Estimating the Product Inhibition Constant from Enzyme Kinetic Equations Using the Direct Linear Plot Method in One-Stage Treatment

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Article Estimating the Product Inhibition Constant from Enzyme Kinetic Equations Using the Direct Linear Plot Method in One-Stage Treatment

Pedro L. Valencia 1,* , Bastián Sepúlveda 2, Diego Gajardo 2 and Carolina Astudillo-Castro 3 1 Department of Chemical and Environmental Engineering, Universidad Técnica Federico Santa María, P.O. Box 110-V, Valparaíso 2390123, Chile 2 Department of Mathematics, Universidad Técnica Federico Santa María, Valparaíso 2390123, Chile; [email protected] (B.S.); [email protected] (D.G.) 3 School of Food Engineering, Pontiﬁcia Universidad Católica de Valparaíso, Valparaíso 2360100, Chile; [email protected] * Correspondence: [email protected]

Received: 15 June 2020; Accepted: 8 July 2020; Published: 1 August 2020

Abstract: A direct linear plot was applied to estimate kinetic constants using the product’s competitive inhibition equation. The challenge consisted of estimating three kinetic constants, Vmax, Km, and Kp, using two independent variables, substrates, and product concentrations, in just one stage of mathematical treatment. The method consisted of combining three initial reaction rate data and avoiding the use of the same three product concentrations (otherwise, this would result in a mathematical indetermination). The direct linear plot method was highly superior to the least-squares method in terms of accuracy and robustness, even under the addition of error. The direct linear plot method is a reliable and robust method that can be applied to estimate Kp in inhibition studies in pharmaceutical and biotechnological areas.

Keywords: direct linear plot; median method; product inhibition; kinetic constants; non-parametric; distribution-free method

1. Introduction The direct linear plot (DLP) is a graphic method to estimate kinetic constants from enzymatic reactions based on the median as a statistic. The method was developed by Eisenthal and Cornish-Bowden in 1974 [1]. The accuracy and robustness of this method were tested during the estimation of Vmax and Km from the Michaelis–Menten equation [1,2]. As in this article, all of the recent studies regarding the application of DLP are based on the estimation of these two parameters from the Michaelis–Menten equation. The application of DLP to equations with more than two parameters has not been studied until now [3]. This median-based method was applied herein to estimate three kinetic constants from the substrate-uncompetitive inhibition equation. The results indicated that DLP was not only applicable to equations with more than two parameters, but it was reliable and robust when compared to the least-squares (LS) method. In the present article, we explored the application of DLP to the product-competitive inhibition equation, which is a diﬀerent type of problem, as will be exposed. Some attempts to estimate inhibition constants from enzyme kinetics using DLP have involved estimating apparent kinetic constants and using secondary plots to estimate the values of these inhibition constants. The application of DLP to estimate inhibition constants was ﬁrst proposed by Eisenthal and Cornish-Bowden [1]. The literature contains outdated examples of this application because the use of DLP to estimate inhibition constants in one-stage has not been assessed [4–8]. In the literature, DLP has been used to estimate the apparent kinetic

Catalysts 2020, 10, 853; doi:10.3390/catal10080853 www.mdpi.com/journal/catalysts Catalysts 2020, 10, x FOR PEER REVIEW 2 of 11 Catalysts 2020, 10, 853 2 of 10

assessed [4–8]. In the literature, DLP has been used to estimate the apparent kinetic constants Vmax and Km; however, studies have used secondary plots to estimate the inhibition constant Kp. The constants Vmax and Km; however, studies have used secondary plots to estimate the inhibition constant present article studied the application of DLP to estimate the product inhibition constant in one-stage Kp. The present article studied the application of DLP to estimate the product inhibition constant in one-stagetreatment, treatment, avoiding avoidingthe estimation the estimation of apparent of apparent kinetic constants kinetic constants and secondary and secondary plots. As plots. the proper As the propercharacterization characterization of inhibition of inhibition in pharmaceutical in pharmaceutical and biotechnological and biotechnological applications applications is a major is concern, a major concern,the importance the importance of this proposal of this proposal is that it isopens that itup opens the possibility up the possibility of using ofa reliable using a and reliable robust and method robust to estimate product inhibition constants. The purpose of this study is to estimate Vmax, Km, and Kp from method to estimate product inhibition constants. The purpose of this study is to estimate Vmax, Km, the competitive inhibition equation in just one stage of calculations, avoiding the apparent constants and Kp from the competitive inhibition equation in just one stage of calculations, avoiding the apparent constantsand secondary and secondaryplots. This plots.means Thisthe values means of the the values product of inhibition the product constants inhibition must constants be obtained must from be obtainedthe DLP method, from the which DLP method, has never which been has tried never before. been The tried problem before. Theis outlined problem as is follows. outlined as follows. The equation for competitive product inhibition involves three parameters—Vmax, Km, and Kp— The equation for competitive product inhibition involves three parameters—Vmax, Km, and Kp—and threeand variables—substratethree variables—substrate concentration, concentration, product concentration,product concentration, and initial reaction and initial rate (Equation reaction (1)).rate (Equation (1)). VmaxS v = 𝑉𝑆 (1) 𝑣= K K + S + 𝐾m P m Kp (1) 𝐾 + 𝑆 + 𝑃 𝐾 This is a diﬀerent type of problem compared to previous articles where the substrate-uncompetitive This is a different type of problem compared to previous articles where the substrate- inhibition equation involved three parameters—Vmax, Km, and KS—but only two variables—substrate uncompetitive inhibition equation involved three parameters—Vmax, Km, and KS—but only two concentration and initial reaction rate (Equation (2)) [3]. variables—substrate concentration and initial reaction rate (Equation (2)) [3]. V𝑉 S𝑆 v = max (2) 𝑣= S2 Km + S + 𝑆 (2) 𝐾 + 𝑆 K+s 𝐾 TheThe experimental dataset dataset required required is istraditionally traditionally used used in the in initial the initial reaction reaction rate method, rate method, which whichdenotes denotes a series a seriesof initial of initial rates ratesvs. substrate vs. substrate conc concentrationsentrations at constant at constant product product concentrations, concentrations, as asindicated indicated in inthe the scheme scheme of ofFigure Figure 1.1 .

P = P1 P = P2 P = Pm S v S v S v S1 v11 S1 v21 S1 vm1 … … … … … …

Sn v1n Sn v2n Sn vmn

FigureFigure 1.1. SchemeScheme ofof thethe traditionaltraditional experimentalexperimental designdesign based on thethe initialinitial reactionreaction raterate methodmethod toto estimateestimate inhibitioninhibition constantsconstants inin enzymeenzyme kinetics.kinetics.

EquationEquation (1)(1) cancan bebe rearrangedrearranged toto EquationEquation (3)(3) inin thethe followingfollowing form:form:

𝑣𝑃𝐾 v P Km 𝑣𝐾 − 𝑆𝑉 + ij i = − 𝑣𝑆 (3) vijKm SjVmax + 𝐾 = vijSj (3) − Kp − The definition of vij is the rate of product concentration Pi and substrate concentration Sj. As the The deﬁnition of v is the rate of product concentration P and substrate concentration S . As the estimation of three parametersij is required, three data pairs andi three equations are neededj to solve estimation of three parameters is required, three data pairs and three equations are needed to solve the the matrix system in Equation (4). matrix system in Equation (4). 𝐾  𝑣 −𝑆 𝑣𝑃 𝑉   −𝑣𝑆  v S v P Km v S  𝑣ii0 −𝑆i0 ii𝑣0i𝑃 ⎛ ⎞ = −𝑣ii0 𝑆i0  (4)  −  𝐾   −   v S v P  Vmax  =  v S  (4)  𝑣jj0 −𝑆j0 𝑣jj0j𝑃   −𝑣jj0 𝑆j0   −  K𝐾   −   v S v P  ⎝ m ⎠  v S  kk0 − k0 kk0 k Kp − kk0 k0 Likewise, the definition of vii’ = v(Pi, Si’), i.e., i values are between 1 and m (the number of product concentrations), and i’ values are between 1 and n (the number of substrate concentrations). Alternatively, the solution for Equation (4) is presented in Equation (5) in algebraic form.

𝑣𝑣𝑣(𝑃𝑆 −𝑆 + 𝑃𝑆 −𝑆 + 𝑃(−𝑆 + 𝑆)) 𝑉 = (5) 𝑃 −𝑃𝑆𝑣𝑣 + (𝑃 −𝑃)𝑆𝑣𝑣 + 𝑃 −𝑃𝑆𝑣𝑣

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Likewise, the deﬁnition of vii’ = v(Pi, Si’), i.e., i values are between 1 and m (the number of product concentrations), and i’ values are between 1 and n (the number of substrate concentrations). Alternatively, the solution for Equation (4) is presented in Equation (5) in algebraic form.

vii vjj vkk (Pk(Si Sj )+Pi(Sj Sk )+Pj( Si +Sk )) Vmax = 0 0 0 0 − 0 0 − 0 − 0 0 (Pj Pi)S v v +(Pi P )S v v +(Pj Pi)S v v − k0 ii0 jj0 − k j0 ii0 kk0 − i0 jj0 kk0

PiSj Sk vii (vkk vjj )+PkSi Sj vkk (vjj vii )+PjSi Sk vjj (vii vkk ) Km = 0 0 0 0 − 0 0 0 0 0 − 0 0 0 0 0 − 0 (5) (Pi Pj) S v v +v ( PiS v +P S v +PjS v P S v ) − k0 ii0 jj0 kk0 − j0 ii0 k j0 ii0 i0 jj0 − k i0 jj0 S S v (v v )+S S v (v v )+S S v (v v ) Km j0 k0 ii0 jj0 − kk0 i0 j0 kk0 ii0 − jj0 i0 k0 jj0 kk0 − ii0 K = p (Pi Pj)S v v +v ( PiS v +P S v +PjS v P S v ) − k0 ii0 jj0 kk0 − j0 ii0 k j0 ii0 i0 jj0 − k i0 jj0 The rank of the matrix in Equation (4) must be 3 for the system to have a solution. For the case when three product concentrations are taken from the same set, Pi = Pj = Pk, the system has no solution because the rank of the resulting matrix is 2. This problem can be easily checked by observing the denominators of Equation (5) when replacing Pi, Pj, and Pk with the same value of product concentration. The strategy proposed to solve this system is to take at least two different values of product concentration. In this way, the total number of feasible data combinations will be calculated using Equation (6). In consequence, calculations of the kinetic constants Vmax, Km, and Kp will be obtained from the combination of two P values from the same dataset, one P from a different one, and the combination of three different P values from three datasets. A list of Vmax, Km, and Kp values from the total combinations indicated in Equation (6) will be obtained. Finally, the estimators of Vmax, Km, and Kp correspond to the median values from the list. ! ! m n n · m (6) 3 − 3 ·

This article aims to compare the quality of the kinetic constant estimations from the product inhibition equation between the DLP and the LS methods.

2. Results and Discussion As in our previous article [3], this problem consists of estimating three kinetic constants, but in this new case, three experimental variables will be used. Experimental data of initial rate perturbed with a random error of variance σ2 were simulated in a dataset of 1000 runs. The dataset, according to Table1, considers the values of the kinetic constants 1, 1 and 10 for Vmax, Km, and Kp, respectively, and the initial rates vs. the substrate concentrations at different product concentrations were plotted in Figure2.

Table 1. Experimental design of substrate and product concentrations to calculate initial reaction rates.

P1 = 0 P2 = 0.5Kp P3 = Kp P4 = 2Kp P5 = 4Kp S v S v S v S v S v

0.1 v11 0.1 v21 0.1 v31 0.1 v41 0.1 v51 0.2 v12 0.2 v22 0.2 v32 0.2 v42 0.2 v52 0.5 v13 0.5 v23 0.5 v33 0.5 v43 0.5 v53 1.0 v14 1.0 v24 1.0 v34 1.0 v44 1.0 v54 2.0 v15 2.0 v25 2.0 v35 2.0 v45 2.0 v55 5.0 v16 5.0 v26 5.0 v36 5.0 v46 5.0 v56 10.0 v17 10.0 v27 10.0 v37 10.0 v47 10.0 v57

These data were used to estimate the kinetic constants using the DLP and LS methods. An example of the dataset obtained for the values of the kinetic constants Vmax, Km, and Kp with a normal distribution of error and variance 0.01 is plotted in Figure3. Catalysts 2020, 10, 853 4 of 10 Catalysts 2020, 10, x FOR PEER REVIEW 4 of 11

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Figure 2. Initial reaction rates vs. substrate concentration for the different product concentrations according to the experimental design presented in Table 1. Data plotted without error for illustration purposes. The kinetic constant values for Vmax, Km, and Kp are 1, 1, and 10, respectively. FigureFigure 2.2. InitialInitial reactionreaction ratesrates vs.vs. substratesubstrate concentrationconcentration forfor thethe didifferentﬀerent productproduct concentrationsconcentrations These data were used to estimate the kinetic constants using the DLP and LS methods. An accordingaccording toexampleto thethe experimentalexperimental of the dataset obtained designdesign for presentedpresented the values of inin the TableTable kinetic1 .1. constants Data Data plotted plotted Vmax, Km without, andwithout Kp with error error a normal for for illustration illustration purposes.purposes. TheThedistribution kinetickinetic of constant constanterror and variance valuesvalues 0.01 forfor isV V plottedmaxmax,, KK inm,m Figure,and and K 3.Kp pareare 1, 1, 1, 1, and and 10, 10, respectively. respectively.

Figure 3. Three-dimensionalFigure 3. Three-dimensional projection projection of of the thedirect direct lin linearear plot plot(DLP) (DLP)values obtained values according obtained to according to the experimental design presented in Table 1. The kinetic constant values for Vmax, Km, and Kp are 1, 1, the experimentaland design10, respectively. presented The median in of Table each kinetic1. The cons kinetictant (red dots) constant is projected values on the for planes.V maxSome, Km, and Kp are 1, 1, and 10, respectively.data are outside Theof the medianlayers for illustration of each purposes kinetic only. constant (red dots) is projected on the planes.

Some data are outsideThe number of of the kinetic layers constant for illustration values plotted purposes in Figures 3 only.and 4 was 6370. This amount of data was obtained avoiding the use of the same three product concentrations during the calculations with The numberEquation of kinetic(4). The dispersion constant of th valuese data was plottedenormous compared in Figures to the3 valueand of4 thewas kinetic 6370. constant, This amount of as can be observed in Figure 4. This is a typical result of the application of the DLP method to the data was obtainedestimation avoiding of kinetic the constants. use of Th theis behavior same can three be observed product in previous concentrations publications [1,3,9]. during Many the calculations with Equationpoints (4). were The left dispersion out of the layers of just the for data illustrati wason purposes; enormous however, compared the vast majority to the of data value are of the kinetic constant, as caninside be observedof layers corresponding in Figure to4 .98%, This 99% is and a typical 99% for V resultmax, Km and of K thep, respectively. application In terms of theof DLP method to the estimation of kinetic constants. This behavior can be observed in previous publications [1,3,9]. ManyFigure points 3. were Three-dimensional left out of the layersprojection just of for the illustration direct linear purposes; plot (DLP) however, values obtained the vast majorityaccording ofto data are inside of layers corresponding to 98%, 99% and 99% for Vmax, Km and Kp, respectively. In terms of the experimental design presented in Table 1. The kinetic constant values for Vmax, Km, and Kp are 1, 1, dispersion,and 10, 56% respectively. of calculated The medianVmax values of each are kinetic in the cons rangetant (red 0.95–1.05; dots) is 22% projected of calculated on the planes. Km values Some are in thedata range are 0.95–1.05outside of andthe layers 24% offor calculated illustrationK purposesp values only. are in the range 9.5–10.5. The amount of data plotted in Figures3 and4 was obtained to estimate the kinetic constants Vmax, Km, and Kp by calculating the medianThe number for each of ofkinetic them. constant This is thevalues procedure plotted neededin Figures in routine3 and 4 experimentswas 6370. This to amount determine of data the inhibitionwas obtained constant avoidingKp. Thethe use statistical of the same evaluation three pr ofoduct DLP requiresconcentrations the repetition during the of this calculations procedure with to avoidEquation bias (4). and The erroneous dispersion conclusions. of the data A was total enormous of 1000 experimental compared to runs the value were carriedof the kinetic out to evaluateconstant, andas can compare be observed the quality in Figure of the 4. estimations This is a typical between result the of DLP theand application the LS methods. of the DLP A comparisonmethod to the of theestimation lower values of kinetic of the constants. sum of squaredThis behavior residuals can be (SSR observed) between in previous DLP and publications LS is plotted [1,3,9]. in Figure Many5. points were left out of the layers just for illustration purposes; however, the vast majority of data are inside of layers corresponding to 98%, 99% and 99% for Vmax, Km and Kp, respectively. In terms of

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dispersion, 56% of calculated Vmax values are in the range 0.95–1.05; 22% of calculated Km values are in the range 0.95–1.05 and 24% of calculated Kp values are in the range 9.5–10.5. The amount of data Catalysts 2020plotted, 10, 853 in Figures 3 and 4 was obtained to estimate the kinetic constants Vmax, Km, and Kp by calculating 5 of 10 the median for each of them. This is the procedure needed in routine experiments to determine the inhibition constant Kp. The statistical evaluation of DLP requires the repetition of this procedure to The DLP methodavoid bias obtained and erroneous lower conclusions. values of ASSR total forof 1000 all experimental the values ofruns variance were carried in the out rangeto evaluate from 0.001 to and compare the quality of the estimations between the DLP and the LS methods. A comparison of 0.020 when both procedures—calculation of the median of Km/Kp and the median of Kp—were carried the lower values of the sum of squared residuals (SSR) between DLP and LS is plotted in Figure 5. out. The calculation of the median of K /K was slightly superior for almost all variances explored. The DLP method obtained lower valuesm ofp SSR for all the values of variance in the range from 0.001 For example,to 0.020 at variancewhen both0.010, procedures—c the calculationalculation of of the the median median of Km of/KpK andm/K thep obtained median of aK frequencyp—were of 0.8 comparedcarried to 0.78 out. for The calculation calculation of the the median median of K ofm/Kpp was. This slightly was superior not significant for almost toall concludevariances that one procedureexplored. was better For example, than the at other.variance However, 0.010, the calculation when both of the calculations median of Km were/Kp obtained compared a frequency regarding the of 0.8 compared to 0.78 for calculation of the median of Kp. This was not significant to conclude that SSR between them, the result depended on the variance, as shown in Figure6a. The calculation of one procedure was better than the other. However, when both calculations were compared regarding the medianthe of SSRKm between/Kp obtained them, the a lowerresult dependSSR ated variance on the variance, values as lowershown thanin Figure 0.010. 6a. The The calculation calculation of the median ofofK ptheo ffmedianered lowerof Km/K valuesp obtained of aSSR lowerat SSR variances at variance higher values thanlower0.010. than 0.010. The The eff ectcalculation of the of ratio Kp/Km the median of Kp offered lower values of SSR at variances higher than 0.010. The effect of the ratio was slightly superior to the calculation of the median of Kp, as shown in Figure6b. However, this was Kp/Km was slightly superior to the calculation of the median of Kp, as shown in Figure 6b. However, not significantly different. In consequence, which procedure is better, in terms of the SSR, depends on this was not significantly different. In consequence, which procedure is better, in terms of the SSR, the experimentaldepends error.on the experimental error.

(a) (b)

Figure 4. DLPFigure for 4. DLP the competitivefor the competitive inhibition inhibition equation equation to estimate estimate VmaxV, maxKm, ,andKm K,p and withK realp with values real of values of 1, 1, and 10, respectively, considering a normal distribution of error with variance 0.01. (a) Plot of Vmax 1, 1, and 10, respectively, considering a normal distribution of error with variance 0.01. (a) Plot of Vmax vs. Km; (b) plot of Vmax vs. Kp. The median values (red dots) are projected on each axis. Some data are vs. K ;(b) plot of V vs. K . The median values (red dots) are projected on each axis. Some data are m outside of layersmax for illustrationp purposes only. Data inside of layers is 98%, 99% and 99% for Vmax, Km outside ofand layers Kp, respectively. for illustration purposes only. Data inside of layers is 98%, 99% and 99% for Vmax, Catalysts 2020, 10, x FOR PEER REVIEW 6 of 11 Km and Kp, respectively.

(a) (b)

Figure 5. Frequency (fi) of the lower sum of squared residuals (SSR) obtained between DLP and least- Figure 5. Frequency (fi) of the lower sum of squared residuals (SSR) obtained between DLP and squares (LS) methods as a function of the variance (σ2) of the2 normal distribution of error (εi). The least-squares (LS) methods as a function of the variance (σ ) of the normal distribution of error (εi). total number of experiments was 1000. (a) DLP applied to the median of Km/Kp; (b) DLP applied to the K K The total numbermedian of of Kp. experiments The dotted line represents was 1000. the ( afrequency) DLP applied 0.5, where to both the methods, median LS of andm DLP,/ p ;(obtainedb) DLP applied to the medianthe same of K statisticalp. The quality. dotted The line results represents showed that the DLP frequency obtained 0.5,a lower where SSR than both LS methods,approximately LS and DLP, obtained thein 80% same of the statistical estimations. quality. The results showed that DLP obtained a lower SSR than LS approximately in 80% of the estimations.

(a) (b)

Figure 6. Frequency (fi) of the lower SSR obtained between both the calculation of the median of Km/Kp and the median of Kp. (a) Results of frequency as a function of the variance (σ2) of the normal

distribution of error (εi). (b) Results of frequency as a function of the ratio Km/Kp. The interpretation is the same as Figure 5.

The relative error of the estimated kinetic constants by LS and DLP methods was plotted for 1000 experimental runs in Figure 7. The DLP always resulted in a lower accumulation of relative error during estimations of the three kinetic constants. The product inhibition constant Kp accumulated the highest relative error when estimated by both LS and DLP methods. However, the accumulated relative error was less than half with DLP compared to LS. This result shows that DLP is a very accurate method to estimate the competitive inhibition constant. The product inhibition constant Kp estimated with DLP method accumulated a lower relative error than the LS method even at higher variance σ2 (experimental error), as shown in Figure 8. The behavior patterns of both the median of Km/Kp and the median of Kp were repeated. The median of Km/Kp was slightly more accurate at variances lower than 0.010. At higher values, the median of Kp was more accurate.

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Catalysts 2020, 10, 853 6 of 10 the highest relative error when estimated by both LS and DLP methods. However, the accumulated relative error was less than half(a with) DLP compared to LS. This result shows(b) that DLP is a very accurate method to estimate the competitive inhibition constant. The product inhibition constant Kp estimated Figure 5. Frequency (fi) of the lower sum of squared residuals (SSR) obtained between DLP and least- 2 with DLP methodsquares accumulated (LS) methods as aa function lower relativeof the variance error (σ2) than of thethe normal LS distribution method evenof error at (ε higheri). The variance σ (experimentaltotal error), number as shownof experiments in Figure was 1000.8. ( Thea) DLP behavior applied to patternsthe median of of K bothm/Kp; (b the) DLP median applied to of theK m/Kp and the median of Kp. The dotted line represents the frequency 0.5, where both methods, LS and DLP, obtained median of Kp were repeated. The median of Km/Kp was slightly more accurate at variances lower than the same statistical quality. The results showed that DLP obtained a lower SSR than LS approximately 0.010. At higherin 80% values, of the estimations. the median of Kp was more accurate.

(a) (b)

Figure 6. Frequency (fi) of the lower SSR obtained between both the calculation of the median of Km/Kp Figure 6. Frequency (fi) of the lower SSR obtained between both the calculation of the median of and the median of Kp. (a) Results of frequency as a function of the variance (σ2) of the 2normal Km/Kp and the median of Kp.(a) Results of frequency as a function of the variance (σ ) of the normal distribution of error (εi). (b) Results of frequency as a function of the ratio Km/Kp. The interpretation is distributionthe of same error as Figure (εi). ( 5.b ) Results of frequency as a function of the ratio Km/Kp. The interpretation is Catalyststhe same 2020, as 10 Figure, x FOR 5PEER. REVIEW 8 of 11 The relative error of the estimated kinetic constants by LS and DLP methods was plotted for 1000 experimental runs in Figure 7. The DLP always resulted in a lower accumulation of relative error during estimations of the three kinetic constants. The product inhibition constant Kp accumulated the highest relative error when estimated by both LS and DLP methods. However, the accumulated relative error was less than half with DLP compared to LS. This result shows that DLP is a very accurate method to estimate the competitive inhibition constant. The product inhibition constant Kp estimated with DLP method accumulated a lower relative error than the LS method even at higher variance σ2 (experimental error), as shown in Figure 8. The behavior patterns of both the median of Km/Kp and the median of Kp were repeated. The median of Km/Kp was slightly more accurate at variances lower than 0.010. At higher values, the median of Kp was more accurate.

FigureFigure 7. Relative7. Relative error error (e i()ei for) for the the estimations estimations of of kinetic kinetic constantsconstants VVmax, ,KKmm, and, and KpK byp by LS LS and and DLP DLP methodsmethods accumulated accumulated during during iterations iterations ((ii)) ofof 10001000 experimentalexperimental runs runs wi withth a anormal normal distribution distribution of of 2 2 errorerror (εi )(ε andi) and variance variance (σ (σ) 0.010.) 0.010. The TheK Kpp waswas estimatedestimated by DLP using using the the median median of of KKm/mK/pK (redp (red line) line) andand the the median median of ofK pK(bluep (blue line). line).

The effect of outliers was analyzed by adding fixed error values to the initial rate v34 (S = 1; P = 10). The added error ranged from 0.01 to +0.01. The results are shown in Figure9. The estimated values − of Vmax by LS were constant due to an artifact of the estimation method. The estimated Vmax was obtained by choosing the highest value from the five non-linear regressions corresponding to the five datasets of product concentrations. The error was added to the initial rate value corresponding to the third dataset of product concentration; in consequence, the highest value of Vmax was not perturbed, and it was always the same value. The DLP method estimated Km and the Kp with more accuracy and with less perturbation than the LS method. Although the LS method estimated Kp with more accuracy at some high positive values of fixed error, the results clearly showed that DLP was more robust than the LS method. The perturbation generated by the added error is plotted in Figure 10. The perturbation of Vmax was omitted due to the artifact during its estimation by the LS method. The results indicated that the estimation of kinetic constants Km and Kp was much less perturbed when the DLP method was used. Almost all the perturbations were maintained below 1%, compared to the LS method, where perturbations were in the range from 4% to 8%.

Figure 8. Total relative error (ei) accumulated for estimations of the product inhibition constants Kp by LS and DLP methods after 1000 experimental runs plotted against variance (σ2) of the normal

distribution of error (εi). The Kp was estimated by DLP using the median of Km/Kp (red bars) and the

median of Kp (yellow bars).

Figure 9. Effect of the fixed error added to the initial rate v34 on the estimated values of the kinetic

constants Vmax, Km, and Kp by LS and DLP methods. Real kinetic constant value (dotted line), estimated

values by LS (black line) and DLP (red and blue lines). The Kp was estimated by DLP using the median of Km/Kp (red line) and the median of Kp (blue line).

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Figure 7. Relative error (ei) for the estimations of kinetic constants Vmax, Km, and Kp by LS and DLP methods accumulated during iterations (i) of 1000 experimental runs with a normal distribution of 2 error (εi) and variance (σ ) 0.010. The Kp was estimated by DLP using the median of Km/Kp (red line) and the median of Kp (blue line). Figure 7. Relative error (ei) for the estimations of kinetic constants Vmax, Km, and Kp by LS and DLP Catalysts 2020, 10, 853 methods accumulated during iterations (i) of 1000 experimental runs with a normal distribution of 7 of 10 error (εi) and variance (σ2) 0.010. The Kp was estimated by DLP using the median of Km/Kp (red line)

and the median of Kp (blue line).

Figure 8. Total relative error (ei) accumulated for estimations of the product inhibition constants Kp by Figure 8.FigureTotal 8. relativeTotal relative error error (ei) (e accumulatedi) accumulated for for estimations estimations of the of product the product inhibition inhibition constants constants Kp by Kp LS and DLP methods after 1000 experimental runs plotted against variance (σ2) of the normal 2 2 by LS andLS and DLP DLPdistribution methods methods of error after after (εi). 1000 The1000 Kp experimental wasexperiment estimatedal by runs DLP runs usingplotted plotted the medianagainst against of Kvariancem/Kp variance(red bars)(σ ) andof ( σ the) ofnormal the normal distributiondistribution of errormedian of (errorofε iK).p (yellow The(εi). TheK bars).p wasKp was estimated estimated byby DLP using using the the median median of K ofm/KKp m(red/Kp bars)(red and bars) theand the median of Kp (yellow bars). median of Kp (yellow bars).

Figure 9. Effect of the fixed error added to the initial rate v34 on the estimated values of the kinetic

constants Vmax, Km, and Kp by LS and DLP methods. Real kinetic constant value (dotted line), estimated values by LS (black line) and DLP (red and blue lines). The Kp was estimated by DLP using the median of Km/Kp (red line) and the median of Kp (blue line). Figure 9.FigureEﬀect 9. Effect of the of ﬁxed the fixed error error added added to to the theinitial initial rate vv3434 onon the the estimated estimated values values of the ofkinetic the kinetic constants Vmax, Km, and Kp by LS and DLP methods. Real kinetic constant value (dotted line), estimated constants Vmax, Km, and Kp by LS and DLP methods. Real kinetic constant value (dotted line), values by LS (black line) and DLP (red and blue lines). The Kp was estimated by DLP using the median estimated values by LS (black line) and DLP (red and blue lines). The Kp was estimated by DLP using of Km/Kp (red line) and the median of Kp (blue line). the medianCatalysts of Km 2020/Kp, 10(red, x FOR line) PEER andREVIEW the median of Kp (blue line). 9 of 11

(a) (b)

Figure 10. Perturbation percentage on the estimations of Km and Kp by LS and DLP methods caused Figure 10. Perturbation percentage on the estimations of Km and Kp by LS and DLP methods caused by by adding fixed error to the initial rate v34 at different values of variance (σ2). (a) Km. (b) Kp. The 2 adding fixed errorperturbation to the initial corresponds rate tov34 theat difference different between values the minimum of variance and the (σ maximum). (a) K mvalue.(b obtained) Kp. The perturbation corresponds torespect the difference to the original between value of the the kinetic minimum constant (Equation and the (12)). maximum value obtained respect to the

original value3. Method of the kinetic constant (Equation (12)).

The DLP method3.1. Experimental exhibited Design the same pattern of behavior than that observed during application to the uncompetitive inhibitionThe set of substrate equation and product [3]. LowerconcentrationsSSR andwas designed estimation by first errors setting the compared real values toof the LS method kinetic constants Vmax, Km, and Kp. The chosen values were 1 for Vmax and Km, and five different Kp were observedvalues, during including estimations 10, 20, 50, 100 of and the 200, competitive to study the effect inhibition on the estimation constant. of the kinetic Again, constants. the DLP method demonstrated accuracyThese values and correspond robustness to the duringdimensionless the kinetic estimation constants of in kinetic the case constants. of Vmax and TheKm. possibility of Dimensionless Vmax is the limit of S/(S + Km) when S tends to infinity. Dimensionless Km is obtained using the DLP method to estimate inhibition constants in pharmaceutical and biotechnological studies when the kinetic equation is written in the form S/Km/(S/Km + 1). In the case of Kp, the dimensionless was demonstrated.value Thecorresponds performance to Kp/Km. As of indicated the DLP by Cornish-Bowden method applied et al. [2,10], to the thecompetitive unitary values of inhibition Vmax equation was diﬀerent comparedand Km involved to the no applicationloss of generality of because the substrate-uncompetitive they finally depend on the units of inhibition measurement. equation The previously values of substrate and product concentrations are listed in Table 1, corresponding to n = 7 and m = studied. The addition5, according of to a variablethe nomenclature product in the concentrationscheme of Figure 1. caused mathematical indetermination during the calculations. TheThe problem chosen values was of substrate overcome concentrations by avoiding were based using on their the distribution same three around product the Km concentrations value from 0.1 Km to 10 Km, with three values below Km and three values above Km, as recommended by Cornish-Bowden [11]. In the case of product concentrations, the choice was based on a Kp value- centered distribution. The experimental design consisted of 35 reaction rates (vij), which were calculated using the product-competitive inhibition equation (Equation (1)) assigning relative errors from a normal population of error εi with variance σ2 (Equation (7)) to simulate the experimental error.

𝑉𝑆 𝑣 = (1 + 𝜀) 𝐾 (7) 𝐾 + 𝑆 + 𝑃 𝐾

The calculated reaction rates (vij) were used as experimental values to estimate the kinetic constants Vmax, Km, and Kp. The estimations were carried out by DLP, using Equation (4), and by the LS method. The number of possible combinations of three initial rate values (vij) to calculate the kinetic constants Vmax, Km, and Kp was 6370, according to Equation (4) and (5). The median for each kinetic constant was obtained from this list. This procedure corresponded to one experiment to calculate the estimators of the kinetic constants, which is equivalent to an empirical procedure based on the initial rates method. A total of 1000 experiments were run to compare the accuracy of both methods and the quality of parameter estimations. An algorithm was written in Python software to perform all the calculations presented in this article. The algorithm can be downloaded from the repository indicated in the Supplementary Material. Estimation of Kp was carried out by two optional procedures: (i) calculating Km/Kp for each of the 1000 experiments, calculating the median for Km/Kp,

Catalysts 2020, 10, 853 8 of 10 in Equations (4) and (5). This obligated a reduction in the number of combinations from 6545 to 6370, where 175 combinations resulted in the indetermination of the matrix (Equation (4)). Another difference is the dependence of Kp on the value of Km, which can be observed in Equation (5), where the calculation consisted of the Km/Kp value, and Kp was calculated after obtaining Km. A significant implication of this is the transmission of the Km error to Kp. Interestingly, based on the observations, this effect was not important. In general, the estimation of Kp by DLP was more accurate and robust than estimation by the LS method. The application of DLP to the uncompetitive inhibition equation exhibited a higher accuracy just in some cases compared to the LS method. In terms of robustness, the DLP was highly superior to the LS method. Presently, in this study, a higher superiority in terms of accuracy and robustness was observed. Despite the higher complexity of the competitive inhibition equation, requiring one more independent variable than the substrate-uncompetitive inhibition equation, the results were better. The transmission of error from Km to Kp could be the key. This error is surely amplified when transmitted from one constant to another, and based on the observations, the effect was better softened by DLP compared to the LS method. As Cornish-Bowden and Endrenyi commented [10], the median method cannot readily be generalized to equations of more than two parameters. This sentence became true when we found a mathematical issue during the application of the DLP to the product’s inhibition equation, where an indeterminate system of equations (Equation (5)) is generated when the same value for the product concentration was used. However, this problem can be solved by selecting the proper combination of data. We, again, demonstrated that DLP can be applied to equations with more than two parameters. Presently, we also demonstrated the application to equations with more than two variables. We conclude that the DLP method is better than the LS method to estimate the product’s competitive inhibition constant in terms of accuracy and robustness. We now count on a new tool to estimate the product-competitive inhibition constant with reliable results.

3. Method

3.1. Experimental Design The set of substrate and product concentrations was designed by first setting the real values of kinetic constants Vmax, Km, and Kp. The chosen values were 1 for Vmax and Km, and five different Kp values, including 10, 20, 50, 100 and 200, to study the effect on the estimation of the kinetic constants. These values correspond to the dimensionless kinetic constants in the case of Vmax and Km. Dimensionless Vmax is the limit of S/(S + Km) when S tends to infinity. Dimensionless Km is obtained when the kinetic equation is written in the form S/Km/(S/Km + 1). In the case of Kp, the dimensionless value corresponds to Kp/Km. As indicated by Cornish-Bowden et al. [2,10], the unitary values of Vmax and Km involved no loss of generality because they finally depend on the units of measurement. The values of substrate and product concentrations are listed in Table1, corresponding to n = 7 and m = 5, according to the nomenclature in the scheme of Figure1. The chosen values of substrate concentrations were based on their distribution around the Km value from 0.1 Km to 10 Km, with three values below Km and three values above Km, as recommended by Cornish-Bowden [11]. In the case of product concentrations, the choice was based on a Kp value-centered distribution. The experimental design consisted of 35 reaction rates (vij), which were calculated using the product-competitive inhibition equation (Equation (1)) assigning relative errors from a normal 2 population of error εi with variance σ (Equation (7)) to simulate the experimental error.

VmaxSij vij = (1 + εi) (7) K + S + Km P m ij Kp j

the LS method. The number of possible combinations of three initial rate values (vij) to calculate the kinetic constants Vmax, Km, and Kp was 6370, according to Equations (4) and (5). The median for each kinetic constant was obtained from this list. This procedure corresponded to one experiment to calculate the estimators of the kinetic constants, which is equivalent to an empirical procedure based on the initial rates method. A total of 1000 experiments were run to compare the accuracy of both methods and the quality of parameter estimations. An algorithm was written in Python software to perform all the calculations presented in this article. The algorithm can be downloaded from the repository indicated in the Supplementary Material. Estimation of Kp was carried out by two optional procedures: (i) calculating Km/Kp for each of the 1000 experiments, calculating the median for Km/Kp, and finally obtaining the estimated Kp using the corresponding Km value, and (ii) calculating the Kp for each of the 1000 experiments and calculating the median of Kp (see Equation (5)). In the case of the app LS method, a non-linear regression was applied to Equation (8), where the apparent Km (Km ) was app estimated. According to the experimental design, five values of Vmax and Km were obtained, one for each product concentration. As Vmax is not affected by competitive inhibition, a higher value was app chosen as the estimated Vmax through the LS method. The Km values were plotted against the product concentrations in a secondary plot, and the values of Km and Kp were obtained by linear regression of Equation (9). VmaxS v = app (8) Km + S

app Km Km = Km + P (9) Kp

The effect of the Kp/Km ratio was also evaluated. The values of Kp considered were 10, 20, 50, 100, and 200. The Kp/Km ratios corresponded to the same values, considering that Km was 1. As the values of Kp were modified, the experimental values of product concentration (P) were changed to maintain the same values of apparent Km, according to Equation (9), among the different datasets to avoid negative effects in the experimental design.

3.2. Accuracy Test The sum of squared residuals (SSR) was calculated for each of the 1000 experiments for both methods, DLP and LS, to evaluate their accuracies, according to Equation (10).

Xn SSR = (v vˆ ) (10) i − i i=1

th th where vi is the i value of the experimental rate (Equation (7)), and vˆi is the i value of the predicted rate calculated according to Equation (1). The relative error (ei) from the diﬀerence between the real (θ) and the estimated value (θˆ) was calculated for all kinetic constants according to Equation (11).

θˆ θ ei = − (11) θ

The number of experiments (out of 1000) in which the LS resulted in a lower SSR and ei was 2 calculated and compared with DLP for diﬀerent values of the variance σ for error εi.

3.3. Eﬀect of Outliers

The sensitivity of both methods to outliers was tested by changing the initial rate value v34 from Table1. This value corresponds to the substrate concentration equal to Km and the product concentration equal to Kp. The initial rate v was changed by varying ﬁxed values of ε from 0.1 to 34 i − +0.1. The rest of the experimental initial rates were calculated as mentioned above using Equation (6) Catalysts 2020, 10, 853 10 of 10

2 with random εi with σ = 0.01. The perturbation generated on the estimated constant was calculated according to Equation (12). θû θˆ %Perturbation = − l 100 (12) θ · where θû and θˆl are the upper and lower estimated values of the real constant θ in the range of the fixed error.

Supplementary Materials: The following are available online at https://github.com/bastiansepulveda/Inhibation- Equation-by-Product-Algorithms.git, algorithms for the analysis of DLP applied to the product’s competitive inhibition equation. Author Contributions: Conceptualization, P.L.V. and B.S.; methodology, P.L.V., B.S. and D.G.; software, D.G. and B.S.; validation, P.L.V., D.G., B.S. and C.A.-C.; formal analysis, P.L.V. and C.A.-C.; writing—original draft preparation, P.L.V. and C.A.-C. All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. Acknowledgments: Pedro Valencia would like to thank Pedro Gajardo for the opportunity of presenting the direct linear plot problem in his class Laboratory of Modeling. Conﬂicts of Interest: The authors declare no conﬂict of interest.

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