A mechanism for multiphasic uptake of solutes in plants

A molecular mechanism is proposed for unitary multiphasic uptake in which the carrier has two binding sites for the substrate. The first site binds n-1 molecules of substrate, and then one additional substrate molecule can become bound at the second binding site. Only this last molecule is transported in the operation of the carrier molecule. In the free state, the carrier can be activated to successive states with increasing affinities for the substrate in the two binding sites. The mechanism is resolved for the steady state conditions, obtaining a simple uptake rate equation, which fits the experimental data. Methods for determining the parameters of the equation are presented. Evidence other than kinetics is discussed for the mechanism. The mechanism also provides a physiological interpretation for multiphasic uptake: the active transport mechanisms (energy-requiring mechanisms) are prevented from operating at high substrate concentrations, thus preventing a waste of energy by the cells.     

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.     

Phosphate‐limited growth of the marine diatom Thalassiosira weissflogii (Bacillariophyceae): evidence of non‐monod growth kinetics1

The marine diatom Thalassiosira weissflogii (Grunow) G. A. Fryxell & Hasle was grown in a chemostat over a series of phosphate‐limited growth rates. Ambient substrate concentrations were determined from bioassays involving picomolar spikes of ³³P‐labeled phosphate, and maximum uptake rates were determined from analogous bioassays that included the addition of micromolar concentrations of unlabeled phosphate and tracer concentrations of ³³P. The relationship between cell phosphorus quotas and growth rates was well described by the Droop equation. Maximum uptake rates of phosphate spikes were several orders of magnitude higher than steady state uptake rates. Despite the large size of the T. weissflogii cells, diffusion of phosphate through the boundary layer around the cells had little effect on growth kinetics, in part because the cellular N:P ratios exceeded the Redfield ratio at all growth rates. Fitting the Monod equation to the experimental data produced an estimate of the nutrient‐saturated growth rate that was ~50% greater than the maximum growth rate observed in batch culture. A modified hyperbolic equation with a curvature that is a maximum in magnitude at positive growth rates gave a better fit to the data and an estimate of the maximum growth rate that was consistent with observations. The failure of the Monod equation to describe the data may reflect a transition from substrate to co‐substrate limitation and/or the presence of an inducible uptake system.     

Modeling substrate inhibition of microbial growth

This article presents a general equation for substrate inhibition of microbial growth using a statistical thermodynamic approach. Existing empirical models adapted from enzyme kinetics, for example, the Haldane-Andrews equation, often criticized for not being physically based for microbial growth, are shown to derive from the general equation in this article, and their empirical parameters are shown to be well defined physically. Three sets of experimental data from the literature are used to test the modeling abilities of the general equation to represent experimental data. The results are compared with those obtained by fitting the same data set to a widely used empirical model existing in the literature. The general equation is found to represent all three experimental data sets better than the alternative model tested. In addition, a graphical method existing in enzyme kinetics is successfully adapted and further developed to determine the number of inhibition sites of a basic functional unit of a bacterial cell. (c) 1996 John Wiley & Sons, Inc.     

Evaluation of steady-state kinetic parameters for enzymes solubilized in water-in-oil microemulsion systems.

1. Equations are derived for the steady-state kinetics of substrate conversion by enzymes confined within the water-droplets of water-in-oil microemulsion systems. 2. Water-soluble substrates initially confined within droplets that do not contain enzyme are assumed to be converted into product only after they enter enzyme-containing droplets via the inter-droplet exchange process. 3. Hyperbolic (Michaelis-Menten) kinetics are predicted when the substrate concentration is varied in microemulsions of fixed composition. Both kcat. and Km are predicted to be dependent on the size and concentration of the water-droplets in the microemulsion. 4. The predicted behaviour is shown to be supported by published experimental data. A physical interpretation of the form of the rate equation is presented. 5. The rate equation for an oil-soluble substrate was derived assuming a pseudo-two-phase (oil & water) model for the microemulsion. Both kcat. and Km are shown to be independent of phi aq. Km is larger than the aqueous solution value by a factor approximately equal to the oil/water partition coefficient of the substrate. The validity of the rate equation is confirmed by published data.     

A dynamic mathematical model of the chemostat

A number of experimental studies on the dynamic, behavior of the chemostat have shown that the specific growth rate does not, instantaneously adjust to changes in the concentration of limiting substrate in the chemostat following disturbances in the steady state input limiting substrate concentration or in the steady state dilution rate. Instead of an instantaneous response, as would be predicted by the Monod equation, experimental studies have shown that the specific growth rate experiences a dynamic lag in responding to the changes in the concentration of limiting substrate in the culture vessel. The observed dynamic lag has been recognized by researchers in such terms as an inertial phenomenon and as a hysteresis effect, but as yet a systems engineering approach has not been applied to the observed data. The present paper criticizes the use of the Monod equation as a dynamic relationship and offers as an alternative a dynamic equation relating specific growth rate to the limiting substrate concentration in the chemostat. Following the development of equations, experimental methods of evaluating parameters are discussed. Dynamic responses of analog simulations (incorporating the newly derived equations) are compared with the dynamic responses predicted by the Monod equation and with the dynamic responses of experimental chemostats.     

Experimental evidence for allosteric modifier saturation as predicted by the bi-substrate Hill equation

The cooperative enzyme reaction rates predicted by the bi-substrate Hill equation and the bi-substrate Monod-Wyman-Changeux (MWC) equation when allosterically inhibited are compared in silico. Theoretically, the Hill equation predicts that when the maximum inhibitory effect at a certain substrate condition has been reached, an increase in allosteric inhibitor concentration will have no effect on reaction rate, that is the Hill equation shows allosteric inhibitor saturation. This saturating inhibitory effect is not present in the MWC equation. Experimental in vitro data for pyruvate kinase, a bi-substrate cooperative enzyme that is allosterically inhibited, are presented. This enzyme also shows inhibitor saturation, and therefore serves as experimental evidence that the bi-substrate Hill equation predicts more realistic allosteric inhibitor behaviour than the bi-substrate MWC equation.     

产弹性蛋白酶细菌的发酵动力学研究

基于14L的发酵罐分批发酵实验数据,建立了发酵过程菌体生长、产物生成及基质消耗随时间变化的数学模型。Logistic方程、Luedeking—Piret方程能够很好地分别描述产弹性蛋白酶菌体生长;发酵产酶过程和基质消耗过程。并将3个动力学模型的预测值和实验值进行了比较,所建立的分批发酵动力学模型能较好地反映弹性蛋白酶分批发酵过程。     

Theory of cell age distribution in suspension culture with feed and/or drain

A mixed problem for the M'Kendrick-von Foerster equation satisfied by the number density function in terms of the age of the viable cells in a suspension culture with feed and/or drain is solved, and a method of calculating the number density function and time-dependent generation time from observed data of cell number and cell mass is presented. This theory is adequate to analyze the growth of cells that undergo binary fission. The equation of mass balance follows as a natural consequence of this treatment. The equation of substrate balance in consideration of the effect of cell volume is derived rigorously.     

Extended monod kinetics for substrate, product, and cell inhibition

A generalized form of Monod kinetics is proposed to account for all kinds of product, cell, and substrate inhibition. This model assumes that there exists a critical inhibitor concentration above which cells cannot grow, and that the constants of the Monod equation are functions of this limiting inhibitor concentration. Methods for evaluating the constants of this rate form are presented. Finally the proposed kinetic form is compared with the available data in the literature, which unfortunately is very sparse. In all cases, this equation form fitted the data very well.     

Sucrose Synthetase of Sweet Potato Roots

The effect of concentration of each substrate in the reaction catalyzed by sucrose synthetase isolated from sweet potato roots was determined. For the sucrose synthesizing reaction, UDP-glucose(ADP-glucose)+fructose→sucrose+UDP(ADP), the substrate saturation curves for UDP-glucose, ADP-glucose and fructose were hyperbolic in shape and the reaction was strongly inhibited by UDP competitively. On the other hand, the substrates for the reversal of sucrose synthetase reaction, sucrose+UDP(ADP)→UDP-glucose(ADP-glucose)+fructose, exhibited a sigmoidal shaped saturation curve which was deviated from the Michaelis-Menten equation. The plot of data according to the empirical Hill equation gives a values greater than 1.0 for every substrate examined in the latter case. In view of these experimental data, the major role of sucrose synthetase is postulated in that this enzyme is involved in the breakdown of sucrose in sweet potato root tissues instead of the sucrose synthesizing reaction. The molecular weight of the enzyme was determined to be about 540,000 by the Sephadex gel filtration chromatography.     

Novel differential elimination method for determining kinetic coefficients under substrate self-inhibition

A differential elimination method (DEM) is developed to determine the kinetic coefficients for substrate self-inhibition. Finite differentiation of the equation eliminates either K_I or K_S, which enables the equation to be linearized so that [^(\textq)] {\hat{\text{q}}} , K_S, and K_I can be estimated without using nonlinear least square regression (NLSR). The DEM options that eliminate K_I or K_S computed the parameter values exactly when the data did not contain any errors. If one-point or random errors were not too large, both DEM options worked as well as NLSR when data were acquired with geometric intervals for substrate concentration. The DEM was more accurate for fitting the data for the smallest and largest values of S, but relatively weaker in estimating the observed maximum substrate utilization rate, q_max. The estimates for S_max, the concentration at which the maximum specific substrate utilization rate is observed, were relatively invariant among the methods, even when K_S and K_I differed. When the intervals were arithmetic (i.e., equal intervals of substrate concentration) and the data contained errors, the DEM and NLSR estimated the parameters poorly, indicating that collecting data with an arithmetic interval greatly increases the risk of poor parameter estimation. Parameter estimates by DEM fit very well experimental data from nitrification or photosynthesis, which were taken with geometric intervals of substrate concentration or light intensity, but fit poorly phenol-degradation data, which were obtained with arithmetic substrate intervals. Besides providing a reasonable substitute for NLSR, the DEM also can be used as a tool to diagnose the quality of experimental data by comparing its estimates between the DEM options, or, more rigorously, to those from NLSR.     

Accurate kinetic parameter estimation during progress curve analysis of systems with endogenous substrate production

We have identified an error in the published integral form of the modified Michaelis–Menten equation that accounts for endogenous substrate production. The correct solution is presented and the error in both the substrate concentration, S, and the kinetic parameters V_m, K_m, and R resulting from the incorrect solution was characterized. The incorrect integral form resulted in substrate concentration errors as high as 50% resulting in 7–50% error in kinetic parameter estimates. To better reflect experimental scenarios, noise containing substrate depletion data were analyzed by both the incorrect and correct integral equations. While both equations resulted in identical fits to substrate depletion data, the final estimates of V_m, K_m, and R were different and K_m and R estimates from the incorrect integral equation deviated substantially from the actual values. Another observation was that at R = 0, the incorrect integral equation reduced to the correct form of the Michaelis–Menten equation. We believe this combination of excellent fits to experimental data, albeit with incorrect kinetic parameter estimates, and the reduction to the Michaelis–Menten equation at R = 0 is primarily responsible for the incorrectness to go unnoticed. However, the resulting error in kinetic parameter estimates will lead to incorrect biological interpretation and we urge the use of the correct integral form presented in this study. Biotechnol. Bioeng. 2011;108: 2499–2503. © 2011 Wiley Periodicals, Inc.     

Analysis of cell growth in a polymer scaffold using a moving boundary approach

Two mathematical models of chondrocyte generation and nutrient consumption are developed to analyze the behavior of cell growth in a biodegradable polymer matrix. Substrate reaction and diffusion are analyzed in two regions: one consisting of cells and nutrients and the other consisting of only nutrients. A pseudo-steady state approximation for the transport of nutrients in these two regions is utilized. The rate of growth is determined by a moving boundary equation that equates the rate at which the interfacial region between the cells and the void space moves to a substrate dependent growth reaction. The change in the location of this interfacial region with time therefore depicts the rate at which the cells propagate. The two limiting cases discussed in this article represent extremes in how the cells will grow in the polymer matrix; one case assumes that cells grow inward from the external boundary, and the other case assumes that cells grow parallel to the external boundary. The results of both models are compared to experimental data found in the literature. It is found through these comparisons that the model parameters, including the unit cell spacing parameter L, the metabolic rate constant k, the growth rate constant k(G), and external mass transfer coefficient, K, may vary as the thickness of the polymer matrix is changed, however, unrealistic and large changes in the diffusion coefficients were required to account for the full range of experimental data. Furthermore, these results suggest modification of the functional form of the growth kinetics to include substrate or product inhibition, or death terms. Based upon diffusion/reaction concepts, these models for cell growth in a biodegradable polymer give bounds for the upper and lower limits of the cellular growth rate and nutrient consumption in a polymer matrix and will aid in the development of more extensive models. (c) 1997 John Wiley & Sons, Inc. Biotechnol Bioeng 56: 422-432, 1997.     

A comparison of two methods for fitting the integrated Michaelis–Menten equation (Short Communication)

The methods of Atkins & Nimmo (1973) and Fernley (1974) for fitting the integrated Michaelis-Menten equation were compared by using the same sets of simulated experimental data. The method of Fernley (1974) is to be preferred because it gives precise and unbiased estimates of the Michaelis-Menten parameters over a wide range of substrate concentrations. However, the estimates may not be symmetrically distributed, especially at low substrate concentrations.     

Regulation of Tension in the Skinned Crayfish Muscle Fiber : II. Role of Calcium

Tension outputs were measured in skinned crayfish muscle fibers exposed to solutions variously buffered for both Mg-adenosine triphosphate (ATP) and Ca. Two types of data are shown, relating tension and substrate concentration with different levels of Ca present, or tension and calcium concentration at different levels of substrate. The data are fitted by curves calculated from a general equation for substrate inhibition. The equation is based on the schema that both tension and relaxation are induced by the substrate and that the relaxing effect of excess substrate is repressed by calcium. The physiological findings of the present work are similar to data obtained by others on biochemical model systems of the contractile proteins.     

Steady-state kinetic studies of the negative co-operativity and flip-flop mechanism for Escherichia coli alkaline phosphatase.

1. A study of variations in experimental error of velocity measurement with substrate concentration for alkaline phosphatase reveals that the standard error is not constant or strictly proportional to velocity, but obeys a more complex dependence. 2. By using an approach based on error estimates at each individual substrate concentration, we show that the double-reciprocal plots in general are curved, necessitating a high-degree rate equation. The curves are analysed according to a recent classification of possible curve shapes for the 3:3 function, which is shown to be the lowest-degree rate equation satisfying the experimental data. 4. Other workers have supposed the enzyme to follow Michaelis-Menten kinetics, and it is shown that this assumption is approximately true at low temperatures in the absence of phosphate. 5. A study of the effects of phosphate concentration, pH and temperature on the kinetics shows that there is a gradual alteration in curve shape with these experimental variables, resulting in an apparent reduction in degree under certain special conditions, and particularly at low temperature. 6. It is shown that the steady-state kinetics do not require a flip-flop or half-of-sites reactivity mechanism as claimed, and a mechanism is proposed, a rate equation calculated and an analysis attempted. 7. An analysis of the product-inhibition effects for a linked two-sited Uni Bi enzyme is given. Alterations of asymptotic double-reciprocal slopes and limiting (1/nu) intercepts with products is discussed, and it is shown how the theory of product inhibition can be extended to complex kinetic situations to extract information as to molecular mechanism. 8. Deviations from Michaelis-Menten kinetics are expressed in terms of the magnitude of the appropriate Sylvester resultants.     

Relationship between substrate concentration,growth rate,and respiration rate of Escherichia coli in continuous culture

Summary Using a continuous flow technique the relationship between growth rate and substrate concentration was investigated with glucose as the limiting factor of a culture of Escherichia coli. Graphical and numerical analysis of the experimental data demonstrated that the application of the Michaelis-Menten equation produced erroneous results, whereas, the constants obtained from the Teissier equation were in agreement with the experimental data. On this basis, new equations defining the steady state cell and substrate concentration in continuous flow cultures were developed and tested against experimental data.Comparison of the specific growth rates, substrate uptake rates and oxygen consumption rates demonstrated that all were directly proportional to each other and could be related to each other by mathematical equations. Specifically it was shown that as the growth rate increased from 0.06 to k _m =0.76 the substrate uptake rate increased from 134 to 1420 mg glucose per gram cell weight per hour and the oxygen consumption rate increased from 48.6 to 505 mg O₂ per gram cell weight per hour. Independent of the growth rate 37% of the carbohydrate consumed were oxidized. The yield factor varied from 0.44 at low growth rates to 0.54 at high growth rates. Analysis of the growth rate-substrate uptake rate relationship indicated that a minimum substrate uptake rate of 55 mg glucose per gram cell weight per hour existed below which cell reproduction would cease. This was supported by the fact that steady state conditions could not be maintained in the culture at D values below 0.02 when the substrate supply rate decreased below 45 mg glucose per gram cell weight per hour.Material contained in this paper was submitted as a thesis in partial fulfillment of the requirements for the Ph. D. degree of Dr. R. S. Lipe.     

The time course of hydrolysis of a beta-lactam antibiotic by intact gram-negative bacteria possessing a periplasmic beta-lactamase.

An equation is derived from first principles for describing the change in concentration with time of a beta-lactam antibiotic in the presence of intact Gram-negative bacteria possessing a beta-lactamase located in the periplasmic space. The equation predicts a first-order decline in beta-lactam concentration of the form [S] = [Si]e lambda t, where [S] is the exogenous concentration of beta-lactam, [Si] is the value of [S] at time zero, t is the time from mixing of cells and antibiotic and lambda (less than 0) is the decay constant. The value of lambda is exactly described by the theory in terms of experimentally measurable quantities. Quantitative data concerning cephaloridine hydrolysis by intact cells of Haemophilus influenzae agreed well with the theory, as did data concerning the uptake of 2-nitrophenyl galactoside by intact cells of Escherichia coli. Cephalosporin C hydrolysis by intact cells of Pseudomonas aeruginosa did not progress as predicted by the theory. The theory is applicable to any substrate which is acted on by an enzyme that is located solely in the periplasmic space and that obeys the Michaelis-Menten equation of enzyme kinetics.