#
The Removal of Time–Concentration Data Points from Progress Curves Improves the Determination of K_{m}: The Example of Paraoxonase 1

^{*}

## Abstract

**:**

_{m}(Michaelis constant) from progress curves have been developed in recent decades. In the present article, we compare different approaches on a set of experimental measurements of lactonase activity of paraoxonase 1 (PON1): (1) a differential-equation-based Michaelis–Menten (MM) reaction model in the program Dynafit; (2) an integrated MM rate equation, based on an approximation of the Lambert W function, in the program GraphPad Prism; (3) various techniques based on initial rates; and (4) the novel program “iFIT”, based on a method that removes data points outside the area of maximum curvature from the progress curve, before analysis with the integrated MM rate equation. We concluded that the integrated MM rate equation alone does not determine kinetic parameters precisely enough; however, when coupled with a method that removes data points (e.g., iFIT), it is highly precise. The results of iFIT are comparable to the results of Dynafit and outperform those of the approach with initial rates or with fitting the entire progress curve in GraphPad Prism; however, iFIT is simpler to use and does not require inputting a reaction mechanism. Removing unnecessary points from progress curves and focusing on the area around the maximum curvature is highly advised for all researchers determining K

_{m}values from progress curves.

## 1. Introduction

_{m}has been the main kinetic parameter used to quantify the affinity of a given enzyme for its substrate. Traditionally, K

_{m}was determined by measuring the initial rate of the enzymatic reaction at several different substrate concentrations and then plotting the initial rates and substrate concentrations onto one of the linearized transformations of the Michaelis–Menten (MM) equation. These include the linear Lineweaver-Burk, Eadie-Hofstee, and Hanes-Woolf plots, each of which has its own (dis)advantages in terms of precision. With the aid of nonlinear regression methods and computers, it later also became possible to fit an MM model equation directly onto the initial rates without prior linearization of the data. With this, all points are given equal weight during fitting. However, all these approaches share a downside, namely, that only one (initial) rate–concentration data point is available from an individual enzymatic reaction performed in the laboratory.

_{0}); when the progress curve reaches its plateau, this term cannot be calculated anymore [2].

_{m}[7]. Such K

_{m}estimates can be considerably far from the true values; see also Figure A1 in Appendix A. A similar problem occurs when K

_{m}is small compared to [S]

_{0}, as the initial part of the progress curve is almost straight, and any information about K

_{m}can be lost in the noise.

_{m}values.

_{m}and V

_{max}. Based on their equation, we have developed an iterative method that determines the area of maximum curvature based on estimates of K

_{m}and V

_{max}and then proceeds to calculate K

_{m}and V

_{max}again from the points within this area. This process continues until the calculated area of maximum curvature remains constant [7]. To calculate K

_{m}and V

_{max}with this iterative method, we selected the integrated MM equation (based on the Lambert W function) and developed a short script in Python that calculates the area of maximum curvature from a progress curve and the resulting K

_{m}and V

_{max}values from this area. The working name of this script is hereafter referred to as iFIT; it is accessible at http://i-fit.si/ (last accessed 29 December 2021).

_{m}values and other kinetic parameters are known [8]. However, although noise is also simulated in such analyses, it is unlikely that this simulated approach fully encapsulates the variety of noise that can occur in an experimental setting. Hence, analysis of simulated progress curves can lead to overly optimistic appraisals of the software used.

## 2. Results

#### 2.1. The Nonenzymatic Constant for Dihydrocoumarin

_{NE}was derived from the 10 reactions performed in the absence of the enzyme. The rate constant was calculated according to Equation (1):

_{NE}values were then averaged, resulting in an k

_{NE}value of (0.084 ± 0.003) min

^{−1}, i.e., t

_{1/2}= (8.2 ± 0.3) min. Consequently, rePON1 lactonase activity was corrected for spontaneous hydrolysis of the substrate.

#### 2.2. Calculating K_{m} and V_{max} from the Initial Rates

_{0}= k

_{NE}· [S]

_{0}and subtracted from the appropriate initial rate of the entire reaction (which comprises the enzymatic and nonenzymatic reactions). Such a calculation increases the precision of the v

_{0}values.

_{0}values are imprecise because they account for neither (1) small errors in pipetting and (2) the small delay between the beginning of the reaction and the beginning of the measurement. Hence, we used adjusted [S]

_{0}values, hereafter denoted as [S]

_{0}*. The adjusted [S]

_{0}* that corresponds to the initial rate is the substrate concentration calculated retroactively from the progress curve (see equation in Section 4.4, paragraph 2). This can be achieved when the enzyme-catalyzed reaction is irreversible or when the ratio of reactants and products at equilibrium is known.

_{0}* values were then used throughout our data analysis due to their accuracy, except for the calculations described in the rightmost two columns of Table 1. The differences between the initially assumed and adjusted substrate concentrations ranged from 0.38% (90.34 vs. 90 μM) to 58% (12.47 vs. 30 μM) (Table 2).

_{0}= V

_{max}· [S]/(K

_{m}+ [S]). The best-fit values for K

_{m}and V

_{max}were 31 ± 6 µM (95% CI: 31 ± 11 µM) and 99 ± 4 µM/min (95% CI: 99 ± 8 µM/min), respectively (Figure 1a).

_{m}and V

_{max}values were 33 ± 4 µM (95% CI: 33 ± 8 µM) and 99 ± 3 µM/min (95% CI: 99 ± 6 µM/min), respectively. The correlation coefficient and R-squared were R = 0.84 and R

^{2}= 0.71, respectively (Figure 1b).

_{m}and V

_{max}values were 34 ± 8 µM (95% CI: 34 ± 16 µM) and 100 ± 1 µM/min (95% CI: 100 ± 1 µM/min), respectively. The correlation coefficient and R-squared were R = 0.98 and R

^{2}= 0.95, respectively (Figure 1c).

_{m}and V

_{max}were 35 ± 2 µM (95% CI: 35 ± 5 µM) and 100 ± 3 µM/min (95% CI: 100 ± 6 µM/min), respectively. The correlation coefficient and R-squared were R = 0.96 and R

^{2}= 0.93, respectively (Figure 1d).

_{m}and V

_{max}without accounting for the nonenzymatic reaction and without adjusting the concentration of S

_{0}was also tested. All the K

_{m}and V

_{max}values are shown in Table 1.

#### 2.3. Calculating Kinetic Parameters with the Integrated MM Equation in Prism: The Entire Curve

_{m}values was assessed with the Kolmogorov–Smirnov test. The value of the K-S test statistic (D) was 0.103, and the p value was 0.879, indicating that the data did not differ significantly from normally distributed data.

_{m}and V

_{max}obtained from the fit depended on the substrate concentration (Table 2, Prism, entire curve). The average of all 30 measurements of K

_{m}for rePON1 was 36 ± 11 µM (95% CI: 36 ± 4 µM); however, these average values are unreliable because the output K

_{m}is dependent on substrate concentration. The average of all 30 V

_{max}values was 109 ± 15 µM/min (95% CI: 109 ± 6 µM/min).

#### 2.4. Calculating Kinetic Parameters with Dynafit

_{1}, k

_{−1}, and k

_{2}must be inserted by the user after the program models the values of these parameters, and the user must calculate K

_{m}from the results. It is also possible for Dynafit to model K

_{m}and V

_{max}directly; however, modeling microscopic parameters allows for the nonenzymatic constant (k

_{NE}= 0.084 min

^{−1}) to be included in the model, which is the main benefit of Dynafit.

_{m}values was assessed with the Kolmogorov–Smirnov test. The value of the K-S test statistic (D) was 0.129, and the p value was 0.656, indicating that the data did not differ significantly from normally distributed data. The kinetic parameters K

_{m}and V

_{max}obtained from the fit did not depend on the substrate concentration (Table 2, Dynafit). Thus, the average of the 30 K

_{m}values was calculated from the progress curves and was 27 ± 4 µM (95% CI: 27 ± 1.4 µM). The average of all 30 V

_{max}values was 91 ± 12 µM/min (95% CI: 91 ± 4 µM/min).

#### 2.5. Calculating Kinetic Parameters with iFIT

_{Q}selected for fitting will be roughly proportional to K

_{m}/V

_{max}. Since our progress curves with DHC as a substrate had a low K

_{m}compared to V

_{max}(Table 2), only a small number of points from the progress curve were ultimately analyzed by iFIT. In cases where a progress curve began to be measured immediately and reached its plateau, iFIT will generally trim points both from the left and the right side.

_{m}and V

_{max}obtained from the fit do not depend on the substrate concentration (Table 2, iFIT). Thus, the average of the 29 K

_{m}values was calculated and was 27 ± 5 µM (95% CI: 27 ± 2 µM). The average of all 30 V

_{max}values was 96 ± 11 µM/min (95% CI: 96 ± 4 µM/min). The normality of the distribution of output K

_{m}values was assessed with the Kolmogorov–Smirnov test. The value of the K-S test statistic (D) was 0.114, and the p value was 0.807, indicating that the data did not differ significantly from normally distributed data.

#### 2.6. The Relationship between Substrate Concentration and Calculated K_{m}

_{m}values determined from the progress curves (Table 2) was the positive correlation between K

_{m}and the adjusted substrate concentration ([S]

_{0}*). The relationship between [S]

_{0}* and K

_{m}for all three programs is shown in Figure 4. The relationship was linear and varied among the three programs used for fitting. With Dynafit and iFIT, the correlation was weak (or no correlation was seen), and K

_{m}hardly increased with increases in [S]

_{0}*. When straight lines were fitted onto the relationship between K

_{m}and [S]

_{0}*, they resulted in the following relationships: K

_{m}= 0.0227 · [S]

_{0}* + 23.35 µM, with R = 0.443 and R

^{2}= 0.196 (for iFIT) and K

_{m}= 0.0173 · [S]

_{0}* + 23.51 µM, with R = 0.525 and R

^{2}= 0.276 (for Dynafit). With Prism (the entire curve), the correlation was strong: K

_{m}= 0.0842 · [S]

_{0}* + 21.48 µM, with R = 0.887 and R

^{2}= 0.787 (Figure 4).

#### 2.7. The Relationship between Different Programs for Calculating K_{m} from Progress Curves

_{m}and [S]

_{0}*, the correlations among the results of the three tools for progress curve analysis were also analyzed (Figure 5). The iFIT–Dynafit and Prism (the entire curve)–Dynafit correlations were strong (R = 0.83 versus R = 0.82 and R

^{2}= 0.69 versus R

^{2}= 0.67, respectively), while the Prism (the entire curve)–iFIT correlation was moderate (R = 0.66 and R

^{2}= 0.43). The K

_{m}values for iFIT were on average slightly higher than those for Dynafit (the slope of the line was 1.15). The K

_{m}values for Prism (the entire curve) were higher than those for both Dynafit and iFIT (the slopes of the lines were 2.36 and 1.36, respectively).

## 3. Discussion

_{m}value for the reaction between PON1 and DHC has seldom been reported in studies so far: it has been reported, e.g., for human serum PON1 [15,20], for rat serum PON1 [21], and for rePON1 [16]. Certain specific challenges that DHC presents as a substrate, e.g., its rate constant of nonenzymatic hydrolysis (t

_{1/2}= 8.25 min), have hardly been mentioned in the literature at all. We therefore wish to contribute to a more thorough understanding of the kinetic parameters of PON1-induced DHC breakdown, which should be useful to all future researchers investigating the lactonase activity of PON1 in experimental as well as clinical contexts.

_{m}value that has been reported in the published literature for DHC and rePON1 G2E6 is 129 ± 8 µM [16]. There is a large discrepancy between this value and our results (K

_{m}was 27, 27, and 31 µM for Dynafit, iFIT, and the MM diagram, respectively; the Prism (the entire curve) average is not included because of the dependence of the output K

_{m}on substrate concentration). Khersonsky and Tawfik [16] did not account for the nonenzymatic hydrolysis of DHC in water, which we have shown to be substantial, and used a relatively high ionic strength for their buffer supplemented with the detergent tergitol. In PON1, we observed that high ionic strength produces higher K

_{m}values when working with DHC. Additionally, not accounting for nonenzymatic hydrolysis produces higher apparent K

_{m}values when calculating K

_{m}from initial rates.

_{0}values instead of the adjusted ones and not subtracting the nonenzymatic reaction from the initial rates resulted in K

_{m}values more than twice as high as those acquired when S

_{0}and v

_{0}were properly adjusted (Table 1). The K

_{m}value calculated without accounting for the nonenzymatic reaction and without adjusting the concentration of S

_{0}was 74 ± 11 μM. This value is closer to the reported value from the literature [16]. Unfortunately, this means that we have no consensus on a specific K

_{m}value against which to compare our results.

_{0}* values are those calculated retroactively from the progress curves. They are usually smaller than the initially assumed concentrations because they do not include the amount of substrate that was consumed before the start of the measurement; this is especially relevant for very small substrate concentrations. Most researchers working with initial rates do not record the entire progress curve and thus never know how far off their substrate concentrations might have been from the initially assumed concentrations; this is another advantage of working with entire progress curves, regardless of which analysis we perform afterwards. Nevertheless, once we record the entire progress curve, it makes more sense to directly extract the kinetic parameters from the progress curve rather than use the initial rates approach.

_{m}that we used (four based on the MM diagram and two based on progress curves; the approach of fitting the entire curve in Prism is not included here for the abovementioned reasons) reveals that the average K

_{m}value ranges between 27 µM for Dynafit and 35 µM for the Lineweaver–Burk method. The results can be grouped into two clusters, with very similar average results acquired with Dynafit and iFIT (27 µM in both cases) and by the four methods based on the MM diagram (31–35 µM). If we consider all these approaches as equally valid, we can conclude that the K

_{m}of rePON1 for DHC is approximately 30 µM. The clusters are less pronounced for the V

_{max}values. The average V

_{max}values are 99–100 μM/min for all initial rate approaches, 96 μM/min for iFIT, and 91 μM/min for Dynafit.

_{m}can be considered equally reliable. The integrated MM equation in GraphPad Prism (hereafter referred to as only “Prism” in the Discussion) is particularly problematic because the increasing substrate concentration has a major effect on the K

_{m}output value. The major reason for this is most likely the nonenzymatic hydrolysis of the substrate, a first-order reaction. Hence, as we increase substrate concentration, nonenzymatic hydrolysis will play a proportionally increasing role in the total reaction and will have the greatest impact at the beginning of the progress curve. When fitting the model function to the entire progress curve, the consequence will be that the model function will no longer fit well to the area of maximum curvature (see Figure A1 in Appendix A). DHC is unusual due to its rate of nonenzymatic hydrolysis; however, many other substrates also decompose spontaneously in the absence of an enzyme. This can be especially problematic if researchers are not aware of the substrate’s properties.

_{m}. Intuitively, we might expect Prism and iFIT to produce similar results (since they both use the same integrated MM equation that does not directly account for side reactions) and Dynafit to produce different results (since it uses a system of differential equations and accounts for nonenzymatic hydrolysis). When calculating K

_{m}and V

_{max}in Dynafit, the user must calculate the three microscopic parameters, k

_{1}, k

_{−1}, and k

_{2}, separately and then calculate K

_{m}from them according to Equation (2).

_{m}that is more precise than working with entire progress curves. The precision of iFIT, i.e., the small dispersal of output K

_{m}values around the mean, is evident, as the standard deviations of the calculated results were much smaller than those for the integrated MM equation in Prism (Table 2). Furthermore, the increase in K

_{m}with substrate concentration was also much smaller. Even more notably, we observed that if we removed the six output K

_{m}values obtained from the progress curves with the lowest substrate concentration (Figure 4), the correlation between substrate concentration and K

_{m}disappeared entirely.

_{m}values [2,8,10]. In some cases, this also led to recommendations regarding the ideal concentration range of the substrate; however, this range differs between different fitting methods. For example, Duggleby and Clarke [2] suggested a [S]

_{0}value around 2.5 · K

_{m}to minimize error, while they claimed that their acquired K

_{m}values were independent of substrate concentration. For the method we used in iFIT, the recommendation (by Stroberg and Schnell) that [S]

_{0}should be in the order of magnitude of K

_{m}is especially pertinent.

_{m}(which depends on the V

_{max}value from which the area of maximum curvature is also calculated). If possible, it is advised to not go below this [S]

_{0}, as this would deprive iFIT of some of the points it uses for analysis.

^{−1}cm

^{−1}, it is difficult to measure reactions with <15 µM DHC (as the noise obscures the signal) or >800 µM DHC (as the upper limits of most spectrometers’ sensitivities are reached). We observed from our data that the K

_{m}values calculated from progress curves with a low substrate concentration were slight outliers in iFIT. We also observed that iFIT can encounter issues with progress curves measured at a very low substrate concentration. Consequently, the optimum substrate concentration for DHC is between 2 and 4× K

_{m}. For other enzyme–substrate reactions, we would like to amend Stroberg and Schnell’s recommendation and suggest using at least 2× K

_{m}as the initial substrate concentration in order to seize the full potential of the data-point-reduction approach that we present in the current article.

_{m}.

_{m}than the integrated MM equation in Prism. The implication of both of these sets of results is that users should not be afraid to replace the whole-curve approach with iFIT.

_{m}and V

_{max}, then of course iFIT cannot be used for progress curve analysis. Similar considerations hold for reaction mixtures with several enzymes, none of which contribute predominantly to the overall reaction. Before using iFIT, researchers should check whether MM kinetics is a reasonable approximation for the behavior of their enzyme.

## 4. Materials and Methods

#### 4.1. Chemicals

#### 4.2. Recombinant PON1 Expression and Purification

_{2}. The culture was grown at 37 °C for 17 h. Then, 500 mL of LB medium containing 100 µg/mL ampicillin, 25 µg/mL chloramphenicol, and 1 mM CaCl

_{2}was inoculated with 5 mL of overnight culture and grown at 37 °C to an OD600 of 0.7. The expression of the rePON1 variant was induced by adding 1 mM isopropyl β-D-1-thiogalactopyranoside, and the culture was grown at 25 °C for 17 h. The cells were harvested by centrifugation at 6000× g for 15 min, and the pellet was stored overnight at −20 °C. The cells were resuspended in 30 mL of lysis buffer (50 mM Tris, pH = 8.0, 1 mM CaCl

_{2}, and 0.1 mM dithiothreitol supplemented with 1 µM pepstatin A, 1 mM phenylmethylsulfonyl fluoride, and 0.03% n-dodecyl-β-D-maltopyranoside (C12-maltoside)) and lysed by sonification. The lysate was centrifuged at 10,000× g for 10 min, and the supernatant was stirred for 1 h at 4 °C. After centrifugation at 20,000× g for 20 min, the soluble fraction was treated with ammonium sulphate (55%, w/v, at 0 °C). The precipitate was centrifuged at 10,000× g for 15 min, resuspended, and dialyzed twice against lysis buffer supplemented with 0.01% C12-maltoside. After dialysis, the protein was added to nitrilotriacetic acid resin, and the mixture was gently shaken overnight at 4 °C. The resin was first washed with lysis buffer with 0.03% C12-maltoside and then with 10 and 20 mM imidazole in lysis buffer with 0.03% C12-maltoside. It was finally eluted with 150 mM imidazole in lysis buffer with 0.03% C12-maltoside. Fractions with the highest rePON1 activity were pooled, dialyzed, and further purified by ion-exchange chromatography. The protein was applied to a 5 mL HighTrap Q HP column (GE Healthcare, City, Marlborough, MA, USA) with a linear gradient from 26% to 33% of buffer B (20 mM Tris, pH = 8.0, 1 mM CaCl

_{2}, 0.1 mM dithiothreitol, 0.03% C12-maltoside, 1 M NaCl) in buffer A (buffer B without 1 M NaCl). Fractions with the highest rePON1 activity were analyzed on an 11% SDS-PAGE gel, pooled, dialyzed against buffer A, and concentrated. Finally, sodium azide (0.02%) was added, and the protein was stored at −70 °C.

#### 4.3. Measurements of the Lactonase Activity of RePON1

_{2}at 25 °C using a Genesys 10S spectrophotometer (Thermo Fisher Scientific, Waltham, MA, USA). A total of 10 µL of rePON1 was added to the buffer, corresponding to a final concentration of 0.01 µM. After the addition of 20 µL of DHC, the increase in absorbance was measured at 270 nm (ε = 1310 M

^{−1}cm

^{−1}). Absorbance measurements were obtained every second. The entire progress curve was recorded, and the recording was stopped when the curve plateaued for at least 1 min. Each progress curve was recorded in three independent experiments, resulting in a total of 30 measurements. The rates (v

_{0}) of spontaneous hydrolysis for all 10 substrate concentrations were also measured. Consequently, rePON1 activity was corrected for spontaneous hydrolysis of each substrate.

#### 4.4. Determination of K_{m} of RePON1

_{m}and V

_{max}from the progress curves. Three different programs for progress curve analysis as well as the classical approach of measuring initial rates were used. When working with initial rates, three different versions of data linearization were used; the MM equation was also fitted directly onto the initial rate data points.

#### 4.4.1. Initial Rate Measurements

_{0}) values can differ based on how many points they are calculated from [25]. To acquire the most precise possible estimate, v

_{0}was not calculated by fitting a straight line directly onto the beginning of the curve, as this would only give an intermediate value for the reaction rate during the early part of the reaction. Instead, we calculated v

_{0}at t = 0 from the first derivative of the progress curve, as described by Hasinoff [26].

_{0}, the following were compared: (1) the MM analysis provided in the GraphPad Prism package, (2) the Lineweaver–Burk method, (3) the Eadie–Hofstee method, and (4) the Hanes–Woolf method. The necessary parameters (K

_{m}, V

_{max}, and the quality-of-fit estimates) were calculated in Prism in all four cases.

#### 4.4.2. Numerical Integration of Differential Equations in Dynafit

_{m}can then be calculated according to Equation (2):

_{m}and V

_{max}, i.e., the rate constants k

_{1}, k

_{−1}, or k

_{2}, Dynafit is the only program that can also include additional kinetic parameters in the model. Most importantly, it can include a kinetic constant for nonenzymatic substrate hydrolysis (which we denoted as k

_{NE}), an important factor in several PON1-mediated enzymatic reactions.

#### 4.4.3. An Integrated MM Rate Equation in Prism: The Entire Curve

_{m}and V

_{max}as well as the errors for these parameters. Unfortunately, there is no automatic (scripted) way within Prism to exclude experimental points from the fit. Unnecessary points are especially problematic for curves that continue for a long time after plateauing. Consequently, imperfect fits are made at the expense of the area of maximum curvature (Figure A1 in Appendix A), which contains the most information about K

_{m}and V

_{max}.

#### 4.4.4. An Integrated MM Rate Equation in iFIT: The Area of Highest Curvature

_{Q}) over which the progress curve exhibits the maximal curvature based on K

_{m}and V

_{max}values was presented by Stroberg and Schnell (Equation (5)) [7]:

_{m}and V

_{max}from the high-curvature regions of progress curves. iFIT uses a script in Python to remove data points on both sides of the maximum-curvature region and fits the rest of the progress curve with Equation (4).

_{m}and V

_{max}from the entire available progress curve using the integrated Michaelis–Menten equation described above. From these, it then calculates the width of the area of maximum curvature, denoted as t

_{Q}, and removes all points from the progress curve, which are more than t

_{Q}/2 away from the point of inflection. From the remaining points, it then calculates K

_{m}and V

_{max}with the integrated MM equation again, uses these to calculate t

_{Q}, and so on, until the calculated t

_{Q}converges to the same value, i.e., the selected experimental time–concentration points are the same in two successive iterations. If this does not happen within 100 iterations, the program reports all 100 steps at the end of the calculation. The iterations may not converge when there is excessive noise in the experimental data. In such cases, the program might oscillate between two or more t

_{Q}values, or t

_{Q}might drift outside the actual area of experimental data points, in which case there will not be an output result.

#### 4.5. Statistics

_{m}and V

_{max}are expressed as mean ± standard deviation. Additionally, the correlation coefficients, the 95% confidence intervals, and R-squared goodness-of-fit measures for linear regression models are also shown for each group of data. For all the groups of output values, the normality of their distribution was calculated with the Kolmogorov–Smirnov test. Differences were considered statistically significant at p < 0.05 (GraphPad Prism).

## 5. Conclusions

_{m}, continues to be highly underutilized in studies that measure human PON1 activity in various medical conditions. This is particularly the case with lactonase activity; not a single study presented in Petrič et al. [19] measured K

_{m}in human patients. Furthermore, it has been suggested that K

_{m}should be more routinely included by researchers when comparing interindividual enzymatic activity to detect differences that might not be noticed in a simple comparison of a specific activity [28]. By providing a framework for measuring the K

_{m}of PON1 for DHC, we hope that we have facilitated future studies of such kinds on PON1.

_{m}values from progress curves consider such a method, which selects points from the curve’s maximum curvature area, as this increases the precision of their fitted K

_{m}values.

## Supplementary Materials

_{m}for Data set 1 that were calculated by Prism, iFIT and Dynafit for penicillin amidase, Table S2: The values of K

_{m}for Data set 2 that were calculated by Prism, iFIT and Dynafit for rat butyrylcholinesterase.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Figure A1.**The integrated Michaelis–Menten equation being fitted in Prism to the same progress curve (blue line) that is shown in Figure 3. The model function (red line) appears to be a good fit for the progress curve. However, once we zoom into the area of maximum curvature, it is clear that the fit is no longer as good, which results in an erroneous K

_{m}value.

## References

- Yun, S.L.; Suelter, C.H. A simple method for calculating Km and V from a single enzyme reaction progress curve. BBA—Enzymol.
**1977**, 480, 1–13. [Google Scholar] [CrossRef] - Duggleby, R.G.; Clarke, R.B. Experimental designs for estimating the parameters of the Michaelis-Menten equation from progress curves of enzyme-catalyzed reactions. Biochim. Biophys. Acta (BBA)—Protein Struct. Mol. Enzymol.
**1991**, 1080, 231–236. [Google Scholar] [CrossRef] - Schnell, S.; Mendoza, C. Closed Form Solution for Time-dependent Enzyme Kinetics. J. Theor. Biol.
**1997**, 187, 207–212. [Google Scholar] [CrossRef] - Goličnik, M. On the Lambert W function and its utility in biochemical kinetics. Biochem. Eng. J.
**2012**, 63, 116–123. [Google Scholar] [CrossRef] - Kuzmič, P. Program DYNAFIT for the Analysis of Enzyme Kinetic Data: Application to HIV Proteinase. Anal. Biochem.
**1996**, 237, 260–273. [Google Scholar] [CrossRef] [PubMed] - Bevc, S.; Konc, J.; Stojan, J.; Hodošček, M.; Penca, M.; Praprotnik, M.; Janežič, D. {ENZO}: A Web Tool for Derivation and Evaluation of Kinetic Models of Enzyme Catalyzed Reactions. PLoS ONE
**2011**, 6, e22265. [Google Scholar] [CrossRef] - Stroberg, W.; Schnell, S. On the estimation errors of KM and V from time-course experiments using the Michaelis–Menten equation. Biophys. Chem.
**2016**, 219, 17–27. [Google Scholar] [CrossRef] - Cho, Y.; Lim, H. Comparison of various estimation methods for the parameters of Michaelis—Menten equation. Transl. Clin. Pharmacol.
**2018**, 26, 39–47. [Google Scholar] [CrossRef][Green Version] - Nikolova, N.; Tenekedjiev, K.; Kolev, K. Uses and misuses of progress curve analysis in enzyme kinetics. Cent. Eur. J. Biol.
**2008**, 3, 345–350. [Google Scholar] [CrossRef][Green Version] - Zavrel, M.; Kochanowski, K.; Spiess, A.C. Comparison of different approaches and computer programs for progress curve analysis of enzyme kinetics. Eng. Life Sci.
**2010**, 10, 191–200. [Google Scholar] [CrossRef] - Marchegiani, F.; Marra, M.; Spazzafumo, L.; James, R.W.; Boemi, M.; Olivieri, F.; Cardelli, M.; Cavallone, L.; Bonfigli, A.R.; Franceschi, C. Paraoxonase Activity and Genotype Predispose to Successful Aging. J. Gerontol. A Biol. Sci. Med. Sci.
**2006**, 61, 541–546. [Google Scholar] [CrossRef] [PubMed][Green Version] - Seres, I.; Paragh, G.; Deschene, E.; Fulop, T.; Khalil, A. Study of factors influencing the decreased HDL associated PON1 activity with aging. Exp. Gerontol.
**2004**, 39, 59–66. [Google Scholar] [CrossRef] [PubMed] - She, Z.G.; Chen, H.Z.; Yan, Y.; Li, H.; Liu, D.P. The human paraoxonase gene cluster as a target in the treatment of atherosclerosis. Antioxid. Redox Signal.
**2012**, 16, 597–632. [Google Scholar] [CrossRef] [PubMed] - Aharoni, A.; Gaidukov, L.; Yagur, S.; Toker, L.; Silman, I.; Tawfik, D.S. Directed evolution of mammalian paraoxonases PON1 and PON3 for bacterial expression and catalytic specialization. Proc. Natl. Acad. Sci. USA
**2004**, 101, 482–487. [Google Scholar] [CrossRef][Green Version] - Billecke, S.; Draganov, D.; Counsell, R.; Stetson, P.; Watson, C.; Hsu, C.; La Du, B.N. Human serum paraoxonase (PON1) isozymes Q and R hydrolyze lactones and cyclic carbonate esters. Drug Metab. Dispos.
**2000**, 28, 1335–1342. [Google Scholar] - Khersonsky, O.; Tawfik, D.S. Structure-reactivity studies of serum paraoxonase PON1 suggest that its native activity is lactonase. Biochemistry
**2005**, 44, 6371–6382. [Google Scholar] [CrossRef] - Harel, M.; Aharoni, A.; Gaidukov, L.; Brumshtein, B.; Khersonsky, O.; Meged, R.; Dvir, H.; Ravelli, R.B.; McCarthy, A.; Toker, L.; et al. Structure and evolution of the serum paraoxonase family of detoxifying and anti-atherosclerotic enzymes. Nat. Struct. Mol. Biol.
**2004**, 11, 412–419. [Google Scholar] [CrossRef] - Ceron, J.J.; Tecles, F.; Tvarijonaviciute, A. Serum paraoxonase 1 (PON1) measurement: An update. BMC Vet. Res.
**2014**, 10, 74. [Google Scholar] [CrossRef] - Petrič, B.; Kunej, T.; Bavec, A. A Multi-Omics Analysis of PON1 Lactonase Activity in Relation to Human Health and Disease. OMICS
**2021**, 25, 38–51. [Google Scholar] [CrossRef] - Goličnik, M.; Bavec, A. Evaluation of the paraoxonase-1 kinetic parameters of the lactonase activity by nonlinear fit of progress curves. J. Enzym. Inhib. Med. Chem.
**2020**, 35, 261–264. [Google Scholar] [CrossRef][Green Version] - Sierra-Campos, E.; Valdez-Solana, M.; Avitia-Domínguez, C.; Campos-Almazán, M.; Flores-Molina, I.; García-Arenas, G.; Téllez-Valencia, A. Effects of Moringa oleifera Leaf Extract on Diabetes-Induced Alterations in Paraoxonase 1 and Catalase in Rats Analyzed through Progress Kinetic and Blind Docking. Antioxidants
**2020**, 9, 840. [Google Scholar] [CrossRef] [PubMed] - Goličnik, M. Alternative algebraic rate-integration approach for progress-curve analysis of enzyme kinetics. Eng. Life Sci.
**2012**, 12, 104–108. [Google Scholar] [CrossRef] - Stojan, J. Rapid Mechanistic Evaluation and Parameter Estimation of Putative Inhibitors in a Single-Step Progress-Curve Analysis: The Case of Horse Butyrylcholinesterase. Molecules
**2017**, 22, 1248. [Google Scholar] [CrossRef] [PubMed][Green Version] - Bavec, A.; Knez, D.; Makovec, T.; Stojan, J.; Gobec, S.; Goličnik, M. Exploring the aryl esterase catalysis of paraoxonase-1 through solvent kinetic isotope effects and phosphonate-based isosteric analogues of the tetrahedral reaction intermediate. Biochimie
**2014**, 106, 184–186. [Google Scholar] [CrossRef] - Marsillach, J.; Richter, R.J.; Costa, L.G.; Furlong, C.E. Paraoxonase-1 (PON1) status analysis using non-organophosphate substrates. Curr. Protoc.
**2021**, 1, e25. [Google Scholar] [CrossRef] - Hasinoff, B. A First-Derivative Method for the Analysis of Michaelis Enzyme Progress Curves. Can. Inst. Food Sci. Technol. J.
**1986**, 19, 181–185. [Google Scholar] [CrossRef] - Ben-David, M.; Elias, M.; Filippi, J.J.; Duñach, E.; Silman, I.; Sussman, J.L.; Tawfik, D.S. Catalytic versatility and backups in enzyme active sites: The case of serum paraoxonase 1. J. Mol. Biol.
**2012**, 418, 181–196. [Google Scholar] [CrossRef] - Veskoukis, A.S.; Paschalis, V.; Kyparos, A.; Nikolaidis, M.G. Administration of exercise-conditioned plasma alters muscle catalase kinetics in rat: An argument for in vivo-like K(m) instead of in vitro-like V(max). Redox Biol.
**2018**, 15, 375–379. [Google Scholar] [CrossRef]

**Figure 1.**Determination of K

_{m}of rePON1 from the initial rates. (

**a**) The Michaelis–Menten diagram: K

_{m}= 31 ± 6 µM. (

**b**) The Eadie–Hofstee diagram: K

_{m}= 33 ± 4 µM, R = 0.84, R

^{2}= 0.71. (

**c**) The Woolf–Hanes diagram: K

_{m}= 34 ± 8 µM, R = 0.98, R

^{2}= 0.95. (

**d**) The Lineweaver–Burk diagram: K

_{m}= 35 ± 2 µM, R = 0.96, R

^{2}= 0.93. The [S]

_{0}* are the initial substrate concentrations calculated from the progress curves.

**Figure 2.**The fit of kinetic progress curve data at different substrate concentrations. Symbols represent absorbance readings converted into concentrations at the given reaction time. For clarity, only one progress curve per experiment and only 1/10 of the data points per progress curve are shown. Smooth lines represent least-square model curves generated by the fit of Equation (4) with the parameter values obtained by the modified model shown in Table 2 (Prism, entire curve). The concentrations shown on the right are the initially assumed ones; the actual (adjusted) concentrations in the cuvettes were calculated from the progress curves and used in all subsequent calculations. The 3rd parallel run is shown. The curves do not originate from the point (0,0); hence, the curves’ plateaus are higher than the S

_{0}* values displayed in Table 2.

**Figure 3.**The fit of kinetic progress curve data by iFIT. The entire curve is displayed on the (

**left**), whereas only points from the area of highest curvature were included in the fit (

**right**). The blue line represents absorbance readings converted into concentration at the given reaction time. Only one progress curve is shown, for clarity. The red line represents the least-square model curve generated by fitting with Equation (4) after the iteration process, with the parameter values obtained by the modified model shown in Table 2 (iFIT). The progress curve is the 400 µM curve from the 3rd parallel run shown in Figure 2 and Table 2.

**Figure 4.**The relationship between adjusted substrate concentration ([S]

_{0}*) and K

_{m}for Dynafit (circles), iFIT (squares), and Prism (the entire curve) (triangles) for all 30 measurements. Lines represent least-square model curves generated by a linear fit. The correlation coefficients and R-squared of the best-fit lines are R = 0.443 and R

^{2}= 0.196 (iFIT), R = 0.525 and R

^{2}= 0.276 (Dynafit), and R = 0.887 and R

^{2}= 0.787 (Prism: the entire curve), respectively. The concentrations displayed were calculated retrospectively from the progress curves (Table 2).

**Figure 5.**The relationship between output K

_{m}values for the same progress curves analyzed by three different programs: (

**a**) iFIT vs. Dynafit, (

**b**) Prism (the entire curve) vs. Dynafit, and (

**c**) Prism (the entire curve) vs. iFIT. On each graph, all K

_{m}calculations are shown. The linear fits of the relationships are as follows: (

**a**) (iFIT K

_{m}) = 1.15 · (Dynafit K

_{m}) − 3.47 µM; (

**b**) (Prism (the entire curve) K

_{m}) = 2.36 · (Dynafit K

_{m}) − 26.26 µM; and (

**c**) (Prism (the entire curve) K

_{m}) = 1.36 · (iFIT K

_{m}) − 0.17 µM. The R and R

^{2}values for all three comparisons are shown in (

**d**).

**Figure 6.**The Michaelis–Menten reaction scheme, with the nonenzymatic decay of substrate added in the bottom line. The formation of the enzyme–substrate complex is the “slow” step of the overall reaction, whereas the formation of product is the “fast” step. If k

_{−1}= 0, the reaction proceeds according to the Van Slyke–Cullen mechanism.

**Table 1.**The K

_{m}and V

_{max}values for all four approaches for determining the kinetic parameters from the initial rates. The left side of the table shows the K

_{m}and V

_{max}values that were acquired by calculating substrate concentration from the progress curve (i.e., [S]

_{0}*) and by accounting for the nonenzymatic hydrolysis of substrate. The right side shows the K

_{m}and V

_{max}values when these two factors were not accounted for. All values are shown with standard deviations.

Method | [S]_{0} * Adjusted, Nonenzymatic Reaction Accounted for | [S]_{0} Not Adjusted, Nonenzymatic Reaction Not Accounted for | ||
---|---|---|---|---|

K_{m} (µM) | V_{max} (µM/Min) | K_{m} (µM) | V_{max} (µM/Min) | |

MM diagram | 31 ± 6 | 99 ± 4 | 74 ± 11 | 138 ± 6 |

Eadie–Hofstee | 33 ± 4 | 99 ± 3 | 62 ± 7 | 130 ± 5 |

Woolf–Hanes | 34 ± 8 | 100 ± 1 | 76 ± 10 | 138.0 ± 0.2 |

Lineweaver–Burk | 35 ± 2 | 100 ± 3 | 76 ± 5 | 137 ± 6 |

**Table 2.**The values of [S]

_{0}*, K

_{m}, and V

_{max}for all 30 rePON1 measurements that were calculated by three programs for progress curve analysis: Dynafit, Prism (i.e., the integrated MM equation: the entire curve), and iFIT (i.e., the integrated MM equation: the area of the highest curvature). The [S]

_{0}values in the leftmost column are the initially assumed concentrations. The adjusted [S]

_{0}values ([S]

_{0}*) were calculated retroactively from the progress curves. AVG: average; STDEV: standard deviation.

[S]_{0}* (μM) | K_{m} (μM) | V_{max} (μM/Min) | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

[S]_{0} (μM) | Dynafit | Prism (Entire Curve) | iFIT | Dynafit | Prism (Entire Curve) | iFIT | Dynafit | Prism (Entire Curve) | iFIT | |

1st parallel run | 30 | 12.50 | 12.47 | 12.49 | 23.36 | 22.89 | 20.20 | 133.30 | 133.90 | 122.20 |

60 | 37.54 | 37.47 | 24.81 | 23.53 | 24.09 | 24.08 | 91.00 | 96.49 | 96.28 | |

90 | 90.23 | 90.34 | 90.10 | 29.86 | 34.27 | 33.50 | 96.50 | 109.40 | 107.21 | |

120 | 109.70 | 109.80 | 109.48 | 26.30 | 30.99 | 27.52 | 96.10 | 109.60 | 102.38 | |

150 | 154.93 | 154.90 | 154.64 | 30.72 | 37.69 | 30.75 | 101.40 | 118.70 | 107.24 | |

200 | 184.80 | 184.90 | 184.17 | 29.54 | 38.34 | 28.27 | 107.60 | 127.20 | 110.39 | |

250 | 247.30 | 246.60 | 246.01 | 27.70 | 38.61 | 26.78 | 104.20 | 127.00 | 108.06 | |

300 | 278.10 | 277.10 | 276.54 | 26.39 | 43.05 | 29.45 | 80.00 | 105.00 | 89.03 | |

350 | 340.72 | 339.50 | 338.87 | 29.84 | 55.11 | 32.78 | 79.50 | 110.40 | 88.87 | |

400 | 347.10 | 345.10 | 344.23 | 26.34 | 47.49 | 28.51 | 84.40 | 113.90 | 92.67 | |

2nd parallel run | 30 | 23.19 | 23.18 | 23.18 | 21.57 | 22.73 | 15.79 | 81.70 | 87.55 | 72.04 |

60 | 58.65 | 58.69 | 58.61 | 24.85 | 27.82 | 29.46 | 83.50 | 92.89 | 94.94 | |

90 | 83.31 | 83.36 | 83.22 | 29.79 | 34.09 | 36.89 | 82.00 | 94.00 | 96.71 | |

120 | 123.27 | 123.30 | 123.01 | 24.97 | 30.59 | 25.32 | 82.40 | 96.41 | 87.58 | |

150 | 143.25 | 143.20 | 142.84 | 25.77 | 32.37 | 26.04 | 83.50 | 98.83 | 88.56 | |

200 | 193.57 | 193.30 | 192.90 | 27.64 | 37.18 | 27.18 | 89.00 | 108.30 | 93.61 | |

250 | 277.90 | 277.30 | 276.78 | 27.60 | 42.33 | 26.68 | 96.00 | 121.40 | 100.30 | |

300 | 309.90 | 309.40 | 308.78 | 24.78 | 42.54 | 24.10 | 92.70 | 120.20 | 96.85 | |

350 | 360.88 | 359.60 | 358.99 | 26.14 | 44.36 | 24.73 | 104.40 | 135.10 | 108.77 | |

400 | 354.96 | 353.80 | 385.88 | 25.42 | 43.10 | 26.17 | 108.00 | 138.10 | 113.57 | |

3rd parallel run | 30 | 19.19 | 19.18 | 19.18 | 18.71 | 19.68 | 18.53 | 72.10 | 77.13 | 74.25 |

60 | 42.93 | 42.95 | 42.90 | 23.13 | 24.88 | 16.27 | 93.30 | 100.80 | 85.69 | |

90 | 63.25 | 63.28 | 63.17 | 21.72 | 23.91 | 22.58 | 88.10 | 96.82 | 94.31 | |

120 | 96.75 | 96.79 | 25.42 | 29.83 | 81.90 | 94.01 | ||||

150 | 118.02 | 118.00 | 117.78 | 24.40 | 29.25 | 26.90 | 85.00 | 98.25 | 93.52 | |

200 | 149.98 | 150.00 | 149.69 | 28.49 | 36.23 | 30.89 | 84.40 | 101.10 | 93.59 | |

250 | 200.12 | 199.80 | 199.40 | 26.29 | 35.61 | 32.59 | 91.80 | 111.20 | 107.15 | |

300 | 248.88 | 248.50 | 248.02 | 29.93 | 45.42 | 27.77 | 84.60 | 108.90 | 88.29 | |

350 | 292.84 | 292.00 | 291.46 | 26.62 | 44.72 | 25.81 | 83.80 | 110.40 | 87.69 | |

400 | 364.62 | 364.40 | 363.11 | 40.45 | 72.72 | 39.44 | 88.70 | 125.00 | 96.45 | |

AVG | 26.58 | 36.40 | 27.07 | 91.03 | 108.93 | 96.49 | ||||

STDEV | 3.84 | 11.02 | 5.38 | 12.01 | 14.95 | 11.15 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Petrič, B.; Goličnik, M.; Bavec, A. The Removal of Time–Concentration Data Points from Progress Curves Improves the Determination of *K _{m}*: The Example of Paraoxonase 1.

*Molecules*

**2022**,

*27*, 1306. https://doi.org/10.3390/molecules27041306

**AMA Style**

Petrič B, Goličnik M, Bavec A. The Removal of Time–Concentration Data Points from Progress Curves Improves the Determination of *K _{m}*: The Example of Paraoxonase 1.

*Molecules*. 2022; 27(4):1306. https://doi.org/10.3390/molecules27041306

**Chicago/Turabian Style**

Petrič, Boštjan, Marko Goličnik, and Aljoša Bavec. 2022. "The Removal of Time–Concentration Data Points from Progress Curves Improves the Determination of *K _{m}*: The Example of Paraoxonase 1"

*Molecules*27, no. 4: 1306. https://doi.org/10.3390/molecules27041306