The use of model-fitting in the interpretation of 'dual' uptake isotherms

Abstract. Published data of the concentration dependence of the uptake rate (uptake isotherms) of K⁺ Na⁺, Cl^?, SO2^?₄, and L-lysine in barley roots, and glucose and 3-O-methylglucose in potato tuber tissue, were re-examined. In as much as these isotherms yield non-linear, concave upward Eadie-Hofstee plots, they might have been termed ‘dual’ isotherms. In addition, all these isotherms have been considered to display discontinuous transitions in gradient. The following models that yield continuous isotherms were fitted to the isotherms: (1) the sum of a Michaelis-Menten term and a linear term; (2) the sum of two Michaelis-Menten terms; (3) the sum of two Michaelis-Menten terms and a linear one. Goodness of fit was judged from: (i) the weighted mean square of deviates; (ii) the standard errors of the kinetic parameters; (iii) the algebraic significance of the terms; (iv) a Rankits plot of the residuals; (v) a Runs test on the residuals. For the precise and detailed isotherms of SO^2? uptake, only model (3) gave a fit that was satisfactory in all respects. There appeared to be no reason to consider these isotherms as multiphasic. The same conclusion was reached for the L-lysine uptake isotherms. For the other isotherms the results were less conclusive. Thai for K⁺ and Na+ could, at any rate, be described satisfactorily by a continuous model, the best fit being obtained with model (2). The uptake isotherms of Cl^? and 3-0-methylglucose could be best described by model (2), and that of glucose by model (3), only the result of the Runs test being unsatisfactory. It is concluded that there is hardly any evidence that the presumed ‘jumps’ or discontinuities or inflections in the gradient of uptake isotherms are not due to experimental error in the data. It is suggested that many uptake isotherms may be described by model (3), although the reason for this is still incompletely understood.     

Multiple or multiphasic uptake mechanisms in plants?

Abstract Borstlap (1983) has alleged that (a) there are no abrupt changes in curves for the concentration dependence of solute uptake in plants, and (b) many such uptake isotherms may be described by the sum of two Michaelis-Menten terms and a linear term. These claims are considered in detail in connection with the recent finding (Nandi, Pant & Nissen, 1987) that phosphate uptake by corn roots increased more rapidly within the higher phases, i.e. at high external phosphate concentrations, but also levelled off faster than predicted from Michaelis-Menten kinetics. Similar deviations are, in retrospect, also found for uptake of other solutes and result in fewer phases at high external solute concentrations. The simplified and strikingly similar multiphasic patterns in the present paper show that (a) the abrupt changes in published isotherms are not due to error in the data, and (b) uptake isotherms cannot be adequately represented by the sum of two Michaelis-Menten terms and a linear term, or by similar continuous functions, if sufficiently detailed and precise data are used. These findings are not consistent with the existence of multiple uptake mechanisms, including free diffusion, in the plasmalemma. Uptake occurs instead by a single, multiphasic mechanism for each solute or group of related solutes. The similarities in the multiphasic patterns indicate, furthermore, that influx of the various solutes may be coupled.     

Validity of the multiphasic concept of ion absorption in plants

Borstlap (Naturwissenschaften 68: 41–43, 1981; Plant Cell Environ. 4: 189–195, 1981) has alleged that the concept of multiphasic uptake of solutes in plants is invalid. This claim is considered in detail and shown not to be justified. Contrary to Borstlap, the concentration-dependence of uptake cannot be adequately represented by the sum of two Michaelis-Menten terms and a linear term, or by similar continuous functions, if sufficiently detailed and precise data are used. Published evidence for the multiphasic concept, including the existence of discontinuities and separate uptake and transition sites, remains valid.     

The effect of energization on the apparent Michaelis–Menten constant for oxyge in nmitochondrial respiration

Lineweaver-Burk plots of 1/v against 1/[O(2)] for rat liver mitochondrial respiration with succinate or ascorbate+NNN'N'-tetramethyl-p-phenylenediamine as substrates are non-linear. In state 3u (uncoupled by trifluoromethoxycarbonyl cyanide phenylhydrazone) such plots tend to be concave upward, whereas in state 4 (energized) the plots were concave downward. The apparent K(m) for oxygen is larger in state 4 than in state 3u, despite the higher turnover in the latter system. It is postulated that at least one reversible reaction occurs between cytochrome c and cytochrome c oxidase, whose rate is increased on energization (reversed electron transfer); a model including such a reaction is proposed which accounts semiquantitatively for the observations.     

Multiphasic uptake of potassium by corn roots: no linear component

Concentration-dependence data (LV Kochian, WJ Lucas Plant Physiol 1982 70: 1723-1731; [1983] 73: 208-215; [1985] 429-436; LV Kochian, J Xin-zhi, WJ Lucas [1985], 79: 771-776) for potassium uptake by corn roots have been claimed to be resolvable into a saturable and a linear component representing two separate uptake mechanisms. A comprehensive analysis shows, however, that the data are not compatible with this concept. They can, in contrast, be precisely represented by multiphasic isotherms, in agreement with many findings for uptake of potassium and other solutes. Solute uptake in plants is proposed to be mediated by multistate entities having carrier-like properties at low external solute concentrations and channel-like properties at high concentrations.     

Liver purine nucleoside phosphorylase in Camelus dromedarius: purification and properties

1. Purine nucleoside phosphorylase (purine nucleoside:orthophosphate ribosyl transferase, EC 2.4.2.1) was purified to electrophoretic homogeneity from the liver of Camelus dromedarius. 2. The enzyme appears to be a dimer with a 44,000 subunit mol. wt and displays non-linear kinetics with concave downward curvature in double reciprocal plots with respect to both inosine and orthophosphate as variable substrates. 3. The effect of thiol compounds on the enzyme activity and of pH on kinetic parameters is reported.     

Uptake of Nitrate by Mutants of Arabidopsis thaliana, Disturbed in Uptake or Reduction of Nitrate

The uptake of nitrate by wildtype plants and chlorate-resistant mutants of Arabidopsis thaliana (L.) Heynh. was studied by intermittent or continuous measurement of the nitrate concentration of the ambient solution. The uptake rate in the wildtype and the nitrate reductase-less mutant B 25 showed a dual-phase relation ship with concentration. Each phase showed Michaelis-Menten kinetics although multiphasic patterns within each phase could not be excluded. A dual-phase relationship was also found in the uptake mutant B I. Here, however, phase II did not follow Michaelis-Menten kinetics and uptake rate of nitrate in the phase II concentration range was considerably lower in the B 1 mutant than in the wildtype. It is concluded that the mutation in B I has disturbed phase II of the nitrate uptake, without affecting phase I, which leads to the suggestion that uptake of nitrate in Arabidopsis is mediated by at least two independent uptake mechanisms. The nitrate uptake rate showed an optimum at pH 8, and it was not stimulated by the presence of calcium. Ammonium had different effects on nitrate uptake: a direct effect, when it was present during the uptake of nitrate, resulting in a release of nitrate and a reduced rate of uptake, and an indirect inhibitory effect, possibly caused by assimilation products of ammonium, which is most pronounced after growth on ammonium as the sole nitrogen source or in long-lasting uptake experiments in the presence of ammonium. Chlorate also showed a multiple effect, an inhibiting one which proved to be competitive and, at very low concentrations of chlorate, a stimulating one. Evidence was obtained that chlorate and nitrate arc taken up by the same carrier.     

A comparison of methods for estimating the kinetic parameters of two simple types of transport process

Sets of experimental data, with known characteristics and error structures, have been simulated for the Michaelis-Menten equation plus a second term, either for linear transport or for competitive inhibition. The Michaelis-Menten equation plus linear term was fitted by several methods and the accuracy and the precision of the parameter estimates from the several methods were compared. The model-fitting methods were: three for least-squares non-linear regression, computer versions of two graphical methods and of two non-parametric methods. The most precise and accurate method was that of D.W. Marquardt (J. Soc. Ind. Appl. Math. 11 (1963) 431–441). The Michaelis-Menten equation with competitive inhibition was also fitted by several methods, viz., two for least-squared non-linear regression, a non-parametric method and four variants of the Preston-Schaeffer-Curran plot (Preston, R.L. et al. (1974) J. Gen. Physiol. 64, 443–467). The most precise and accurate of these was the non-linear regression method of W.W. Cleland (Adv. Enzymol. 29 (1967) 1–32). For both these models, the various graphical methods and non-parametric methods gave poor results and are not recommended.     

Kinetic studies of the enzymatic isomerization of xylose

d-Xylose isomerase catalyses the conversion of the common pentose, d-xylose, to its keto-isomer, d-xylose. This reaction is of interest because many microorganisms that are unable to metabolize d-xylose can utilize d-xylulose. The kinetics of a commonly used immobilized whole-cell isomerase, Sweetzyme Q, have been determined from initial rate studies on the forward and reverse reactions. The effect of pH, temperature, and substrate and product concentration on enzyme activity have all been examined. Reaction rates were modelled with the Michaelis-Menten equation. Using constants determined from Lineweaver-Burk plots, the rate equation accurately simulated experimental conversion data.     

Oxygen diffusion in a spherical cell with nonlinear oxygen uptake kinetics

The oxygen diffusion in a spherical cell is analyzed in the present work. An oxygen uptake kinetics of the Michaelis-Menten type is employed. The oxygen uptake kinetics predicts the oxygen uptake rates which agree fairly well with the observed data. It has been found that difference between the predicted steady state oxygen tension distribution using the previous simplified oxygen uptake kinetics and that using the present non-linear kinetics is very significant.     

External and internal diffusion in heterogeneous enzymes systems

The intrusion of diffusion in heterogeneous enzyme reactions, which follow. Michaelis-Menten kinetics, is quantitatively characterized by dimensionless parameters that are independent of the substrate concentration. The effects of these parameters on the overall rate of reaction is illustrated on plots commonly employed in enzyme kinetics. The departure from Michaelis-Menten kinetics due to diffusion limitations can be best assessed by using Hofstee plots which are also suitable to distinguish between internal and external transport effects. A graphical method is described for the evaluation of the reaction rate as a function of the surface concentration of the substrate from measured data.     

Kinetics of galactose uptake by perfused rat livers: applicability of a family of models

Mathematical models for general substrate uptake mechanisms in the liver have been used to describe the kinetics of galactose removal. In this study sets of galactose uptake rates to galactose concentration relations obtained by perfusion of ten livers of 200 g rats were examined. The rate of galactose uptake (ν) was related to the galactose concentration in the sinusoids (calculated as the logarithmic mean of in- and outlet concentration, ?). In all experiments a saturation pattern emerged, but the resulting 1/ν, versus $\text{1/}\text{c}\text{?}$ plots were all markedly convex, discarding simple Michaelis-Menten kinetics. Data were therefore examined in the light of a family of kinetic models, including the following modifications: substrate inhibition, porto-systemic shunting, and allosterism. The kinetic constants were assessed by iterative procedures, aiming at linearization of the double reciprocal plots. The two latter models were found to fit the experimental data by entering a shunting of 61% of the hepatic blood flow, or two active sites, respectively. Since this degree of shunting is improbable the results speak in favour of allosterism. The work gives an example of whole-liver kinetic considerations when simple Michaelis-Menten is insufficient.     

Mechanism of sulfate uptake by excised maize roots

The mechanism of uptake of sulfate by excised maize roots (Zea mays L. cv. Ganga-5) was investigated. It was found that in the concentration range l × 10^?9 M ? 5 × 10^?2 M, the sulfate uptake could be represented by a single multiphasic isotherm having four phases. Each phase covered a limited concentration range and obeyed Michaelis-Menten kinetics, which indicates mediation in sulfate uptake by a structure which changes characteristics (kinetic constants) at certain discrete salts concentrations (transition points).     

Comparison of Substrate Affinities Among Several Rumen Bacteria: a Possible Determinant of Rumen Bacterial Competition

Five rumen bacteria, Selenomonas ruminantium, Bacteroides ruminicola, Megasphaera elsdenii, Streptococcus bovis, and Butyrivibrio fibrisolvens were grown in continuous culture. Estimates of substrate affinities were derived from Lineweaver-Burk plots of dilution rate versus substrate concentration. Each bacterium was grown on at least four of the six substrates: glucose, maltose, sucrose, cellobiose, xylose, and lactate. Wide variations in substrate affinities were seen among the substrates utilized by a species and among species for the same substrate. These wide differences indicate that substrate affinity may be a significant determinant of bacterial competition in the rumen where soluble substrate concentrations are often low. Growth of these bacteria in continuous culture did not always follow typical Michaelis-Menten kinetics. Inflated theoretical maximum growth rates and non-linear Lineweaver-Burk plots were sometimes seen. Maintenance energy expenditures and limitation of growth rate by factors other than substrate concentration (i.e., protein synthesis) are discussed as possible determinants of these deviations.     

Multiphasic Uptake of Ammonium by Soybean Roots

Uptake of ammonium by intact soybean (Glycine max Merr. cv. Amsoy) plants can be represented by 3 phases of a single, multiphasic mechanism in the range 1.78 × 10^-5-3.57 × 10^-3M. Each phase covers a limited concentration range and obeys Michaelis-Menten kinetics. The multiphasic pattern of NH₄⁺ uptake is remarkably consistent at all stages of soybean growth (20, 40, 60 and 80 days).     

Separation of lactic acid from acetic acid using a four-zone SMB

A simulated moving bed (SMB) process has been developed to separate l-(+)-lactic acid from acetic acid, a major impurity in the fermentation broth of Lactobacillus rhamnosus. Poly(4-vinylpyridine) resin (PVP) was selected as the adsorbent. Adsorption isotherms and mass transfer parameters of the organic acids were estimated from single-column frontal tests. Experimental results show that the Langmuir isotherms obtained from the frontal tests can be used in the design of an SMB process to achieve 99.9% purity and over 93% yield of lactic acid. The column profiles and effluent histories, however, deviate from rate model predictions based on the Langmuir isotherms. They agree more closely with the predictions based on a modified Langmuir isotherm for lactic acid. The standing wave design method for systems with modified Langmuir isotherms is developed in this study. Rate model simulations show that the process based on the modified design method can achieve high purity (>99.9%) and high yield (>99.9%). For this nonlinear system, accurate isotherm model and model parameters are needed in the design, and the zone flow rates must be closely monitored and controlled in order to ensure high purity and high yield in the SMB process.     

Ion Uptake in Higher Plants and KCl Stimulation of Plasmalemma Adenosine Triphosphatase: Comparison of Models

Criteria for the acceptance or rejection of the dual, the cooperative, and the multiphasic model of ion uptake are given and are used to evaluate the models on the basis of previously published analyses. In addition, mathematical representations of the models are fitted to concentration-dependence data for various ions and plant tissues by a general curve fitting program. The calculated parameters are evaluated for biological relevance, and the fit is compared by statistical analysis. The kinetics of ion uptake in higher plants are found to be consistent with the concept of multiphasic uptake mechanisms, while the dual and the cooperative model must be rejected. KCl stimulation of plasmalemma-bound ATPases is also shown to obey multiphasic kinetics, thus strengthening the correlation between ion uptake and membrane-bound ATPases.     

Dose responses of auxins

Published data for the most widely used bioassays for auxins as. well as other auxin-induced responses, have been analyzed to determine whether the response is ultrasensitive, hyperbolic or subsensitive, i.e. whether the change from 10 to 90% of maximal response requires less than, equal to or more than an 81-fold increase in external auxin concentration. Auxin-dependent callus growth is most often ultrasensitive. Other responses, including the curvature of Avena coleoptiles, may show simple Mi-chaelis-Menten kinetics. In the majority of cases, however, the response is sub-sensitive, often very markedly so. The finding of subsensitive responses is consistent with the proposal by A. J. Trewavas that plant development is regulated by changes in the sensitivity to plant growth substances.
Detailed data for subsensitive dose responses, particularly auxin-induced inhibition of root growth, can often be precisely represented by bi- or multiphasic isotherms, with the phases separated by sharp transitions. The relationship to auxin uptake and the relationship, if any, between carriers and receptors for auxins remain to be elucidated.     

Pig heart fumarase really does exhibit negative kinetic co-operativity at a constant ionic strength.

The kinetics of the action of fumarase on L-malate and fumarate were investigated at constant ionic strength. This was done to evaluate reports that fumarase follows simple Michaelis-Menten kinetics. However, when pH, buffer concentration and ionic strength are all maintained at constant values, the Lineweaver-Burk plots exhibit pronounced downward curvature, characteristic of negative kinetic co-operativity.