Optimal Design of a Series of GSTR's Performing Reversible Reactions Catalyzed by Soluble Enzymes: A Theoretical Study

The condition for the minimum overall reactor volume of a given number of CSTR's in series is theoretically determined for a reversible, single reactant-single product (Uni-Uni) enzyme catalyzed reaction. The reactor network is assumed to operate in steady-state, isothermal conditions with a single phase and a constant activity of biocatalyst. The method is based on a mathematical analysis of the discrete substrate concentration profile along the CSTR's assuming complete micromixing. The algebraic equations describing the critical loci are obtained for the general case, the mathematical proof that these equations define a minimum is presented, and an exact solution arising from an asymptotic situation is found. An approximate analytical method of optimization based on the aforementioned critical behavior is reported and its validity and usefulness discussed. The formulae introduced can be used in more general situations as tools for getting the approximate range where the optimal overall volume of the series of CSTR's lies. Hence, the reasoning developed is important for the preliminary CSTR design and relevant in the initial steps of the more involved methods of numerical optimization. Finally, the enzymatic conversion of fumarate to L-malate is examined as a model system in order to assess the usefulness and applicability of the analysis developed.     

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.     

Optimal design for CSTR's in series performing enzymatic lactose hydrolysis

Analytical expressions are derived for the optimal design (based on minimum overall reactors volume) of a series of N CSTR's performing enzymatic lactose hydrolysis. It is assumed that lactose hydrolysis obeys Michaelis-Menten kinetics with competitive product (galactose) inhibition and no enzyme deactivation occurs. The optimum design of a cascade of ideally mixed reactors are compared with equal size reactors and with plug flow reactor required for a given overall degree of lactose conversion. The effect of operating parameters such as temperature, lactose initial (feed) concentration and conversion, enzyme and product initial concentration on the optimal overall holding time are also investigated. Optimization results for a series of N CSTR's up to five are obtained and compared with plug flow reactor.     

A heuristic approach for the economic optimization of a series of CSTR's performing Michaelis-Menten reactions

The estimation of the size of each reactor of a series of CSTR's performing a Michaelis-Menten reaction in the liquid phase can be obtained to advantage via an optimization technique leading to the minimum overall capital cost. The cost scaleup is assumed to be described by a power rule on the equipment capacity. Various contributions are lumped into the exponent, thus leading to values above unity. The analytical development leading to the optimal intermediate concentrations of substrate according to the foregoing criterion is presented. A short-cut method based on an empirical expression that approximates the numerical solution is reported. This correlation is found to be exact at the asymptotic behaviors, and to give accurate results within an acceptable error level for the range with physical interest. Therefore, it is particularly useful during the predesign steps of equipment for the biochemical industry.     

Optimal design of a number of membrane reactors in series using Michaelis-Menten one-substrate reversible kinetics

Summary Exact analytical expressions are derived for the optimal design (minimum overall reaction volume) of N perfectly mixed membrane reactors in series carrying out an enzyme catalysed Michaelis-Menten, one-substrate/one-product reversible reaction. The equations enable the direct calculation of the smallest total reactor volume needed for a given overall conversion degree. Results show that when substrate rejection is present, membrane reactors perform better compared with continuous stirred tank reactors.     

Optimum design of a series of CSTRs performing reversible Michaelis-Menten kinetics: a rigorous mathematical study

The optimum design of a given number of CSTRs in series performing reversible Michaelis-Menten kinetics in the liquid phase assuming constant activity of the enzyme is studied. In this study, the presence of product in the feed stream to the first reactor, as well as the effect of the product intermediate concentrations in the downstream reactors on the reaction rate are investigated. For a given number of N CSTRs required to perform a certain degree of substrate conversion and under steady state operation and constant volumetric flow rate, the reactor optimization problem is posed as a constrained nonlinear programming problem (NLP). The reactor optimization is based on the minimum overall residence time (volume) of N reactors in series. When all the reactors in series operate isothermally, the constrained NLP is solved as an unconstrained NLP. And an analytical expression for the optimum overall residence time is obtained. Also, the necessary and sufficient conditions for the minimum overall residence time of N CSTRs are derived analytically. In the presence of product in the feed stream, the reversible Michaelis-Menten kinetics shows competitive product inhibition. And this is, because of the increase in the apparent rate constant K^' _m that results in a reduction of the overall reaction rate. The optimum total residence time is found to increase as the ratio (‚₀) of product to substrate concentrations in the feed stream increases. The isomerization of glucose to fructose, which follows a reversible Michaelis-Menten kinetics, is chosen as a model for the numerical examples.     

On the optimum distribution of enzyme feed in a cascade of CSTR's performing an enzyme-catalyzed reaction with deactivation

A search for the optimum fractional distribution of an enzyme-rich stream to the various reactors of a cascade of CSTR's was implemented. A theoretical analysis, laid out in dimensionless form and based on the assumptions that the system is operated under steady state conditions, the enzyme undergoes first order deactivation, and the reaction catalyzed by the enzyme follows Michaelis-Menten kinetics, is reported. The objective function utilised is the minimisation of the overall volume of the cascade, and analytical expressions are obtained for the concentration of active enzyme and substrate in the outlet stream from each reactor. It is found that the best option is to add the whole enzyme-rich stream to the first reactor in the cascade irrespective of the operating and kinetic parameters of the system.     

To what extent is a constant volume design worse than a minimum volume design for a series of CSTR's?

The balance equations for substrate in a cascade of CSTR's undergoing an enzyme-catalyzed reaction following Michaelis-Menten kinetics are developed in dimensionless form. Analytical expressions relating the intermediate concentrations are independently obtained for the cases of minimum overall volume and constant volume. The fractional deviations between the overall volumes following these two design criteria are calculated and presented for several values of the relevant parameters. For situations of practical interest, the fractional deviation is below 10%. Increasing values of the Michaelis-Menten parameter, K _m(or decreasing values of the number of reactors in the cascade, N) lead to lower values of the maximum deviation; this maximum deviation is attained at lower conversions of substrate when K _mis increased or N decreased.List of Symbols C _{S, i}mol.m^–3 concentration of substrate at the outlet of the i-th reactor - C _* ^{S, i} normalized concentration of substrate at the outlet of the i-th reactor - C _* ^{S, i, eq} normalized concentration of substrate at the outlet of the i-th reactor using the design criterion of constant volume - C _* ^{S, i, opt} normalized concentration of substrate at the outlet of the i-th reactor using the design criterion of minimum overall volume - C _{S, 0} mol.m^–3 concentration of substrate at the inlet to the first reactor - Da _i Damköhler number for the i-th reactor - Da _eq constant Damköhler number for each reactor of the cascade - Da _{tot, eq} overall Damköhler number for the cascade assuming equal-sized reactors - Da _{tot, min} minimum overall Damköhler number for the cascade - Er fractional deviation between the overall volumes using the two different design criteria - K _mmol. m^–3 Michaelis-Menten constant - K _* ^M dimensionless Michaelis-Menten constant - N number of reactors of the cascade - Q m³. s^–1 volumetric flow rate - V _im³ volume of the i-th reactor - v _max mol. m^–3. s^–1 reaction rate under saturation conditions of the enzyme with substrate - V _{tot, opt} m³ minimum overall volume of the cascade - V _{tot, eq} m³ overall volume of the cascade assuming equal-sized reactors     

Determination of Urea Permeability in Red Cells by Minimum Method : A test of the phenomenological equations

A new method has been developed for measuring the permeability coefficient, ω, of small nonelectrolytes. The method depends upon a mathematical analysis of the time course of cell volume changes in the neighborhood of the minimum volume following addition of a permeating solute to an isosmolal buffer. Coefficients determined by the minimum volume method agree with those obtained using radioactive tracers. ω for urea in human red cells was found to decrease as the volume flow, J_v, into the cell increased. Such behavior is entirely unexpected for a single uniform rate-limiting barrier on the basis of the linear phenomenological equations derived from irreversible thermodynamics. However, the present findings are consonant with a complex membrane system consisting of a tight barrier on the outer face of the human red cell membrane and a somewhat less restrictive barrier behind it closer to the inner membrane face. A theoretical analysis of such a series model has been made which makes predictions consistent with the experimental findings.     

A horizontal bioreactor for ethanol production by immobilized cells

A mathematical model which describes ethanol formation in a horizontal tank reactor containing Saccharomyces cerevisiae immobilized in small beads of calcium alginate has been developed. The design equations combine flow dynamics of the reactor as well as product formation kinetics. The model was verified for 11 continuous experiments, where dilution rate, feed glucose concentration and bead volume fraction were varied. The model predicts effluent ethanol concentration and CO₂ production rate within the experimental error. A simplification of the model is possible, when the feed glucose concentration does not exceed 150 kg/m³. The simplification results in an analytical solution of the design equation and hence can easily be applied for design purposes as well as for optimization studies.     

A fuzzy logic controller.

This paper describes a fuzzy sets method which is very useful for handling uncertainties and essential for knowledge acquisition of a human expert. Kinetics of a reactor is often complex and not trivial to describe by mathematical equations. Reactor control by traditional control technology is therefore difficult. A novel technology is presented. In the following a fuzzy inference (approximate reasoning) is used for decision making in analogy to human thinking, facilitating a more sophisticated control. Readers of this paper do not need any advanced mathematics beyond the four basic operations in arithmetic (+, -, x, divided by) and using the maximum and minimum values. This fuzzy inference is introduced to construct a fuzzy logic controller which is suitable for a nonlinear, multivariable and time variant system applied to a bioreactor.     

Equilibrium position, kinetics, and reactor concepts for the adipyl-7-ADCA-hydrolysis process

One of the building blocks of cephalosporin antibiotics is 7-amino-deacetoxycephalosporanic acid (7-ADCA). It is currently produced from penicillin G using an elaborate chemical ring-expansion step followed by an enzyme-catalyzed hydrolysis. However, 7-ADCA-like components can also be produced by direct fermentation. This is of scientific and economic interest because the elaborate ring-expansion step is performed within the microorganism. In this article, the hydrolysis of the fermentation product adipyl-7-ADCA is studied. Adipyl-7-ADCA can be hydrolyzed in an equilibrium reaction to adipic acid and 7-ADCA using glutaryl-acylase. The equilibrium reaction yield is described as a function of pH, temperature, and initial adipyl-7-ADCA concentration. Reaction rate equations were derived for adipyl-7-ADCA-hydrolysis using three (pH-independent) reaction rate constants and the apparent equilibrium constant. The reaction rate constants were calculated from experimental data. Based on the equilibrium position and reaction rate equations the hydrolysis reaction was optimized and standard reactor configurations were evaluated. It was found that equilibrium yields are high at high pH, high temperature and low-initial adipyl-7-ADCA concentration. The course of the reaction could be described well as a function of pH (7-9), temperature (20-40 degrees C) and concentration using the reaction rate equations. It was shown that a series of CSTR's is the best alternative for the process.     

Enzyme activity maintenance in packed-bed reactors via continuous enzyme addition

An operational scheme for using immobilized enzymes in packed-bed reactors that permits operation at a constant throughput rate and constant product quality is described. The scheme used columns operated in series with continuous enzyme addition to compensate for enzyme decay. A mathematical technique was developed to determine the enzyme addition rate, enzyme usage, and enzyme volume in the column system. Operation of columns in series is compared to operation where the flow rate is decreased to compensate for a loss of enzyme activity for both zero-and first-order decay. The analyses indicated that columns in series resulted in better enzyme utilization but larger reactor volumes than parallel reactors with decreasing flow rate.     

Continuous production of L-tert-leucine in series of two enzyme membrane reactors

The L-tert-leucine synthesis was performed continuously in series of two enzyme-membrane reactors by reductive amination of trimethylpyruvate with leucine dehydrogenase. The necessary “native” cofactor NADH is regenerated with the aid of a second enzyme, formate dehydrogenase. Considering detailed kinetic studies of initial reaction rates under conditions relevant to the process a kinetic model was developed. The model shows that the overall reaction rate is strongly inhibited by the reaction product. The reactor's models combine the mass balances and proposed kinetic equations. The model adequacy was verified by using it to simulate the experiments and by comparing experimental and computed conversion, space-time yield and enzyme consumption. The calculations for the three reactor's types (batch, single CSTR and a cascade of two CSTRs in series) were compared. The results showed that a single CSTR is no favourable reactor configuration due to the very strong product inhibition. Space-time yield drops from 560 g litre^?1 day^?1 in a batch reactor to 110 g litre^?1 day^?1 in a single CSTR at the highest conversion of 98%. At the conversion of 95% the difference in biocatalyst costs between batch and two CSTR in series is negligible. Therefore the use of two enzyme membrane reactors in series was proposed. The modelling in this work shows that the optimisation of the quantity of the enzyme used results in a minimisation of the biocatalyst costs.     

Identification of quantitative trait nucleotides that regulate cancer growth: a simulation approach

A general growth model derived from basic cellular properties can be used to describe the dynamic process of cancer growth with mathematical equations. It has been recognized that cancer growth is under genetic control, with a multitude of interacting genes each segregating in a Mendelian fashion and displaying environmental sensitivity. In this article, we integrate the mathematical aspects of the pervasive growth model into a statistical framework for the identification of quantitative trait nucleotides that underlie cancer growth. This integrative framework is constructed with a single nucleotide polymorphism-based haplotype blocking analysis. Simulation studies have been performed to demonstrate the usefulness of the model. The proposed model provides a generic platform model for testing and detecting specific DNA sequence variants that regulates the timing of cancer emergence, growth and differentiation.     

Non-destructive biomass estimation of herbaceous plant individuals: A transferable method between contrasted environments

Monitoring plant growth at the individual level in arrays of environmental conditions is key to understanding plant functioning with strong implications for ecophysiology, population biology and community ecology. This requires non-destructive methods for repeated estimates of individual plant biomass in time. Although allometric equations have been widely used for trees and shrubs, there is currently no general approach for herbaceous species that can be applied across habitats, plant architecture, life stage and leading to transferable equations between contrasted environments. Here we propose a method based on three biometric measurements of the minimum volume occupied by aboveground plant organs. A total of 36 equations were fitted and compared for twelve species of temperate grasslands, corresponding to various volume shapes, scaling functions (linear or power) and including (or not) a life stage effect. The accuracy of the selected equations was compared to similar attempts reported in the literature. We further assessed the across-site transferability of the best allometric equations. The goodness-of-fit of the best equations selected for each species was high (̄R² = 0.83). The type of selected equations was species-specific, emphasising the benefits of considering a wide range of plant volume shapes and both linear and power functions. Using a comprehensive assessment of allometric equation transferability, we found that site effects could be neglected for eleven out of twelve species. Biomass equations based on the minimum volume proved accurate. The proposed method is easy to implement in any type of habitat, copes with various plant architectures and reduces risks of error measurement compared to previously developed approaches. The method further allows, for the first time, to use a single equation for monitoring the growth trajectory of herbaceous plant individuals in contrasted environments.     

Mathematical modelling of immobilized living yeast cell reactor for sugar bioconversion to ethanol

In the present work, a mathematical model was developed regarding the immobilized living yeast cell reactor for sugar bioconversion to ethanol. The model, composed of a system of ordinary differential equations (ODEs) enables the computation of the paramters involved in the steady state reactor behaviour. Comparing the values computed through the integration of this mathematical model with the experimental data, it has been shown its capacity to describe sufficiently accurate the steady state behaviour of the continuous fixed film bioconversion reactor.     

Modeling of antibiotics production in magneto three-phase airlift fermenter

A mathematical model is developed to describe the performance of a three-phase airlift reactor utilizing a transverse magnetic field. The model is based on the complete mixing model for the bulk of liquid phase and on the Michaelis-Menten kinetics. The model equations are solved by the explicit finite difference method from transient to steady state conditions. The results of the numerical simulation indicate that the magnetic field increases the degree of bioconversion. The mathematical model is experimentally verified in a three-phase airlift reactor with P. chrysogenum immobilized on magnetic beads. The experimental results are well described by the developed model when the reactor operates in the stabilized regime. At relatively high magnetic field intensities a certain discrepancy in the model solution was observed when the model over estimates the product concentration.     

Pulse waves in prestressed arteries

In order to better understand the effect of initial stress in blood flow in arteries, a theoretical analysis of wave propagation in an initially inflated and axially stretched cylindrical thick shell is investigated. For simplicity in the mathematical analysis, the blood is assumed to be an incompressible inviscid fluid while the arterial wall is taken to be an isotropic, homogeneous and incompressible elastic material. Employing the theory of small deformations superimposed on a large initial field the governing differential equations of perturbed solid motions are obtained in cylindrical polar coordinates. Considering the difficulty in obtaining a closed form solution for the field equations, an approximate power series method is utilized. The dispersion relations for the most general case of this approximation and for the thin tube case are thoroughly discussed. The speeds of waves propagating in an unstressed tube are obtained as a special case of our general treatment. It is observed that the speeds of both waves increase with increasing inner pressure and axial stretch.