Enzyme reactions at the surface of living cells. I. Electric repulsion of charged ligands and recognition of signals from the external milieu

The dynamic behaviour of a polyelectrolyte-bound enzyme is studied when diffusion of substrate or diffusion of product is coupled to electric repulsion and to Michaelis-Menten enzyme reaction. The definition of the classical concepts of electric partition coefficients and Donnan potential of a polyelectrolyte membrane has been extended under global non-equilibrium conditions. This extension is permissible when a strong repulsion exists of substrate and product by the fixed negative charges of the membrane. Coupling between product diffusion, electric repulsion and enzyme reaction at constant advancement may result in a hysteresis loop of the partition coefficient as the product concentration is increased in the reservoir. This hysteresis loop vanishes as the rate of product diffusion increases. No hysteresis loop may occur when electric repulsion effects are coupled to substrate diffusion and reaction. The existence of multiple values of the partition coefficient for a fixed concentration of product implies that the membrane may store short-term memory of the former product concentration present in the external milieu. The occurrence of hysteresis generated by coupling enzyme reaction, product diffusion, electric partition effects at constant advancement of the reaction may be viewed as a sensing device of product concentration in the external milieu. Surprisingly, non-linearities required to generate this sensing device come from electrostatic effects and not from enzyme kinetics.     

Kinetic properties of enzyme populations in vivo: alkaline phosphatase of the Escherichia coli periplasm.

Studies were conducted to determine the role that diffusion may play in the in vivo kinetics of the Escherichia coli periplasmic enzyme, alkaline phosphatase (AP, encoded by the gene pho A). Passive diffusion of solutes, from solution into the periplasm, is thought to occur mainly through porins in the outer membrane. The outer membrane therefore serves as a diffusion barrier separating a population of periplasmic enzymes from bulk substrate. E. coli strains containing a plasmid with the pho A gene linked to the lac promoter were used in this study in order to vary the amount of enzyme per cell. Alkaline phosphatase assays were conducted with intact cells, and the substrate concentration at half-maximum velocity (normally the Km for the enzyme) was determined as a function of enzyme concentration per cell. The results showed that diffusion of substrate to the enzyme caused as much as a 1000-fold change in this parameter, compared to that of purified enzyme. This suggested that diffusion was the rate-limiting step of the enzymatic reaction in these cells. In agreement with this type of reaction, Eadie-Hofstee and Lineweaver-Burk plots were not linear. At their extremes, these plots represented two types of kinetics. At high substrate concentration, equilibrium of substrate between bulk solution and the periplasm was achieved, and the kinetic properties conformed to Michaelis-Menten. At low substrate concentrations, there were a large number of free (unbound) enzymes, and each substrate molecule that entered the periplasm, through the diffusion barrier, resulted in product formation.(ABSTRACT TRUNCATED AT 250 WORDS)     

微球载体固定化纤维素酶的反应动力学模型研究

建立了固定化纤维素酶的反应动力学模型,该模型以米氏方程为基础并考虑了产物竞争性抑制的影响。在此模型的基础上进行模拟,系统研究了产物竞争性抑制、内扩散限制、溶液中的宏观底物浓度、载体大小等因素对球形载体内部的底物、产物浓度分布和效率因子的影响。产物竞争性抑制的存在将增加载体颗粒内的底物浓度,对效率因子的影响较小。底物内扩散系数或者反应体系中底物浓度增大时,载体颗粒内的底物浓度和效率因子都将增大。载体粒径增大,载体颗粒内的底物浓度和效率因子均减小。     

Kinetics studies in a continuous steady state hollow fiber membrane enzyme reactor

Porous hollow cellulose fibers have been used to separate a nonflowing enzyme solution of alkaline phosphatase from a continuous flow of substrate. The porosity of the hollow fiber membrane allows the substrate and product to diffuse freely through the membrane while restricting the permeation of the enzyme. The resulting “immobilized” enzyme system has been shown to behave as a continuous reactor—converting p-nitrophenylphosphate to p-nitrophenol. By varying the concentrations, flow rate, etc., either diffusion or enzyme kinetics can be studied. The continual influx of product and removal of substrate at steady state allows the study of kinetics of relatively short half-life enzymes and unstable systems.     

Influence of substrate and product diffusion on the heterogeneous kinetics of enzymic reversible reactions

When a reversible reaction is catalyzed by a surface bound enzyme, the diffusion of both substrate and product can considerably modify the kinetic properties of the reaction. According to this theoretical analysis, the enzyme activity is decreased due to the presence of substrate and product concentration gradients in the enzyme microenvironment, and the relative kinetic importance of the two diffusion steps mainly depend on the value of a dimensionless criterion inversely proportional to the equilibrium constant. Moreover diffusional effects increase with increasing bound enzyme activity, but decrease with increasing substrate and product concentration. Analytical expressions are established for the limiting values of substrate and product concentrations in the enzyme microenvironment, as well as for the increase in half-maximal-activity substrate concentration in the presence of substrate and product diffusional limitations.     

Effectiveness factors for substrate and product inhibition

The transformation technique of Na and Na (Math. Biosci., 6 , 25, 1970) is extended to convert boundary-value problems involving the steady-state diffusion equation for spherical immobilized enzyme particles exhibiting substrate and product inhibition to initial-value problems. This allows a study of the influence of external mass transfer resistances on the effectiveness factors. It also considerably reduces the number of calculations required to investigate the effect of changes in the kinetic parameters on the overall rate of reaction. The existence of multiple steady states for substrate inhibition kinetics in spherical catalyst particles is illustrated and a criterion for uniqueness of steady states is developed. Effectiveness factors for competitive and noncompetitive product inhibition increase with increasing value of the Sherwood number for the substrate and increasing value of the ratio of substrate to product effective diffusivities within the particle.     

Role of substrate inhibition kinetics in enzymatic chemical oscillations.

Two chemical kinetic models are investigated using standard nonlinear dynamics techniques to determine the conditions under which substrate inhibition kinetics can lead to oscillations. The first model is a classical substrate inhibition scheme based on Michaelis-Menten kinetics and involves a single substrate. Only when this reaction takes place in a flow reactor (i.e., both substrate and product are taken to follow reversible flow terms) are oscillations observed; however, the range of parameter values over which such oscillations occur is so narrow it is experimentally unobservable. A second model based on a general mechanism applied to the kinetics of many pH-dependent enzymes is also studied. This second model includes both substrate inhibition kinetics as well as autocatalysis through the activation of the enzyme by hydrogen ion. We find that it is the autocatalysis that is always responsible for oscillatory behavior in this scheme. The substrate inhibition terms affect the steady-state behavior but do not lead to oscillations unless product inhibition or multiple substrates are present; this is a general conclusion we can draw from our studies of both the classical substrate inhibition scheme and the pH-dependent enzyme mechanism. Finally, an analysis of the nullclines for these two models allows us to prove that the nullcline slopes must have a negative value for oscillatory behavior to exist; this proof can explain our results. From our analysis, we conclude with a brief discussion of other enzymes that might be expected to produce oscillatory behavior based on a pH-dependent substrate inhibition mechanism.     

Theoretical analysis of intrinsic reaction kinetics and the behavior of immobilized enzymes system for steady-state conditions

Mathematical modeling of immobilized enzymes under different kinetics mechanism viz. simple Michaelis–Menten, uncompetitive substrate inhibition, total competitive product inhibition, total non-competitive product inhibition and reversible Michaelis–Menten reaction are discussed. These five kinetic models are based on reaction diffusion equations containing non-linear terms related to Michaelis–Menten kinetics of the enzymatic reaction. Modified Adomian decomposition method is employed to derive the general analytical expressions of substrate and product concentration for all these five mechanisms for all possible values of the parameters Φ_S (Thiele modulus for substrate), Φ_P (Thiele modulus for product) and α (dimensionless inhibition degree). Also we have presented the general analytical expressions for the mean integrated effectiveness factor for all values of parameters. Analytical results are compared with the numerical results and also with the limiting case results, which are found to be good in agreement.     

Evaluation of intrinsic immobilized kinetics in hollow fiber reactor systems.

Immobilized cell and enzyme hollow fiber reactors have been developed for a variety of biochemical and biomedical applications. Reported mathematical models for predicting substrate conversion in these reactors have been limited in accuracy because of the use of free-solution kinetic parameters. This paper describes a method for determining the intrinsic kinetics of enzymes immobilized in hollow fiber reactor systems using a mathematical model for diffusion and reaction in porous media and an optimization procedure to fit intrinsic kinetic parameters to experimental data. Two enzymes, a thermophilic beta-galactosidase that exhibits product inhibition and L-lysine alpha-oxidase, were used in the analysis. The intrinsic kinetic parameters show that immobilization enhanced the activity of the beta-galactosidase while decreasing the activity of L-lysine alpha-oxidase. Both immobilized enzymes had higher Km values than did the soluble enzyme, indicating less affinity for the substrate. These results are used to illustrate the significant improvement in the ability to predict substrate conversion in hollow fiber reactors.     

Evaluation of the effectiveness factor along immobilized enzyme fixed-bed reactors: Design of a reactor with naringinase covalently immobilized into glycophase-coated porous glass

A design equation is presented for packed-bed reactors containing immobilized enzymes in spherical porous particles with internal diffusion effects and obeying reversible one-intermediate Michaelis-Menten kinetics. The equation is also able to explain irreversible and competitive product inhibition kinetics. It allows the axial substrate profiles to be calculated and the dependence of the effectiveness factor along the reactor length to be continuously evaluated. The design equation was applied to explain the behavior of naringinase immobilized in Glycophase-coated porous glass operating in a packed-bed reactor and hydrolyzing both p-nitrophenyl-alpha-L-rhamnoside and naringin. The theoretically predicted results were found to fit well with experimentally measured values.     

Regulation of ketogenesis. Mitochondrial acetyl-CoA acetyltransferase from rat liver: initial-rate kinetics in the presence of the product CoASH reveal intermediary plateau regions

The analysis of the initial-rate kinetics of the liver mitochondrial acetyl-CoA acetyltransferase (acetoacetyl-CoA thiolase) in the direction of acetoacetyl-CoA synthesis under product inhibition was performed. 1. Acetyl-CoA acetyltransferase shows a hyperbolic response of reaction velocity to changes in acetyl-CoA concentrations with an apparent Km of 0.237 +/- 0.001 mM. 2. CoASH is a (non-competitive) product inhibitor with a Kis of 22.6 microM and shifts the apparent Km for acetyl-CoA to the physiological concentration of this substrate in mitochondria (S0.5 = 1.12 mM in the presence of 121 microM CoASH). 3. CoASH causes a transformation of the Michaelis-Menten kinetics into initial-rate kinetics with four intermediary plateau regions. 4. The product analogue desulpho-CoA triggers a negative cooperativity as to the dependence of the reaction velocity on the acetyl-CoA concentration. These product effects drastically desensitize the acetyl-CoA acetyltransferase in its reaction velocity response to the acetyl-CoA concentrations and simultaneously extend the substrate dependence range. Thus a control of acetoacetyl-CoA synthesis by the substrate is established over the physiological acetyl-CoA concentration range. We suggest that this control mechanism is the key in establishing the rates of ketogenesis.     

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.     

Immobilized electric eel acetylcholinesterasemii. II. Flow kinetics of acetylcholinesterase chemically attached to nylon tubing.

Acetylcholinesterase has been attached covalently to the inner surface of nylon tubing. An experimental study has been carried out on the flow kinetics; solutions of acetylthiocholine at various concentrations were passed through tubing at various flow rates, and measurements made of the rates of formation of product. The results were analyzed in the light of the theoretical treatment of Kobayashi and Laidler, four different methods of analysis being employed. It is found that at lower substrate concentrations and flow rates the reactions are largely diffusion controlled. The Km(app) values are substantially higher than the Km value for diffusion-free conditions, but approach it as the flow rate is increased, when the diffusion layer becomes less important. The results are entirely consistent with the Kobayaski-Laidler theory, and provide guidelines for the design of open tubular heterogeneous enzyme reactors, both for industrial and analytic purposes.     

Intramolecular and intermolecular enzymatic modulation of ion channels in excised membrane patches.

A calcium-activated potassium channel in posterior pituitary nerve terminals was modulated by phosphorylation and dephosphorylation. Nearly every patch of membrane containing this channel also contained both membrane bound protein phosphatase and membrane-bound protein kinase. By examining the statistical and kinetic nature of phosphorylation and dephosphorylation in excised patches, it was possible to evaluate two contrasting models for these enzymatic reactions. One of these models treated catalysis as an intermolecular process in which the enzyme and substrate are separate molecular species that diffuse and encounter one another during collisions. The second model treated catalysis as an intramolecular process in which the enzyme and substrate reside within a stable macromolecular complex. The study began with a Poisson analysis of the distribution of channel number in patches, and of the number of protein phosphatase-free and protein kinase-free patches. Subsequent kinetic analysis of dephosphorylation yielded an estimate of the mean number of protein phosphatase molecules per patch that was similar to the value obtained from Poisson analysis. Because these two estimates were independent predictions based on the intermolecular model, their agreement supported this model. Analysis of channel number in protein phosphatase-free patches and of the rarity of patches showing partial but incomplete rundown provided additional support for the intermolecular model over the intramolecular model. Furthermore, dephosphorylation exhibited monotonic kinetics with a rate well below the diffusion limit. Thus, several different lines of analysis support the intermolecular model for dephosphorylation, in which the protein phosphatase must encounter its substrate to effect catalysis. In contrast to the monotonic kinetics of dephosphorylation, the phosphorylation reaction exhibited sigmoidal kinetics, with a rate that depended on membrane potential. Voltage dependence is an unlikely property for a kinetic step involving encounters resulting from diffusion. Furthermore, the velocity of the phosphorylation reaction exceeded the diffusion limit, and this observation is inconsistent with the intermolecular model. Thus, both intermolecular and intramolecular enzymatic mechanisms operate in the modulation of the calcium-activated potassium channel of the posterior pituitary. These studies provide a functional characterization of the interactions between enzyme and substrate in intact patches of cell membrane.     

Theory of the kinetics of reactions catalyzed by enzymes attached to membranes

A theoretical treatment has been worked out for the kinetics of solid-supported enzyme systems, with diffusive and electrostatic effects taken into account. A utilization factor, defined as the ratio of the actual reaction rate to the rate of substrate consumption in the outer solution, is defined, and equations to evaluate the utilization factor are given for five kinetic conditions: (a) Michaelis-Menten behavior, (b) substrate inhibition, (c) product inhibition (competitive), (d) product inhibition (noncompetitive), and (e) product inhibition (anticompetitive). When the solid-supported enzymes obey a Michaelis-Menten relationship, an equation for the apparent Michaelis constant is given and a criterion for insignificant diffusion effects is shown. A substrate-inhibited enzyme reaction may display multiple steady-state behavior, and a criterion for uniqueness is presented. In the case of product-inhibited enzyme reactions, the utilization factor is always less than that which corresponds to a Michaelis-Menten relationship. Equations to evaluate the apparent Michaelis and inhibition constants are given.     

Determination of relative rate constants for in vitro RNA processing reactions by internal competition

Studies of RNA recognition and catalysis typically involve measurement of rate constants for reactions of individual RNA sequence variants by fitting changes in substrate or product concentration to exponential or linear functions. A complementary approach is determination of relative rate constants by internal competition, which involves quantifying the time-dependent changes in substrate or product ratios in reactions containing multiple substrates. Here, we review approaches for determining relative rate constants by analysis of both substrate and product ratios and illustrate their application using the in vitro processing of precursor transfer RNA (tRNA) by ribonuclease P as a model system. The presence of inactive substrate populations is a common complicating factor in analysis of reactions involving RNA substrates, and approaches for quantitative correction of observed rate constants for these effects are illustrated. These results, together with recent applications in the literature, indicate that internal competition offers an alternate method for analyzing RNA processing kinetics using standard molecular biology methods that directly quantifies substrate specificity and may be extended to a range of applications.     

Kinetic studies of alpha-galactosidase-containing mold pellets on PNPG hydrolysis.

Little is known about techniques for applying untreated microbial cells containing enzymes directly to industrial processes as a biocatalyst. The kinetic behavior of alpha-galactosidase-containing spherical pellets which are formed naturally under given conditions in a submerged culture of Mortierella vinacea was studied on the hydrolysis of PNPG (p-nitrophenyl-alpha-D-galactopyranoside). The effect on intraparticle diffusion on the overall reaction rate was assessed by the use of an effectiveness factor, which was calculated by the approximate solution to the equation derived from the mass balance within a pellet. The experimental effectiveness factors were found to be represented as a single function of the modified Thiele modulus, including such parameters as pellet size, enzyme concentration in the pellet, and substrate concentration. As the diffusional effect became more significant, the marked substrate inhibition as seen for a free enzyme disappeared gradually. The effect of product inhibition on the pellets was much weaker than that for a free enzyme at a given substrate concentration. In the region of diffusion controlled reaction, it was found that the rate is proportional to the square root of the enzyme concentration in the pellet. In addition, similarly to what was reported previously for a free enzyme,the reaction in a batch system was found to be approximately representable as simple first-order kinetics in which the rate constant was dependent on the initial substrate concentration.     

Product Inhibition in Native-State Proteolysis

The proteolysis kinetics of intact proteins by nonspecific proteases provides valuable information on transient partial unfolding of proteins under native conditions. Native-state proteolysis is an approach to utilize the proteolysis kinetics to assess the energetics of partial unfolding in a quantitative manner. In native-state proteolysis, folded proteins are incubated with nonspecific proteases, and the rate of proteolysis is determined from the disappearance of the intact protein. We report here that proteolysis of intact proteins by nonspecific proteases, thermolysin and subtilisin deviates from first-order kinetics. First-order kinetics has been assumed for the analysis of native-state proteolysis. By analyzing the kinetics of proteolysis with varying concentrations of substrate proteins and also with cleavage products, we found that the deviation from first-order kinetics results from product inhibition. A kinetic model including competitive product inhibition agrees well with the proteolysis time course and allows us to determine the uninhibited rate constant for proteolysis as well as the apparent inhibition constant. Our finding suggests that the likelihood of product inhibition must be considered for quantitative assessment of proteolysis kinetics.     

Estimation of intrinsic kinetic constants for pore diffusion-limited immobilized enzyme reactions

A simple method is presented that establishes intrinsic rate parameters when slow pore diffusion of substrate limits immobilized enzyme reactions that obey Michaelis-Menten kinetics. The Aris-Bischoff modulus is employed. Data at high substrate concentrations, where the enzyme would be saturated in the absence of diffusion limitation, and at low substrate concentrations, where effectiveness factors are inversely proportional to reaction modulus, are used to determine maximum rate and Michaelis constant, respectively. Because Michaelis-Menten and Langmuir-Hinshelwood kinetics are formally identical, this method may be used to estimate intrinsic rate parameters of many heterogeneous catalysts. The technique is demonstrated using experimental data from the hydrolysis of maize dextrin with diffusion-limited immobilized glucoamylase. This system yields a Michaelis constant of 0.14%, compared to 0.11% for soluble glucoamylase and 0.24% for immobilized glucoamylase free of diffusional effects.     

Quantitative treatment of a bioreactor employing a recombinant cell system — A theoretical analysis

Summary A theoretical analysis on the product distribution in a bioreactor employing a recombinant cell system has been made. Simulated results of a batch or plug flow reactor based on Monod's kinetics show that a higher initial concentration of plasmidless cell should be avoided to prevent rapid consumption of substrate. This enables one to get maximum concentration of metabolite, the desired product.