Complementary variational principles for diffusion problems with Michaelis-Menten kinetics

The theory of complementary variational principles is used to obtain maximum and minimum principles for diffusion problems with Michaelis-Menten kinetics. In an illustrative calculation we obtain an extremely accurate variational solution in good agreement with the numerical solution of McElwain (1978).     

Deviation from Michaelis-Menten kinetics for fumarase.

Withdrawal of prolactin or of insulin from the circulation of lactating rats leads, within 3h, to increased inactivation by phosphorylation of mammary-gland pyruvate dehydrogenase. Prolactin may act by priming the tissue to respond directly to normal concentrations of circulating insulin and by this means be responsible for the increased activation of the enzyme during the course of normal lactation.     

The design of stirred reactors with hollow fiber catalysts for Michaelis-Menten kinetics

A method of calculating the conversion in Rony's hollow fiber reactor is outlined. It, is assumed that, the kinetics are of Michaelis-Menten type and that diffusion within the hollow fiber, as well as through its wall, should be taken into account. The normalization of the Thiele modulus suffices to unify the treatment of internal diffusion and the pseudosteady state hypothesis is found to be valid under almost all conditions.     

Optimal design for CSTR's in series using reversible Michaelis-Menten reactions

An analytical expression is derived for the optimal design of a series of CSTR's performing reversible Michaelis-Menten kinetics in the liquid phase. The optimal design is based on minimum overall volume ofN reactors in series required to achieve a certain degree of substrate conversion. The reversible Michaelis-Menten equation is also able to explain competitive product inhibition and irreversible Michaelis-Menten kinetics. The reversible Michaelis-Menten kinetics covers three types of enzymatic reactions depending on the values of the rate constant for the forward (k _s) and reverse (k _p) reactions. An optimum design is obtained in the three cases ofK _s=K_p, K_s>K_p andK _s<K_p. The minimum overall reactors volume is compared with the more convenient equal-sized CSTR's. The effect of enzyme deactivation on the minimum overall reactors volume is investigated. The performance of a series of CSTR's is compared with plug-flow reactor. Glucose isomerization which exhibits reversible Michaelis-Menten kinetics is used as a model system for optimization.     

On oxygen diffusion in a spherical cell with Michaelis-Menten oxygen uptake kinetics

This paper demonstrates that there is one and only one solution to a non-linear singular two-point boundary-value problem which describes oxygen diffusion in a spherical cell. Previous authors have calculated numerical results that differ substantially. Numerical computations using the multiple shooting method support the results of McElwain.     

An error estimation of Michaelis-Menten (Monod)-type kinetics

Due to research on biochemistry and genetic engineering, mathematical models of microbial growth have become more complicated but Michaelis-Menten or Monod type expressions have still been used for conversion rates, uptake rates, etc. It is worth examining the error that can be caused by these quasi-steady-state-hypotheses. This paper presents a simple but very effective rationale function that describes the error of the quasi-steady-state hypothesis in enzyme kinetics. A simplified fermentation kinetic model was used for comparison of microbial growth but no analytical error function has been found for batch cultivation. In the case of continuous fermentation the error can be given in an analytical form. Many simulations, based on real SCP experiments, show a significant effect of the quasi-steady-state hypothesis. Since the rate constants of intracellular events are not really known, we have to be very careful when taking into account Michaelis-Menten type expressions in the building of complicated models. Correspondence to: L. Szigeti     

Use of autocatalytic kinetics to obtain composition of lignocellulosic materials

Non-isothermal thermogravimetric analysis (TGA) data of biomasses and pulps originating from non-wood and alternatives materials (i.e., Tagasaste or rice straw) have been fitted with refined models, which include autocatalytic kinetics. Data sets were obtained for different experimental conditions, such as variations of heating rate and atmosphere, i.e., inert (pyrolysis) versus oxidative atmosphere (combustion). Besides the access to classical kinetic parameters (pre-exponential factor, activation energy, and reaction order), the improved data analysis enabled the determination of the chemical composition of the samples (cellulose, hemicellulose, extractives, lignin). The latter compared very well with those obtained by conventional methods (chemical analysis, HPLC). Given the reduced environmental impact and rapidness of the method, potential applications for research related to new biomasses and industrial processes can be foreseen.     

Approximate Michaelis-Menten kinetics displayed in a stochastic model of cell-mediated cytotoxicity

A nonlinear continuous-time Markov chain describing a two-step process of cytolytic cells binding to target and the subsequent lysis and release of label is shown to have kinetics which resemble standard enzyme-substrate kinetics. The Michaelis-Menten saturation function is found as a special case resulting when the target population is in excess. A comparison theorem for the pseudo-steady-state distribution Π is constructed to enable examination of that distribution whose expected value E and variance V satisfy - K_mE + (C_T − E)(T_T − E) + V = 0, where K_m is the Michaelis half-saturation constant and C_T and T_T are the initial populations of the two cell types. Using Π as an initial condition, the release of label process is examined. The main result is that the fraction of specific release, ƒ, has the approximate form when T_t is large, so that a nonlinear regression procedure is appropriate for the determination of the parameters.     

Classification of stability behavior of bioreactors with wall attachment and substrate-inhibited kinetics

The biodegradation of pollutants in continuous operation when the microbial population exhibits wall attachment is studied. The proposed model for wall attachment assumes two morphological forms of the microbial cell connected by metamorphosis reactions with first order exchange kinetics. An analysis of the stability of the bioreactor, carried out using elementary principles of the singularity theory and continuation techniques, allows for classification in the multidimensional parameter space of the various stability behaviors exhibited by the reactor model, for both substrate-inhibited and Monod kinetics. The analysis also shows the enhanced stability behavior of the bioreactor due to wall attachment.     

An autocatalytic reaction as a model for the kinetics of the aggregation of beta-amyloid

Alzheimer's disease is the commonest form of senile dementia, affecting almost 20 million people worldwide. This neurodegenerative disorder is characterized by amyloid deposition in senile plaques, composed primarily of fibrils of an aggregated peptide, beta-amyloid. Fibrillation of beta-amyloid is a nucleation-dependent polymerization process, which is controlled by two kinetics parameters: the nucleation rate and the elongation or growth rate. As the kinetics of fibrillation is strongly dependent on the presence of trace amounts of fibrils, we suggest that the aggregation of beta-amyloid is a model of autocatalytic reaction. A mathematical analysis, permitting quantitative monitoring of the kinetics of fibrillogenesis of beta-amyloid, nucleation, and elongation constants, is presented. The model was checked by applying it to the aggregation of the fragment 1-40 of the beta-amyloid. Understanding of these rate constants may facilitate the study of the effect of substances used for controlling fibril creation and growth. The disaggregating effect of dodecyl trimethylammonium bromide, a cationic surfactant, was easily quantified by means of the model.     

Upper and lower bounds from the maximum principle. Intracellular diffusion with Michaelis-Menten kinetics

Analytical bounding functions for diffusion problems with Michaelis-Menten kinetics were recently presented by Anderson and Arthurs, 1985 (Bull. math. Biol. 47, 145–153). Their methods, successful to some extent for a small range of parameters, has the disadvantage of providing a weak upper bound. The optimal approach for the use of one-line bounding kinetics is presented. The use of two-line bounding kinetics is also shown, in order to give, sufficient accuracy in those cases where the one-line approach does not provide satisfactory results. The bounding functions provide excellent upper and lower bounds on the true solution for the entire range of kinetic and transport parameters.     

Feedback stabilization of fed-batch bioreactors: non-monotonic growth kinetics

This paper deals with the design of a feedback controller for fed-batch microbial conversion processes that forces the substrate concentration C(S) to a desired setpoint, starting from an arbitrary (initial) substrate concentration when non-monotonic growth kinetics apply. This problem is representative for a lot of industrial fermentation processes, with the baker's yeast fermentation as a well-known example. It is assumed that the specific growth rate mu is function of the substrate concentration only. A first approach exploits the availability of on-line measurements of both the substrate and biomass concentration. A second approach is merely based on on-line measurements of the biomass concentration, which provide an estimate for the specific growth rate. After a reformulation of the substrate concentration setpoint into a specific growth rate setpoint, it is demonstrated that the fed-batch process can still be stabilized around any desired operating point along the non-monotonic kinetics.     

Clean-in-place systems for industrial bioreactors: Design,validation and operation

Summary Guidelines for design, validation and operation of clean-in-place systems for industrial fermentation plant are presented. Design of vessels, surface finishes, materials of construction, types and locations of valves are some of the considerations addressed. Requisite levels of turbulence for cleaning of pipes and vessels are discussed as well as typical cleaning sequences. Recommendations for validation of cleaning are presented and the significance of design of cleaning systems in ensuring satisfactory validation is pointed out. To the extent possible, validation of cleaning should be carried out with real process soil or soil closely simulating actual fermentation broths.     

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).