Theoretical analysis of intrinsic reaction kinetics and the behavior of immobilized enzymes system for steady-state conditions

Mathematical modeling of immobilized enzymes under different kinetics mechanism viz. simple Michaelis–Menten, uncompetitive substrate inhibition, total competitive product inhibition, total non-competitive product inhibition and reversible Michaelis–Menten reaction are discussed. These five kinetic models are based on reaction diffusion equations containing non-linear terms related to Michaelis–Menten kinetics of the enzymatic reaction. Modified Adomian decomposition method is employed to derive the general analytical expressions of substrate and product concentration for all these five mechanisms for all possible values of the parameters Φ_S (Thiele modulus for substrate), Φ_P (Thiele modulus for product) and α (dimensionless inhibition degree). Also we have presented the general analytical expressions for the mean integrated effectiveness factor for all values of parameters. Analytical results are compared with the numerical results and also with the limiting case results, which are found to be good in agreement.     

Effectiveness Factors and Conversion in a Biocatalytic Membrane Reactor

Analytical expressions of the effectiveness factor of a biocatalytic membrane reactor, and its asymptote as the Thiele modulus becomes large, are presented. The evaluation of the effectiveness factor is based on the solution of the governing equations for solute transport in the two regions of the reactor, i.e. the lumen and the matrix (with the biofilm immobilized in the matrix). The lumen solution accounts for both axial diffusion and radial convective flow, while the matrix solution is based on Robin-type boundary conditions. The effectiveness factor is shown to be a function of the Thiele modulus, the partition coefficient, the Sherwood number, the Peclet number, and membrane thickness. Three regions of Thiele moduli are defined in the effectiveness factor graphs. These correspond with reaction rate limited, internal-diffusion limited, and external mass transfer limited solute transport. Radial convective flows were shown to only improve the effectiveness factor in the region of internal diffusion limitation. The assumption of first order kinetics is shown to be applicable only in the Thiele modulus regions of internal and external mass transfer limitation. An iteration scheme is also presented for estimating the effectiveness factor when the solute fractional conversion is known. The model is validated with experimental data from a membrane gradostat reactor immobilised with Phanerochaete chrysosporium for the production of lignin and manganese peroxidases. The developed model and experimental data allow for the determination of the Thiele modulus at which the effectiveness factor and fractional conversion are optimal.     

Numerical solutions of the Lamm equation. II. Equilibrium sedimentation

The Lamm equation has been solved numerically for conditions corresponding to equilibrium runs for a nonlinear concentration dependence of the form s/s₀ = (1 + kc)^?1. It is shown that the approach to equilibrium is very close to being exponential (in time) as in the case k = 0. We also compare results for the nonlinear case given above with results obtained for linear c-dependence of the form s/s₀ = 1 – kc. For relatively high speeds the time required to attain equilibrium may be greatly underestimated by use of the latter approximation. Finally, we present analytic approximations for the concentration distribution at equilibrium and as a function of time.     

Mode analysis of a tubular structure of coupled non-linear oscillators for digestive-tract modelling

A commonly accepted mathematical model for the slow-wave electrical activity of the gastro-intestinal tract of humans and animals comprises a set of interconnected nonlinear oscillators. Using a van der Pol oscillator with third-power conductance characteristics as the unit oscillator a number of structures have been analysed using a matrix Krylov-Bogolioubov method linearisation. The mode analysis of one-dimensional chains and two-dimensional arrays has been reported. In this paper the method has been extended to consider a tubular structure which is relevant to modelling small-intestinal rhythms. It is shown that this structure is capable of producing stable single models, non-resonant double modes and degenerated modes. General expressions are obtained for anm×n structure and examples given of two special conditions of 3×4 (i.e. odd numbers of oscillators in a ring) and 4×3 cases. The analytical results obtained for these two cases have been vertified experimentally using an electronic implementation of coupled van der Pol oscillators. Results obtained using fifth-power non-linear oscillators are summarised.     

Kinetic behaviour of α-galactosidase produced by Absidia griseola: a comparison between free and immobilized forms of the enzyme

In this research the characteristics of free (partially purified) and immobilized (mould pellets of Absidia griseola) -galactosidase have been investigated. Inhibition studies of the enzyme showed that p-nitrophenol and sucrose do not have any inhibitory effect on the enzyme, but that galactose is a competitive inhibitor. In the immobilized form, inhibition was lower than in the free enzyme and the level of inhibition decreased as the temperature increased. The activity and stability of free and immobilized enzyme were investigated with respect to temperature, and the results showed that the optimal temperature range of the free enzyme was 45–50 °C, while the immobilized enzyme had an optimum at 55–60 °C. The optimum pH for the free enzyme was 6.0 and the value was decreased to 5.0 by immobilizing. The experimental effectiveness factors were found to be represented as a single function of the modified Thiele modulus, including parameters such as pellet size, enzyme concentration in the pellets and substrate concentration. Both experimental and theoretical data concerning effectiveness factors are nearly the same.     

A stochastic model for then-compartment irreversible system

This paper deals with a stochasticn-compartment irreversible system with a non-homogeneous Poisson input and arbitrary residence time for each of the compartments. Results relating to the number of particles present in each of the compartments as well as the total number of particles present in the system at any time are derived. Further, explicit expressions for the auto covariance function for each compartment and the cross-covariance function between any two compartments with a given time lag are obtained. As a particular case, then-compartment irreversible system is analyzed with homogeneous Poisson input and exponential residence time distribution for each of the compartments. The possible applications of the model are discussed.     

Kinetics of γ-benzyl-L-glutamate NCA's polymerizations and molecular-weight distributions on corresponding polymers

The kinetics of γ-benzyl-L -glutamate NCA's polymerizations in dioxane as solvent are discussed. The partial orders respective to [A], the NCA concentration and [I₀], the initial initiator concentration are given; the rate constants are deduced and a mechanism is proposed to justify a rate of polymerization V_p = k[A][I₀]². The dependence of the rate constants on the conformation of the growing chains is demonstrated; the acceleration is attributed to the ordered structures favorable to hydrogen bonding. The kinetics of aging have also been examined and discussed; it is shown that they cannot modify the kinetics of polymerization. The DP _n were obtained on the same samples before and after debenzylation; it is proved that at any concentration, DP _n ? [A₀]/[I₀]. The molecular-weight distributions were obtained by chromatography and a polydispersity lower than 1.3 was deduced for each sample. The trimodal distribution, appearing as soon as [A₀]/[I₀] is larger than 3, is attributed to the existence of the three structures σ, β, and α. The weight fraction of each of the structures was correlated to the kinetics of polymerization.     

Does a latitudinal gradient in seedling survival favour larger seeds in the tropics?

The mean size of seeds produced by plants at the equator is two to three orders of magnitude higher than the mean size of seeds produced by plants at 60°. We compiled data from the literature to assess the possibility that this latitudinal gradient in seed size allows species to cope with more difficult seedling establishment conditions in tropical environments. We found no relationship between latitude and seedling survival through 1 week (P = 0.27, n = 112 species). There was also no evidence that a larger seed mass is required to gain a given level of seedling survival in tropical environments (P = 0.37, n = 112 species), and no relationship between latitude and the duration of the juvenile period (P = 0.57, n = 132 species). Thus, our results are not compatible with the idea that seedling establishment is more difficult in the tropics.     

Equilibria in structured populations

The existence of a stable positive equilibrium state for the density of a population which is internally structured by means of a single scalar such as age, size, etc. is studied as a bifurcation problem. Using an inherent birth modulus n as a bifurcation parameter it is shown for very general nonlinear model equations, in which vital birth and growth processes depend on population density, that a global unbounded continuum of nontrivial equilibrium pairs (n, ) bifurcates from the unique (normalized) critical point (1, 0). The pairs are locally positive and conditions are given under which the continuum is globally positive. Local stability is shown to depend on the direction of bifurcation. For the important case when density dependence is a nonlinear expression involving a linear functional of density (such as total population size) it is shown how a detailed global bifurcation diagram is easily constructed in applications from the graph of a certain real valued function obtained from an invariant on the continuum. Uniqueness and nonuniqueness of positive equilibrium states are studied. The results are illustrated by several applications to models appearing in the literature.This research was done while the author was on leave at the Lehrstuhl für Biomathematik, Universität Tübingen, Auf der Morgenstelle 10, 7400 Tübingen 1, Federal Republic of Germany     

Yeast alcohol dehydrogenase immobilized in a glutaraldehyde–albumin matrix: Kinetics and cofactor diffusional effects*

This study was carried out to define how the overall rate of reaction would be influenced by different degrees of diffusional resistance to cofactor transport within an oxidoreductase membrane matrix. To accomplish this, 0.7–6.6μM yeast alcohol dehydrogenase was immobilized in an albumin matrix crosslinked with 2.5 or 5.0% glutaraldehyde to give 102–1685 μM thick membranes. The enzyme half-life was at least doubled at pH 7.5 or 8.8 on immobilization. Values of the kinetic constants for the soluble and immobilized enzyme were determined at 25°C and pH 8.8 over the range of 0.01–1.0M bulk solution concentration of ethanol as substrate and 140–1000μM bulk solution concentration of nicotinamide adenine dinucleotide (NAD⁺) as cofactor, to give essentially single substrate kinetics in NAD⁺. Equilibrium partitioning of ethanol and NAD⁺ between the solution and membrane was measured and used in the data analysis. The four kinetic constants for the soluble enzyme agreed with literature values; and all increased with immobilization of the enzyme. The Michaelis constants for NAD⁺ and for ethanol were greater for the immobilized enzyme. The diffusional resistance to NAD⁺ transport, presented in terms of the Thiele modulus, showed that the overall rate of reaction was decreased by about 50% even at values of the modulus as low as 2.0.     

Pesticide ecotoxicological effect factors and their uncertainties for freshwater ecosystems

Background, aim, and scope  Characterization factors for ecotoxicity in the Life Cycle Impact Assessment (LCIA) are used to convert emissions into ecotoxicological impacts. Deriving them involves a fate and an effect analysis step. The fate factor quantifies the change in environmental concentration per unit of emission, while the effect factor quantifies the change in impact on the ecosystem per unit of environmental concentration. This paper calculates freshwater ecotoxicological effect factors for 397 pesticides belonging to 11 pesticide-specific toxic modes of action (TMoA), such as acetylcholinesterase inhibition and photosynthesis inhibition. Moreover, uncertainties in the effect factors due to uncertain background concentrations and due to limited toxicity data are quantified. Methods  To calculate median ecotoxicological effect factors (EEFs), toxic pressure assessments were made, based on the species sensitivity distribution—and the multisubstance potentially affected fraction—concept. The EEF quantifies an estimate of the fraction of species that is probably affected due to a marginal change in concentration of a pesticide. EEFs were divided into a TMoA-specific and a chemical-specific part, which were calculated on the basis of physicochemical properties, emissions, and toxicity data. Propagation of parameter uncertainty in the EEFs and the TMoA- and chemical-specific parts was quantified by Monte Carlo simulation and results were reported as 90% confidence intervals. Results  Median EEFs range from 2·10⁻³ to 7·10⁶ l/g. Uncertainty in the TMoA-specific part is dominated by uncertainty in the TMoA-specific spread in species sensitivity and by uncertainty in the effective toxicity of a TMoA. Uncertainty in the chemical-specific part of the EEFs depends on the number of species for which toxicity data are available to calculate average toxicity (n _s) and ranges from a median uncertainty of 2.6 orders of magnitude for n _s = 2 to one order of magnitude for n _s ≥ 4. The TMoA-specific effect factor for systemic fungicides shows the largest uncertainty range. For seven TMoAs, uncertainty ranges of the TMoA-specific effect factor are less than two orders of magnitude. For the other four TMoAs, the EEF uncertainty range is between two and eight orders of magnitude. For the chemical-specific part of the EEFs, we found that variation in uncertainty readily decreases for pesticides for which toxicity data are available for at least three species. Discussion  The same parameters that contributed most to uncertainty were found for pesticides as were found before for high-production-volume chemicals. However, uncertainty in concentrations of pesticides was lower. TMoA-specific factors obtained with the applied nonlinear method differ up to nine orders of magnitude from the factor of 0.5, which is used in the linear method. With the applied method, a distinction in EEFs can be made among different TMoAs. Conclusions   Ecotoxicological effect factors are presented, including overviews of their uncertainty ranges and the main contributors to uncertainty. The applied nonlinear method provides the possibility to quantify parameter uncertainty in the TMoA-specific part of the ecotoxicological effect factor, which is helpful to get more insight in how uncertainty in ecotoxicological characterization factors can be reduced. Recommendations and perspectives  The calculated uncertainty ranges can be included in life cycle assessment (LCA) case studies, which allows for better interpretation of LCA results obtained with the EEFs. To put the uncertainty in effect factors into perspective within LCIA, more information on the uncertainty in fate factors should be derived. Electronic supplementary material  The online version of this article (doi:) contains supplementary material, which is available to authorized users.     

Dynamical machinery of a biochemical clock

Closed positive feedback loops of catalytic reactions between macromolecules, or hypercycles, provide a kinetic mechanism whereby each Species serves to catalyze selfreproduction of its successor in the loop. Hypercycles of five members or more evolve into limit cycles characteristic of a biochemical clock. Computer study of the coupled non-linear differential equations which describe these systems shows that the periodT _n of then-species limit cycle is given byT _n=nτ_n, where τ_n is an elemental repeat period reflecting translational time invariance. Analytic solutions of the equations are developed so that the time evolution of elementaryn-hypercycles can be traced in dynamical detail. It is shown that the magnitude of τ_n is, to good approximation, a linear function ofn. For a givenn, τ_n is a very sensitive function of the relative concentration a given member of the loop has at the time its predecessor dominates the state of the hypercycle. These concentrations decrease with increasingn. Aroundn=15 they become so small that elementary hypercycles become unstable against disruptive concentration fluctuations. Species concentrations for more realistic hypercycles tend not to be as small, so that the present estimate of a maximum number of components is a lower bound.     

Mass transfer limitations in diffusion-limited isotropic hollow fiber bioreactors

Diffusional mass transfer limitations in hollow fiber bioreactors – with densely packed whole cells in its extracapillary space to perform biotransformation reactions – have been studied theoretically using a steady-state diffusion and reaction model. Simple analytical expressions have been derived to calculate the radial and axial concentration profiles for zero- and first-order kinetics, as well as to plot effectiveness factor versus Thiele modulus plots for first-order kinetics. The influence of the magnitude of the effective diffusion coefficients, the thickness of the isotropic membrane as well as the size of the annular cell region have been assessed to optimise the reactor performance.     

Growth of Saccharomycopsis fibuligera in a continuous-stirred-tank fermentor

Growth rates and amylase production rates were determined for the yeast Saccharomycopsis fibuligera grown on a simulated potato processing waste in a continuous-stirred-tank fermentor. S. fibuligera formed large multicellular flocs in the fermentor, and cell growth was reduced at low dilution rates because of mass-transfer resistance. The average Thiele modulus, which is the measure of extent of substrate diffusion, had a value ranging from ?_av = 2.2 for D = 0.10 to 0.3 for D = 0.40. Growth rates were described by the Monod equation modified to include mass-transfer effects. This modified Monod equation was used to predict growth rates from measured floc-size distribution. Experimentally determined growth rates were in close agreement with these predicted values.     

Kinetic resolution of racemic glycidyl butyrate using a multiphase membrane enzyme reactor: experiments and model verification

A laboratory-scale multiphase hollow fiber membrane reactor was employed to investigate the lipase-catalyzed enzymatic resolution of racemic glycidyl butyrate. A mathematical formulation was feveloped to simulate the performance of this system. Model parameters were determined independently (except the effective rate constant, k(s)) and incorporated in the model simulations. In this study, two modes of operation are considered: subtractive resolution, in which the unreacted substrate is recovered in the organic stream; and product recovery, where the optically pure product of the enzymatic reaction is recovered in the aqueous stream. Good agreement was obtained between theoretical predictions and experimental results under a variety of conditions. The effect of mass transport limitations on the performance of this system was investigated. An increase in enzyme loading resulted in a higher Thiele modulus due to an elevated rate constant as well as a concomitant decrease in the effective diffusivity. Optical purity decreased in both subtractive resolution and product recovery at higher Thiele modulus with the effect being more pronounced in the product recovery mode. Finally, normalized plots were established to describe the effect of enzyme immobilization on both the effective enzymes activity and effective diffusivity. (c) 1993 Wiley & Sons, Inc.     

A Derivative Matching Approach to Moment Closure for the Stochastic Logistic Model

Continuous-time birth-death Markov processes serve as useful models in population biology. When the birth-death rates are nonlinear, the time evolution of the first n order moments of the population is not closed, in the sense that it depends on moments of order higher than n. For analysis purposes, the time evolution of the first n order moments is often made to be closed by approximating these higher order moments as a nonlinear function of moments up to order n, which we refer to as the moment closure function. In this paper, a systematic procedure for constructing moment closure functions of arbitrary order is presented for the stochastic logistic model. We obtain the moment closure function by first assuming a certain separable form for it, and then matching time derivatives of the exact (not closed) moment equations with that of the approximate (closed) equations for some initial time and set of initial conditions. The separable structure ensures that the steady-state solutions for the approximate equations are unique, real and positive, while the derivative matching guarantees a good approximation, at least locally in time. Explicit formulas to construct these moment closure functions for arbitrary order of truncation n are provided with higher values of n leading to better approximations of the actual moment dynamics. A host of other moment closure functions previously proposed in the literature are also investigated. Among these we show that only the ones that achieve derivative matching provide a close approximation to the exact solution. Moreover, we improve the accuracy of several previously proposed moment closure functions by forcing derivative matching.     

Calculations on O2 transfer in capsules with animal cells for the determination of maximum capsule size without O2 limitation

The size of the capsules without O2 limitation and maximum allowable capsule size for different cell densities were calculated by the observable modulus (modified Thiele modulus). When hybridoma (ATCC HB-8852) at 2.5 3 × 105cells/ ml was encapsulated and grown, the cell density reached to 2.6 × 107cells/ml in the 7th day of incubation. The average diameter of the capsule was 2.1mm. The maximum cell density obtained from experiment agreed with the calculated cell density for the given size of the capsules. © Rapid Science Ltd. 1998     

Thermal transitions in mixtures of polydeoxyribodinucleotides

Oligo d(C-A) and oligo d(T-G) of known average lengths, prepared by a combination of chemical and enzymatic procedures, have been mixed in 0.02 M and 0.07 M Na⁺, and absorbance has been studied as a function of increasing temperature. The transitions have been analyzed for the temperature of maximum slope T_m, the breadth of the transition, the value of the slope at T_m, and the maximum hyperchromicity. Linear expressions have been found relating the inverse of the length in nucleotide units (n^?1) of the shorter oligomer, irrespective of its identity, to T_m and also to the transition breadth. From a difference in slope between the T_m versus n^?1 expressions for the two molarities, the entropy and enthalpy of melting have been calculated as a function of n^?1.     

Assessing the Risk of Primary Amoebic Meningoencephalitis from Swimming in the Presence of Environmental Naegleria fowleri

Free-living Naegleria fowleri amoebae cause primary amoebic meningoencephalitis (PAM). Because of the apparent conflict between their ubiquity and the rarity of cases observed, we sought to develop a model characterizing the risk of PAM after swimming as a function of the concentration of N. fowleri. The probability of death from PAM as a function of the number of amoebae inhaled is modeled according to results obtained from animals infected with amoeba strains. The calculation of the probability of inhaling one or more amoebae while swimming is based on a double hypothesis: that the distribution of amoebae in the water follows a Poisson distribution and that the mean quantity of water inhaled while swimming is 10 ml. The risk of PAM for a given concentration of amoebae is then obtained by summing the following products: the probability of inhaling n amoebae × the probability of PAM associated with inhaling these n amoebae. We chose the lognormal model to assess the risk of PAM because it yielded the best analysis of the studentized residuals. Nonetheless, the levels of risk thereby obtained cannot be applied to humans without correction, because they are substantially greater than those indicated by available epidemiologic data. The curve was thus adjusted by a factor calculated with the least-squares method. This provides the PAM risk in humans as a function of the N. fowleri concentration in the river. For example, the risk is 8.5 × 10⁻⁸ at a concentration of 10 N. fowleri amoebae per liter.     

Effect of Selection on a Quantitative Characteristic

In this paper we study the effect of selection procedures on certain parameters of the distribution function (d.f.) (such as mean, percentile etc.) of a quantitative characteristic X, in successive generations when the d.f. is governed by a single locus. We study the changes in gene frequencies under the truncation and genotype selection procedures by obtaining approximations to the gene frequencies, since exact expressions are not available. Using these approximations for the gene frequencies, we compute the selection differentials of X for different values of n, the number of generations. We also obtain the limiting distributions as n → ∞ and compute the number of generations required for the above parameter(s) of the d.f. to reach a value close enough to the limiting value.