An explicit solution for progress curve analysis in systems characterized by endogenous substrate production

The Lambert W function was used to explicitly relate substrate concentration S, to time t, and the kinetic parameters V (m), K (m), and R in the modified Michaelis-Menten equation that accounts for endogenous substrate production. The applicability of this explicit formulation for kinetic parameter estimation by progress curve analysis was demonstrated using a combination of synthetic and experimental substrate depletion data. Synthetic substrate depletion data were generated using S (0) values of 1, 2, and 3 μM and V (m), K (m), and R values of 1.0 μM h(-1), 1.0 μM, and 0.1 μM h(-1), respectively, and contained 5% normally distributed error. Experimental data were obtained from two previously published studies on hydrogen depletion in four experimental systems. In all instances, experimental data were well described by the explicit solution presented in this study. Differential equation solution and iterative S estimation are eliminated with the explicit solution approach, thereby simplifying progress curve analysis in systems characterized by endogenous substrate production.     

An explicit expression for determining cometabolism kinetics using progress curve analysis

We present an explicit expression for describing the kinetics of cometabolic biotransformation of environmental pollutants. This expression is based on the Lambert W function and explicitly relates the substrate concentration, S, to time, t, the two experimentally measured variables. This explicit relationship simplifies kinetic parameter estimation as differential equation solution and iterative estimation of the substrate concentration are eliminated. The applicability of this new expression for nonlinear kinetic parameter estimation was first demonstrated using noise containing synthetic data where final estimates of the kinetic parameters were very close to their actual values. Subsequently 1.1.1-trichloroethane degradation data at initial concentrations of 750 and 375 μM were described using the explicit expression resulting in r and K(s) estimates of 0.26 μM/mg d and 28.08 μM and 0.30 μM/mg d and 28.70 μM, respectively, very similar to 0.276 μM/mg d and 31.2 μM, respectively, that were reported in the original study. The new explicit expression presented in this study simplifies estimation of cometabolic kinetic parameters and can be easily used across all computational platforms thereby providing an attractive alternative for progress curve analysis.     

Simultaneous estimation ofV max,Km, and the rate of endogenous substrate production (R) from substrate depletion data

The nonlinear and 3 linearized forms of the integrated Michaelis-Menten equation were evaluated for their ability to provide reliable estimates of uptake kinetic parameters, when the initial substrate concentration (S₀) is not error-free. Of the 3 linearized forms, the one where t/(S₀–S) is regressed against ln(S₀/S)/(S₀–S) gave estimates ofV _max and K_m closest to the true population means of these parameters. Further, this linearization was the least sensitive of the 3 to errors (±1%) in S₀. Our results illustrate the danger of relying on r² values for choosing among the 3 linearized forms of the integrated Michaelis-Menten equation. Nonlinear regression analysis of progress curve data, when S₀ is not free of error, was superior to even the best of the 3 linearized forms. The integrated Michaelis-Menten equation should not be used to estimateV _max and K_m when substrate production occurs concomitant with consumption of added substrate. We propose the use of a new equation for estimation of these parameters along with a parameter describing endogenous substrate production (R) for kinetic studies done with samples from natural habitats, in which the substrate of interest is an intermediate. The application of this new equation was illustrated for both simulated data and previously obtained H₂ depletion data. The only means by whichV _max, K_m, and R may be evaluated from progress curve data using this new equation is via nonlinear regression, since a linearized form of this equation could not be derived. Mathematical components of computer programs written for fitting data to either of the above nonlinear models using nonlinear least squares analysis are presented.     

A comparison of the parameter estimating procedures for the Michaelis-Menten model

The performance of four parameter estimating procedures for the estimation of the adjustable parameters in the Michaelis-Menten model, the maximum initial rate Vmax, and the Michaelis-Menten constant Km, including Lineweaver & Burk transformation (L-B), Eadie & Hofstee transformation (E-H), Eisenthal & Cornish-Bowden transformation (ECB), and Hsu & Tseng random search (H-T) is compared. The analysis of the simulated data reveals the followings: (i) Vmax can be estimated more precisely than Km. (ii) The sum of square errors, from the smallest to the largest, follows the sequence H-T, E-H, ECB, L-B. (iii) Considering the sum of square errors, relative error, and computing time, the overall performance follows the sequence H-T, L-B, E-H, ECB, from the best to the worst. (iv) The performance of E-H and ECB are on the same level. (v) L-B and E-H are appropriate for pricesly measured data. H-T should be adopted for data whose error level are high. (vi) Increasing the number of data points has a positive effect on the performance of H-T, and a negative effect on the performance of L-B, E-H, and ECB.     

Influence of vasoconstriction, temperature, and flow on the kinetics of serotonin uptake in rabbit lungs

In the process of estimating the kinetic parameters of the pulmonary endothelial serotonin (5-HT) uptake, it is critically important to distinguish the effects of hemodynamic changes from endothelial injury. Therefore, the effects of changes in flow rate (1.7-5.0 ml/s), hemodynamics (vasoconstriction by norepinephrine), and temperature (39 vs. 33 degrees C) were investigated in isolated rabbit lungs. Indicator-dilution data were expressed in terms of the Michaelis-Menten equation for the two 5-HT uptake pathways in the preparation. The maximum uptake velocity (Vmax1) and the 5-HT concentration at half-maximum velocity (Km1) of the first pathway as well as the first-order constant (Vmax2/Km2) of the linear part of the second pathway were determined. Neither vasoconstriction nor flow variations had any effect on Km1, whereas increasing the flow rate caused extensive recruitment, with a concomitant increase in Vmax1 and Vmax2/Km2. Furthermore, all the kinetic parameters were significantly decreased at the lower temperature. We conclude that Km1 is independent of organ hemodynamics (vasoconstriction and flow) but susceptible to changes in 5-HT uptake capacity caused by a change in temperature. Vmax1 and Vmax2/Km2 respond to alterations in 5-HT uptake capacity and perfused organ volume. These are prerequisites to apply kinetic modeling as a method for the investigation of pulmonary endothelial function and integrity.     

Progress curve analysis of adenosine deaminase-catalyzed reactions

The kinetic constants of the adenosine deaminase-catalyzed conversion of adenosine to inosine were found to be readily obtainable by analyzing the progress curve of a single reaction. A novel inhibitor, 9-(1-hydroxymethyl-3-methylbutyl)adenine, was studied to test the validity of the progress curve method with this enzyme. Estimates of kinetic constants determined by this method were compared to those determined by the conventional initial velocity analysis. The Km and Vmax values for adenosine and the Ki value for the inhibitor were estimated to be 26.1 microM, 1.27 mumol/min/unit of enzyme, and 0.48 microM, respectively, by the initial velocity method, and 29.3 microM, 1.27 mumol/min/unit of enzyme, and 0.52 microM, respectively, by the progress curve analysis. The inhibitor was shown to act competitively with substrate by both methods of analysis. The progress curve experiments were very simple to perform and the constants were calculated (with an interfaced microcomputer) within a few minutes of the completion of each assay.     

Application of progress curve analysis to in situ enzyme kinetics using 1H NMR spectroscopy

The steady-state kinetics of enzymes in tissues, cells, and concentrated lysates can be characterized using high-resolution nuclear magnetic resonance spectroscopy; this is possible because almost invariably there are differences in the spectra of substrates and products of a reaction and these spectra are obtainable even from optically opaque samples. We used 1H spin-echo NMR spectroscopy to study the hydrolysis of alpha-L-glutamyl-L-alanine by cytosolic peptidases of lysed human erythrocytes. Nonlinear regression of the integrated Michaelis-Menten expression onto the progress-curve data yielded, directly, estimates of Vmax and Km for the hydrolase; a procedure for analyzing progress curves in this manner was adapted and compared with a commonly used procedure which employs the Newton-Raphson algorithm. We also performed a sensitivity analysis of the integrated Michaelis-Menten expression; this yielded equations that indicate under what conditions estimates of Km and Vmax are most sensitive to variations in experimental observables. Specifically, we showed that the most accurate estimates of the steady-state parameters from analysis of progress curves are obtained when the initial substrate concentration is much greater than Km. Furthermore, estimates of these parameters obtained by such an analysis are most sensitive to data obtained when the reaction is 60-80% complete, having started with the highest practicable initial substrate concentration.     

Effects of liver blood flow on hepatic uptake kinetics of galactose in anesthetized cats: parallel tube model

Hepatic galactose uptake in cats anesthetized with pentobarbital was determined during (i) steady-state infusions at several doses, (ii) rapidly increasing infusion rates at different blood flows, and (iii) prolonged infusion of a single dose at different blood flows. The hepatic venous long-circuit technique was used to allow frequent sampling of arterial, portal, and hepatic venous blood without depletion of the animal's blood volume and to allow measurement and alteration of total hepatic blood flow. Uptake was shown to follow Michaelis-Menten kinetics and was consistent with the "parallel tube model." The kinetic parameters Vmax and Km could be determined under steady-state and nonsteady-state conditions and were independent of hepatic blood flow over the range 60-150% of control flow. Mean Vmax was 80 mumol/(min X 100 g liver) and mean Km was 215 microM. Vmax declined by 50% when flow was reduced to half normal. It is concluded that the parallel tube model can be used to describe and predict hepatic galactose kinetics in anesthetized cats, although other models may fit the data equally well.     

Kinetic analysis of the Ca2+-dependent, membrane-bound, macrophage phospholipase A2 and the effects of arachidonic acid

The kinetics of the Ca2+-dependent, alkaline pH optimum, membrane-bound phospholipase A2 from the P388D1 macrophage-like cell line were studied using various phosphatidylcholine (PC) and phosphatidylethanolamine (PE) substrates. This enzyme exhibits "surface dilution kinetics" toward PC in Triton X-100 mixed micelles, and the "dual phospholipid model" was found to adequately describe its kinetic behavior. With substrate in the form of sonicated vesicles, the dual phospholipid model should give rise to Michaelis-Menten type kinetics. However, the hydrolysis of dipalmitoyl-PC, 1-palmitoyl-2-oleoyl-PC, and 1-stearoyl-2-arachidonoyl-PC vesicles exhibited two distinct activities. Below 10 microM, the data appeared to follow Michaelis-Menten behavior, while at higher concentrations, the data could best be fit to a Hill equation with a Hill coefficient of 2. These PCs had Vmax values for the low substrate concentration range of 0.2-0.6 nmol min-1 mg-1 and Km values of 1-2 microM. At the high substrate concentration range, the Vmax values were between 5 and 7 nmol min-1 mg-1. PC containing unsaturated fatty acids had an apparent Km, determined from the Hill equation, of about 15 microM, while the apparent Km of dipalmitoyl-PC was 0.6 microM. When 70% glycerol was included in the assays, a single Michaelis-Menten curve was obtained for both dipalmitoyl-PC and 1-stearoyl,2-arachidonoyl-PC. Possible explanations for these kinetic results include reconstitution of the membrane-bound phospholipase A2 in the phospholipid vesicle or the enzyme has tow distinct phospholipid binding function. The kinetics for both dipalmitoyl-PC and dipalmitoyl-PE hydrolysis in vesicles was very similar, indicating that the enzyme does not greatly prefer one of these head groups over the other. The enzyme also showed no preference for arachidonoyl containing phospholipid. Enzymatic activity toward PC containing saturated fatty acids was linear to about 15% hydrolysis while the hydrolysis of PC containing unsaturated fatty acids was linear to only about 5%. This loss of linearity was due to inhibition by released unsaturated fatty acids. Arachidonic acid was found to be a competitive inhibitor of dipalmitoyl PC hydrolysis with a K1 of 5 microM. This tight binding suggests a possible in vivo regulatory role for arachidonic acid. Three compounds of the arachidonic acid cascade, prostaglandin F2 alpha, 6-keto-prostaglandin F1 alpha, and thromboxane B2, showed no inhibition of enzymatic activity.     

The analysis of enzyme progress curves by numerical differentiation, including competitive product inhibition and enzyme reactivation

A new method for analyzing steady-state enzyme kinetic data is presented. The technique, which is based on the numerical differentiation of the complete reaction curve, has several advantages over initial velocity and integrated Michaelis-Menten equation methods. The differentiated data are fit to the differential equation describing the appropriate kinetic scheme. This approach is particularly valuable in cases of strong competitive product inhibition and of changing concentrations of active enzyme. The method assumes a reversible reaction and is applicable to a very wide variety of steady-state kinetic schemes. A particular advantage of this approach over integrated methods is that it is independent of [S0] and hence of errors in [S0]. The combination of complete progress curve and computer analysis makes this approach very efficient with respect to both time and materials. Running on an IBM PC XT or equivalent microcomputer with an 8087 coprocessor, the analyses are very fast, the complete process usually being complete in a minute or two. The utility of the technique is demonstrated by application to both simulated and real data. We show that the differentiation of the progress curve for the ribonuclease-catalyzed hydrolysis of 2',3'-cyclic cytidine monophosphate reveals strong product inhibition by 3'-CMP, and this product inhibition accounts for the large discrepancies reported in the literature for the value of Km for this substrate. The method was also applied to determine the rate of reactivation of beta-lactamase which had been reversibly inactivated by cloxacillin. Since large numbers of data points are required for the numerical differentiation the method has become practical only with the advent of computer-acquired data systems.     

Estimation of the initial velocity of enzyme-catalysed reactions by non-linear regression analysis of progress curves.

Most methods for studying the kinetic properties of an enzyme involve the determination of initial velocities. When the reaction progress curve shows significant curvature due to depletion of the substrate, accumulation of inhibitory products or instability of the enzyme, estimation of the initial velocity is a subjective and inexact process. Two methods have been suggested [Cornish-Bowden (1975) Biochem. J. 144, 305-312; Boeker (1982) Biochem J. 203, 117-123] that attempt to eliminate this subjective element. The present study offers a third alternative, which is based on fitting a reparameterized form of the integrated Michaelis-Menten equation to the progress curves by non-linear regression. This method yields estimates and standard errors of the initial velocity and of the time to reach 50% reaction. No prior knowledge of the apparent product concentration at zero time or infinite time is required, since both of these quantities are also estimated from the data. It is shown that this method yields reliable estimates of the initial velocity under a wide range of circumstances, including those where the two previously published methods perform poorly.     

Applicability of the integral form of the Michaelis-Menten equation for the kinetic study of transport ATPases

Applicability of the integrated form of the Michaelis-Menten equation to kinetic analysis of transport ATPases has been shown during continuous pH-metric recording of their activity. Two values of Km for both Na, K-ATPase and Ca-ATPase have been found to be consistent with the reported data. Both values of Km for Na, K-ATPase change with temperature, i. e. at 37 degrees, 26 degrees and 15 degrees C they are as follows: Km1--21, Km2--171; Km1--3.32, Km2--47; and Km1--1,2, Km2--20 microM, respectively. This method of determination of Km and V for transport ATPases compares favourably with the previously used methods in resulting efficiency.     

In situ kinetic analysis of glyoxalase I and glyoxalase II in Saccharomyces cerevisiae.

The kinetics of glyoxalase I [(R)-S-lactoylglutathione methylglyoxal-lyase; EC 4.4.1.5] and glyoxalase II (S-2-hydroxyacylglutathione hydrolase; EC 3.1.2.6) from Saccharomyces cerevisiae was studied in situ, in digitonin permeabilized cells, using two different approaches: initial rate analysis and progress curves analysis. Initial rate analysis was performed by hyperbolic regression of initial rates using the program HYPERFIT. Glyoxalase I exhibited saturation kinetics on 0.05-2.5 mM hemithioacetal concentration range, with kinetic parameters Km 0.53 +/- 0.07 mM and V (3.18 +/- 0.16) x 10(-2) mM.min(-1). Glyoxalase II also showed saturation kinetics in the SD-lactoylglutathione concentration range of 0.15-3 mM and Km 0.32 +/- 0.13 mM and V (1.03 +/- 0.10) x 10(-3) mM.min(-1) were obtained. The kinetic parameters of both enzymes were also estimated by nonlinear regression of progress curves using the raw absorbance data and integrated differential rate equations with the program GEPASI. Several optimization methods were used to minimize the sum of squares of residuals. The best parameter fit for the glyoxalase I reaction was obtained with a single curve analysis, using the irreversible Michaelis-Menten model. The kinetic parameters obtained, Km 0.62 +/- 0.18 mM and V (2.86 +/- 0.01) x 10(-2) mM.min(-1), were in agreement with those obtained by initial rate analysis. The results obtained for glyoxalase II, using either the irreversible Michaelis-Menten model or a phenomenological reversible hyperbolic model, showed a high correlation of residuals with time and/or high values of standard deviation associated with Km. The possible causes for the discrepancy between data obtained from initial rate analysis and progress curve analysis, for glyoxalase II, are discussed.     

Uptake of γ-Aminobutyric Acid by Brain Tissue Preparations: A Reevaluation

The kinetic constants Km and Vmax for the uptake of gamma-aminobutyric acid (GABA) by various preparations from rat cerebral cortex were determined by means of Eadie-Hofstee plots and computer analysis. The Km values were much greater in 0.1-mm slices than in synaptosomal preparations, and the Km value increased further with the thickness of the slices. The apparent high Km values in slices were probably due to depletion of the GABA concentration in the extracellular fluid as the exogenous GABA ran the gauntlet of competing uptake sites on its way to sites deep within the slice, thereby bringing about a requirement for higher GABA concentrations in the incubation medium in order to maintain the internal GABA levels at the "Km level." Evidence was obtained for three GABA uptake systems with Km values (in synaptosomes) of 1.1 microM, 43 microM, and 3.9 mM, respectively. In contrast, only two uptake systems for D-aspartate were detected, with Km values of 1.8 microM and 1.8 mM, respectively. The implications of the findings in the study with respect to previous data in the literature are discussed.     

Kinetics of pulmonary angiotensin-converting enzyme and 5'-nucleotidase in vivo

Angiotensin-converting enzyme and 5'-nucleotidase line the luminal surface of pulmonary microvascular endothelium and participate in the synthesis and/or degradation of potent vasoactive substances. We applied Michaelis-Menten kinetics in simultaneous estimations of apparent constants Km and Amax (product of Vmax and microvascular plasma volume) of these two enzymes for the substrates 3H-labeled benzoyl-Phe-Ala-Pro and 14C-labeled 5'-AMP, respectively, in vivo. Values of angiotensin-converting enzyme for benzoyl-Phe-Ala-Pro (Km = 10-11 microM; Amax = 12-13 mumol X min-1) were somewhat higher than published estimates in vitro and changed predictably in response to the known enzyme inhibitor captopril. Kinetic values of 5'-nucleotidase for 5'-AMP (Km = 3-4 microM; Amax = 3-4 mumol/min) were substantially lower than those reported in vitro but also responded predictably to the competitive inhibitor of 5'-nucleotidase, adenosine 5'-[alpha, beta-methylene]diphosphate. These data offer in vivo estimates of enzyme kinetics that are useful in revealing enzyme behavior in their normal physiological environment and provide means of evaluating the action of pharmacological, physiological, and pathological modulators of enzyme activity, in vivo.     

Kinetics of serotonin uptake in isolated rabbit lungs

The single-pass, bolus-injection method was used to study the effect of serotonin (5-HT) concentration on the extraction of 5-HT by isolated perfused rabbit lungs. The extraction pattern suggested that an uptake model, which includes multiple parallel uptake processes, provided a better representation of the data than the simple Michaelis-Menten equation, which has commonly been used to fit the saturable uptake data in previous studies. In particular, the rabbit lung data could be fit with two such parallel pathways. Since the 5-HT uptake could virtually be completely blocked by imipramine, both pathways can be considered to be carrier-mediated processes. The high-affinity pathway was saturable within the range of concentrations studied, with a Km and Vmax of approximately 0.84 microM and 0.21 nmol.s-1.g wet lung wt-1. The Km for the low-affinity pathway was larger than concentrations for which accurate uptake measurements are practical in the perfused organ. Thus, for the low-affinity pathway, only Vmax/Km was identifiable. Vmax/Km values for the high- and low-affinity pathways were approximately 2.87 and 0.35 ml/s, respectively. The results suggest that it will be worthwhile to investigate the behavior of these uptake parameters in response to changes in lung physiology and endothelial function in future studies.     

Nonlinear estimation of Monod growth kinetic parameters from a single substrate depletion curve

Monod growth kinetic parameters were estimated by fitting sigmoidal substrate depletion data to the integrated Monod equation, using nonlinear least-squares analysis. When the initial substrate concentration was in the mixed-order region, nonlinear estimation of simulated data sets containing known measurement errors provided accurate estimates of the mu max, Ks, and Y values used to create these data. Nonlinear regression analysis of sigmoidal substrate depletion data was also evaluated for H2-limited batch growth of Desulfovibrio sp. strain G11. The integrated Monod equation can be more convenient for the estimation of growth kinetic parameters, particularly for gaseous substrates, but it must be recognized that the estimates of mu max, Ks, and Y obtained may be influenced by the growth rate history of the inoculum.     

Assimilation of alpha-glutamyl-peptides by human erythrocytes. A possible means of glutamate supply for glutathione synthesis.

Human erythrocytes are essentially impermeable to glutamate and yet there is a continual requirement for the amino acid for glutathione synthesis. In addition, the intracellular glutamate concentration is approximately five times that of plasma. We present evidence that glutamate enters the red cell as small peptides which are rapidly hydrolysed by cytoplasmic peptidase(s) and that with the estimated physiological levels of plasma glutamyl-peptides the rate of inward flux would be adequate to maintain the glutamate pool at its observed level. Experimentally, we used 1H spin-echo n.m.r. spectroscopy to follow peptide hydrolysis, since peptide spectra are different from those of the free amino acids and the spin-echo sequence enables the monitoring of reactions in concentrated lysates and whole cell suspensions. Thus, the system was studied under near-physiological conditions. Weighted non-linear regression analysis of progress curves using the integrated Michaelis-Menten equation was used to obtain estimates of Km and Vmax. for the hydrolysis of alpha-L-glutamyl-L-alanine and L-alanyl-alpha-L-glutamate in lysates and whole cell suspensions; the values for lysates were Km = 3.60 +/- 0.29 and 5.4 +/- 0.4 mmol/l and Vmax. = 120 +/- 4 and 46.7 +/- 1.7 mmol/h per 1 of packed cells respectively. In whole cell suspensions the rate of peptide hydrolysis was much slower and dominated by the transmembrane flux-rate. The estimates of the steady-state kinetic parameters for the transport were Kt = 2.35 +/- 0.41 and 11.2 +/- 1.0 mmol/l and Vmax. = 3.26 +/- 0.13 and 19.7 +/- 0.7 mmol/h per 1 of packed cells respectively for the previously mentioned peptides. Using the n.m.r. procedure we failed to detect any glutaminase activity in whole cells or lysates; thus, we exclude the possibility that glutamate gains entry to the cell as glutamine which is subsequently hydrolysed by glutaminase.     

Nonlinear estimation of Monod growth kinetic parameters from a single substrate depletion curve.

Monod growth kinetic parameters were estimated by fitting sigmoidal substrate depletion data to the integrated Monod equation, using nonlinear least-squares analysis. When the initial substrate concentration was in the mixed-order region, nonlinear estimation of simulated data sets containing known measurement errors provided accurate estimates of the mu max, Ks, and Y values used to create these data. Nonlinear regression analysis of sigmoidal substrate depletion data was also evaluated for H2-limited batch growth of Desulfovibrio sp. strain G11. The integrated Monod equation can be more convenient for the estimation of growth kinetic parameters, particularly for gaseous substrates, but it must be recognized that the estimates of mu max, Ks, and Y obtained may be influenced by the growth rate history of the inoculum.