Kinetic behavior of penicillin acylase immobilized on acrylic carrier

The usefulness of Lilly's kinetic equation to describe penicillin G hydrolysis performed by immobilized penicillin acylase onto the acrylic carrier has been shown. Based on the experimental results characteristic kinetic constants have been estimated. The effect of noncompetitive inhibition of 6-amino penicillanic acid has not been found. Five components of reaction resistance have been defined. These components were also estimated for the reaction of the native enzyme as well as the Boehringer preparation.List of Symbols C _E g/m³ enzyme concentration - C _P,C _Q mol/m³ product concentrations - C _S mol/m³ substrate concentration - C _SO mol/m³ initial substrate concentration - K _A mol/m³ constant which defines the affinity of a substrate to the enzyme - K _iS mol/m³ substrate inhibitory constant - K _iP mol/m³ PhAA inhibitory constant - K _iQ mol/m³ 6-APA inhibitory constant - k ₃ mol/g/min constant rate of dissociation of the active complex - R(1) concentrational component of reaction resistance - R(2) resistance component derived from substrate affinity - R(3) resistance component due to the inhibition of the enzyme by substrate - R(4) resistance component due to the inhibition of the enzyme by PhAA - R(5) resistance component due to inhibition of the enzyme by 6-APA - r = dC_s/dt mol/m³ min rate of reaction - t min reaction time - (i) relative resistance of reaction     

Strategy for the simulation of batch reactors when the enzyme-catalyzed reaction is accompanied by enzyme deactivation

The balance equations pertaining to the modelling of batch reactors performing an enzyme-catalyzed reaction in the presence of enzyme deactivation are developed. The functional form of the solution for the general situation where both the rate of the enzyme-catalyzed reaction and the rate of enzyme deactivation are dependent on the substrate concentration is obtained, as well as the condition that applies if a maximum conversion of substrate is sought. Finally, two examples of practical interest are explored to emphasize the usefulness of the analysis presented.List of Symbols C _E mol/m³ concentration of active enzyme - C _E,O mol/m³ initial concentration of active enzyme - C _S mol/m³ concentration of substrate - C _S,O mol/m³ initial concentration of substrate - C _S,min mol/m³ minimum value for the concentration of substrate - k 1/s first order rate constant associated with conversion of enzyme/substrate complex into product - k ₁ 1/s first order deactivation constant of enzyme (or free enzyme) - k ₂ 1/s first order deactivation constant of enzyme in enzyme/substrate complex form - K _m mol/m³ Michaelis-Menten constant - p mol/(m³s) time derivative of C _S - q mol/m³ auxiliary variable - t s time elapsed after reactor startup Greek Symbols 1/s univariate function expressing the dependence of the rate of enzyme deactivation on C _S - mol/m³ dummy variable of integration - mol/m³ dummy variable of integration - 1/s univariate function expressing the dependence of the rate of substrate depletion on C _S - m³/(mol s) derivative of with respect to C _S     

Process of penicillin G hydrolysis catalyzed by penicillin acylase immobilized on acylic carrier

The usefulness of penicillin acylase immobilized onto butyl acrylate — ethyl glycol dimethacrylate (called in this paper acrylic carrier) in penicillin G hydrolysis performed in a stirred tank reactor is shown. The enzyme-acrylic carrier preparation does not deteriorate its own properties in the mixing condition of slurry reactor. The experiments were carried out in a batch and a continuous stirred tank reactor as well as continuous stirred tank reactors in series. It was found to be a satisfactory agreement between experimental and predicted results. It also indicated the optimal substrate concentration range which provides the most effective enzyme operation. A superiority of the three reactors in series over the batch reactor is shown.List of Symbols C_E g/m³ equivalent enzyme concentration - C_SO mol/m³ initial penicillin G concentration - K_A mol/m³ substrate affinity constant - K_iS mol/m³ substrate inhibitory constant - K_iP mol/m³ PhAA inhibitory constant - K_iQ mol/m³ 6-APA inhibitory constant - k₃ mol/g min constant rate of dissotiation of the active complex - r mol/m³ rate of reaction - t min. reaction time - t_j min. maintenance time - degree of conversion - _B, _F dimensionless time - min. residence time - PA penicillin acylase - PG penicillin G - PhAA phenylacetic acid - 6-APA 6-aminopenicillanic acid     

Cost minimization in the predesign of an enzymatic CSTR: an overall approach

The balance equations pertaining to the modelling of a CSTR performing an enzyme-catalyzed reaction in the presence of enzyme deactivation are developed. Combination of heuristic correlations for the size-dependent cost of equipment and the purification-dependent cost of recovery of product with the mass balances was used as a basis for the development of expressions relating a (suitably defined) dimensionless economic parameter with the optimal outlet substrate concentration under the assumption that overall production costs per unit mass of product were to be minimized. The situation of Michaelis-Menten kinetics for the substrate depletion and first order kinetics for the deactivation of enzyme (considering that the free enzyme and the enzyme in the enzyme/substrate complex deactivate at different rates) was explored, and plots for several values of the parameters germane to the analysis are included.List of Symbols C _E mol m^–3 concentration of active enzyme - C _E,0 mol m^–3 initial concentration of active enzyme - C _p mol m^–3 concentration of product of interest - C _s mol m^–3 concentration of substrate - C _s,0 mol m^–3 initial concentration of substrate - I $ capital cost of equipment - k _d s^–1 deactivation constant of free enzyme - k _d s^–1 deactivation constant of enzyme in enzyme/substrate complex - K _m mol m^–3 Michaelis-Menten constant - K _m dimensionless counterpart of K _m - k _r s^–1 rate constant associated with conversion of enzyme/substrate complex into product - M _w kg mol^–1 molecular weight of product of interest - P $ kg^–1 cost of recovery of product of interest in pure form - Q m³s^–1 volumetric flow rate - V m³ volume of reactor - X $ kg^–1 global manufacture cost of product of interest in pure form - X dimensionless counterpart of X Greek Symbols ₁ $ m^–1.8 constant - ₂ $ m^–3 constant - t s useful life of CSTR - ₀ ratio of initial concentrations of enzyme and substrate - ratio of deactivation constant of free enzyme to rate constant of depletion of substrate - ratio of deactivation constants - univariate function expressing the dependence of the rate of enzyme deactivation on C _S - univariate function expressing the dependence of the rate of substrate depletion on C _S - dimensionless economic parameter     

Optimal temperature and concentration profiles in a cascade of CSTR's performing Michaelis-Menten reactions with first order enzyme deactivation

A necessary condition is found for the intermediate temperatures and substrate concentrations in a series of CSTR's performing an enzyme-catalyzed reaction which leads to the minimum overall volume of the cascade for given initial and final temperatures and substrate concentrations. The reaction is assumed to occur in a single phase under steady state conditions. The common case of Michaelis-Menten kinetics coupled with first order deactivation of the enzyme is considered. This analysis shows that intermediate stream temperatures play as important a role as intermediate substrate concentrations when optimizing in the presence of nonisothermal conditions. The general procedure is applied to a practical example involving a series of two reactors with reasonable values for the relevant five operating parameters. These parameters are defined as dimensionless ratios involving activation energies (or enthalpy changes of reaction), preexponential factors, and initial temperature and substrate concentration. For negligible rate of deactivation, the qptimality condition corresponds to having the ratio of any two consecutive concentrations as a single-parameter increasing function of the previous ratio of consecutive concentrations.List of Symbols C _E,0 mol.m^–3 Initial concentration of active enzyme - C _E,i mol.m^–3 Concentration of active enzyme at the outlet of the i-th reactor - C _S,0 mol.m^–3 Initial concentration of substrate - C _S,i mol.m^–3 Concentration of substrate at the outlet of the i-th reactor - Da _i Damköhler number associated with the i-th reactor ((V _i.k_v,0.C_E,0)/(Q.C_S,0)) - Da _min Minimum value of the overall Damköhler number - Da _tot Overall Damköhler number - E _d J.mol^–1 Activation energy of the step of deactivation of the enzyme - E _m J.mol^–1 Standard enthalpy change of the step of binding of substrate to the enzyme - E _v J.mol^–1 Activation energy of the step of enzymatic transformation of substrate - i Integer variable - j Dummy integer variable - k Dummy integer variable - k _d,i s^–1 Kinetic constant associated with the deactivation of enzyme in the i-th reactor (k _d,o·exp{–E _d/(R.T _i}) - k _d,0 s^–1 Preexponential factor of the kinetic constant associated with the deactivation of the enzyme - K _m,i mol.m^–3 Equilibrium constant associated with the binding of substrate to the enzyme in the i-th reactor, (k _m,o·exp{–E _m}(R.T _i}) - K _m,0 mol.m^–3 Preexponential factor of the Michaelis-Menten constant associated with the binding of substrate to the enzyme - k _v,i s^–1 Kinetic constant associated with the transformation of the substrate by the enzyme in the i-th reactor (k _v,o·exp{–E _v/(R.T _i})) - k _v,0 s^–1 Preexponential factor of the kinetic constant associated with the transformation of the substrate by the enzyme - N Number of reactors in the series - Q m³.s^–1 Volumetric flow rate of reacting liquid through the reactor network - R J.K^–1.mol^–1 Ideal gas constant - T _i K Absolute temperature at the outlet of the i-th reactor - T ₀ K Initial absolute temperature - V _i m³ Volume of the i-th reactor - v _max mol.m^–3.s^–1 Maximum rate of reaction under saturation conditions of substrate - x _i Normalized concentration of substrate (C_S,i/C_{S, 0}) - x _i,opt Optimum value of the normalized concentration of substrate - y _i Dimensionless temperature (exp{–T ₀/T _i}) - y _i,opt Optimum value of the dimensionless temperature Greek Symbols Dimensionless preexponential factor associated with the Michaelis-Menten constant (K _m,0/C_s,0) - Dimensionless activation energy of the step of enzymatic transformation of substrate (E _v/R.T₀)) - Dimensionless standard enthalpy change of the step of binding of substrate to the enzyme (E _m/(R.T₀)) - Dimensionless activation energy of the step of deactivation of the enzyme (E _d/(R.T₀)) - Dimensionless deactivation preexponential factor ((k _d,0.C_S,0)/(k_v,0.C_E,0)     

Optimum temperature policy for batch reactors performing biochemical reactions in the presence of enzyme deactivation

A necessary condition is found for the optimum temperature policy which leads to the minimum reaction time for a given final conversion of substrate in a well stirred, enzymatic batch reactor performing an enzyme-catalyzed reaction following Michaelis-Menten kinetics in the presence of first order enzyme decay. The reasoning, which is based on Euler's classical approach to variational calculus, is relevant for the predesign steps because it indicates in a simple fashion which temperature program should be followed in order to obtain the maximum advantage of existing enzyme using the type of reactor usually elected by technologists in the fine biochemistry field. In order to highlight the relevance and applicability of the work reported here, the case of optimality under isothermal operating conditions is considered and a practical example is worked out.List of Symbols C _E mol.m^–3 concentration of active enzyme - C _E ^* dimensionless counterpart of C_E - C _E,0 mol.m^–3 initial concentration of active enzyme - C _E,b mol.m^–3 final concentration of active enzyme - C _E,opt ^* optimal dimensionless counterpart of C_E - C _smol.m^–3 concentration of substrate - C _S ^Emphasis>/* dimensionless counterpart of C_S - C _S,0mol.m^–3 initial concentration of substrate - C _S,bmol.m^–3 final concentration of substrate - E enzyme in active form - E ₃ ^* dimensionless counterpart of E_a,3 - E _a,1J.mol^–1 activation energy associated with k₁ - E _a,3J.mol^–1 activation energy associated with k₃ - E _d enzyme in deactivated form - ES enzyme/substrate complex - k ₁ s^–1 kinetic constant associated with the enzyme-catalyzed transformation of substrate - k _1,0 s^–1 preexponential factor associated with k₁ - k ₂ mol^–1.m³s^–1 kinetic constant associated with the binding of substrate to the enzyme - k _–2 s^–1 kinetic constant associated with the dissociation of the enzyme/substrate complex - K _2,0 mol.m^–3 constant value of K₂ - K _2,0 ^* dimensionless counterpart of K_2,0 - k ₃ s^–1 kinetic constant associated with the deactivation of enzyme - k _3,0 s^–1 preexponential factor associated with k₃ - k _3,0 ^* dimensionless counterpart of k_3,0 - P product - R J.K^–1.mol^–1 ideal gas constant - S substrate - t s time since start-up of reaction - T K absolute temperature - T ^* dimensionless absolute temperature - T _i,opt ^* optimal dimensionless isothermal temperature of operation - T _opt ^* optimal dimensionless temperature of operation - t _b s time of a batch - t _b ^* dimensionless counterpart of t_b - t _b,min ^* minimum value of the dimensionless counterpart of t_b Greek Symbols dimensionless counterpart of C_E,0 - dimensionless counterpart of C_E,b - dummmy variable of integration - dummy variable of integration - auxiliary dimensionless variable - ^* dimensionless variation of k₁ with temperature - _i ^* dimensionless value of k₁ under isothermal conditions - _opt ^* optimal dimensionless variation of k₁ with temperature     

Intraparticle mass-transfer resistance and apparent time stability of immobilized yeast alcohol dehydrogenase

The stability and, consequently, the lifetime of immobilized enzymes (IME) are important factors in practical applications of IME, especially so far as design and operation of the enzyme reactors are concerned. In this paper a model is presented which describes the effect of intraparticle diffusion on time stability behaviour of IME, and which has been verified experimentally by the two-substrate enzymic reaction. As a model reaction the ethanol oxidation catalysed by immobilized yeast alcohol dehydrogenase was chosen. The reaction was performed in the batch-recycle reactor at 303 K and pH-value 8.9, under the conditions of high ethanol concentration and low coenzyme (NAD⁺) concentration, so that NAD⁺ was the limiting substrate. The values of the apparent and intrinsic deactivation constant as well as the apparent relative lifetime of the enzyme were calculated.The results show that the diffusional resistance influences the time stability of the IME catalyst and that IME appears to be more stabilized under the larger diffusion resistance.List of Symbols C _A, C_B, C_E mol · m^–3 concentration of coenzyme NAD⁺, ethanol and enzyme, respectively - C _p mol · m³ concentration of reaction product NADH - d _p mm particle diameter - D _eff m² · s^–1 effective volume diffusivity of NAD⁺ within porous matrix - k _d s^–1 intrinsic deactivation constant - K _A, K_A, K_B mol · m^–3 kinetic constant defined by Eq. (1) - K _A ^x mol · m^–3 kinetic constant defined by Eq. (5) - r _A mol · m^–3 · s^–1 intrinsic reaction rate - R m particle radius - R _v mol · m^–3 · s^–1 observed reaction rate per unit volume of immobilized enzyme - t _E s enzyme deactivation time - t _r s reaction time - V mol · m^–3 · s^–1 maximum reaction rate in Eq. (1) - V ^x mol · m^–3 · s^–1 parameter defined by Eq. (4) - V _f m³ total volume of fluid in reactor - w _s kg mass of immobilized enzyme bed - factor defined by Eqs. (19) and (20) - kg · m^–3 density of immobilized enzyme bed - unstableness factor - effectiveness factor - Thiele modulus - relative half-lifetime of immobilized enzyme Index o values obtained with fresh immobilized enzyme     

Reactor design for the enzymatic isomerization of glucose to fructose

A comprehensive methodology is presented for the design of reactors using immobilized enzymes as catalysts. The design is based on material balances and rate equations for enzyme action and decay and considers the effect of mass transfer limitations on the expression of enzyme activity. The enzymatic isomerization of glucose into fructose with a commercial immobilized glucose isomerase was selected as a case study. Results obtained are consistent with data obtained from existing high-fructose syrup plants. The methodology may be extended to other cases, provided sound expressions for enzyme action and decay are available and a simple flow pattern within the reactor might be assumed.List of Symbols C kat/kg specific activity of the catalyst - D m²/s substrate diffusivity within the catalyst particle - Dr m reactor diameter - d d operating time of each reactor - E kat initial enzyme activity - E _i kat initial enzyme activity in each reactor - F m³/s process flowrate - F _i m³/s reactor feed flowrate at a given time - F ₀ m³/s initial feed flowrate to each reactor - H number of enzyme half-lives used in the reactors - K mole/m³ equilibrium constant - K _S mole/m³ Michaelis constant for substrate - K _P mole/m³ Michaelis constant for product - K _m mole/m³ apparent Michaelis constant f(K, K _s, K_p, s₀) - k mole/s · kat reaction rate constant - k _d d^–1 first-order thermal inactivation rate constant - L m reactor height - L _r m height of catalyst bed - N _R number of reactors - P _i kg catalyst weight in each reactor - p mole/m³ product concentration - R m particle radius - R _P ratio of minimum to maximum process flowrate - r m distance to the center of the spherical particle - s mole/m³ substrate concentration - s _0i mole/m³ substrate concentration at reactor inlet - s ₀ mole/m³ bulk substrate concentration - s mole/m³ apparent substrate concentration - T K temperature - t d time - t _i d operating time for reactor i - t _s d time elapsed between two successive charges of each reactor - V m³ reactor volumen - V _m mole/m³ s maximum apparent reaction rate - V _p mole/m³ s maximum reaction rate for product - V _R m³ actual volume of catalyst bed - V _r m³ calculated volume of catalyst bed - V _S mol/m³ s maximum reaction rate for substrate - v mol/m³ s initial reaction rate - v _i m/s linear velocity - v _m mol/m³ s apparent initial reaction rate f(K_m, s,V_m) - X substrate conversion - X _eq substrate conversion at equilibrium - =s/K dimensionless substrate concentration - ₀=s₀/K bulk dimensionless substrate concentration - _eq=s_eq/K dimensionless substrate concentration at equilibrium - local effectiveness factor - mean integrated effectiveness factor - Thiéle modulus - =r/R dimensionless radius - _s kg/m³ hydrated support density - substrate protection factor - s residence time     

Flow dynamics of immobilized enzyme reactors

Summary Enzymic conversion of glucose to fructose was carried out in a packed bed and in a fluidized bed reactor. The flow dynamics of these two flow systems, loaded with two different types of immobilized loaded with two different types of immobilized glucose isomerase particles, were studied. The theoretical RTD curve calculated from the axial dispersed plug flow model equation was matched to the experimental RTD curve by an optimization technique. The effect of fluid velocity on the extent of liquid dispersion was established. Theoretical predictions on the conversion of glucose to fructose were calculated using three mathematical models, namely, a plug flow model, a continuous stirred tank reactor (CSTR) model and an axial dispersed plug flow model. The experimental results showed that the axial dispersed plug flow model was superior in predicting the performance of both the packed bed and fluidized bed reactor.Abbreviations C Dimensionless concentration - D Dispersion coefficient [cm²/sec] - d _p Mean particle diameter [cm] - E Enzyme concentration [mol/gm] - F Fructose concentration [mol/cm³] - F _e Equilibrium fructose concentration [mol/cm³] - G Glucose concentration [mol/cm³] - G _e Equílibrium glucose concentration [mol/cm³] - G _o Initial glucose concentration [mol/cm³] - Reduced glucose concentration [mol/cm³] - K Equilibrium constant - K _mf Forward reaction rate constant [mol/cm³] - K _mr Reserve reaction rate constant [mol/cm³] - K _m Rate constant [mol/cm³] - L Total length of the reactor bed [cm] - l Length [cm] - Q Flow rate [cm³/s] - r Rate of reaction based on volume of substrate - u Superficial liquid velocity [cm/s] - v Interstitial liquid velocity [cm/s] - V Reactor bed volume [cm³] - V _mf Forward reaction rate constant [mol/s·g enzyme] - V _mr Reserve reaction rate constant [mol/s·g enzyme] - z Dimensionless distance along the reactor - Density [g/cm²]     

Biological elimination of volatile xenobiotic compounds in biofilters

Biofiltration is a technique which is frequently applied for the odour abatement of waste gases. This technique is based on the ability of microorganisms (generally bacteria, and to a small extent moulds and yeasts) to degrade several organic as well as inorganic compounds to mineral end-products, like water and carbon dioxide. In the case of biofiltration, microorganisms are attached to suited packing materials in the filter, which contain the inorganic nutrients necessary for microbial growth. In order to make biofiltration applicable on a larger scale in process industry, it is necessary to find microorganisms able to eliminate compounds which are strange to life, the so-called xenobiotics. At the Eindhoven University of Technology microorganisms were isolated for the elimination of a number of xenobiotics, e.g. aromatic compounds and chlorinated hydrocarbons.Both the microkinetics of the biodegradation in aqueous batch systems and the macrokinetics of biofilters were studied. Special attention was paid to the influence of the superficial gas velocity and the organic load on the filter bed's elimination capacity. Also the discontinuous operation of biofilters is discussed.Nomenclature C ₁],C _1,0,C _í mol/m³ liquid phase concentration - K _s mol/m³ saturation constant - Y kg/kmol yield coefficient - t, t d time - _md^–1 maximum growth rate - X, X ₀ g/m³ microorganism concentration - P _gPa partial vapour pressure - H Pa m³/mol Henry's law constant - m (-) distribution coefficient - C _g,C _g,max mol/m³ gas phase concentration - R J (mol K) gas constant - T K temperature - (-) gas phase- and liquid phase volume ratio - m³/(m² h) superficial gas velocity - H m height of the filter bed - h m height in the filter bed In memory of Dick van Zuidam     

A graphical method of determining the Michaelis-Menten constant free of external mass transfer resistance for immobilized enzymes in a packed bed reactor

Summary A graphical method of determining the Michaelis-Menten constant free of the external mass transfer resistance for a packed bed immobilized enzyme system was illustrated with examples from 3 different enzyme reactions. The intercept at the ordinate obtained by the straight line extrapolation of data points in the plot of apparent K_m value vs. the reciprocal of superficial velocity in column allowed an easy calculation of K_m free of external mass transfer resistance. An asymptotic value of apparent K_m value at infinite zero superficial velocity was ascribed to the fact that the mass transfer coefficient k_L, approached a definite value at this condition.Nomenclature K_m Michaelis-Menten constant, M/L³ - K_m' K_m free of external mass transfer resistance in a given ionic strength, M/L³ - Km" apparent K_m with external mass transfer resistance, M/L³ - S substrate concentration, M/L³ - S_o initial substrate concentration, M/L³ - k₂ rate constant, t^-1 - E enzyme concentration in support, M/L³ - void volume per unit volume of reactor, dimensionless - u superficial velocity of substrate, L/t - K_L mass transfer coefficient in liquid film, L/t - a external surface area of support per unit volume of reactor, L^-1 - ratio of average channeling length to particle diameter, dimensionless - d_p diameter of support particle, L - X fractional conversion of substrate, dimensionless - H partition coefficient, dimensionless - k a constant, 3 k₂E(1-)d_p/4 - T space time, t - N molecular flux, M/L²t - r radius of immobilized enzyme particle, L     

On the appropriateness of use of a continuous formulation for the modelling of discrete multireactant systems following Michaëlis–Menten kinetics

The possibility of solving the mass balances to a multiplicity of substrates within a CSTR in the presence of a chemical reaction following Michaelis-Menten kinetics using the assumption that the discrete distribution of said substrates is well approximated by an equivalent continuous distribution on the molecular weight is explored. The applicability of such reasoning is tested with a convenient numerical example. In addition to providing the limiting behavior of the discrete formulation as the number of homologous substrates increases, the continuous formulation yields in general simpler functional forms for the final distribution of substrates than the discrete counterpart due to the recursive nature of the solution in the latter case.List of Symbols C{N. M} mol/m³ concentration of substrate containing N monomer residues each with molecular weight M - {N, M} normalized value of C{N. M} - C {M} mol/m³ da concentration of substrate of molecular weight M - _in normalized value of C {M} at the i-th iteration of a finite difference method - {M} normalized value of C {M} - C ₀{N.M} mol/m³ inlet concentration of substrate containing N monomer residues each with molecular weight M - {N ·M} normalized value of C₀{N. M} - _{0 i} normalized value of C ₀ {M} at the i-th iteration of a finite difference method - C ₀ {M} mol/m³ da initial concentration of substrate of molecular weight M - C _tot mol/m³ (constant) overall concentration of substrates (discrete model) - C _tot mol/m³ (constant) overall concentration of substrates (continuous model) - D deviation of the continuous approach relative to the discrete approach - i dummy integer variable - I arbitrary integration constant - j dummy integer variable - k dummy integer variable - K _m mol/m³ Michaëlis-Menten constant for the substrates - l dummy integer variable - M da molecular weight of substrate - M normalized value of M - M da maximum molecular weight of a reacting substrate - N number of monomer residues of a reacting substrate - N maximum number of monomer residues of a reacting substrate - N total number of increments for the finite difference method - Q m³/s volumetric flow rate of liquid through the reactor - S inert product molecule - S _i substrate containing i monomer residues - V m³ volume of the reactor - v _max mol/m³ s reaction rate under saturating conditions of the enzyme active site with substrate - v _max{N. M} mol/m³ s reaction rate under saturating conditions of the enzyme active site with substrate containing N monomer residues with molecular weight M - _max{N · M} dimensionless value of v_max{N. M} (discrete model) - _max{M} dimensionless value of v _max {M} (continuous model) - mol/m³ s molecular weight-averaged value of v_max (discrete model) - mol.da/m³s molecular weight-averaged value of v_max (continuous model) - v _max {M} mol.da/m³s reaction rate under saturating conditions of the enzyme active site with substrate with molecular weight M - _max {M} dimensionless value of v_max{M} - _{max, (i)} dimensionless value of v_max{M} at the i-th iteration of a finite difference method - v _max mol/m³ s reference constant value of v _max Greek Symbols dimensionless operating parameter (discrete distribution) - dimensionless operating parameter (continuous distribution) - M da (average) molecular weight of a monomeric subunit - M selected increment for the finite difference method - auxiliary corrective factor (discrete model)     

Determination of the optimum operating time for batch isothermal performance of enzyme-catalyzed multisubstrate reactions

This communication consists of a mathematical analysis encompassing the maximization of the average rate of monomer production in a batch reactor performing an enzymatic reaction in a system consisting of a multiplicity of polymeric substrates which compete with one another for the active site of a soluble enzyme, under the assumption that the form of the rate expression is consistent with the Michaelis-Menten mechanism. The general form for the functional dependence of the various substrate concentrations on time is obtained in dimensionless form using matrix terminology; the optimum batch time is found for a simpler situation and the effect of various process and system variables thereon is discussed. The reasoning developed here emphasizes, in a quantitative fashion, the fact that the commonly used lumped substrate approaches lead to nonconservative decisions in industrial practice, and hence should be avoided when searching for trustworthy estimates of optimum operation.List of Symbols O 1/s row vector of zeros - a 1/s row vector of rate constants k _i(i = 2,...,N) - A 1/s matrix of rate constants k _i and k_–i (i=2,...,N) - b 1/s row vector of rate constant k ₂ and zeros - C mol/m³ molar concentration of S - C mol/m³ vector of molar concentrations of C _i (i=0, 1, 2, ..., N) - C ₀ mol/m³ column vector of initial molar concentrations of C _i(i=0, 1, 2,.., N) - C _–01 mol/m³ column vector of initial molar concentrations of C _i(i=2,..., N) - C _{E, tot} mol/m³ total molar concentration of enzyme molecules - C _i mol/m³ molar concentration of S _i (i=0,1,2,...,N) - C _{i, o} mol/m³ initial molar concentration of S _i(i=0, 1, 2, ..., N) - E enzyme molecule - I identity matrix - K 1/s matrix of lumped rate constants - k _i 1/s pseudo-first order lumped rate constant associated with the formation of S _i -1 (i=1, 2, ...,N) - k _{cat, i} 1/s first order rate constant associated with the formation of S _i-1 (i=1, 2, ..., N) - K _m mol/m³ Michaelis-Menten constant - L number of distinct eigenvalues - M _i multiplicity of the i-th eigenvalue - N maximum number of monomer residues in a single polymeric molecule - r ₁ mol/m³ s rate of formation of S ₀ - r _i mol/m³ s rate of release of S _i -1 - r _opt maximum average dimensionless rate of production of monomer S₀ - S lumped, pseudo substrate - S₁ inert moiety - S _i substrate containing i monomer residues, each labile to detachment as - S₀ by enzymatic action (i=1,2,...,N) - t s time elapsed since startup of batch reaction - t _lag s time interval required for cleaning, loading, and unloading the batch reactor - t _opt s time interval leading to the maximum average rate of monomer production - v _ij s^1-j eigenvectors associated with eigenvalue imi (i=1, 2, ..., L; j =1, 2, ..., M_i) Greek Symbols _ij mol/m³ arbitrary constant associated with eigenvalue _i (i=1, 2, ..., L; j=1, 2, ..., M _i ) - 1/s generic eigenvalue - _i 1/s i-th eigenvalue     

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     

Optimization of the microbial synthesis of dihydroxyacetone from glycerol with <Emphasis Type="Italic">Gluconobacter oxydans</Emphasis>

An optimized repeated-fed-batch fermentation process for the synthesis of dihydroxyacetone (DHA) from glycerol utilizing Gluconobacter oxydans is presented. Cleaning, sterilization, and inoculation procedures could be reduced significantly compared to the conventional fed-batch process. A stringent requirement was that the product concentration was kept below a critical threshold level at all times in order to avoid irreversible product inhibition of the cells. On the basis of experimentally validated model calculations, a threshold value of about 60 kg m^-3 DHA was obtained. The innovative bioreactor system consisted of a stirred tank reactor combined with a packed trickle-bed column. In the packed column, active cells could be retained by in situ immobilization on a hydrophilized Ralu-ring carrier material. Within 17 days, the productivity of the process could be increased by 75% to about 2.8 kg m^-3 h^-1. However, it was observed that the maximum achievable productivity had not been reached yet.Abbreviations K _O Monod half saturation constant of dissolved oxygen (kg m^-3) - K _S Monod half saturation constant of substrate glycerol (kg m^-3) - O Dissolved oxygen concentration (kg m^-3) - P Product concentration (kg m^-3) - P _crit Critical product concentration constant (kg m^-3) - S Substrate concentration (kg m^-3) - t Time (s) - X Biomass concentration (dry weight) (kg m^-3) - Y _P/S Yield coefficient of product from substrate - Y _X/S Yield coefficient of biomass from substrate - Growth dependent specific production rate constant (kg m^-3) - Growth independent specific production rate constant (s^-1) - Specific growth rate (s^-1) - _max Maximum specific growth rate constant (s^-1)     

Production of glutamine by micro-gel bead-immobilized Corynebacterium glutamicum 9703-T cells in a stirred tank reactor

The effectiveness of using micro-gel bead-immobilized cells for aerobic processes was investigated. Glutamine production by Corynebacterium glutamicum, 9703-T, cells was used as an example. The cells were immobilized in Sr-alginate micro-gel beads 500 m in diameter and used for fermentation processes in a stirred tank reactor with a modified impeller at 400 min^–1. Continuous production of glutamine was carried out for more than 220 h in this reactor and no gel breakage was observed. As a result of the high oxygen transfer capacity of this system, the glutamine yield from glucose was more than three times higher, while the organic acid accumulation was more than 24 times lower than those obtained with 3.0 mm-gel bead-immobilized cells in an airlift fermentor under similar experimental conditions. During the continuous fermentations there was evolution and proliferation of non-glutamine producing strains which led to a gradual decrease in the productivity of the systems. Although a modified production medium which suppresses cell growth during the production phase was effective in maintaining the productivity, the stability of the whole system was shortened due to high cell deactivation rate in such a medium.List of Symbols C kg/m³ glutamine concentration - C _A mol/m ³ local oxygen concentration inside the gel beads - C _AS mol/m ³ oxygen concentration at the surface of the gel beads - De m²/h effective diffusion coefficient of oxygen in the gel bead - DO mol/m³ dissolved oxygen concentration - F dm³/h medium flow rate - K h^–1 glutamine decomposition rate constant - Km mol/m³ Michaelis Menten constant - QO _2max mol/(kg · h) maximum specific respiration rate - R m radius of the gel beads - r m radial distance - t h time - V _C dm ³ volume of the gel beads - V _L dm ³ liquid volume in the reactor - Vm mol/(m³ · h) maximum respiration rate - X kg/m³ cell concentration - x r/R - y C _A /C_AS - h^–1 cell deactivation rate constant - Thiele modulus defined by R(Vm/De Km) ^1/2 - C _AS /Km - _C kg/(m³-gel · h) specific glutamine formation rate - _c dm³-gel/dm³ V _C /V _L      

Effect of aggregate size in cell cultures of Tagetes patula on thiophene production and cell growth

Summary The effect of the size of Tagetes patula (marigolds) cell aggregates on growth and thiophene production in MS-medium was studied. A heterogeneous aggregate suspension was aseptically divided into 7 fractions, each with a defined aggregate diameter range, with subsequent inoculation of the fractions into MS growth medium. Growth occurred in all aggregate fractions and thiophene production increased with increasing aggregate diameter starting at about 3 mm, an effect possibly due to an increasing lack of oxygen in the aggregate centre. Calculations of oxygen concentration profiles in the aggregates showed namely, that the critical aggregate diameter where the oxygen concentration in the aggregate centre becomes very low, is about 3 mm. Aggregates with a diameter exceeding 1.2 cm showed a decreased thiophene production, however, these aggregates were hollow. The thiophenes produced mainly consisted of 5-(4-hydroxy-1-butenyl)1-2,2-bithienyl, which was excreted into the medium.Nomenclature ID _e effective diffusion coefficient (m²s^-1) - c oxygen concentration (mol m^-3) - c _s substrate concentration at surface (mol m^-3) - c _s.exp experimental value of c _s (mol m^-3) - c _eq substrate concentration at equilibrium (mol m^-3) - r _s consumption rate (mol m^-3 s^-1) - d _crit critical aggregate diameter (m) - d _agg aggregate diameter (m) - L length of aggregate (m) - W width of aggregate (m) - t time (s) - r distance from aggregate centre (m) - R radius of aggregate (m) - R(c) oxygen consumption (mol m^-3 s^-1) - V _c convection velocity (m s^-1) - V _m intrinsic maximum consumption rate (mol kg^-1 s^-1) - K _m intrinsic Michaelis Menten constant (mol m^-3) - V _m apparent maximum consumption rate (mol kg^-1 s^-1) - K _m apparent Michaelis Menten constant (mol m^-3) - * multiplication sign     

Mathematical modeling of mixing in a horizontal rotating tubular bioreactor: “Simple flow” model

The construction of the horizontal rotating tubular bioreactor (HRTB) represents a combination of a thin-layer bioreactor and a biodisc reactor. The bioreactor was made of a plastic tube whose interior was divided by the O-ring shaped partition walls. For the investigation of mixing properties in HRTB the temperature step method was applied. The temperature change in the bioreactor as a response to a temperature step in the inlet flow was monitored by six Pt-100 sensors (t ₉₀ response time 0.08 s and resolution 0.002 °C) which were connected with an interface unit and personal computer. Mixing properties of the bioreactor were modeled using the modified tank in series concept which divided the bioreactor into ideally mixed compartments. A mathematical mixing model with simple flow was developed according to the physical model of the compartments network and corresponding heat balances. Numerical integration of an established set of differential equations was done by the Runge-Kutt-Fehlberg method. The final mathematical model with simple flow contained four adjustable parameters (N1,Ni, F _cr andF _p ) and five fixed parameters.List of Symbols A _u m² inner surface of bioreactor's wall - A _ui m² i-th part of inner surface of bioreactor's wall - A _v m² outlet surface of bioreactor's wall - A _vi m² i-th part of outlet surface of bioreactor's wall - C _p kJ kg^–1 K^–1 heat capacity of liquid - C _pr kJ kg^–1 K^–1 heat capacity of bioreactor's wall - D h^–1 dilution rate - E °C °C^–1 h^–1 error of mathematical model - F _cr dm³s^–1 circulation flow in the model - F _p dm³ s^–1 back flow in the model - F _t dm³s^–1 inlet flow in the bioreactor - I °C intensity of temperature step, the difference in temperature between the temperature of the inlet liquid flow and the temperature of liquid in the bioreactor before the temperature step - K1 Wm^–2K^–1 heat transfer coefficient between the liquid and bioreactor's wall - K2 Wm^–2K^–1 heat transfer coefficient between the bioreactor's wall and air - m _s kg mass of bioreactor's wall - L m length of bioreactor - L _k m wetted perimeter of bioreactor - n min^–1 rotational speed of bioreactor - n _s number of temperature sensors - N1 number of cascades - Ni number of compartments inside the cascade - Nu Nusselt number - Pr Prandtl number - r _u m inner diameter of bioreactor - r _v m outside diameter of bioreactor - Re Reynolds number - s(t) step function - t s time - T °C temperature - T _c °C calculated temperature - T _m °C measured temperature - T _N1,Ni °C temperature of liquid in a defined compartment inside cascade - T _N1,S °C temperature of defined part of bioreactor's wall - T _S °C temperature of bioreactor's wall - T _v °C temperature of liquid in bioreactor - T _z °C temperature of surrounding air - V _t dm³ volume of liquid in the bioreactor Greek Symbols kJm^–1s^–1 K^–1 thermal conductivity of liquid in the bioreactor - kgm^–3 density of liquid in the bioreactor - m²s^–1 kinematic viscosity of liquid in the bioreactor Matrix Coefficient B - C - D - E B+C+D - G1 - G2 - G3 - A _ui - A _vi - Q ₁ - Q ₂ - Q ₃      

Development of reactor model for glucose isomerization catalyzed by whole-cell immobilized glucose isomerase

Based on the kinetic constants determined and the mathematical model of the reactor system developed, the performance of axial flow packed bed continuous enzyme reactor system was studied experimentally and also simulated with the aid of a computer for ultimate objective of optimization of the glucose isomerase reactor system.A reactor model was established analogous to heterogeneous catalytic reactor model taking into account the effect of fluid mass transfer and reversible kinetics. The investigated catalyst system consists of immobilized Streptomyces bambergiensis cells containing the enzyme glucose isomerase, which catalyzes the isomerization of glucose to fructose.List of Symbols A ₀, A ₁, A ₂ parameters in axial dispersion reactor model - c _go, c_g, c_gemol m^–3 glucose concentration at time t=0, at any time and at equilibrium conditions - c _gsmol m^–3 glucose concentration at particle surface - C dimensionless glucose concentration - d _pm particle diameter - d _rm diameter of reactor tube - Da Damkohler number - D _eff m² s^–1 effective glucose diffusion coefficient in Ca-alginate gel beads - k _fm s^–1 film transfer coefficient - K _e equilibrium constant - K _mg, K_mfmol m^–3 Michaelis-Menten constant for glucose and fructose, respectively - K _mmol m^–3 modified Michaelis-Menten constant - K dimensionless parameter - K ^* dimensionless parameter - L m length of reactor tube - Pe Peclet number - Pe _p particle Peclet number - Q m³ s^–1 volumetric flow rate - (-r _g) mol m^–3 s^–1 reaction rate - Re _p Reynolds particle number - Sc Schmidt number - Sh Sherwood number - t s time - v ₀ m s^–1 linear superficial fluid velocity - V _mg, V_mfmol g^–1 s^–1 maximal reaction rate for glucose and fructose, respectively - V _mmol m^–3 s^–1 modified maximal reaction rate for glucose - V _mg ^x mol m^–2 s^–1 maximal reaction rate for glucose - X _g, X_ge glucose conversion and glucose conversion at equilibrium conditions - X normalized conversion - Y dimensionless glucose concentration - void fraction of fixed bed - effectiveness factor of biocatalyst - Pa s kinematic viscosity of substrate - ₁ s first absolute weighted moment - ₂ s² second central weighted moment - _gkg m^–3 substrate density - _pkg m^–3 particle density - ² dimensionless variance of RTD curve - s residence time     

The effect of micro- and macromixing on enzyme reaction in a real CSTR

The effect of micromixing and macromixing on enzyme reaction of Michaelis-Menten type in a real continuously stirred tank reactor (CSTR) is considered. The effect of bypassing of a fraction of feed stream, dead space, initial enzyme concentration and Michaelis-Menten constant on substrate conversion is evaluated. Bypass reduces the substrate conversion significantly compared with other parameters in the case of micro and macromixing. Micromixing predicts higher substrate conversions compared with macromixing. The effect of micro and macromixing on substrate conversion is negligible at low and high conversions.List of Symbols C kmol/m³ concentration of reactant - ¯C kmol/m³ average concentration of reactant - C_A kmol/m³ exit concentration of reactant A - C_Aa kmol/m³ exit concentration of reactant A from active zone - C_AO kmol/m³ initial concentration of reactant A - C_EO kmol/m³ initial enzyme concentration - C_O kmol/m³ initial concentration of reactant - E(t) 1/s exit age distribution function - k 1/s reaction rate constant - M kmol/m³ Michaelis-Menten constant - r kmol/(m³s) rate of reaction - –r_A kmol/(m³s) rate of reaction with respect to A - t s time - v m³/s volumetric feed rate - v_a m³/s volumetric feed rate entering the active zone - v_b m³/s volumetric feed rate entering the bypass stream - V m³ total volume of the vessel - V_a m³ active volume of the vessel - V_d m³ volume of dead space - X_A conversion of A Greek Letters fraction of feed stream bypassing the vessel (v_b/v) - fraction of the total volume as dead space (V_d/V) - (t) 1/s Dirac delta function, an ideal pulse occurring at time t = 0 - s life expectancy of a molecule - 1/s intensity function or escape probability function - s space time or mean residence time