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     

Kinetic behaviour of penicillin acylase stabilized by poly (ethyleneimine)

The paper presents the kinetic evaluations of poly(ethyleneimine)-penicillin acylase preparations. The comparative studies show that the investigated system is much better than the native enzyme, and slightly worse than commercially available Boenringer preparation. Additionally, the high stability of PEI-enzyme system, very easy way of its preparation, high flexibility, and possibility to set the needed enzyme concentration are particularly favourable for use of the membrane bioreactor with PEI-enzyme system immobilized in its volume. Some advantages of the use of such bioreactor are also discussed.List of Symbols C _E IU/m³ activity concentration - C _S mol/m³ substrate concentration - C _P , C_Q mol/m³ products concentration - K _A mol/m³ constant which defines the affinity of a substrate to 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/IU min constant rate of dissociation of the active complex - mol/m³ min rate of reaction This work was supported by Government Committee of Science: Grant KBN # 3 0321 92 01     

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     

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     

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     

A structured model for the kinetics of fungal amylase production

Summary The cell is considered to be divided into nucleic acids, proteinaceous material and storage compounds. The enzyme is believed to be constitutive but repressed by the rate of catabolism. A structured model is developed to describe the growth, amylase production and dissolved oxygen profile in the batch culture ofAspergillus oryzae.Nomenclature CA Conc. of enzyme SKB units/m³ - C_D,C_E,C_G,C_O,C_S Conc. of D.E.G. mass, oxygen, substrate kg/m³ - C_O* Mean oxygen conc. in gas-liquid interface kg/m³ - C_X Total cell conc. kg/m³ - D D-mass (proteins) kg - E E-mass (storage carbon) kg - G G-mass (necleic acids) kg - K_E Rate of usage of E-mass kg/m³/h - K_EO K_E with oxygen limitation kg/m³/h - Q Active fraction of promotor genes - - R_D,R_E,R_G Rate of production of D,E,G-mass kg/m³/h - R_O,R_S Rate of usage of oxygen, substrate kg/m³/h - R_DO,R_EO,R_GO R_D,R_E and R_G with oxygen limitation kg/m³/h - S Substrate (carbon) concentration kg/m³ - t Time h - t_o Time at which C_S = 0 h - K₁,K₂,K₃,K₂₄,K₂₅ Stoichiometric constants - - K₄,K₅,K₆,K₇ Rate constants h^–1 - K₈-K₁₂,K₁₈,K₂₀-K₂₃ Michaelis-Menten constants kg/m³ - K₂₆,K₂₆ Absorption coefficient, K₂₆ at C_X = 0 h^–1 - K₂₇ Empirical constant kg/m³ - K₁₅ Rate of enzyme formation SKB units/kg - K₁₆,K₁₇ Equilibrium constants m³/kg - K₁₉ Decay constant for mRNA h^–1     

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     

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     

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     

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     

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     

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]     

Simultaneous hydrolysis of tri- and tetrasaccharides by industrial mixtures of glucoamylase and α-amylase: kinetics and thermodynamics

The present paper deals with the study of the kinetics and thermodynamics of batch enzymic hydrolysis of multisubstrate media in the presence of more than one glucanase. A kinetic model, similar to the one proposed for competitive inhibition, is presented to describe the competition between two substrates, tri- and tetrasaccharides, for the same enzyme, glucoamylase. A literature research on the most recurrent values of the Michaelis-Menten constant also shows a linear relationship between this parameter and the molecular weight of the sugar substrate for a given glucanase.List of Symbols a dimensionless empirical parameter in Eq. (1) - b dimensionless empirical parameter in Eq. (1) - B dimensionless empirical parameter in Eq. (6) - E kg/m³ enzyme concentration - E _a kcal/mol activation energy - kc kg/m³ competition constant - k _M kg/m³ Michaelis-Menten constant - MW kg·10^–3 molecular weight - r mol/(min·kg) specific hydrolysis rate of glucoamylase - R cal/(°K·mol) ideal gas constant - S kg/m³ substrate concentration - T °K absolute temperature - v kg/(m³·h) hydrolysis rate of glucoamylase Indices 1 values referring to trisaccharides - 2 values referring to tetrasaccharides - 0 starting values - max maximum values     

Modelling of enzyme hydrolysis of maltose in a single pellet of immobilized biocatalyst

The reversible hydrolysis of maltose to glucose by immobilized glucoamylase entrapped in spherical solid particles is studied theoretically. For this purpose a known kinetic model taking into account these reversible reactions and the competitive synthesis of iso-maltose was adopted. The mass transfer limitations in the bulk liquid and in the pores of the particles containing the enzyme are considered, using Fick's law. On the basis of mathematical modelling the optimum conditions for biocatalyst performance are established. An appropriate combination of particle size and initial substrate concentration may lead to reduction of undesirable mass transfer resistance and therefore product inhibition and to an improved selectivity of the biocatalyst with respect of glucose formation.List of Symbols C _i kmoles/m³ current concentration ofi-th component along the radius - C _oi kmoles/m³ bulk concentration ofi-th component - C _i ^* kmoles/m³ concentrations ofi-th component on the pellet surface - D _si ,D _i m²/s internal and molecular diffusion coefficient ofi-th component - W _M kmoles/m³·s reaction rate of maltose hydrolysis - W _IM kmoles/m³·s reaction rate of iso-maltose formation - W _G kmoles/m³·s reaction rate of glucose production - R ₀ m pellet radius - r m current radius of the pellet - t s time coordinate - r ₀ ratio of the time step to the square of the radial coordinate - Re Reynolds number =w·R/v - Sc Schmidt number =v/D - Bi Biot number = R/D - A _j ,B, C _j coefficients in the system of linear equations, Eq. (8) - X _i dimensionless degree of transformation - NR number of independent reactions - N number of division sections of the pellet radius - G kmoles/m³ concentration of glucose - M kmoles/m³ concentration of maltose - IM kmoles/m³ concentration of isomaltose - K _m kmoles/m³ Michaelis constant - V _max kmoles/m³·s maximum reaction rate in Eq. (6) - K _i kmoles/m³ inhibition constant - K _1eq ,K _2eq equilibrium constants in Eq. (6) - , h steps along the time and radial coordinate in the pellet - m/s mass transfer coefficient - dimensionless radius of the pellet - computation accuracy Indices i number of reaction component - j index along the radius of the pellet - k index along the time coordinate This work was accomplished with thanks to the financial support of the Bulgarian National Fund for Scientific Investigations —Grant No. MU-1-BE/93.     

Thyroid Hormone-Induced Alterations in Membrane Structure-Function Relationships: Studies on Kinetic Properties of Rat Kidney Microsomal Na+,K+-ATPase and Lipid/Phospholipid Profiles

The effects of thyroidectomy (Tx) and subsequent treatment with 3,5,3′-triiodothyronine (T₃) or combined replacement therapy (T_R) with T₃ and thyroxine (T₄) on the substrate and temperature kinetics properties of Na⁺,K⁺-ATPase and lipid/phospholipid makeup of rat kidney microsomes were examined. Enzyme activity was somewhat high in the hypothyroid (Tx) animals and increased significantly following T₃ treatment, while T_R treatment caused a decrease. In the Tx and T₃ groups enzyme activity resolved in two kinetic components, while in the T_R group the enzyme showed allosteric behavior up to 0.5 mm ATP concentration. The K _m and V _max values of both the components decreased in Tx animals without affecting the catalytic efficiency. T₃ treatment caused a significant increase in the V _max of both the components, with a significant increase in the catalytic efficiency, while the K _m values were not upregulated. The T_R regimen lowered the K _m and V _max of component II but improved the catalytic efficiency. Thyroid status-dependent changes were also noted in the temperature kinetics of the enzyme. Regression analysis revealed that changes in the substrate and temperature kinetics parameters correlated with specific phospholipid components.     

Sucrose inversion by immobilized yeast cells in a complete mixing reactor

Using whole cell invertase of Saccharomyces pastorianus, entrapped in spherical agar pellets, sucrose hydrolysis was carried out in a continuously fed fluidized bed reactor. The effective rate of reaction determined experimentally for the catalytic pellet was correlated with particle radius (R), intraparticle concentration of enzyme (E_p) and external concentration of substrate (S _R). The results were elucidated by theoretical analysis incorporating internal mass transfer resistance. At high degrees of diffusional resistance, the effectiveness factor was successfully estimted from Bischoff's equation. A dimensionless number, m_A ? R(k₂E_p/K_mD)^0.5(K_m/(K_m + S _R)), was used conveniently to predict the effectiveness factor in those cases wher the intraparticle diffusional effect was less significant. This number was employed to determine critical pellet size for an optimal reaction. The relationship between the properties of the pellet (size and intraparticle enzyme activity) and its apparent kinetic constants (k′₂ and K′_m), estimated according to Lineweaver-Burk, are discussed.     

Immobilization of lactase for the continuous hydrolysis of whey permaete

The production of lactose-based sweeteners is considered very promising. Fungal lactase has been immobilized on crosslinked chitin to develop a process for the continuous hydrolysis of demineralized whey permaete. The optimization of lactase immobilization on chitin and chitosan was performed, activities of 4 · 10⁵ and 2.2 · 10⁵ u/kg at yields of 33 and 23% were obtained for both supports, respectively. The chitin based catalyst was selected for further studies and a procedure was developed for in-situ enzyme immobilization. The kinetic behaviour of the catalyst was determined to propose a kinetic model for the initial rate of lactose hydrolysis. Pseudo steady-state and long term operation of packed bed reactors with chitin-immobilized lactase ranging from small laboratory to pre-pilot unit was carried out. The results are discussed and compared with commercial immobilized lactases. Preliminary economic evaluation for the production of ultrafiltered whey protein and hydrolyzed lactose syrup, within a dairy industry in Chile, was satisfactory in terms of profitability, both for the chitin immobilized lactase developed and for a commercial immobilized lactase.List of Symbols a moles/m³ glucose concentration in Eq. (1) - C _i US$ total annual cost (without considering plant depreciation) - D US$ annual depreciation - F m³/h flowrate - h m³/h volumetric mass transfer coefficient - i moles/m³ galactose concentration in Eqs. (1) and (2) - K _A moles/m³ dissociation constant for glucose in Eq. (1) - K _A moles/m³ dissociation constant for glucose in Eq. (1) - K _I moles/m³ inhibition constant for galactose in Eqs. (1) and (2) - K _m moles/m³ Michaelis constant for substrate in Eqs. (1) and (2) - k _D h^–1 first-order thermal deactivation constant - P kg dry weight of catalyst - PV US$ net present value - R % discounted cash-flow rate of return - s moles/m³ substrate concentration - s₀ moles/m³ feed substrate concentration - S _n US$ annual sales income - TC US$ total capital income - t _1/2 h catalyst half-life - v moles/h · kg initial rate of reaction - V _MAX moles/h · kg maximum reaction rate in Eqs. (1) and (2) - V _MAX moles/h · kg maximum reaction rate in Eq. (1) - ¯V _max moles/h initial rate of reaction - V _R m³ reaction volume free of catalyst particles - X substrate degree of conversion = s₀–s/s₀ - Damkoehler number = ¯V _MAX /h k _m - moles/(m³ · h) reactor productivity in Eq. (3)     

Kinetics of lipase-catalyzed hydrolysis of palm oil in lecithin/izooctane reversed micelles

Candida rugosa lipase has been used to investigate the hydrolysis of palm oil in a lecithin/isooctane reversed micellar system. The reaction obeys Michaelis-Menten kinetics for the initial conditions. Kinetic parameters such as maximum rate and Michaelis constant (K _m) were determined for lipase-catalyzed hydrolysis in n-hexane and isooctane. According to the K _m values, the enzyme affinity towards the substrate was increased in isooctane. The maximum degree of hydrolysis was generally decreased as the initial substrate concentration was increased. This may suggest that the hydrolysis in lecithin reversed micelles should be regarded as a one-substrate first-order reversible reaction. It is shown in this study that the proposed one-substrate first-order kinetic model can serve for the precise prediction of the degree of hydrolysis for a known reaction time or vice versa, when the initial substrate concentration is less than 0.325 mol/dm³. A disagreement with this model was found when the initial substrate concentration was higher than approximately 0.3 mol/dm³. This may be due to the effects of the products on lipase activity or even to the conversion of the reversed micellar system to other systems. Received: 16 May 1997 / Received revision: 22 October 1997 / Accepted: 24 October 1997