Analysis of slow-binding enzyme inhibitors at elevated enzyme concentrations

ANALYTICAL BIOCHEMISTRY Analytical Biochemistry 337 (2005) 211–223 www.elsevier.com/locate/yabio

Analysis of slow-binding enzyme inhibitors at elevated enzyme concentrations Basil Perdicakisa, Heather J. Montgomeryb, J. Guy Guillemetteb, Eric Jervisa,* a

Department of Chemical Engineering, University of Waterloo, Waterloo, Ont., Canada N2L 3G1 b Department of Chemistry, University of Waterloo, Waterloo, Ont., Canada N2L 3G1 Received 29 July 2004 Available online 8 December 2004

Abstract The improvement in the characterization of slow-binding inhibitors achieved by performing experiments at elevated enzyme concentrations is presented. In particular, the characterization of slow-binding inhibitors conforming to a two-step mode of inhibition with a steady-state dissociation constant that is much lower than the initial dissociation constant with enzyme is discussed. For these systems, inhibition is rapid and low steady-state product concentrations are produced at saturating inhibitor concentrations. By working at elevated enzyme concentrations, improved signal-to-noise ratios are achieved and data may be collected at saturating inhibitor levels. Numerical simulations confirmed that improved parameter estimates are obtained and useful data to discern the mechanism of slow-binding inhibition are produced by working at elevated enzyme concentrations. The saturation kinetics that were unobservable in two previous studies of an enzyme inhibitor system were measured by performing experiments at an elevated enzyme concentration. These results indicate that consideration of the quality of the data acquired using a particular assay is an important factor when selecting the enzyme concentration at which to perform experiments used to characterize the class of enzyme inhibitors examined herein.
2004 Elsevier Inc. All rights reserved. Keywords: Slow-binding; Progress curve; Inhibitor; Regression; MATLAB; Oxyhemoglobin; Nitric oxide; Nitric oxide synthase; Enzyme assay; Multiwell plate

Slow-binding inhibitors do not rapidly establish equilibrium with their target enzymes relative to enzymatic turnover of substrate [1] (for reviews of slow- and tight-binding inhibition, see Morrison and Walsh [2] and Szedlacsek and Duggleby [3]). Two common mechanisms for competitive slow-binding inhibition are shown in Fig. 1 [4,5]. In mechanism A, slow-binding inhibition may be due to a low magnitude of k2 or to a low apparent first-order rate constant, k2[I], that will occur if the inhibitor concentration is varied in the region of Ki for a tight-binding inhibitor [6]. For small molecule inhibitors, the diffusion-limited value for k2 is *

Corresponding author. Fax: +1 519 746 4979. E-mail address: [email protected] (E. Jervis).

0003-2697/$ - see front matter
2004 Elsevier Inc. All rights reserved. doi:10.1016/j.ab.2004.11.012

on the order of 109 M
1s
1 [6]. A low value for k2 may be due to barriers preventing the inhibitor from rapidly binding to the active site. In mechanism B (Fig. 1), the enzyme inhibitor EI1 complex is rapidly formed, and then a slow conformational change in the EI complex to the EI* complex occurs. Besides increased potency, one clinical advantage of slow- and tight-bind1 Abbreviations used: iNOS, inducible nitric oxide synthase; 1400W, N-(3-(aminomethyl)benzyl) acetamidine; ADP, adenosine diphosphate; NADPH, nicotinamide adenine dinucleotidephosphate; H4B, (6R)-5,6,7,8-tetrahydrobiopterin; DTT, dithiothreitol; FAD, flavin adenine dinucleotide; FMN, flavin mononucleotide; SOD, superoxide dismutase; BSA, bovine serum albumin; ODE, ordinary differential equation; ONLR, ordinary nonlinear regression; WNLR, weighted nonlinear regression.

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high 1400W concentrations [13,14]. Garvey et al. [14] characterized the in vitro inhibition of human iNOS with 1400W by monitoring the formation of L -[3H]citrulline from L -[3H]arginine and determined that this system conformed to inhibitory mechanism B with k3  k
3. Subsequently, our research group, using the oxyhemoglobin assay [15], also examined the inhibition of purified human iNOS by 1400W [13] and obtained results consistent with those of Garvey et al. [14]. In both studies, data could not be acquired above the apparent Ki value due to rapid inhibition and low product concentrations obtained at higher inhibitor concentrations. The saturation kinetics that were previously unobservable in the iNOS-1400W system using these traditional experimental designs, due to poor signal at high inhibitor concentrations, have now been measured by performing experiments at a higher enzyme concentration. This approach allows for more accurate estimates of Ki and k3 and also provides useful data to discriminate between a two-step inhibitory mechanism and a onestep inhibitory mechanism.

Materials and methods Fig. 1. Two potential mechanisms for slow-binding inhibition.

ing inhibitors that conform to mechanism B is that, unlike classical inhibitors, upstream accumulation of substrate might not affect inhibitor effectiveness in vivo [2]. A review of recent publications illustrates the continuing importance and applicability of the formulations developed by Cha [4,5] for the analysis of slow-binding EI complexes [7–11]. This article focuses on EI systems conforming to mechanism B, where there is no initial tight-binding inhibitory effect but where k3 is much greater than k
3. Sculley and Morrison [12] noted that the investigation of the enzyme kinetics of inhibitors that conform to mechanism B with k3  k
3 is difficult and proposed a method for data analysis. Note, however, that depending on the magnitude of k
3, this method may require enzyme preparations that remain stable for several hours to obtain estimates for k3 and k
3. In this article, we evaluate the utility of initiating reactions by the addition of inhibitor at high enzyme concentrations to estimate Ki and k3 and to discriminate between mechanisms A and B in Fig. 1. The utility of performing experiments at elevated enzyme concentrations was confirmed by numerical simulation. Theoretical predictions were also verified experimentally using inducible nitric oxide synthase (iNOS) and the potent iNOS selective inhibitor N-(3-(aminomethyl)benzyl) acetamidine (1400W). Previous attempts to characterize the iNOS-1400W system encountered difficulty due to the rapid onset of iNOS inhibition at

Materials 1400W was synthesized as described previously [13]. All other reagents were purchased from Sigma–Aldrich (Oakville, Ont., Canada) and used without further purification. Protein expression and purification Human iNOS enzyme carrying a deletion of the first 70 amino acids and an amino terminal polyhistidine tail was coexpressed with calmodulin in Escherichia coli and purified using metal-chelating chromatography followed by 2 0 ,5 0 -adenosine diphosphate (ADP) column chromatography [13,16]. Enzyme kinetics Kinetic parameters characterizing inhibition of iNOS by 1400W were calculated from progress curve data obtained using the oxyhemoglobin capture assay [15,17,18] that yields results consistent with the direct radioactive assay that monitors the formation of L -[3H]citrulline from L -[3H]arginine [19]. Reactions were initiated by the addition of L -arginine and monitored on a 96-well plate reader (Spectramax 190, Molecular Devices, Sunnyvale, CA, USA) at 26 C. The plate reader was controlled using SOFTmax PRO software (version 3.0, Molecular Devices). Nitric oxide-mediated oxidation of HbO2 was monitored at 401 nm (De401 nm = 0.101 ± 0.005 OD401 nm/nm). The increase in absor-

Slow-binding enzyme inhibitors / B. Perdicakis et al. / Anal. Biochem. 337 (2005) 211–223

bance that occurred on oxidation of HbO2 to MetHb was linear (R2 = 0.97) over the range of hemoglobin concentrations tested (0–15 lM) [15]. Reaction mixtures (100 ll) contained 50 mM Tris–HCl (pH 7.5), 1% (v/v) glycerol, 500 lM nicotinamide adenine dinucleotidephosphate (NADPH), 5 lM (6R)-5,6,7,8-tetrahydrobiopterin (H4B), 1 mM CaCl2, 16 lM dithiothreitol (DTT), 1 lM flavin adenine dinucleotide (FAD), 1 lM flavin mononucleotide (FMN), 100 U/ml superoxide dismutase (SOD), 50 U/ml catalase, 10 lM bovine HbO2, 0.2 mg/ml bovine serum albumin (BSA), 25 lM L -arginine, and 95 nM iNOS (kcat @ 690 ± 60 nm of nitric oxide min
1 mg iNOS
1, apparent Km (L -arginine) = 3.4 ± 0.4 lM). H4B was prepared in 20 mM DTT [18]. Solutions (10 ll) containing 950 nM of enzyme, 50 mM Tris–HCl (pH 7.5), 10% (v/v) glycerol, 500 lM NADPH, 10 lM FAD, 10 lM FMN, 50 lM H4B, 160 lM DTT, 100 U/ml SOD, and 50 U/ml catalase were pipetted into the desired wells. Reactions were initiated by the addition of a 90-ll solution containing substrate (L -arginine) and various amounts of inhibitor ranging from 0 lM (for uninhibited reactions) to 50 lM. Similar 10-ll solutions were prepared for blank wells measuring background signal, but with 833 lM 1400 W to fully inhibit enzyme activity. Blank well data was subtracted from progress curve data. Blank wells containing 833 lM 1400W and iNOS produced values similar to those produced by wells containing no iNOS or no substrate. The wells were allowed to mix for 3 s prior to monitoring. The time elapsed prior to monitoring the reaction was recorded [20]. Reactions were monitored for 5 min at 6- to 7-s intervals. Simulation of enzyme kinetic data Enzyme kinetic data were simulated in the computer program MATLAB (version 6.1, Mathworks, Boston, MA, USA) by numerical integration of the ordinary differential equations (ODEs) characterizing an enzymesingle substrate inhibitor system conforming to mechanism B in Fig. 1. Due to the presence of kinetic rate constants that vary over several orders of magnitude in this system, the ODE solver, ODE15s, was used to integrate the resulting stiff set of ODEs [21]. Normally distributed noise was added to model output using the MATLAB function randn. Product concentration data were simulated with k1 = k2 = 0.1 (nM s)
1 [22], kcat = 1 s
1, Km = 4 lM, Ki = 3 lM, k3 = 0.07 s
1, and k
3 = 0 s
1. Theory and data analysis Time-dependent inhibition of iNOS by 1400W was studied by the addition of iNOS to solutions containing L -arginine and 1400W. Nonlinear regression analysis

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was performed using MATLAB. Progress curve data were fit to Eq. (1) using the MATLAB function lsqcurvefit [2,3]: ½P  ¼

vs t þ ðvo
vs Þð1
e
kobs t Þ þ d; k obs

ð1Þ

where [P] is the product concentration (lM), vo is the initial rate of product formation (or velocity) in the presence of inhibitor (lM s
1), vs is the steady-state rate of product formation in the presence of inhibitor (lM s
1), kobs is the exponential decay constant (s
1), t is the time (s), and d is the offset at time zero (lM). Using the method of analysis advocated by Morrison and Walsh [2], Ki was calculated by fitting vo/vuh versus inhibitor concentration data to Eq. (2): v0 1  ¼
; vuh K i 1 þ ½S þ 1 Km

ð2Þ

where vuh is the velocity of the uninhibited reaction (lM s
1). Initial velocities for uninhibited data were estimated by fitting product concentration versus time data to the integrated Michaelis–Menten equation [23] by nonlinear regression [20]. In the absence of useful vs data (see below), k3 was determined by nonlinear regression of Eq. (3): k obs ¼ k
3 þ

k 3 ½I
 : K i 1 þ K½Sm þ ½I

ð3Þ

Duggleby et al. [24] commented that kobs data will not be affected when the time at which the reaction is initiated is not accurately known or by inaccuracies in the characterization of background signal drift, and concluded that a fit of kobs versus inhibitor concentration data to Eq. (3) was therefore the most robust method to estimate the required parameters. Hereafter in this article, estimation of Ki using Eq. (2) followed by estimation of k3 using Eq. (3) is referred to as calculation method 1, whereas estimation of Ki and k3 directly from Eq. (3) is referred to as calculation method 2. If experiments are performed at inhibitor concentrations in the region of the Ki value for slow-binding inhibitors conforming to mechanism B with k3  k
3, then the steady-state velocity, vs, will approach zero. Sculley and Morrison [12] noted that when k3  k
3, estimation of all three parameters will require two experiments. For the first experiment, the inhibitor concentration is varied in the vicinity of its Ki value and Eq. (2) is used to estimate Ki. The second experiment involves preincubating inhibitor with enzyme near the steadystate equilibrium value. Data are then fit to the appropriate model to estimate k3 and k
3 [12]. One potential drawback of this method is that stable enzyme may be required for several hours to perform the second experiment, depending on the magnitude of k
3. Garvey et al. [14] demonstrated that the iNOS enzyme decays faster

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than the conformationally altered iNOS-1400W complex, and they were only able to estimate k
3 as being the inherent loss of iNOS activity (i.e. 7 · 10
5 s
1). In this article, efforts were made to improve the characterization of Ki and k3 by analyzing experiments performed in the region of the Ki value. Estimation of k
3 could be accomplished using a second experiment as outlined previously [6,24]. In the analysis presented here, k
3 was set to zero because k3 was assumed to be much greater than k
3. With k
3 set to zero, the expected value for the vs responses is also zero. Therefore, the vs responses obtained from fits of product concentration versus time data to Eq. (1) were not analyzed. Furthermore, only competitive inhibition is discussed. Data was fit to selected models by ordinary nonlinear regression (ONLR) or weighted nonlinear regression (WNLR). The variance of six replicate vo/vuh or kobs responses (WNLR1) and the square of standard error of the vo and kobs responses (WNLR2) were used in the WNLR algorithms.

Results Effect of enzyme concentration on progress curves Numerical simulation of progress curve data obtained in the presence of a saturating concentration of a slow-binding inhibitor was performed at various enzyme concentrations (Fig. 2). Simulation parameters are provided in the Fig. 2 caption. Due to the slow establishment of equilibrium between enzyme and inhibitor, progress curves obtained at various enzyme concentrations in the presence of slow-binding inhibitors do not conform to Selwyn
s test (i.e., progress curves are not superimposable when plotted against Eo Æ t) [25]. Fig. 2 demonstrates the improved signal achieved using higher enzyme concentrations. If it is assumed that assay noise remains relatively constant over the range of product concentrations shown in Fig. 2, as observed using the oxyhemoglobin assay, then higher signal-to-noise ratios will also be realized. In fact, for progress curves with vs approximately equal to zero, the product concentration at steady-state is equal to vo/kobs [3] and scales linearly with the initial enzyme concentration. This improvement allows for the collection of data at saturating inhibitor concentrations that might not be attainable at lower enzyme concentrations. Fitting of progress curve data to Eq. (1) Prior to examining the effects of enzyme concentration on the estimation of Ki and k3, studies were performed to investigate the estimation of the vo and kobs responses from product concentration versus time data under various conditions and experimental designs.

Fig. 2. Numerical simulation of progress curves obtained in the presence of a saturating concentration of inhibitor at various enzyme concentrations. Progress curve data were simulated by numerical integration of the ordinary differential equations characterizing inhibition mechanism B in Fig. 1. The input inhibitor and substrate concentrations were 50 and 25 lM, respectively. The simulated enzyme concentrations were 100 (n), 75 (s), 50 (*), 25 (), 10 (·), 5 (h), and 1 (+) nM. Other simulation parameters are noted in the Materials and methods section.

Product concentration data were simulated at an enzyme concentration of 95 nM, a substrate concentration of 25 lM, and six inhibitor concentrations ranging from 10 to 50 lM. Product concentration data were simulated with a standard deviation (rP) of 25 nM. Progress curves were simulated at 7-s intervals for 250 s. Under these conditions substrate depletion varied from 10% to approximately 2% at low and high inhibitor concentrations, respectively. vs, vo, and kobs values (n = 1250) were calculated at each inhibitor concentration, and the coefficient of variation (cov) of each of the response populations was tabulated. Finally, 100 times the theoretical standard error of the responses, as calculated by a first-order Taylor series truncation [26] divided by the theoretical response, was also calculated (covse). Fig. 3 demonstrates that the coefficients of variation in both the vo and kobs responses increase at higher inhibitor concentrations. The coefficients of variation of the vo and kobs responses are approximately 2–14% at the low and high inhibitor concentrations, respectively. This observation indicates that the responses become less precise with increasing inhibitor concentrations, suggesting that weighted regression should be used to calculate Ki and k3 from Eqs. (2) and (3). Fig. 3 also indicates that the coefficients of variation of the vo and kobs responses calculated using the theoretical standard error of these responses are in excel-

Slow-binding enzyme inhibitors / B. Perdicakis et al. / Anal. Biochem. 337 (2005) 211–223

Fig. 3. Coefficients of variation of vo and kobs responses as a function of inhibitor concentration. Progress curves (n = 1250) were simulated at each inhibitor concentration by numerical integration of the ordinary differential equations characterizing inhibition mechanism B in Fig. 1. The input enzyme and substrate concentrations were 100 nM and 25 lM, respectively. Product concentration data were simulated with random errors. The standard deviation of the normally distributed error was equal to 25 nM. Other notable simulation parameters are provided in the Materials and Methods section. The cov () and covse (—) for the vo response data, and the cov (n) and covse (___) for the kobs response data, are shown.

lent agreement with those of the response populations obtained from numerical simulation. These results indicate that weighting fits of vo and kobs responses versus inhibitor concentration data to Eqs. (2) and (3) [24] with the standard error of the responses is an efficient method [27] of estimating the required parameters. Simulations also showed that the systematic error in measuring the kobs response was larger than the systematic error in measuring the vo response, particularly at lower inhibitor concentrations. This error is likely due to the increased depletion of substrate that occurs at lower inhibitor concentrations. One assumption of Eq. (1) is that negligible depletion of substrate occurs. In practice, some depletion of substrate is required to fit a progress curve to Eq. (1). For the system examined, 10% depletion of substrate [28,29] resulted in a 6% error in the mean of the simulated kobs values but less than 1% error in the mean of the simulated vo values. One explanation as to why the kobs response is more susceptible than the initial velocity response to systematic error arising from increased depletion of substrate is that estimation of the kobs response is more heavily dependent on the final asymptote of a progress curve, where substrate depletion is more prominent. This hypothesis was supported by numerical simulations showing that 30% substrate depletion gave an error in the vo response of less than 3%, whereas it gave an error in the kobs response of approximately 17%. Simulations were also performed to examine the effect of increasing amounts of experimental lag time on the estimation of vo and kobs. At an inhibitor concentration of 50 lM, the coefficients of variation for the vo and

215

kobs responses increased from approximately 10% for both responses to 46 and 22%, respectively, when the experimental lag time was increased from 0 to 21 s. The larger increase in the coefficient of variation for the vo response, as compared with the kobs response, with increasing experimental lag time is intuitive because the vo response is defined as the progress curve velocity at time equal to zero. Estimations of the vo and kobs responses using progress curves where the data are equally spaced in time or product concentration were also compared [2,30]. Progress curves were simulated at an inhibitor concentration of 50 lM. Product concentration versus time data were analyzed by using either 19 data points separated by at least 50 nM of product concentration or 36 data points equally spaced by 7-s time intervals. Based on this limited study, it was concluded that there was no significant advantage to spacing data by product concentration. This result is in agreement with that obtained by Duggleby and Clarke [30], who concluded that analyzing data equally spaced in time was the best method for estimating Km and Vm from uninhibited progress curves. Estimation of Ki and k3 at different enzyme concentrations Simulations were performed to examine the effect of estimating Ki and k3 at two different enzyme concentrations and under two different assay error structures. Product concentration data were simulated at an enzyme concentration of 95 nM, a substrate concentration of 25 lM, and inhibitor concentrations of 10, 15, 25, 30, 40, and 50 lM. Product concentration data were also simulated at an enzyme concentration of 10 nM, a substrate concentration of 4 lM, and inhibitor concentrations of 1.0, 3.6, 6.0, 7.2, 9.6, and 12.0 lM. Product concentration data were simulated with a constant standard deviation of either 25 or 2.5 nM. Progress curves were simulated at 7-s intervals for 250 s with no experimental lag time. Based on the results of the simulations described above, vo and kobs responses were estimated by fitting progress curves with data points equally spaced in time to Eq. (1). Ki and k3 were calculated using both calculation methods 1 and 2 by ONLR and WNLR. Six replicates were simulated at each inhibitor concentration. With the advent of multiwell plate readers, it is now reasonable to perform several replicates of an experiment to estimate the underlying error structure. Six replicates were simulated so that the variance associated with each measurement could be estimated accurately [31,32]. The variance of six replicate vo and kobs responses (WNLR1) and the square of standard error of the vo and kobs responses (WNLR2) were used for WNLR. In practice, using the variance of replicate vo and kobs responses to WNLR fits may be more appropriate given that this

2.6 3.2 100 2.8 3.5 99 9.1 12.1 34 6.6 7.1 34 7.2 8.0 32 7.0 7.6 34 9.0 9.8 40 13.8 18.8 38 Case D: [E] = 10 nM, rP = 2.5 nM 5.5 4.4 3.6 6.3 5.3 4.3 63 81 100 Mean absolute percentage error pffiffiffiffiffiffiffiffiffiffiffi ffi 100 MSD=input value Efficiency

8.3 9.4 42

19.1 25.2 100 >100 >100 7 >100 >100 0 23.9 31.8 81 >100 >100 13 >100 >100 0 34.6 39.3 64 >100 >100 0 Case C: [E] = 10 nM, rP = 25 nM >100 >100 23.4 >100 >100 29.0 0 23 100 Mean absolute percentage error pffiffiffiffiffiffiffiffiffiffiffi ffi 100 MSD=input value Efficiency

>100 >100 12

3.0 3.1 57 3.2 3.3 55 1.7 1.8 100 3.7 3.7 47 4.0 4.1 44 1.9 1.9 91 10.9 10.9 3 8.3 8.3 4 Case B: [E] = 95 nM, rP = 2.5 nM 0.3 0.6 0.6 0.4 0.7 0.6 100 55 61 Mean absolute percentage error pffiffiffiffiffiffiffiffiffiffiffi ffi 100 MSD=input value Efficiency

11.1 11.2 3

2.6 3.2 100 2.8 3.5 92 4.9 6.3 55 6.1 6.3 34 6.3 6.6 34 5.5 5.9 40 8.2 9.4 30 8.1 10.0 36 Case A: [E] = 95 nM, rP = 25 nM 3.2 2.9 2.5 3.7 3.4 2.9 72 88 100 Mean absolute percentage error pffiffiffiffiffiffiffiffiffiffiffi ffi 100 MSD=input value Efficiency

7.8 9.1 34

WNLR2 WNLR1 ONLR WNLR2 WNLR1

Calculation method 1

ONLR WNLR2 WNLR1

Calculation method 2

ONLR WNLR2 WNLR1

Calculation method 1

ONLR

k3 calculation Ki calculation Weighting scheme

method will incorporate other possible sources of experimental error [27]. Numerical simulations (n = 250) were performed for case A in Table 1, and 100 simulations were performed for the remaining cases. The mean absolute percentage error of the calculated Ki and k3 values, 100 times pffiffiffiffiffiffiffiffiffiffiffi ffi the square root of the mean squared deviation ( MSDÞ of the populations over the simulation input values, and the efficiency of each method relative to the best method for each type of error [33] are reported in Table 1. Case A in Table 1 indicates that the most efficient method for estimating Ki is using calculation method 1 with weighting scheme WNLR2 and that the most efficient method for estimating k3 is using calculation method 2 with weighting scheme WNLR2. The good performance of weighting scheme WNLR2 is in agreement with the results shown in Fig. 3. Although weighting scheme WNLR2 is the most efficient, it performs only marginally better than weighting scheme WNLR1. When the standard deviation of the product concentration data is lowered to 2.5 nM, the performance of calculation method 1 improves, whereas the performance of calculation method 2 does not improve noticeably (case B in Table 1). This occurs because calculation method two relies exclusively on the kobs response to estimate Ki and k3, whereas calculation method 1 uses both the vo and kobs responses. As noted previously, the systematic error in estimating kobs is greater than that in estimating vo when substrate depletion approaches 10%. In case B in Table 1, the error in the product concentration data has been reduced 10-fold compared with the error in case A. The reduced error in the product concentration results in minimal scatter of the regressed vo and kobs values. The coefficients of variation of the vo and kobs data for case B are less than 1% for all responses and are 10-fold less than the corresponding coefficients of variation in case A (e.g., Fig. 3). In case A (rP = 25 nM), the higher random noise in the regressed kobs responses partially masks the systematic error in estimating the kobs response, whereas in case B (rP = 2.5 nM), the systematic error is more apparent with the reduced noise. Weighting schemes WNLR1 and WNLR2 are not the most efficient methods in case B because they more heavily weight the vo and kobs responses with higher systematic errors. Once again, the decrease in efficiency that results is more apparent for calculation method 2 than for calculation method 1. In spite of the minor systematic error in estimating the vo and kobs responses, it should be noted that the mean absolute percentage errors in estimating Ki and k3 in cases A and B are within generally acceptable bounds. In case C in Table 1, simulations were performed at a low enzyme concentration and a high assay noise level. A comparison of case A (high enzyme concentration and high noise) with case C clearly indicates the benefit

Calculation method 2

Slow-binding enzyme inhibitors / B. Perdicakis et al. / Anal. Biochem. 337 (2005) 211–223

Table 1 Comparison of the calculation of Ki and k3 using various calculation methods, weighting schemes, and enzyme concentrations

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Slow-binding enzyme inhibitors / B. Perdicakis et al. / Anal. Biochem. 337 (2005) 211–223

217

of increasing the enzyme concentration for the calculation of Ki and k3. In case C, calculation methods 2 and 1 used in conjunction with weighting schemes ONLR and WNLR1 fail completely due to the reduced signal-to-noise ratios. Surprisingly, weighting scheme WNLR2 remains robust under the reduced signal-tonoise ratios, but the coefficients of variation in the regressed Ki and k3 values increase by nearly an order of magnitude compared with those in case A. In case D, simulations were performed at a low enzyme concentration and a low assay noise level. As expected, estimates of Ki and k3 are much improved at a low enzyme concentration under low assay noise (case D) as opposed to high assay noise (case C). Experimental results In two previous reports, the saturation kinetics of the iNOS-1400W system could not be measured due to low steady-state product concentrations and the rapidity of inhibition that occurred at higher inhibitor concentrations [13,14]. In this article, experiments were performed to demonstrate that saturation kinetics could be measured by working at higher enzyme concentrations. Time-dependent inhibition of iNOS by 1400W was studied by the addition of iNOS to reaction solutions containing L -arginine and 1400W. Progress curve data were obtained at an iNOS concentration of 95 nM and an L -arginine concentration of 25 lM. A family of progress curves obtained under these conditions at 1400W concentrations ranging from 0 to 30 lM is shown in Fig. 4. Values for vo, vs, and kobs were estimated by nonlinear regression of data to Eq. (1) for each inhibited progress curve. As expected from previous results [13,14], the calculated vs responses were not significantly different from zero. These results are typical for progress curves initiated with enzyme when k3  k
3 at concentrations of a slow-binding inhibitor in the region of the Ki value. Statistical analysis of a fit of vo/vuh and kobs data versus 1400W concentration to Eqs. (2) and (3) A fit of vo/vuh versus 1400W concentration data to Eq. (2) and a fit of kobs versus 1400W concentration data to Eq. (3) are shown in Figs. 5 and 6, respectively. Because the vs responses were not significantly different from zero (i.e., inhibited progress curves exhibited horizontal asymptotes), k
3 was set equal to zero in the analysis. The vo/vuh response versus 1400W concentration data clearly decrease with increasing inhibitor concentrations (Fig. 5). The kobs response versus 1400W concentration data also saturate at the higher 1400W concentrations (Fig. 6). Both of these observations indicate that the inhibition of iNOS by 1400W conforms to mechanism

Fig. 4. Plots of data collected using the oxyhemoglobin assay on a multiwell plate reader corrected for secondary interferences and signal offset at time zero. A family of progress curves for experiments performed on a 96-well plate is shown. Progress curves were obtained at an iNOS concentration of 95 nM, L -arginine concentration of 25 lM, and at 1400W concentrations of 0 (,), 10 (s), 15 (*), 20 (), 25 (·), and 30 (h) lM are shown. The concentration of NADPH was 500 lM, and the HbO2 concentration was 10 lM. Data from blank wells are also shown (D). Lines (-) indicate a fit of data to Eq. (1), the integrated Michaelis–Menten equation, or a straight line by robust regression for inhibited progress curves, uninhibited progress curves, or blank well data, respectively.

B of Fig. 1, in agreement with previous results [13,14]. By performing experiments at higher enzyme concentrations, data were obtained at higher 1400W concentrations, and the decrease in the vo/vuh response with increasing 1400W concentrations, as well as in the saturation of the kobs response, was more readily observed than in previous experiments performed at a lower enzyme concentration [13]. Therefore, by performing experiments at a higher enzyme concentration, more conclusive evidence that the mode of inhibition conformed to mechanism B, as opposed to mechanism A, was obtained. Based on the results shown in Table 1, a Ki value equal to 2.7 ± 0.2 lM was calculated using calculation method 1 in conjunction with weighting scheme WNLR1 (Fig. 5), and a k3 value equal to 0.07 ± 0.01 s
1 was calculated using calculation method 2 in conjunction with weighting scheme WNLR1 (Fig. 6). The apparent Km value for L -arginine for the iNOS preparation used in this study was 3.4 ± 0.4 lM. The residual populations shown in Figs. 5 and 6 exhibited the appropriate linear normal probability plots [34] and passed the Lillefors test [35] for normality. These results indicate that the residual populations are normally distributed, in agreement with regression theory. The estimated values of Ki and k3 did not change appreciably when calculation method 1 or calculation method 2 was employed, when alternate weighting

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Fig. 5. Statistical analysis of a fit of vo/vuh versus 1400W concentration data to Eq. (2) by nonlinear regression. Experiments were performed on a 96well plate at iNOS and L -arginine concentrations of 95 nM and 25 lM, respectively. The 1400W concentration was varied between 0 and 50 lM. Reactions were initiated by the addition of enzyme. The concentration of NADPH was 500 lM, and the HbO2 concentration was 10 lM. vo values were calculated from a fit of product concentration versus time data to Eq. (1), and vuh values were obtained from fits of uninhibited progress curves. vo/vuh versus inhibitor concentration data were then fit to Eq. (2). Top left plot: vo/vuh versus 1400W concentration data (s) and nonlinear fit of data (-, Ki = 2.7 ± 0.2). Top right plot: residuals versus predicted vo/vuh values. Middle left plot: residuals versus 1400W concentration. Middle right plot: normalized weighting factors versus predicted vo/vuh values. Bottom right plot: variance of vo/vuh replicates versus 1400W concentration. Bottom left plot: expected value of residuals conforming to a normal distribution (s) versus observed value and linear fit to data (-).

schemes were implemented, or when data collected during separate days were pooled. Furthermore, although there was more than 10% substrate depletion at the 5and 10-lM 1400W concentrations, the regressed values of Ki and k3 were not significantly affected by the inclusion of these data points. Numerical simulations indicated that the vo/vuh and kobs responses obtained by nonlinear regression of Eq. (1) to product concentration data, calculated by numerical integration of the governing ODEs, would be expected to increase by only 0.02 and 0.0024 s
1 from the values calculated using Eqs. (2) and (3), respectively, due to depletion of substrate at a 1400W concentration of 5 lM. Given the scatter in the vo/vuh and kobs responses shown in Figs. 5 and 6, it is not surprising that this systematic error is not

detectable. The reported values for Ki and k3 were based on nonlinear regression of data in which less than 10% substrate depletion occurred (i.e., 5- and 10-lM 1400W concentrations were excluded from the analysis). No lack of fit [34] was calculated at a 95% confidence level for the fit of kobs (P = 0.17), or the vo/vuh (P = 0.12) data versus 1400W concentration data, even when the 5and 10-lM 1400W concentrations were included in the analysis. Drawback of performing experiments at elevated enzyme concentration A drawback of using a higher enzyme concentration is that experiments have to be performed at a higher

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Fig. 6. Statistical analysis of a fit of kobs versus 1400W concentration data to Eq. (3) by nonlinear regression. Experiments were performed as indicated in the Fig. 5 caption. kobs values were calculated from a fit of product concentration versus time data to Eq. (1). kobs versus 1400W concentration data were then fit to Eq. (3). Top left plot: kobs versus 1400W concentration data (s) and nonlinear fit of data (-, k3 = 0.07 ± 0.01 s
1). Top right plot: residuals versus predicted kobs values. Middle left plot: residuals versus 1400W concentration. Middle right plot: normalized weighting factors versus predicted kobs values. Bottom right plot: variance of kobs replicates versus 1400W concentration. Bottom left plot: expected value of residuals conforming to a normal distribution (s) versus observed value and linear fit to data (-).

substrate concentration so that significant depletion of substrate does not occur in the presence of inhibitor, which is one of the assumptions of the model [4]. When experiments are performed at elevated substrate concentrations, the accuracy of the calculated Ki values is more dependent on the Km values than is the case when experiments are performed at lower substrate concentrations (Fig. 7). This occurs because apparent Ki values calculated using Eqs. (2) or (3) are converted into estimates of Ki by dividing by (1+ [S]/Km). For example, underestimation of the Km value by 60% would result in approximately a 50% error in estimating the Ki value when experiments are performed at a substrate concentration equal to the Km value. Conversely, an error of more than 150% in estimating the Ki value would result in a substrate concentration equal to 10 times the Km value. Fig. 7 also indicates that any reasonable errors in esti-

mating the actual substrate concentration when experiments are performed at elevated substrate concentrations do not affect the error in estimating the Ki value to the same extent as they affect the error in estimating the Km value. In practice these disadvantages have to be weighed against the advantage of improved signal-to-noise ratios obtained by performing experiments at elevated enzyme concentrations.

Discussion The motivation for performing this research was the difficulty encountered in characterizing the inhibition of iNOS by 1400W in a previous study [13]. In that work, experiments were performed at an iNOS concentration of 10 nM and a substrate concentration equal to two

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Fig. 7. Percentage errors in Ki as a function of the percentage errors in Km and substrate concentrations. The percentage errors in Ki as a function of the percentage errors in Km at substrate concentrations equal to the actual Km values (—) and 10 times the Km value () are shown. The percentage errors in Ki as a function of the percentage errors in Km when there is also a 10% increase (-Æ-) or a 10% decrease (Æ ÆÆ) in the nominal substrate concentration at an ideal substrate concentration of 10 times the Km value are also shown.

times the Km value. Although a slight downward trend in the vo/vuh response was observed, this response exhibited a large degree of scatter. The kobs response exhibited some curvature at higher 1400W concentrations, indicative of mechanism B. This result was in agreement with a previous study [14]. Eq. (3) is a rectangular hyperbola similar to the Michaelis–Menten equation. However, unlike fitting initial velocity versus substrate concentration data to the Michaelis–Menten equation, where initial velocities are easier to estimate at saturating substrate concentrations, kobs values are more difficult to measure at saturating inhibitor concentrations. Due to the rapidity of inhibition and the decrease in product formation at higher 1400W concentrations, data could be obtained only at inhibitor concentrations below the apparent Ki value, as was also observed by Garvey et al. [14]. Based on the difficulty in characterizing this system, new experimental designs were sought to improve the quality of the data collected. Although there are several methods of characterizing the inhibition of enzymes by slow-binding inhibitors [36–38], this article focused on implementing the formulations developed by Cha [4,5] due to their widespread use [7–11]. Two options were considered to obtain saturation data: increase the substrate concentration or increase the enzyme concentration. Ideally, experiments are to be performed at a substrate concentration near the Km value so as to lessen the effect of any errors in the Km estimate in calculating the Ki value (Fig. 7). For competitive slow-binding inhibitors, if experiments are performed under conditions where the steady-state velocity response is approximately zero, and the sub-

strate concentration is several-fold greater than the Km value, the best that can be achieved is less than a twofold increase in maximal product concentration and the initial velocity at comparable levels of inhibition. For enzyme inhibitor systems that conform to Selwyn
s test [25], increasing enzyme concentration will only reduce the amount of time that inhibited progress curves must be monitored prior to achieving a set product concentration provided that the enzyme remains stable during the assay. Slow-binding inhibitors do not conform to Selwyn
s test [6] because the establishment of a steady-state equilibrium is slow in relation to substrate turnover. Fig. 2 demonstrates the increase in product concentration that can be achieved by working at elevated enzyme concentrations. However, the advantages gained by working at elevated enzyme concentrations will depend on the error structure of the enzyme assay. If the error in measuring product concentration increases with increasing product concentrations or substrate concentrations [23], then the benefits of improved product concentration signal will be reduced. No increase in the error of product concentration data was observed at elevated substrate or product concentrations using the oxyhemoglobin assay. The analysis of slowbinding inhibitors at elevated enzyme concentrations is most beneficial when implemented on assays with similar homoscedastic (i.e., constant absolute) error in product concentration data. The increase in the coefficients of variation of the simulated vo and kobs with increasing inhibitor concentration, as exhibited in Fig. 3, indicates the difficulty in accurately estimating the vo and kobs responses as the inhibition becomes more rapid at higher inhibitor concentrations (e.g., Fig. 4). The derivation of Eq. (1) assumes negligible depletion of substrate and free inhibitor. If higher enzyme concentrations are employed, then the substrate concentration must also be raised to ensure that these assumptions are adhered to in the experiment. Performing experiments at elevated enzyme and substrate concentrations will maximize signal-to-noise ratios and significantly reduce substrate depletion in inhibited progress curves in comparison with experiments performed at high enzyme concentrations and at substrate concentrations in the vicinity of the Km value. By raising the substrate and enzyme concentrations, the steady-state product concentration in case A versus case C (Table 1) increased 14fold at comparable levels of inhibition. In the derivation of Eq. (1), it is substrate depletion in the inhibited progress curves that is assumed to be negligible; no assumptions are made regarding substrate depletion in uninhibited progress curves. In practice, it is advantageous to collect data while the uninhibited progress curve remains linear so as to easily distinguish between nonlinearity in inhibited progress curves due to slowbinding inhibition and that due to substrate depletion [2]. An advantage of working at elevated substrate con-

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centrations is that the inherent linearity of uninhibited progress curves is increased [30,39]. For example, for the uninhibited progress curves simulated in case A in Table 1, the substrate depletion after 250 s was more than 70%, but a fit of uninhibited product concentration versus time data to a straight line gave an R2 value of 0.99. The drawback of working under these conditions is an increased reliance on an accurate estimate of the Km value, or an apparent Km value [6], provided that other substrates are present at saturating conditions (Fig. 7). The apparent Km value for the iNOS preparation was calculated by performing experiments on three separate days. The calculated coefficient of variation for the estimated Km values was 12%, indicating that an accurate estimate of the Km value is achievable for the iNOS-L -arginine system. For some systems, it might not be possible to raise the substrate concentration due to substrate inhibition [12]. The derivation of Eq. (1) also assumes negligible depletion of free inhibitor (i.e., no tight-binding effects between enzyme and inhibitor). Tight binding occurs when the free inhibitor concentration is significantly reduced due to binding with enzyme. Quantitatively, this effect has been defined to occur when the ratio of the total enzyme concentration over the Ki value is greater than 0.1 [40] or when the total inhibitor concentration that results in a reduction in uninhibited enzyme activity is similar to the total enzyme concentration [3,6]. If the enzyme concentration is raised to such an extent that tight-binding effects occur between inhibitor and enzyme, then Eqs. (1)–(3) are no longer valid and models that account for tight-binding effects must be used to analyze the resulting data [12,38]. A comparison of case A (high enzyme concentration and high assay noise) with case C (low enzyme concentration and high assay noise) in Table 1 shows the advantages of working at an elevated enzyme concentration. At the lower enzyme concentration, poor results are obtained due to low signal-to-noise ratios, but much improved results are obtained by increasing signal-tonoise ratios simply by performing experiments at higher enzyme concentrations. A comparison of case A with case D (low enzyme concentration and low assay noise) indicates that the results obtained between the two cases are in qualitative agreement. For example, the mean absolute percentage errors in Ki and k3 derived using calculation methods 1 and 2 and weighting scheme WNLR2 are very similar in case A and case D. This indicates that the apparent sensitivity of an assay for analyzing slow-binding inhibitors may be increased in proportion to the enzyme concentration used in the experiment. Therefore, if saturation data cannot be obtained when experiments are performed at a given enzyme concentration [14], then the enzyme concentration may be altered to increase the signal gen-

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erated in an experiment, thereby increasing signal-tonoise ratios and improving the apparent sensitivity of the assay. Table 1 indicates that consideration of the quality of the data obtained using a particular assay is an important factor in selecting the appropriate enzyme concentration at which to analyze slow-binding enzyme inhibitors of the type considered in this article. A further benefit of analyzing slow-binding inhibitors at elevated enzyme concentrations is that the effects of experimental drift or secondary interferences will be minimized. Duggleby et al. [24] commented that this effect would influence the vo response more heavily than the kobs response. If experiments are performed at higher enzyme concentrations, then the vo response will increase linearly with increasing enzyme concentrations [4,5]. Thus, the ratio of the vo response over the experimental drift will increase unless the drift also increases to a similar extent at higher enzyme concentrations [39,41]. In practice, the vs response may be included in Eq. (1) even if a slow-binding inhibitor is analyzed with vs approximately equal to zero because this will not be known a priori. Furthermore, any experimental drift in the enzyme assay that is not perfectly corrected for cannot be accounted for in nonlinear regression fits of product concentration versus time data to Eq. (1) if vs is set to zero. The ratio of the maximal kobs value over the calculated k3 value characterizing 1400W inhibition of iNOS in our previous study was just over 30% [13]. The ratio of the maximal kobs value observed over the calculated k3 value for the work performed by Garvey et al. [14] was less than 45%. By increasing the iNOS concentration from 10 to 100 nM, the ratio of the maximal kobs value observed over the calculated k3 value was increased to more than 70%, and experiments were performed at 2.3 times the apparent Ki value as opposed to 0.6 times the apparent Ki value (Fig. 6). Because data could be obtained at higher inhibitor concentrations, the saturation in the kobs response was much more evident in this study than in previous work [13,14]. In addition to improving the confidence in estimating k3, obtaining data at higher kobs values also improves the confidence of the Ki estimate using calculation method 2 because this value is defined as the inhibitor concentration that results in a kobs value of (k
3 + k3/2). The ratios of Ki over Km calculated in two previous studies and in this study are 0.9 [14], 1.0 [13], and 0.8, respectively, and are in good agreement. The ratios of the k3 over the estimated kcat values calculated in a previous study and in this study are 0.043 [13] and 0.045, respectively, also in close agreement. As anticipated, a second benefit of performing the experiments at higher iNOS concentrations was an improvement in the vo/vuh response (Fig. 5). The decrease in the vo/vuh response with increasing 1400W concentrations is clearly indicative of mechanism B. Ki

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values derived using calculation method 1 (i.e., using the vo/vuh response) or method 2 (i.e., using the kobs response) were in excellent agreement. A significant discrepancy in the Ki values obtained using calculation methods 1 and 2 will be observed if the uninhibited velocity, vuh, is not properly estimated [23,42], particularly at higher enzyme concentrations. Complete characterization of enzyme inhibitors conforming to mechanism B requires estimates of Ki, k3, and k
3. Determination of k
3 is difficult if enzyme species inactivate at rates that are much greater than k
3. Under these conditions, it might only be possible to estimate that k
3 is less than the inherent loss of uninhibited enzyme activity [14] as opposed to the actual calculation of k
3 [6,24]. For such enzyme inhibitor systems, the methods presented will allow for accurate estimation of Ki and k3 and determination of the inhibitory mechanism. This data may then be combined with the best estimate of k
3 obtained from experimental designs in which enzyme is preincubated with inhibitor to complete the characterization of the enzyme inhibitor system. The analysis of slow-binding inhibitors conforming to mechanism B with k3  k
3 is significantly improved by performing experiments at elevated enzyme concentrations, as confirmed by numerical simulation and experiments with the iNOS-1400W system. At elevated enzyme concentrations, data may be collected at higher inhibitor concentrations and more conclusive evidence of whether a system conforms to mechanism A or B may be obtained. Furthermore, it is possible to combine data obtained at lower enzyme and inhibitor concentrations with data obtained at higher enzyme and inhibitor concentrations (Eqs. (2) and (3)). Increasing the enzyme concentration at saturating inhibitor concentrations provides a simple method for extending the range of inhibitor concentrations that yield useful estimates of the vo and kobs responses while allowing previously collected data to be incorporated into the estimation of Ki and k3. This aspect of the methods developed should be appealing to researchers who might encounter low signal-to-noise ratios only after having invested significant time in performing experiments at nonsaturating inhibitor concentrations.

[2]

[3] [4] [5] [6]

[7]

[8]

[9]

[10]

[11]

[12]

[13]

[14]

[15] [16]

[17]

Acknowledgments

[18]

This research was supported by Grant 203286 to Eric Jervis and Grant 183521 to J. Guy Guillemette from the Natural Sciences and Engineering Research Council of Canada.

[19]

References

[21]

[1] S.R. Stone, J.F. Morrison, Mechanism of inhibition of dihydrofolate reductases from bacterial and vertebrate sources by various

[20]

[22]

classes of folate analogues, Biochim. Biophys. Acta 869 (1986) 275–285. J.F. Morrison, C.T. Walsh, The behavior and significance of slowbinding enzyme inhibitors, Adv. Enzymol. Relat. Areas Mol. Biol. 61 (1988) 201–301. S.E. Szedlacsek, R.G. Duggleby, Kinetics of slow and tightbinding inhibitors, Methods Enzymol. 249 (1995) 144–171. S. Cha, Tight-binding inhibitors: I. Kinetic behavior, Biochem. Pharmacol. 24 (1975) 2177–2185. S. Cha, Erratum, Biochem. Pharmacol. 25 (1976) 1561. J.F. Morrison, S.R. Stone, Approaches to the study and analysis of the inhibition of enzymes by slow and tight-binding inhibitors, Comments Mol. Cell. Biophys. 2 (1985) 347–368. M. Kapoor, C.C. Reddy, M.V. Krishnasastry, N. Surolia, A. Surolia, Slow-tight binding inhibition of enoyl-acyl carrier protein reductase from Plasmodium falciparum by triclosan, Biochem. J. 381 (2004) 719–724. A.R. Rezaie, Kinetics of factor Xa inhibition by recombinant tick anticoagulant peptide: both active site and exosite interactions are required for a slow- and tight-binding inhibition mechanism, Biochemistry 43 (2004) 3368–3375. V. Singh, W. Shi, G.B. Evans, P.C. Tyler, R.H. Furneaux, S.C. Almo, V.L. Schramm, Picomolar transition state analogue inhibitors of human 5 0 -methylthioadenosine phosphorylase and X-ray structure with MT-immucillin-A, Biochemistry 43 (2004) 9–18. R. Rawat, A. Whitty, P.J. Tonge, The isoniazid-NAD adduct is a slow, tight-binding inhibitor of InhA, the Mycobacterium tuberculosis enoyl reductase: adduct affinity and drug resistance, Proc. Natl. Acad. Sci. USA 100 (2003) 13881–13886. J. Pandhare, C. Dash, M. Rao, V. Deshpande, Slow tight binding inhibition of proteinase K by a proteinaceous inhibitor: conformational alterations responsible for conferring irreversibility to the enzyme-inhibitor complex, J. Biol. Chem. 278 (2003) 48735–48744. M. Sculley, J. Morrison, The determination of kinetic constants governing the slow, tight-binding inhibition of enzyme-catalysed reactions, Biochim. Biophys. Acta 874 (1986) 44–51. H.J. Montgomery, B. Perdicakis, D. Fishlock, G.A. Lajoie, E. Jervis, J.G. Guillemette, Photo-control of nitric oxide synthase activity using a caged isoform specific inhibitor, Bioorg. Med. Chem. 10 (2002) 1919–1927. E.P. Garvey, J.A. Oplinger, E.S. Furfine, R.L. Kiff, F. Laszlo, B.J.R. Whittle, R.G. Knowles, 1400W is a slow, tight binding, and highly selective inhibitor of inducible nitric-oxide synthase in vitro and in vivo, J. Biol. Chem. 272 (1997) 4959–4963. S.S. Gross, Microtiter plate assay for determining kinetics of nitric oxide synthesis, Methods Enzymol. 268 (1996) 159–168. D.K. Ghosh, M.B. Rashid, B. Crane, V. Taskar, M. Mast, M.A. Misukonis, J.B. Weinberg, N.T. Eissa, Characterization of key residues in the subdomain encoded by exons 8 and 9 of human inducible nitric oxide synthase: a critical role for Asp-280 in substrate binding and subunit interactions, Proc. Natl. Acad. Sci. USA 98 (2001) 10392–10397. J. Dawson, R.G. Knowles, A microtiter-plate assay of nitric oxide synthase activity, Mol. Biotechnol. 12 (1999) 275–279. J.M. Hevel, M.A. Marletta, Nitric-oxide synthase assays, Methods Enzymol. 233 (1994) 250–258. H.J. Montgomery, V. Romanov, J.G. Guillemette, Removal of a putative inhibitory element reduces the calcium-dependent calmodulin activation of neuronal nitric-oxide synthase, J. Biol. Chem. 275 (2000) 5052–5058. B. Perdicakis, H.J. Montgomery, J.G. Guillemette, E. Jervis, Validation and characterization of uninhibited enzyme kinetics performed in multiwell plates, Anal. Biochem. 332 (2004) 122–136. L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite, SIAM J. Sci. Comput. 18 (1997) 1–22. H.W. Blanch, D.S. Clark, Biochemical Engineering, Marcel Dekker, New York, 1997.

Slow-binding enzyme inhibitors / B. Perdicakis et al. / Anal. Biochem. 337 (2005) 211–223 [23] R.G. Duggleby, J.F. Morrison, The analysis of progress curves for enzyme-catalysed reactions by non-linear regression, Biochim. Biophys. Acta 481 (1977) 297–312. [24] R.G. Duggleby, P.V. Attwood, J.C. Wallace, D.B. Keech, Avidin is a slow-binding inhibitor of pyruvate carboxylase, Biochemistry 21 (1982) 3364–3370. [25] M.J. Selwyn, A simple test for inactivation of an enzyme during assay, Biochim. Biophys. Acta 105 (1965) 193–195. [26] J. Reich, C Curve Fitting and Modelling for Scientists and Engineers, McGraw-Hill, New York, 1992. [27] A. Cornish-Bowden, Analysis of Enzyme Kinetic Data, Oxford University Press, Oxford, UK, 1995. [28] S.G. Waley, The kinetics of slow-binding and slow, tight-binding inhibition: the effects of substrate depletion, Biochem. J. 294 (1993) 195–200. [29] E. Giroux, Fitting progress curves in assays of slow-binding enzyme inhibitors, J. Enzymol. Inhib. 5 (1991) 249–257. [30] R.G. Duggleby, R.B. Clarke, Experimental designs for estimating the parameters of the Michaelis–Menten equation from progress curves of enzyme-catalyzed reactions, Biochim. Biophys. Acta 1080 (1991) 231–236. [31] J.H. Ottaway, Normalization in the fitting of data by iterative methods: application to tracer kinetics and enzyme kinetics, Biochem. J. 134 (1973) 729–736. [32] A.C. Storer, M.G. Darlison, A. Cornish-Bowden, The nature of experimental error in enzyme kinetic measurements, Biochem. J. 151 (1975) 361–367.

223

[33] A. Cornish-Bowden, L. Endrenyi, Fitting of enzyme kinetic data without prior knowledge of weights, Biochem. J. 193 (1981) 1005– 1008. [34] N.R. Draper, H. Smith, Applied Regression Analysis, John Wiley, New York, 1998. [35] W.J. Conover, Practical Nonparametric Statistics, John Wiley, New York, 1999. [36] P. Kuzmic, Program DYNAFIT for the analysis of enzyme kinetic data: application to HIV proteinase, Anal. Biochem. 237 (1996) 260–273. [37] R.G. Duggleby, Analysis of enzyme progress curves by nonlinear regression, Methods Enzymol. 249 (1995) 61–90. [38] J.W. Williams, J.F. Morrison, R.G. Duggleby, Methotrexate, a high-affinity pseudosubstrate of dihydrofolate reductase, Biochemistry 18 (1979) 2567–2573. [39] A. Cornish-Bowden, Fundamentals of Enzyme Kinetics, Portland Press, London, 2004. [40] O.H. Straus, A. Goldstein, Zone behavior of enzymes: illustrated by the effect of dissociation constant and dilution on the system cholinesterase–physostigmine, J. Gen. Physiol. 26 (1943) 559– 585. [41] M. Feelisch, D. Kubitzek, J. Werringloer, The oxyhemoglobin assay, in: M. Feelisch, J.S. Stamler (Eds.), Methods in Nitric Oxide Research, John Wiley, New York, 1996, pp. 455–478. [42] R.G. Duggleby, Progress-curve analysis in enzyme kinetics: numerical solution of integrated rate equations, Biochem. J. 235 (1986) 613–615.