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.     

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.     

Transient response of encapsulated enzymes in hollow-fiber reactor

A mathematical model for the transient response of encapsulated enzymes is developed showing the effects of the outer boundary layer, the encapsulating membrane, the partition coefficient, and diffusion with reaction within the encapsulating medium. The model incorporates both first-order kinetics and Michaelis-Menten kinetics for the reaction rate. Using typical hollow-fiber or microcapsule parameters, the model shows that (a) the partition coefficient affects the overall rate only when the rate-limiting step is diffusion through the membrane, (b) the transient overall effectiveness factor rises sharply with time and approaches an asymptotic value for most situations, and (c) the first-order approximation to Michaelis-Menten kinetics is not valid when the initial outside bulk concentration is higher than the Michaelis constant and the overall rate is reaction limited. The model is compared with experimental data using uricase in a hollow-fiber enzyme reactor configuration. Batch assay and CSTUER (continuous-stirred ultrafiltration enzyme reactor) studies were conducted on the free enzyme to provide some of the parameters used in the model. The CSTUER data fit the case of substrate inhibition kinetics with the apparent Michaelis constant approaching zero. The hollow-fiber reactor was conducted with uricase dissolved in both a buffer solution and a concentrated hemoglobin solution. Diffusivities of the solute were measured in both solutions as was the osmotic pressure of the hemoglobin solution. While experimental data for uricase in buffer solution could easily be matched by the model, that in the concentrated hemoglobin solution could not.     

Accuracy of alternative representations for integrated biochemical systems

The Michaelis-Menten formalism often provides appropriate representations of individual enzyme-catalyzed reactions in vitro but is not well suited for the mathematical analysis of complex biochemical networks. Mathematically tractable alternatives are the linear formalism and the power-law formalism. Within the power-law formalism there are alternative ways to represent biochemical processes, depending upon the degree to which fluxes and concentrations are aggregated. Two of the most relevant variants for dealing with biochemical pathways are treated in this paper. In one variant, aggregation leads to a rate law for each enzyme-catalyzed reaction, which is then represented by a power-law function. In the other, aggregation produces a composite rate law for either net rate of increase or net rate of decrease of each system constituent; the composite rate laws are then represented by a power-law function. The first variant is the mathematical basis for a method of biochemical analysis called metabolic control, the latter for biochemical systems theory. We compare the accuracy of the linear and of the two power-law representations for networks of biochemical reactions governed by Michaelis-Menten and Hill kinetics. Michaelis-Menten kinetics are always represented more accurately by power-law than by linear functions. Hill kinetics are in most cases best modeled by power-law functions, but in some cases linear functions are best. Aggregation into composite rate laws for net increase or net decrease of each system constituent almost always improves the accuracy of the power-law representation. The improvement in accuracy is one of several factors that contribute to the wide range of validity of this power-law representation. Other contributing factors that are discussed include the nonlinear character of the power-law formalism, homeostatic regulatory mechanisms in living systems, and simplification of rate laws by regulatory mechanisms in vivo.     

Determination of effective diffusion coefficient of substrate in gel particles with immobilized biocatalyst

A method of determining of the effective diffusion coefficient of substrate in a particle, where the diffusion and consumption of substrate by biocatalytic reaction are present simultaneously, was designed and experimentally verified. The method is based on measuring the overall rate of heterogeneous biocatalytic reaction in particles of varying diameter. The effective diffusion coefficient, D_e, was determined by fitting the measured reaction rates with the solution of the reaction-diffusion equation. The method is tailored for cases where the enzyme reaction is governed by the Michaelis-Menten kinetics. The value of K_m required for the solution of the mathematical model was adopted from the measurement of the kinetics of free cells, whereas the rate parameter, k₂, was optimized together with D_e. As an experimental model, the sucrose hydrolysis catalyzed by Ca-alginate-entrapped yeast cells was examined. The particle diameter varied in the range of 1.2–3.9 mm and the initial reaction rates were measured in a batch-stirred reactor at a sucrose concentration of 100 m . The D_e of sucrose at 30°C was found to be 2.9 · 10⁻¹⁰ m²s⁻¹.     

The effects of chemical kinetics on oxygen delivery to tissue.

A mathematical model has been formulated to analyze the effect of nonequilibrium kinetics on oxygen delivery to tissue. The model takes into account molecular diffusion, facilitated diffusion in the capillary blood, convection, chemical kinetics of O2 with hemoglobin, and the rate of metabolic consumption. A line iterative technique is described to solve numerically the resulting coupled system of nonlinear partial differential equations with physiologically relevant boundary and entrance conditions. With nonequilibrium kinetics the end-capillary PO2 is found to be lower than that in the venous blood. The effect is more pronounced during hypoxia and anemia. It is found that the tissue PO2 at the lethal corner decreases with the decrease in blood velocity, arterial PO2, hemoglobin concentration, P50, and increase in COHb concentration or metabolic rate, while the difference between end-capillary PO2 and venous PO2 increases, which reflects the effect of nonequilibrium kinetics on the delivery of O2 to tissue. Thus, the consideration of venous PO2 as an indicator of tissue PO2 in clinical and experimental studies may be questionable.     

Sexual reproduction and parent-offspring conflict in the RNA "Tumour" virus/host relationship: implications for vertebrate oncogene evolution.

Longmuir and co-workers have reported that respiration of certain tissue slices is approximated by Michaelis-Menten kinetics. From this and other experimental findings, Longmuir proposed that a carrier is involved in tissue oxygen transport. Gold developed a deterministic model to examine this hypothesis. This report presents a stochastic model for a fixed site carrier in a more general framework that includes the stochastic counter-part to Gold's deterministic model as a special case. The kinetics of tissue oxygen consumption predicted by the model are examined for various cases.     

A stochastic model for facilitated transport of oxygen through tissue slices

Longmuir and co-workers have reported that respiration of certain tissue slices is approximated by Michaelis-Menten kinetics. From this and other experimental findings, Longmuir proposed that a carrier is involved in tissue oxygen transport. Gold developed a deterministic model to examine this hypothesis. This report presents a stochastic model for a fixed site carrier in a more general framework that includes the stochastic counter-part to Gold's deterministic model as a special case. The kinetics of tissue oxygen consumption predicted by the model are examined for various cases.     

A general model of reaction kinetics in biological systems

Dynamic mathematical models in biotechnology require, besides the information about the stoichiometry of the biological reaction system, knowledge about the reaction kinetics. Modulation phenomena like limitation, inhibition and activation occur in different forms of competition with the key enzymes responsible for the respective metabolic reaction steps. The identification of a priori unknown reaction kinetics is often a critical task due to the non-linearity and (over-) parameterization of the model equations introduced to account for all the possible modulation phenomena. The contribution of this paper is to propose a general formulation of reaction kinetics, as an extension of the Michaelis-Menten kinetics, which allows limitation/activation and inhibition effects to be described with a reduced number of parameters. The versatility of the new model structure is demonstrated with application examples.     

Mass-transfer limitations for immobilized enzyme-catalyzed kinetic resolution of racemate in a fixed-bed reactor

A mathematical model has been developed for immobilized enzyme-catalyzed kinetic resolution of racemate in a fixed-bed reactor in which the enzyme-catalyzed reaction (the irreversible uni-uni competitive Michaelis-Menten kinetics is chosen as an example) was coupled with intraparticle diffusion, external mass transfer, and axial dispersion. The effects of mass-transfer limitations, competitive inhibition of substrates, deactivation on the enzyme effective enantioselectivity, and the optical purity and yield of the desired product are examined quantitatively over a wide range of parameters using the orthogonal collocation method. For a first-order reaction, an analytical solution is derived from the mathematical model for slab-, cylindrical-, and spherical-enzyme supports. Based on the analytical solution for the steady-state resolution process, a new concise formulation is presented to predict quantitatively the mass-transfer limitations on enzyme effective enantioselectivity and optical purity and yield of the desired product for a continuous steady-state kinetic resolution process in a fixed-bed reactor.     

Constitutive Uptake of Choline Sulfate by Roots and Leaf Slices of Barley

Barley roots take up choline sulfate constitutively via 3 separate, homogeneous mechanisms which obey Michaelis-Menten kinetics and have, respectively, low, high, and very high affinity for the sulfate ester. Leaf slices possess only the low-and the high-affinity mechanism. These are both inhibited by dini-trophenol and NaF but are differently affected by solute analogues. Uptake via the high-affinity mechanism is active; the low-affinity mechanism has also characteristics of facilitated diffusion.     

Simplifying principles for chemical and enzyme reaction kinetics

Tihonov's Theorems for systems of first-order ordinary differential equations containing small parameters in the derivatives, which form the mathematical foundation of the steady-state approximation, are restated. A general procedure for simplifying chemical and enzyme reaction kinetics, based on the difference of characteristic time scales, is presented. Korzuhin's Theorem. which makes it possible to approximate any kinetic system by a closed chemical system, is also reported. The notions and theorems are illustrated with examples of Michaelis-Menten enzyme kinetics and of a simple autocatalytic system. Another example illustrates how the differences in the rate constants of different elementary reactions may be exploited to simplify reaction kinetics by using Tihonov's Theorem. All necessary mathematical notions are explained in the appendices. The most simple formulation of Tihonov's 1st Theorem ‘for beginners’ is also given.     

Investigations of reaction kinetics for immobilized enzymes--identification of parameters in the presence of diffusion limitation

A method is proposed for identification of kinetic parameters when diffusion of substrates is limiting in reactions catalyzed by immobilized enzymes. This method overcomes conventional sequential procedures, which assume immobilization does not affect the conformation of the enzyme and, thus, consider intrinsic and inherent kinetics to be the same. The coupled equations describing intraparticle mass transport are solved simultaneously using numerical methods and are used for direct estimation of kinetic parameters by fitting modeling results to time-course measurements in a stirred tank reactor. While most traditional procedures were based on Michaelis-Menten kinetics, the method presented here is applicable to more complex kinetic mechanisms involving multiple state variables, such as ping-pong bi-bi. The method is applied to the kinetic resolution of (R/S)-1-methoxy-2-propanol with vinyl acetate catalyzed by Candida antarctica lipase B. A mathematical model is developed consisting of irreversible ping-pong bi-bi kinetics, including competitive inhibition of both enantiomers. The kinetic model, which fits to experimental data over a wide range of both substrates (5-95%) and temperatures (5-56 degrees C), is used for simulations to study typical behavior of immobilized enzyme systems.     

A numerical study of oxygen diffusion in a spherical cell with the Michaelis-Menten oxygen uptake kinetics

A numerical method is presented for the solution of reaction diffusion systems in biology. The method is used to re-examine the oxygen diffusion in a spherical cell with the Michaelis-Menten oxygen uptake kinetics.     

Analysis of a nonlinear diffusion problem With Michaelis-Menten kinetics by an integral equation method

A mathematical model for oxygen diffusion in a spherical cell with Michaelis-Menten oxygen uptake kinetics is analyzed by means of an intergral equation method. It is shown that an integral equation formulation can be used to obtain a numerical solution associated with this boundary and initial value problem. Through an illustrative numerical calculation we are able to obtain an accurate solution for both the steady and transient problems. Finally, a comparison is made with the numerical solution of McElwain and the variational solution of Anderson and Arthurs for the steady state and Lin's result concerning the unsteady state.     

Comparative study of the O(2), CO(2) and temperature effect on respiration between "Conference" pear cell protoplasts in suspension and intact pears

The influence of the O(2) and CO(2) concentration and the temperature on the O(2) uptake rate of cool-stored intact pears and pear cell protoplasts in suspension was compared. Protocols to isolate pear cell protoplasts from pear tissue and two methods to measure protoplast respiration have been developed. Modified Michaelis-Menten kinetics were applied to describe the effect of the O(2) and the CO(2) concentration on the O(2) uptake rate and temperature dependence was analysed with an Arrhenius equation. Both systems were described with a non-competitive type of CO(2) inhibition. Due to the inclusion of gas diffusion properties, the Michaelis-Menten constant for intact pears (2.5 mM) was significantly larger than the one for protoplasts in suspension (3 microM), which was in turn larger than the Michaelis-Menten constant obtained in mitochondrial respiration measurements described in the literature. It was calculated that only 3.6% of the total diffusion effect absorbed in the Michaelis-Menten constant for intact pears, could be attributed to intracellular gas diffusion. The number of cells per volume of tissue was counted microscopically to establish a relationship between the pear cell protoplast and intact pear O(2) uptake rate. A remarkable similarity was observed: values of 61.8 nmol kg(-1) s(-1) for protoplasts and 87.1 nmol kg(-1) s(-1) for intact pears were obtained. Also, the inhibitory effect of CO(2) on the respiration rate was almost identical for protoplasts and intact pears, suggesting that protoplast suspensions are useful for the study of other aspects of the respiration metabolism.     

The role of localization in the operation of the mitotic spindle assembly checkpoint

The mitotic spindle assembly checkpoint (^MSAC) is an important regulatory mechanism of the cell cycle, ensuring proper chromosome segregation in mitosis. It delays the transition to anaphase until all chromosomes are properly attached to the mitotic spindle by emitting a diffusible “wait anaphase”-signal from unattached kinetochores. Current models of the checkpoint disregard important spatial properties like localization, diffusion and realistic numbers of kinetochores. To allow for in silico studies of the dynamics of these models in a more realistic environment, we introduce a mathematical framework for quasi-spatial simulation of localized biochemical processes that are typically observed during checkpoint activation and maintenance. The “emitted inhibition” model of the ^MSAC by Doncic et al. (Proc Natl Acad Sci USA 2005; 102:6332–7) assumes instantaneous activation of the diffusible “wait anaphase”-signal upon kinetochore encounter. We modify this model to account for binding kinetics with finite rates and use the developed framework to determine the feasible range of the binding parameters. We find that for proper activation, the binding rate constant has to be fast and above a critical value. Furthermore, this critical value depends significantly on the amount of local binding sites at each kinetochore. The critical values lie in a physiological realistic regime (10⁴–10⁶ M^-1s^-1). We also determine the feasible parameter range for fast checkpoint activation of the “Mad2 template” model, for which the kinetic parameters have recently been studied in vitro by Simonetta et al. (PLoS Biology 2009; 7:1000010). We find critical values for binding and catalysis rate constants, both significantly higher than the measured values. Our results suggest that yet unknown mechanisms at the kinetochores facilitate binding and catalysis in vivo. We conclude that quantitative models of the ^MSAC have to account for the limited availability of binding sites at kinetochores.     

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.     

Comparison of two experimental methods for the determination of Michaelis-Menten kinetics of an immobilized enzyme

For the application of immobilized enzymes, the influence of immobilization on the activity of the enzyme should be Known. This influence can be obtained by determining the intrinsic kinetic parameters of the immobilized enzyme, and by comparing them with the kinetic parameters of the suspended enzyme. This article deals with the determination of the intrinsic kinetic parameters of an agarose-gel bead immobilized oxygen-consuming enzyme: L-lactate 2-monooxygenase. The reaction rate of the enzyme can be described by Michaelis-Menten kinetics. Batch conversion experiments using a biological oxygen monitor, as well as steady-state profile measurements within the biocatalyst particles using an oxygen microsensor, were performed. Two different mathematical methods were used for the batch conversion experiments, both assuming a pseudosteady-state situation with respect to the shape of the profile inside the bead. One of the methods used an approximate relation for the effectiveness factor for Michaelis-Menten kinetics which interpolates between the analytical solutions for zero- and first-order kinetics. The other mathematical method was based on a numerical solution and combined a mass balance over the reactor with a mass balance over the bead. The main difference in the application of the two methods is the computer calculation time; the completely numerical calculation procedure was about 20 times slower than the other calculation procedure.The intrinsic kinetic parameters resulting from both experimental methods were compared to check the reliability of the methods. There was no significant difference in the intrinsic kinetic parameters obtained from the two experimental methods. By comparison of the kinetic parameters for the suspended enzyme with the intrinsic kinetic parameters for the immobilized enzyme, it appeared that immobilization caused a decrease in the value of V(m) by a factor of 2, but there was no significant difference in the values obtained for K(m).