Detection of an enzyme iso-mechanism by means of the kinetics of covalent inhibition

Abstract

Turnover of substrates by many enzymes involves free enzyme forms that differ from the stable form of the enzyme in the absence of substrate. These enzyme species, known as isoforms, have, in general, different physical and chemical properties than the native enzymes. They usually occur only in small concentrations under steady state turnover conditions and thus are difficult to detect. We show in this paper that in one particular case of an enzyme (a class C β-lactamase) with specific substrates (cephalosporins) the presence of an enzyme isoform (E′) can be detected by means of its different reactivity than the native enzyme (E) with a class of covalent inhibitors (phosphonate monoesters). Generation of E′ from E arises either directly from substrate turnover or by way of a branched path from an acyl-enzyme intermediate. The relatively slow spontaneous restoration of E from E′ is accelerated by certain small molecules in solution, for example cyclic amines such as imidazole and salts such as sodium chloride. Solvent deuterium kinetic isotope effects and the effect of methanol on cephalosporin turnover showed that for both E and E′, k_cat is limited by deacylation of an acyl-enzyme intermediate rather than by enzyme isomerization.

Graphical Abstract

1. Introduction

Irreversible (covalent) enzyme inhibitors have long played an important role in enzymology, biochemistry and medicine. Originally it was hoped that targeted examples of these would generally become effective drugs but the side reactions (lack of reagent specificity) and subsequent side effects of many early examples dampened this enthusiasm. More recently, however, the discovery and design of more specific compounds have led to a renaissance of the concept [1–3]. The employment of modern methodologies has been a key to these advances [4–7]. Quantitative measurement of the effectiveness of such compounds as enzyme inhibitors by means of inhibition kinetics studies has been an essential part of these advances (for a recent example, see reference 8). Experimental methods to determine the kinetic constants of covalent enzyme inhibition are of either the direct or indirect type. In the former method (Scheme 1), the concentration of E, often measured by means of enzyme activity, is monitored as a function of time [9]. Alternatively, the concentration of I can be monitored, or, if another product, P_I, is also generated from the reaction, the appearance of this species can be followed. On the other hand, the indirect method employs a reporter substrate S (Scheme 2). If E is added to a mixture of I and S, substrate turnover, monitored either by way of the concentration of S or P, decreases to zero with time as the enzyme becomes progressively inactivated. The kinetic constants for reaction of E and I can be deconvoluted from total progress curves of substrate turnover [10]. The indirect method has been used, for example, to study the inhibition of ß-lactamases [11,12]; one such enzyme is the subject of the present work.

Scheme 1.

Scheme 2.

One would anticipate that the k_i values derived from the two methods would be the same, otherwise the two methods would not be used essentially interchangeably as they seem to have been. We describe in this paper, however, a specific case where the two methods do not yield the same result. This was noticed during our studies of the kinetics of inhibition of the class C ß-lactamase of Enterobacter cloacae P99 by a particular acyl phosphonate [13]. The discrepancy was then found to also occur with several other phosphonates, as described below. Specific phosphonates have been shown to be active site-directed inactivators of the P99 enzyme, generating inert complexes that have the structure of a transition state analogues [14,15].

In order to rationalize the anomalous kinetics observations described above, it was necessary to postulate the presence of a second free enzyme form of the enzyme when in the presence of certain substrates, which reacts with phosphonates at a different rate than the free enzyme in the absence of substrate. This suggests the presence of an iso-mechanism [16] (reviewed in reference 17) where reaction of the native enzyme with substrate leads to generation of a second free enzyme form that can then isomerize back to the original enzyme after substrate turnover. Unequivocal experimental demonstrations of iso-mechanisms were originally quite rare although there are now several well-defined examples [17,18]. In an investigation most closely related to the present work, Cho et al. interpreted their solvent kinetic isotope data for binding of a non-covalent inhibitor to pepsin in the absence and presence of a substrate in terms of the existence of two free enzyme forms [19]. Thus, we attempted to interpret our data in terms of three simple but different iso-mechanisms. First, Scheme 3 shows the simplest and classical iso-mechanism that many/most enzymes actually follow because after release of product the enzyme must isomerize from the form (E′) that binds product to the one (E) that binds substrate and also has the active site functional groups optimally placed for catalysis. The molecular rearrangement that occurs on conversion of E′ to E has been referred to by Rose as the “dark side” of enzyme catalysis [20].

Scheme 3.

An alternative to the classic model is shown below as Scheme 4. In this example, the alternative enzyme form E′ is again generated on “normal” turnover, but in this case E′ is also capable of substrate turnover and inhibition by phosphonate, i.e. must have a functional active site. In the general case here, the steady state turnover parameters for E′ will be different from those of E. Finally, Scheme 5 shows a variant of Scheme 4 where formation of E′ occurs by partitioning of an ES intermediate between normal turnover, regenerating E and an alternative path which leads to E′. Scheme 5 is particularly relevant to serine ß-lactamases because a well-defined acyl-enzyme intermediate often accumulates during turnover (see discussion). The three schemes differ principally in that in Scheme 3, the E′ → E transition occurs during every turnover of substrate, whereas in Schemes 4 and 5, this is not required and almost complete turnover of substrate by E′ is possible; this difference has kinetic consequences enabling experimental distinction between the schemes. In this paper, we describe our experiments on the reaction of the P99 ß-lactamase with phosphonate inhibitors in the presence and absence of typical substrates and their interpretation in terms of Schemes 3 – 5.

Scheme 4.

Scheme 5.

2. Materials and Methods

2.1. Materials

The ß-lactamase of Enterobacter cloacae P99 was obtained from the Centre for Applied Microbiology and Research (Porton Down, Wiltshire, U.K.) and used as supplied. The catalytic properties of this enzyme preparation have been previously documented and the concentrations of stock solutions were obtained spectrophotometrically [21]. Cephalothin, cephalosporin C, 7-aminocephalosporanic acid (ACA), and deacetoxycephalothin were gifts from Eli Lilly and Co. Cefoxitin was a gift from Merck & Co. (see Chart 1 for the substrate structures). Deuterium oxide (99.9 atom % ²H) was purchased from Sigma-Aldrich. The phosphonate inhibitors, p-nitrophenyl N-(methoxycarbonyl)aminomethylphosphonate (1), m-nitrophenyl N-(phenylacetyl)aminomethylphosphonate (2), p-nitrophenyl N-(benzyloxycarbonyl)aminomethylphosphonate (3), and benzoyl N-(benzyloxycarbonyl)aminomethylphosphonate (4) (See Chart 2) were available from previous studies in this laboratory [14,22]. Imidazole and triethanolamine were purchased from Fluka Chemical Co. Thiazolidine, oxazole, 4-picoline and Tris were from Sigma-Aldrich and 2,2,2-trifluoroethylamine from Acros Organics.

Chart 1.

Substrates employed

Chart 2.

Phosphonate inhibitors employed

2.2. Methods

All kinetics measurements, except where otherwise specified, were carried out in 20 mM MOPS (Sigma-Aldrich) buffer at pH 7.5 and 25 °C. Absorption spectra and spectrophotometric reaction rates were obtained from a HP8452A spectrophotometer.

2.3. Irreversible Inhibition Kinetics

Second order rate constants of irreversible inhibition of the P99 ß-lactamase by the phosphonates were determined by two distinct methods, direct and indirect, with the former in two variants. Experiments were carried out at least twice and the means reported in Table 1.

2.3.1. Direct method

Activity measurements

Solutions of the enzyme (ca. 0.1 μM) and phosphonates (1.0 μM) were incubated together. Small aliquots were withdrawn at suitable time intervals and the enzyme assayed spectrophotometrically against 0.2 mM cephalothin at 278 nm. Initial rates of cephalothin hydrolysis decreased smoothly to zero with time. These data were fitted to Scheme 1 by the Dynafit program [23] to obtain second order rate constants (termed k_i^direct). Earlier studies had shown that the phosphonates 1 - 4 inhibited the P99 ß-lactamase in what appeared to be a purely second order fashion at the phosphonate concentrations employed in this study [14, 22].

Spectrophotometric measurements

With the p-nitrophenyl phosphonates 1 and 3, inactivation kinetics could also be studied directly by monitoring p-nitrophenoxide release spectrophotometrically at 400 nm [22]. In the case of the more effective inhibitor 3, reactions were monitored by means of a Durrum D-110 stopped-flow spectrophotometer. Enzyme concentrations were 5.0 μM for these spectrophotometric experiments. Second order rate constants (k_i^direct) were obtained from the absorbance vs. time data as described above.

2.3.2. Indirect method

Second order rate constants of inhibition of the ß-lactamase by the phosphonates could also be determined from spectrophotometric total progress curves of chromophoric substrate turnover in the presence of the inhibitors. The substrates, substrate concentrations, and analytical wavelengths employed were as follows: cephalothin, 200 μM, 278 nm; cephalosporin C, 400 μM, 278 nm; 7-aminocephalosporanic acid (ACA), 2.0 mM, 285 nm; and cefoxitin, 1.0 mM, 292 nm. In each case, the substrate concentration was at least five times its K_m^obs value. For compound 3 with substrates the stopped flow method was used. The spectrophotometric progress curves in the presence and absence of inhibitor were fitted to Scheme 6 by means of the Dynafit program [23]. With the cephalosporin substrates, non-covalent product inhibition was observed in the latter stages of the reaction. Effective K_p values were obtained from fitting of the progress curve for reaction of the same concentration of substrate in the absence of phosphonate; for these fits, K_m values were fixed at those obtained from initial rate analysis – see below). In the case of 4, slow spontaneous hydrolysis of the inhibitor was also taken into account; the relevant rate constant, k₀, was obtained from independent experiments. Thus, on fitting total progress curves in the presence of inhibitor, the only unknown rate parameter to be determined was the second order inactivation constant k_i^indirect.

Scheme 6.

Direct and indirect inactivation rate constants were also obtained by these methods in the presence of a variety of salts and nitrogen bases (see below and Results).

2.4. Steady state hydrolysis kinetics

Steady state parameters, k_cat^obs and K_m^obs, for hydrolysis of the cephalosporin substrates by the P99 ß-lactamase under the conditions specified above were determined spectrophotometrically from initial rate measurements: cephalothin, 260 nm (Δε = 6800 M⁻¹ cm⁻¹); cephalosporin C, 278 nm (Δε = 1200 M⁻¹ cm⁻¹); ACA, 284 nm (Δε = 1890 M⁻¹ cm⁻¹); deacetoxycephalothin, 278 nm (Δε = 2060 M⁻¹ cm⁻¹). Enzyme concentrations were generally 1 - 4 nM except in the case of ACA where 2.0 μM was used. Values of k_cat^obs and K_m^obs were obtained from fitting the initial rate measurements to the Michaelis-Menten equation by Kaleidagraph. Steady state parameters for cephalosporin C hydrolysis were also determined in solutions containing 100 mM imidazole (see below).

2.5. The effects of salts and nitrogen bases on k_cat^obs values for various substrates

The effects of a variety of nitrogen bases (imidazole, oxazole, thiazolidine, 2,2,2-trifluoroethylamine, Tris, triethanolamine and 4-picoline) and salts (sodium chloride and sodium sulphate) on the initial rates of P99 ß-lactamase (10.0 nM) hydrolysis of various substrates at essentially saturating concentrations were determined spectrophotometrically. The substrates, concentrations and wavelengths employed were: cephalothin, 1 mM, 288 nm; cephalosporin C, 4 mM, 320 nm; ACA, 2 mM, 285 nm; deacetoxycephalothin, 0.6 mM, 284 nm. Solutions were prepared at the proper pH by addition of 1M NaOH solution to the acid form of MOPS mixed with the above species, i.e. HCl was not used.

2.6. Solvent deuterium kinetic isotope effect on k_cat^obs

The kinetic isotope effect for the P99 ß-lactamase-catalyzed hydrolysis of cephalothin was determined from spectrophotometrically obtained initial rates of reaction in ¹H₂O and ²H₂O buffers, essentially as previously described [24]. The cephalothin concentration employed was 1 mM and that of enzyme 10 nM. A solvent kinetic isotope effect on this reaction in the presence of imidazole (100 mM) was also determined.

2.7. Effect of methanol on k_cat^obs

The effect of methanol on the initial rates of solvolysis of cephalothin under essentially saturating conditions (1.0 mM) in the presence of the P99 β-lactamase (10 nM) was determined spectrophotometrically at 288 nm in aqueous methanol/MOPS buffer as described previously [21,25]. Methanol concentrations between 0 and 2.5 M were employed. The partition ratio of the acyl-enzyme intermediate between methanolysis and hydrolysis was then determined from least-squares fits of the data as previously described [21]. This ratio was also determined in solutions containing imidazole (100 mM).

3. Results

3.1. Phosphonate inhibition kinetics

The P99 ß-lactamase is rapidly and essentially irreversibly inhibited by specific aryl [14,22] and acyl [13] phosphonates such as 1 - 4 in what appeared to be a direct second order reaction (Scheme 1) whereby the enzyme active site is covalently modified by phosphonylation. Titration data [14] and a crystal structure [15] of the inactivated enzyme show that stoichiometric reaction of one inhibitor molecule with an enzyme molecule leads to the appearance of a single inhibitor molecule covalently bound to the enzyme active site. Although an initial non-covalent inhibitor-binding step may well also occur in these reactions, it was never detected by systematic deviations from the second order fits to the data at the concentrations employed in the experiments described in the above-cited papers or at those employed in the experiments in this paper. Values of the second order rate constant of inhibition, k_i, could be obtained from several different experiments, as described in the Introduction.

The rate constant can be obtained directly from measurements of enzyme activity as a function of incubation time of the inhibitor with the enzyme, for example on reaction of the enzyme with 2 (Figure 1A). Alternatively, in certain cases, the reaction can be more conveniently monitored spectroscopically. This can be readily done, for example, with the phosphonates 1 and 3 where a strongly absorbing leaving group, p-nitrophenoxide, is released on reaction (Scheme 7).

Fig. 1.

(A) Activity of the P99 ß-lactamase (0.10 μM) in the presence of the phosphonate 2 (1.0 μM) as a function of time of incubation. The points (●) represent experimentally determined spectrophotometric initial rates of hydrolysis of cephalothin (0.2 mM). The line represents the best fit of the data to Scheme 1.

(B) A spectrophotometric reaction progress curve depicting release of p-nitrophenoxide from a mixture of the P99 β-lactamase (5.0 μM) and the phosphonate 1 (5.0 μM). The points (◯) are experimental and the fitted line was calculated from Scheme 1.

(C) Spectrophotometric progress curves for hydrolysis of cephalosporin C (0.40 mM), catalyzed by the P99 β-lactamase (4.4 nM) in the absence (◯) and presence (●) of the phosphonate 1 (20 μM). The points are experimental and the lines were calculated from the simultaneous fit of the data to Scheme 2. The presence of product inhibition was included (see text).

Scheme 7.

The results of such an experiment where the ß-lactamase reacts with 1, are shown in Figure 1B. Directly obtained rate constants, k_i^direct, for the inhibitors 1 – 4, using the appropriate method from those described above, are given in Table 1. Previous experience with these inhibitors has shown that the activity and spectrophotometric methods yield the same values for the inactivation rates constants [26,27].

Table 1.

Directly- and Indirectly-Determined Second Order Rate Constants for Inhibition of the the P99 ß-Lactamase by Phosphonates

Inhibitor	kidirect x 10−3 (M−1 s−1)	kiindirect x 10−3 (M−1 s−1)	Substrate	kcatobs (s−1)
1	(3.1 ± 0.2)a	(1.25 ± 0.01)	cephalosporin C	680 ± 10
1	(3.1 ± 0.2)a	(1.12 ± 0.10)	cephalothin	690 ± 30
1	(3.1 ± 0.2)a	(3.0 ± 0.2)	ACA	0.63 ± 0.04
1	(3.1 ± 0.2)a	(3.0 ± 0.3)	cefoxitin	0.31c
2	(5.1 ± 0.3)b	(1.9 ± 0.2)	cephalosporin C	680 ± 10
2	(5.1 ± 0.3)b	(2.2 ± 0.2)	cephalothin	690 ± 30
3	(36 ± 3)a,d	(18.5 ± 0.3)a,e	cephalosporin C	680 ± 10
4	(59 ± 3)b	(12 ± 4)	cephalothin	690 ± 30

^adetermined from nitrophenol release

^bdetermined from activity loss

^cMazzella and Pratt [28]

^ddetermined from stopped flow experiments with 5 μM P99 and 200 μM phosphonate

^edetermined from stopped flow experiments with 5 μM P99, 5000 μM cephalosporin C, and 200 μM phosphonate

Alternatively, in principle at least, the same inactivation rate constant should be available from a competition experiment where the inhibitor competes with a substrate for the enzyme (Scheme 2) and where the substrate concentration as a function of time is monitored. One would expect from such an experiment that the total progress curve for loss of substrate in the presence of inhibitor would display the same initial rate as that in its absence but, with time, the slope of the former curve would decrease with respect to the latter, as the inhibitor progressively inactivated the enzyme. Eventually, if sufficient inhibitor were present, the rate of substrate disappearance in the former case would fall to zero prior to complete loss of substrate because the enzyme would become completely inactivated. Data from such an experiment is shown in Figure 1C. This shows total progress curves for hydrolysis of cephalosporin C by the ß-lactamase in the presence and absence of the phosphonate 1. Clearly the expected inactivation phenomenon has occurred. These data could be fitted to Scheme 6, as described in the Methods to yield values of the inactivation rate constant obtained, as we have termed it, indirectly, k_i^indirect. Values of k_i^indirect obtained in this fashion for various combinations of substrate and inhibitor are also given in Table 1.

The important result evident from the data of Table 1 is that for the best substrates with high k_cat^obs values, cephalosporin C and cephalothin, k_i^indirect, is distinctly smaller than k_i^direct. This is true for all of the phosphonates examined, 1 - 4. In contrast, for the low k_cat^obs substrates, ACA and cefoxitin, k_i^direct and k_i^indirect values are identical at the higher k_i^direct value, within experimental uncertainty. The difference between k_i^direct and k_i^indirect for the good substrates cannot be explained in terms of the simple Schemes 1 and 2.

If it is assumed, as all previous experiments indicate [14,15], that these phosphonate inhibitors react only with the active site of free enzyme species, then the data of Table 1 can be most simply explained by a scheme that involves two free enzyme species which react with phosphonate at different rates. The simplest such reaction sequence would be that of Scheme 3, a classical iso-mechanism [17]. In this scheme, turnover of S by E leads to release of product P and a second free enzyme form E′. Both E and E′ react with I leading to inhibition but E′ is not able to turn over S. E′ spontaneously returns to the active and thermodynamically more stable form of free enzyme E to complete the substrate turnover process.

In a slightly more complicated alternative scheme (Scheme 4), the enzyme form E′ is also able to turn over S, leading to the presence of a separate catalytic cycle. If the reversion of E′ to E is slow compared to the catalytic rate steps in the latter cycle, most substrate turnover will occur by way of this path. Finally, a branched mechanism with two paths of substrate turnover, is described in the introduction as Scheme 5. In this case, an ES intermediate partitions, prior to completing turnover, to an isomeric form E′S that releases the isomeric free enzyme form E′ and product P. Both E and E′ are active enzyme forms that can react with both S and I, leading to substrate turnover and inhibition, respectively. E′ reverts to E eventually, after all substrate is consumed, but many substrate turnovers can, in principle, occur via E′ before that happens. In each of the Schemes 3–5, E and E′, in principle, react with I at different rates, the former with a rate constant that should be the experimentally measured k_i^direct and the latter with a rate constant that should correspond to k_i^indirect. The results of Table 1, interpreted in this way, show that in the present case, for good substrates, k_i^indirect < k_i^direct. The data from experiments such as illustrated in Figure 1C therefore may fit to one or more of Schemes 3–5. If more than one scheme were to fit, other types of experiment would be needed to distinguish between them. The quantitative fitting of the data to each of these schemes is described below in Section 3.3.

3.2. Kinetics of conversion of E′ to E

In their classic study of fumarase, Rose et al. found several examples where enzyme species in the “dark side” of the substrate turnover sequence were connected by steps that were catalyzed by external general acids or bases [20,29]. Informed by these results, we tested whether such species, and others, might affect the conversion of E′ to E that the phosphonate inhibition kinetics described in the previous section had revealed. The effects of a base, imidazole, and two salts, sodium chloride and sodium sulfate on the k_i^direct. and k_i^indirect values for phosphonate 2 are seen in Table 2. These data show that in the presence of 100 mM imidazole the value of k_i^indirect, determined during cephalosporin C turnover, has increased to closely approximate that of k_i^direct. The same effect has been produced by the salts sodium chloride and sodium sulfate, both also at 100 mM concentration. The decrease in the absolute values of the rate constants in the presence of the salts is probably due to a general salt effect on the enzyme. The other important observation from Table 2 is that in order to fit the total progress curves to determine k_i^indirect from Scheme 2 in the presence of the additives, it was necessary to increase the k_cat^obs value of the substrate. The results of experiments to directly show this effect on k_cat^obs are shown in Figures 2A–2C. Clearly, imidazole accelerates the enzyme-catalyzed reaction under k_cat conditions and the effect saturates at high imidazole concentrations. Most likely this derives from a change in rate-determining step promoted by the presence of imidazole. It should be noted that, in the absence of enzyme and substrate, imidazole had a negligible effect on the rate of disappearance of substrate from solution in the time scale of the experiments described above. Similar hyperbolic curves were obtained from the salts, e.g. sodium chloride, Figure 3. In terms of Schemes 3–5, the effect of imidazole and salt catalysis on the E′ → E transition has changed the dominant enzyme form in the steady state for substrate turnover from E′ to E with a resultant change in k_cat^obs.

Table 2.

Effect of imidazole and salts on k_i^direct and k_i^indirect for the inhibitor 2

Effector (100 mM)	kidirect x 10−3 (M−1s−1)a	kiindirect x 10−3 (M−1s−1)b	kcatobs (s−1)c
-	5.1 ± 0.3	1.9 ± 0.2	680 ± 10d
Imidazole	5.2 ± 0.8	4.8 ± 0.6	1980 ± 80e
NaCl	2.9 ± 0.1	3.0 ± 0.2	1290 ± 150
Na2SO4	2.5 ± 0.3	2.7 ± 0.4	970 ± 180

^adetermined by activity loss

^bcephalosporin C was the substrate

^cvalue required to fit the data to Scheme 2

^dK_m^obs = (59 ± 5) μM, k_cat^obs/K_m^obs = (1.16 ± 0.10) x 10⁷ M⁻¹s⁻¹

^eK_m^obs = (155 ± 17) μM, k_cat^obs/K_m^obs = (1.28 ± 0.15) x 10⁷ M⁻¹s⁻¹

Fig. 2.

Activity of the P99 β-lactamase (10.0 nM) as a function of added imidazole concentration. The substrates were (A) cephalosporin C (4.0 mM), (B) cephalothin (1.0 mM), and (C) deacetoxycephalothin (0.6 mM). The points (●) are experimental and the lines calculated from equations 1–3.

Fig. 3.

Activity of the P99 β-lactamase (10.0 nM) against cephalothin (1.0 mM) as a function of added sodium chloride concentration. The points (●) are experimental and the line was calculated from equations 1–3.

Quantitative analysis of the data of Figures 2A – 2C in terms of Schemes 3–5 first required their elaboration to Schemes 8–10, respectively, where the relevant rate constants and the observed product inhibition are included. The pseudo first order rate constant k_E is composed of a contribution from the reaction in the absence of the added ligand L (amine or salt), pseudo first order rate constant k₀, and one from the reaction in its presence, second order rate constant k₅. For the fitting to Schemes 9 and 10, k₀ was assumed to be zero (insignificant compared to k₅[L] and k₄). Steady state rate equations 1 – 3 are presented below for the effect of ligand (L), e.g. imidazole, on k_cat^obs according to Schemes 8 – 10, respectively.

Scheme 8.

Scheme 10.

Scheme 9.

where k_E = k₀ + k₅[L]

$$k_{cat}^{obs}=\frac{k_2{k}_0/k_5+k_2\left[L\right]}{\left[\left(k_2+k_0\right)/k_5+\left[L\right]\right]}$$ (1)

$$k_{cat}^{obs}=\frac{k_4\left(\left[S\right]/K_m\prime \right)/\left(k_2/k_5\right)+k_2\left[L\right]}{\left(\left[S\right]/K_m\prime \right)/\left(k_2/k_5\right)+\left[L\right]}$$ (2)

$$k_{cat}^{obs}=\frac{k_4\left(\left[S\right]/K_m\prime \right)/\left(k_{ES}/k_5\right)+k_2\left[L\right]}{\left(\left[S\right]/K_m\prime \right)/\left(k_{ES}/k_5\right)+\left[L\right]}$$ (3)

The K_m′ values equate to the K_m^obs values of the substrates at zero ligand concentration, which was determined for each substrate from initial rate determinations (for the actual values, see Table 2).

Fitting of the data of Figures 2A – 2C to the equations 1 – 3 (see the solid lines in these figures) yielded the rate parameters listed in Table 3. Consideration of the hyperbolic equations 1 – 3 shows that this data fitting would yield three parameters, two of which relate to the values of k_cat^obs at zero and infinite ligand concentrations and are unremarkable. The third parameter describes the steepness of the curvature between the above two parameters, the ligand concentration at half maximal increase. The equations all yield the same value of this concentration, calculated from the fitted parameters, viz. 4.5 mM for cephalosporin C, 4.4 mM for cephalothin and 120 mM for deacetoxycephalothin.

Table 3.

Kinetic parameters for imidazole catalysis of the hydrolysis of cefems by the P99 ß-lactamase.

Scheme (see main text)	Cephem	Rate parameter value
8	cephalosporin C	k2 = 1900 ± 40 s−1
		k0 = 1800 ± 200 s−1
		k5 = (8.2 ± 1.8) x 105 M−1s−1
	cephalothin	k2 = 2000 ± 180 s−1
		k0 = 760 ± 10 s−1
		k5 = (6.3 ± 1.8) x 105 M−1s−1
	deacetoxycephalothin	k2 = 4800 ± 200 s−1
		k0 = 2980 ± 180 s−1
		k5 = (6.4 ± 1.7) x 104 M−1s−1
9	cephalosporin C	k2 = 1910 ± 40 s−1
		k4 = 940 ± 50 s−1
		k5/kES = 14900 ± 3500 M−1
	cephalothin	k2 = 760 ± 10 s−1
		k4 = 550 ± 10 s−1
		k5/kES = 14800 ± 4300 M−1s−1
	deacetoxycephalothin	k2 = 4800 ± 400 s−1
		k4 = 1840 ± 80 s−1
		k5/kES = 83 ± 25 M−1
10	cephalosporin C	k2 = 1910 ± 40 s−1
		k4 = 940 ± 50 s−1
		kE = (2.8 ± 0.6) x 107 M−1s−1
	cephalothin	k2 = 760 ± 10 s−1
		k4 = 550 ± 10 s−1
		kE = (1.1 ± 0.3) x 107 M−1s−1
	deacetoxycephalothin	k2 = 4800 ± 400 s−1
		k4 = 1840 ± 80 s−1
		kE = (4800 ± 400) M−1s−1

A variety of other nitrogen bases were found to either increase or have no effect on the k_cat^obs values of cephalosporins. Secondary or tertiary amines in 5- or 6-membered rings (imidazole, 4-picoline and thiazolidine) appear to be particularly effective (data for 4-picoline and thiazolidine are not shown). The base form of the amine is certainly catalytic since there would be little 4-picoline (pK_a 5.9) protonated at pH 7.5. The base strength of the amine must also be an issue since oxazole (pK_a 1.0), essentially identical to imidazole in size and shape, was ineffective. Results with imidazole (pKa 7.2) at different pH values, however, suggest that the imidazolium cation may also catalyze the reaction (data not shown), although less effectively than the base form. Acyclic primary amines (2,2,2-trifluoroethylamine (pK_a 5.5) and Tris (pK_a 7.8), and the tertiary amine triethanolamine (pK_a 7.8) were also ineffective at pH 7.5, suggesting that there is considerable structural specificity required of a catalyst of the E′→E conversion.

3.3. Fitting of the phosphonate inhibition data to Schemes 8–10

Given the support for Schemes 8 – 10 now available from the imidazole experiments, attempts were made to fit the phosphonate 1 inhibition data of Figure 1C obtained in the presence of cephalosporin C. To achieve this fitting, k_cat^obs values for cephalosporin C in the absence and presence of 100 mM imidazole were assigned for the three schemes as shown in Table 4. The values of k₁ and k₋₁ were obtained from the K_m^obs value measured by the initial rates method in the presence of 100 mM imidazole and those for k₃ and k₋₃ from the K_m^obs value in the absence of imidazole (k₁ and k₃ values were assumed to have the diffusion limited second order rate constant 200 μM⁻¹s⁻¹). The value of k_−p was calculated from an experimental value of K_p, (see Methods) and an assumed value of 200 μM⁻¹s⁻¹ for k_p. In fitting Scheme 9, a fixed value of 3000 s⁻¹ (> k₂ and k₄) was arbitrarily assigned to k_ES. Changes in this value yielded different k_E values but the k_ES/k_E ratio remained the same. Finally, k_i was assigned the value of k_i^direct, determined experimentally (Table 1). The latter assignment follows from the fact that, in the absence of substrate, E would be the only free enzyme form present in solution. The values of k_i′ and k_E were then allowed to vary in the fitting procedure. It would be anticipated that any good fit to the data would give a k_i′ value quite close to that of k_i^indirect since, in the presence of substrate, the dominant free enzyme form would be E′.

Table 4.

Interpretation of k_cat^obs in terms of Schemes 8 – 10.

Scheme	kcatobs ([imidazole] = 0)	kcatobs ([imidazole] = 100 mM)
8	k2kE/(k2 + kE)	k2
9	k4	k2
10	k4	k2

Attempts to fit the data of Figure 1C to Scheme 8, given the restrictions to the rate constants described above, failed. This is not surprising, in retrospect, because of the high value of the E′ → E rate constant k_E, which has to be the dominant factor in k_cat^obs. The result of this is that only a small amount of E′ would be present in the steady state and essentially all reaction with the phosphonate would occur with E with a rate constant k_i^direct. Thus, the required k_i′ to fit the data would be much smaller than the experimental k_i^indirect and, indeed, a k_i′ value of zero gave the best (poor) fit.

In contrast, both Schemes 9 and 10 gave good fits (Figure 4) to the data of Figure 1C. The fit shown to Scheme 9 (Figure 4A) arises from a k_E value of zero, distinctly smaller than k₄, leading to a build up of E′ in the steady state that would be available for reaction with I (see below for further discussion). The fit also required a k_i′ value of (1.25 ± 0.1) x 10³ M⁻¹s⁻¹, identical to that of k_i^indirect [(1.25 ± 0.01) x 10³ M⁻¹s⁻¹ (Table 1)]. Thus, Scheme 9 is not only able to fit the data but also yielded the required value for k_i^indirect.

Fig. 4.

Spectrophotometric progress curves for hydrolysis of cephalosporin C (0.40 mM), catalyzed by the P99 β-lactamase (4.4 nM) in the absence (◯) and presence (●) of the phosphonate 1 (20 μM). The points are experimental and the lines were calculated from the simultaneous fit of the data to (A) Scheme 9 and (B) Scheme 10.

The fit of the data to Scheme 10 (Figure 4B) was essentially identical to that of Scheme 9. The fit was accommodated with a k_E value of zero, such that the dominant free enzyme in the steady state is again E′. The required k_i′ value for the fit is again (1.25 ± 0.1) x 10³ M⁻¹s⁻¹, which, as noted above, is the same as that of k_i^indirect. Scheme 10, therefore, is also in accord with the experimental observations.

3.4. Solvent deuterium kinetic isotope effects

Values of k_cat^obs for cephalothin were obtained in both ¹H₂O and ²H₂O, in the absence and presence of imidazole (100 mM), and provided in Table 5. These normal isotope effects indicate proton movement in the transition states, as expected for enzyme-catalyzed hydrolysis of acyl-enzyme intermediates [24,30,31].

Table 5.

Solvent deuterium kinetic isotope effects on the hydrolysis of cephalothin, catalyzed by the P99 ß-Lactamase.

[Imidazole] (mM)	H2Okcatobs/D2Okcatobs
0	2.34 ± 0.08
100	1.76 ± 0.03

3.5. The effect of methanol on k_cat^obs

The effect of methanol on k_cat^obs for cephalothin was obtained in the absence and presence of imidazole (100 mM) (Figure 5). As observed previously in the absence of imidazole, k_cat^obs values increased with methanol concentration, an effect that was shown to be due to interception of the acyl-enzyme intermediate by methanol [21,30,32]. The present results show that an analogous increase in k_cat^obs occurs in the presence of imidazole, presumably from an analogous cause. The ratio of the second order rate constants of methanolysis to that of hydrolysis was 23.5 ± 0.3 in the former case and 26.5 ± 0.2 in the latter.

Fig. 5.

Fractional increase in initial rates of total solvolysis of cephalothin (1.0 mM), catalyzed by the P99 β-lactamase (10.0 nM), as a function of added methanol concentration, in the absence (□) and presence (◯) of imidazole (0.10 M). The points are experimental and the lines calculated as described in the text.

4. Discussion

The results described above demonstrate by the method of irreversible inhibition kinetics that the dominant steady state free enzyme form released following turnover of good cephalosporin substrates of the P99 β-lactamase is different from the free enzyme form prior to substrate addition, i.e turnover of these substrates generates an alternative free enzyme form, E′, an isoform, in solution. The isoform, however, appears to retain the functionality of the active site since it, like the native form, is irreversibly inhibited by specific phosphonate monoesters and turns over substrate. Both of these reactions, however, proceed more slowly than those of the native enzyme: values of k_i for phosphonate inhibition and k_cat for cephalosporin turnover are smaller (although k_cat/K_m remains the same, Table 2). It might be noted here, since this point will come up again below, that turnover rates of most substrates by the P99 β-lactamase, including cephalosporins, under saturating conditions are limited by the rate of hydrolysis of a covalent acyl-enzyme intermediate [28,32].

Analysis of the phosphonate inhibition kinetics showed that the two simplest viable candidates for the reaction mechanism are Schemes 9 and 10. These differ in their mode of generation of the alternative free enzyme form E′. In scheme 9, E′ is formed directly on turnover of the substrate S by the native enzyme form E, whereas in Scheme 10, E′ is formed by branching of an ES intermediate on the normal turnover pathway, probably the accumulating acyl-enzyme, to give rise to E′S and release of E′ from this complex (probably by hydrolysis of the alternative acyl-enzyme). In each case, the directly measured k_cat^obs corresponds to k₄, the rate constant of breakdown (hydrolysis) of E′S (Table 4). The rate constant of spontaneous re-isomerization of E′ to E (k₀) must be small with respect to k₄ so that, in the steady state most turnover of S is catalyzed by E′.

It was found that k_cat^obs for substrate turnover was increased by the presence of certain small molecule catalysts in solution. This includes, in particular, specific cyclic nitrogen bases such as imidazole (Figure 2), 4-picoline and thiazolidine (data not shown). These results suggest that a general base catalyzed reaction may be involved in the increase in k_cat. This interpretation is supported by the ineffectiveness of the weak base oxazole, a close structural analogue of imidazole. Structural properties of the base must also be important since the aliphatic amines trifluoroethylamine, Tris, and triethanolamine were ineffective as catalysts of this reaction. These results are interpreted in terms of Schemes 9 and 10 as demonstrating catalysis of the enzyme isomerization step, E′ → E, serving to restore E as the dominant free enzyme form in S turnover and changing k_cat^obs from k₄ (deacylation of E′S) to the larger k₂ (deacylation of ES). The presence of these catalysts also had the related effect of increasing the rate constant of inactivation of the enzyme by phosphonation from that of reaction of E′ with I (k_i^indirect) to the larger value from reaction of E with I (k_i^direct).

The conversion of E′ to E is also catalyzed by inorganic salts such as sodium chloride (Figure 3) and sodium sulfate. This might indicate a protein conformational difference, possibly quite localized, between E′ and E; this effect could also, however, be quite complicated in molecular detail.

The requirement for E′ in the turnover of substrates by the P99 β-lactamase and in the inhibition by phosphonates in the presence of substrates appears to be particularly striking in cephalosporin substrates bearing a good leaving group at the C3′ position, such as cephalothin and cephalosporin C. It is known that in certain cases such leaving groups can be eliminated from accumulating acyl-enzyme intermediates leading to more inert acyl-enzymes [28]. Such species can then be reactivated towards hydrolysis by addition of specific nucleophiles to the C3 exomethylene group [33], in effect reversing the elimination reaction. These reactions cannot, however, be responsible for the phenomena described in this paper. First, turnover of deacetoxycephalothin, lacking a C3′ leaving group, is also catalyzed by imidazole (Figure 2C). Second, and more definitively, it is known that on turnover of cephalosporin substrates with good leaving groups at the C3′ position (e.g. acetoxy, as in cephalothin and cephalosporin C), the leaving group does not depart at the acyl-enzyme stage but, subsequently, in solution after rapid enzyme-catalyzed deacylation of the initial acyl-enzyme [28].

The solvent kinetic deuterium isotope effects (Table 5) and methanolysis kinetics, determined under saturating substrate conditions, are in accord with the k_cat steps, both in the absence (k₄ and in the presence of saturating imidazole (k₂), being acyl transfer reactions (deacylation of the enzyme active site). In both the isotope effects and methanolysis results the perturbations on the two rate constants are different, signifying a real difference between ES and E′S. The methanolysis result, particularly, argues against the E′→E (k_E) step being rate–limiting in either case.

There is no direct evidence from this study to distinguish between Schemes 9 and 10. There is, however, much precedent with poor substrates of class C β-lactamases such as the P99 enzyme, for partition of initially formed acyl-enzyme intermediates into more inert acyl-enzyme forms [28,34,35]. The presence of an E′ has not usually been detected, probably partly at least because the reconversion of E′ to E is faster than the slow deacylation. Indeed, in this study, E′ was not detected in the presence of the poor substrates ACA and cefoxitin (Table 1). On the other hand, one example of a poor substrate (slow deacylation) where E′ was detected by pre- and post-steady state kinetics is that of the third generation cephalosporin cefotaxime and the P99 β-lactamase [36]. In this case, Scheme 10 was concluded to best fit the data. With good substrates, as used in the present study, detection of E′ by direct observation with the substrate alone is technically more difficult. It might be noted that suggestions have been made previously for the existence of slowly equilibrating forms of native class C β-lactamases [37–39] but there seems to be no unequivocal demonstration of them in the absence of substrates, i.e. of acyl-enzymes. Certainly, the reactions of irreversible inactivating agents such as the phosphonate monoesters employed in this work, show no sign of biphasic kinetics because of enzyme heterogeneity, e.g. see Figure 1A, 1B and reference 14.

Thus, in this paper we have described the successful application of a new method of detection of an enzyme isoform generated during the turnover of substrates by a particular enzyme, a class C β-lactamase. The method should be general, given the availability of an efficient irreversible inhibitor. Its success does depend on various unpredictable kinetic parameters but such uncertainties could be offset by variation of substrate and inhibitor. In favorable cases, the results of these experiments could be supplemented by pre- and/or post-steady state experiments [36].

Highlights.

• Kinetics of enzyme inactivation by covalent inhibitors can be different in the presence of substrates

• The above phenomenon can be interpreted in terms of an iso-mechanism

• An example case is inactivation of a class C β-lactamase by phosphonate monoesters

• A general method for detection of iso-mechanisms