to the split‐site model, we can consider ∆G _b(E + S) to be the sum of two “extra” free energy contributions over and above those found in the hypothetical enzyme E': x _B from the spontaneous binding of S_B, and x _R from the nonspontaneous binding of S_R. Combining these relationships, we get

$$∆G_{\mathrm{b}}\left(\mathrm{E}+\mathrm{S}\right)=∆G_{\mathrm{b}}\left({\mathrm{E}}^{\prime }+\mathrm{S}\right)+x_{\mathrm{B}}+x_{\mathrm{R}}=\mathrm{O}+x_{\mathrm{B}}+x_{\mathrm{R}}\approx =-8\text{kcal}/\mathrm{mol},\text{with}{x}_{\mathrm{R}}\le x_{\mathrm{B}},$$ (5)

From the relationships evident in Figure 1 we can derive (see Supporting Information, Section V) the relationship between ∆G ^‡ _cat,E (red, E·S ➔ E·S^‡) and ∆G ^‡ _uncat (purple, S ➔ S^‡):

$${∆G}_{\mathrm{cat},\mathrm{E}}^{‡}={∆G}_{\text{uncat}}^{‡}-\left|∆G_{{\mathrm{E}}^{\prime }\mathrm{S}\left(\mathrm{B}\right)}\right|-x_{\mathrm{R}}$$ (6)

So, ∆G ^‡ _cat,E is lower than ∆G ^‡ _uncat by (|∆G _E'S(B)| + x _R).

Now that we have introduced the split‐site model, we can use it to estimate how far binding energy can go toward surmounting the 10–25 kcal/mol proton transfer energy barriers implicated in Table 1. Let us imagine that enzyme E' uses just one H‐bond to bind S_B, so the activation energy is lowered by |∆G _E'S(B)| = 5 kcal/mol (a typical value for a common H‐bond; see Table 3). If converting E' to a “normal” enzyme (E) takes place by adding two more H‐bonds to the S_B interaction, then x _B = −10 kcal/mol, and using Equation (5), we can calculate x _R:

$$x_{\mathrm{R}}=∆G_{\mathrm{b}}\left(\mathrm{E}+\mathrm{S}\right)-x_{\mathrm{B}}=-8-\left(-10\right)=+2\text{kcal}/\mathrm{mol}$$ (7)

Using Equation (6) we can calculate that the enzyme E can decrease the activation energy of the uncatalyzed reaction by 7 kcal/mol (= 5 + 2).

TABLE 3.

Dependence of H‐bond strength on distance between the heteroatom donor (D—H) and acceptor (A), where D and A are either oxygen or nitrogen. Adapted from References 24, 60, 61, 62

	Strength (kcal/mol)	Distance (Å)	Barrier height a
None	0	≥ 3.7	–
Weak	2–5	3.6–2.8	High
Normal	5–10	2.8–2.6	Moderate
Strong	10–20	2.6–2.4	Low
Very strong	> 20	< 2.4	Zero

^aThis is the energy barrier that the proton must surmount as it jumps from D to A.

For a more favorable binding interaction, we might envision a normal enzyme (E") that binds S_B with three extra H‐bonds compared to E'. In this case, using Equation (7) we can calculate that x _R = −8–(−15) = +7 kcal/mol, and this enzyme can lower E _a by 12 kcal/mol (= 5 + 7). Doing the same set of calculations for triose‐phosphate isomerase substrate binding (K _m = 1.3 mM) gives a range of E _a lowering of 11–16 kcal/mol; indeed, recent empirical valence bond calculations suggest that triose‐phosphate isomerase substrate binding lowers E _a by 11 kcal/mol compared to the uncatalyzed reaction. ⁶³ , ⁶⁴ Given K _m ranging from 100 nM to 100 mM for all enzymes, we can estimate that S_B binding energy could supply 6–18 kcal/mol toward lowering the activation energy.

5. TRANSITION STATE EFFECTS: LOW‐BARRIER HYDROGEN BONDS

Hydrogen bond strength varies with the identity of the heteroatom bonded to the hydrogen (i.e., the H‐bond donor, D), and with orientation and distance. Table 3 shows how H‐bond strength varies with distance for the most common interactions, those involving the N and O heteroatoms. Strong H‐bonds, also known as low‐barrier H‐bonds (LBHBs), though relatively rare, are believed to be important in pushing forward some enzyme‐catalyzed proton transfers that are slow and nonspontaneous under standard conditions ⁶⁵ (but see ²⁵ , ⁶⁶ , ⁶⁷ and references therein). The data in Table 3 show how H‐bond strength decreases with donor (D)‐acceptor (A) distance (as 1/r ³, similar to other dipole–dipole interactions; see Figure S5).

We also note from Table 3 that strong H‐bonds have a low barrier height (hence the name LBHB). Finally, LBHB formation requires, in addition to shorter D···A distances, close pK _a matching: pK _a of DH and AH⁺ must be within 1–2 units of each other. ²⁴ , ⁶⁰ , ⁶¹

As Cleland ²⁴ proposed in 1994, LBHB formation in the transition state can provide much of the 10–25 kcal/mol necessary to drive the slow, nonspontaneous proton transfer reactions listed in Table 1 (but see ²⁵ , ⁶⁶ , ⁶⁷ and references therein for criticism of this hypothesis). Specifically, if a conformational change during the E·S to E·S^‡ conversion brings A closer to D by 0.1–0.5 Å, then a weak H‐bond in the E·S ground state can be converted to a strong LBHB in the E·S^‡ transition state. Cleland ²⁴ and others ⁶⁵ , ⁶⁸ in fact identified possible transition state LBHBs in many enzymes (Table 4). Cleland even proposed that all enzyme‐catalyzed nonspontaneous proton transfers may include LBHB formation in the transition state, ⁶² but so far, experimental evidence is only strong for hydrolases. ²⁴ , ⁶⁰ , ⁶¹

TABLE 4.

Enzymes that may form LBHBs in the transition state. Unless otherwise specified, the H‐bond acceptor is a substrate enolate C—O:⁻ group

Enzyme	LBHB D/A pair	D···A distance (Å)	H‐bond strength (kcal/mol)
Ketosteroid isomerase 24 , 61	tyr14‐OH···−:O‐		> 12
Triose‐Pi isomerase 24 , 61	his95=NH···−:O‐ a
Citrate synthase 24 , 61	his274=NH···−:O‐ a	2.4–2.5	15–20
Orotidylate decarboxylase 24	lys93‐NH3 +···−:O‐
Mandelate racemase 24 , 61	glu317‐COOH···−2:OOC‐ b	2.7	> 7
Aconitase 24	Fe2+·OH2···−2:OOC‐ b	2.5–2.7	12–20
Carboxypeptidase 24 (thermolysin)	glu270(143)‐COO:−···H2O·Zn2+	2.2–2.5	15–25
Chymotrypsin (and other serine proteases) 60 , 61	asp102‐COO:−···HN = his57	2.5–2.7	12–20
Lactate dehydrogenase 61	his195 ≡N:···HO‐lactate
Alcohol dehydrogenase 62	ser48‐O:−···HO‐alcohol	2.5	15

^aEvidence against the existence of these two LBHBs has been presented. ⁶⁹ , ⁷⁰ , ⁷¹ , ⁷²

^bSubstrate dianionic aci‐carboxylate.

In Table 4 we see that the LBHBs that form in the transition state have strengths of 7–25 kcal/mol. Assuming that these started out in the ground state as weak H‐bonds ⁶¹ , ⁶² (2–5 kcal/mol), the strengthening of these H‐bonds in the transition state could supply 5–20 kcal/mol toward lowering the activation energy.

6. CONCLUSIONS

From Table 1 we see that ground state changes in pK _a due to the local environment at an enzyme's active site can sometimes be enough to raise the K _eq for proton transfer by 9–22 orders of magnitude, turning exceedingly nonspontaneous proton transfer steps into ones with ∆G° ≈ 0. More commonly though, ground state pK _a shift effects still leave ∆G° values of +4 to +14 kcal/mol, and for these enzyme‐catalyzed nonspontaneous proton transfer reactions, E _a ranges from 10–25 kcal/mol. The three interactions that enzymes can harness to dramatically lower the remaining activation barriers are: effective molarity, transition state/substrate binding, and transition state formation of LBHBs. We have shown that we can expect activation energy decreases due to these three interactions of: 4–11 kcal/mol from effective molarity; 6–18 kcal/mol from transition state/substrate binding; and 5–20 kcal/mol from transition state LBHB formation. At the lower end of the 10–25 kcal/mol range, the activation barrier could be surmounted by any one of these three interactions; at the upper end, a combination of two of the three interactions could easily supply the required energy.

Taking trypsin as an example, Kollman et al. ⁴⁷ used quantum mechanical free energy calculations to conclude that the enzyme lowers ∆G‡ (compared to the reaction in solution) by 17 kcal/mol: 11.4 kcal/mol from effective molarity (which they referred to as cratic free energy, derived from “aligning the reactive groups into a geometry enabling a facile reaction”), and 6 kcal/mol from substrate/transition state binding interactions at the active site. Kollman et al found strikingly similar effects for catechol O‐methyl transferase: 9–13 kcal/mol of effective molarity stabilization and 5 kcal/mol of substrate/transition state binding stabilization.

In conclusion, although the aqueous proton transfer reactions in Table 1 may seem to be too slow and nonspontaneous to be of use in enzyme catalysis, the enzyme has four very useful “engines” that it can use to drive the reactions forward.

AUTHOR CONTRIBUTIONS

Todd Silverstein: Conceptualization; data curation; formal analysis; investigation; methodology; resources; software; supervision; validation; visualization; writing‐original draft; writing‐review & editing.

Supporting information

Appendix S1: Supplementary Information

Silverstein TP. How enzymes harness highly unfavorable proton transfer reactions. Protein Science. 2021;30:735–744. 10.1002/pro.4037