Prediction of substrate consumption rate, average biofilm density and active thickness for a thin spherical biofilm at pseudo-steady state

In this work, a new approach is proposed to evaluate substrate consumption rate, average biofilm density and active thickness of a spherical bioparticle in a completely mixed fluidized bed system. The substrate consumption rate and average biofilm density are predicted for a given biofilm surface substrate concentration and operational biofilm thickness. A diffusion and reaction model is developed with an effective diffusion coefficient that depends on the average biofilm density. This approach, a first in the literature, predicts the optimum average density of a biofilm to yield the maximum substrate consumption rate within the biofilm. A reasonable correlation was observed between the model prediction and experimental results for substrate consumption rate and average biofilm density for thin and fully active biofilms.     

Continuous cultures limited by a gaseous substrate: Development of a simple, unstructured mathematical model and experimental verification with Methanobacterium thermoautotrophicum

This article presents a simple, unstructured mathematical model describing microbial growth in continuous culture limited by a gaseous substrate. The model predicts constant gas conversion rates and a decreasing biomass concentration with increasing dilution rate. It has been found that the parameters influencing growth are primarily the gas transfer rate and the dilution rate. Furthermore, it is shown that, for correct simulation of growth, the influence of gaseous substrate consumption on the effective gas flow through the system has to be taken into account.Continuous cultures of Methanobacterium thermoautotrophicum were performed at three different gassing rates. In addition to the measurement of the rates of biomass production, product formation, and substrate consumption, microbial heat dissipation was assessed using a reaction calorimeter. For the on-line measurement of the concentration of the growth-limiting substrate, H(2), a specially developed probe has been used. Experimental data from continuous cultures were in good agreement with the model simulations. An increase in gassing rate enhanced gaseous substrate consumption and methane production rates. However, the biomass yield as well as the specific conversion rates remained constant, irrespective of the gassing rate. It was found that growth performance in continuous culture limited by a gaseous substrate is substantially different from "classic" continuous culture in which the limiting substrate is provided by the liquid feed. In this report, the differences between both continuous culture systems are discussed.     

Analytic solution to steady-state radial diffusion of a substrate with first-order reaction kinetics in the tissue of a Krogh's cylinder

It is often useful to calculate the concentration profile for a substrate undergoing reaction in the tissue surrounding a capillary. In this paper, we consider a model geometry consisting of a long straight cylinder of tissue surrounding a capillary. Substrate diffuses radially out of the capillary through the tissue, with consumption of substrate in the tissue directly proportional to substrate concentration (i.e., first-order reaction kinetics). The model is extended to include the case where a cylinder of necrotic tissue surrounds a metabolically active inner tissue cylinder. A simple analytic solution is derived, and concentration profiles are generated for various combinations of parameters. Compared to the case where substrate consumption is independent of concentration, this model predicts much more rapid depletion of substrate near the capillary interface. This can have significant implications for the calculation of the hypoxic fraction (e.g., tissue with pO(2)<0.5-5 mmHg) when tumor oxygenation is modeled. The model also permits calculation of the limiting substrate concentration for cell viability when the reaction rate constant is known and vice versa.     

Iteration model of starch hydrolysis by amylolytic enzymes.

An elaborate computer program to simulate the process of starch hydrolysis by amylolytic enzymes was been developed. It is based on the Monte Carlo method and iteration kinetic model, which predict productive and non-productive amylase complexes with substrates. It describes both multienzymatic and multisubstrate reactions simulating the "real" concentrations of all components versus the time of the depolymerization reaction the number of substrates, intermediate products, and final products are limited only by computer memory. In this work, it is assumed that the "proper" substrate for amylases is the glucoside linkages in starch molecules. Dynamic changes of substrate during the simulation adequately influence the increase or decrease of reaction velocity, as well as the kinetics of depolymerization. The presented kinetic model, can be adapted to describe most enzymatic degradations of a polymer. This computer program has been tested on experimental data obtained for alpha- and beta-amylases.     

Computer Simulation of Radial Immunodiffusion: I. Selection of an Algorithm for the Diffusion Process

Theories of diffusion with chemical reaction are reviewed as to their contributions toward developing an algorithm needed for computer simulation of immunodiffusion. The Spiers-Augustin moving sink and the Engelberg stationary sink theories show how the antibody-antigen reaction can be incorporated into boundary conditions of the free diffusion differential equations. For this, a stoichiometric precipitate was assumed and the location of precipitin lines could be predicted. The Hill simultaneous linear adsorption theory provides a mathematical device for including another special type of antibody-antigen reaction in antigen excess regions of the gel. It permits an explanation for the lowered antigen diffusion coefficient, observed in the Oudin arrangement of single linear diffusion, but does not enable prediction of the location of precipitin lines. The most promising mathematical approach for a general solution is implied in the Augustin alternating cycle theory. This assumes the immunodiffusion process can be evaluated by alternating computation cycles: free diffusion without chemical reaction and chemical reaction without diffusion. The algorithm for the free diffusion update cycle, extended to both linear and radial geometries, is given in detail since it was based on gross flow rather than more conventional expressions in terms of net flow. Limitations on the numerical integration process using this algorithm are illustrated for free diffusion from a cylindrical well.     

Performance of the continuous glucose isomerase reactor system for the production of fructose syrup

Glucose isomerase in the form of heat-treated whole-cell enzyme prepared from Streptomyces phaeochromogenus follows the reversible single-substrate reaction kinetics in isomerization of glucose to fructose. Based on the Kinetic constants determined and the mathematical model of the reactor system developed, the preformance of a plug-flow-type continuous-enzyme reactor system was studied experimentally and also simulated with the aid of a computer for the ultimate objective of optimization of the glucose isomerase reactor system. The enzyme decay function for both the enzyme storage and during the use in the continuous reactor, was found to follow the first-order decay kinetics. When the enzyme decay function is taken into consideration, the ideal homogeneous enzyme reactor kinetics provided a satisfactory working model without further complicatin of the mathematical model, and the results of computer simulation were found to be in good agreement with the experimental results. Under a given set of constraints the performance of the continuous glucose isomerase reactor system can be predicted by using the computer simulation method described in this paper. The important parameters studied for the optimization of reactor operation were enzyme loading, mean space time of the reactor, substrate feed concentration, enzyme decay constants, and the fractional conversion, in addition to the kinetic constants. All these parameters have significant effect on the productivity. Some unique properties of the glucose isomerization reaction and its effects on the performance of the continuous glucose isomerase reactor system have been studied and discussed. The reaction kinetics of glucose isomerase and the effects of both the enzyme loading and the changes in reaction rate within a continuous reactor on the productivity are all found to be of particular importance to this enzyme reactor system.     

Inhibition Kinetics of Phenol Degradation by Pseudomonas aeruginosa from Continuous Culture and Washout Data

Steady states of a continuous culture with an inhibitory substrate were used to estimate kinetic parameters under substrate limitation (chemostat operation). Pure cultures of an indigenous Pseudomonas aeruginosa were grown in continuous culture on phenol, the sole source of carbon and energy, at dilution rates of 0.010 to 0.20 h^{- 1}. Using different dilution rates, several steady states were investigated and the specific phenol consumption rates were calculated. In addition, phenol degradation was investigated by increasing the dilution rate above the critical dilution rate (washout cultivation). The results showed that the specific phenol consumption rate increased with increased dilution rate at steady state and that the degradation by Pseudomonas aeruginosa can be described by simple substrate inhibition kinetics under substrate limitation but cannot be described by simple substrate inhibition kinetics under washout cultivation. Fitting of the steady-state data from continuous cultivation to various inhibition models resulted in the best fit for the Yano and Koga kinetic inhibition model. The r_{s max} value of 0.278 mg/mg/h obtained from the Yano and Koga equation was comparable to the experimentally calculated r_{s max} value of 0.283 mg/mg/h obtained under washout cultivation.     

"Clotspeed," a mathematical simulation of the functional properties of prothrombinase

Prothrombinase is a Ca2+-dependent, 1:1, enzymatic complex of Factor Xa and Factor Va that assembles on the surface of negatively charged phospholipid vesicles or platelets. It catalyzes the proteolytic conversion of prothrombin to the blood-clotting enzyme thrombin. Experimentally determined kinetic parameters, plus Kd and n values for the interaction of substrate, cofactor (Factor Va), and serine protease (Factor Xa) for both phospholipid and each other, were used to develop a model that simulates the functional properties of the enzymatic complex. Through the use of a desk-top computer and a program designated "Clotspeed," the distribution of enzymatic components and substrate between the bulk fluid and phospholipid is determined for a given set of initial concentrations of reaction components. Simulated reaction rates are then calculated from the calculated distributions, fractional binding, and local and bulk concentration of reactants. Predicted behavior includes formal Michaelis-Mentenlike properties for the reaction, increasing apparent Km with increased levels of phospholipid, and apparent inhibition by excess substrate, enzyme, and phospholipid. Inhibition by excess enzyme and phospholipid was demonstrated experimentally in quantitative agreement with predicted results. The model is useful in that it rationalizes well the seemingly unusual properties of prothrombinase in straightforward physical terms, provides a means of rationally choosing experimental conditions to both further test and refine the model, and explores the properties not only of prothrombinase but also other blood-clotting or surface-bound enzymatic complexes.     

Kinetics of synthesesis of polymeric dAT on oligomeric templates

A model is developed to explain the following experimental observations: oligomers of the alternating DNA copolymer dAT serve as templates for the enzymatic synthesis of macromolecular dAT; the rate of polymer synthesis with a template oligomer of given length increases to a maximum and then decreases as the temperature increases; the temperature for the maximum rate of synthesis increases with the length of the template. These observations were interpreted in terms of the repeated “slipping” of the oligomer along the product strand to expose template sites. This communication extends this idea. We suppose that the rate-limiting step in the polymerization is either the attachment of the substrate to the exposed template site or the synthesis of the phosphodiester bond. Then the rate of synthesis is proportional to the equilibrium concentration either of exposed template sites or of template sites occupied by substrate molecules. These concentrations are evaluated in terms of a simple theory of the helix–coil transition. The model qualitatively reproduces the dependence of the rate of synthesis on template length and temperature and predicts rates of synthesis in approximate agreement with experiment.     

Prediction of average biofilm density and performance of a spherical bioparticle under substrate inhibition

In this article, a model was proposed to predict the average performance and biofilm density of a spherical bioparticle under substrate inhibition in a fluidized bed system. The average biofilm density and substrate consumption rates were predicted for a definite biofilm thickness and limiting substrate concentrations. A diffusion and reaction model was developed over the bioparticle with biofilm-density dependent effective diffusion coefficients for maximum substrate consumption theory. This theory predicts the optimum density of a biofilm to yield a maximum substrate consumption rate within the biofilm, developed for the first time with this study and experimentally verified. A good correlation was observed between the model prediction and experimental results for biofilm density and substrate consumption rates. (c) 1997 John Wiley & Sons, Inc. Biotechnol Bioeng 56: 319-329, 1997.     

初始底物浓度对序批式培养光合细菌产氢动力学影响

实验研究了初始底物浓度对序批式培养光合细菌生长、降解及产氢过程的影响,根据最大比生长速率实验数据拟合得到其关于初始底物浓度影响的关联式,并在建立的修正Monod模型基础上建立了光合细菌比生长速率、基质比消耗速率和比产氢速率关于底物初始浓度影响的数学模型,模型预测值与实验结果在光合细菌生长期和稳定期内得到较好吻合,反映了光合细菌生长、降解和产氢过程中受底物初始浓度限制性和抑制性影响的基本规律。分析发现光合细菌生长、降解基质和产氢过程中最适底物浓度为50 mmol/L,初始底物浓度低于或高于该浓度时,光合细菌生长、降解及产氢过程都受到限制性或抑制性影响,且抑制性影响较限制性影响效果更明显;底物比消耗速率受初始底物浓度影响较小。     

Continuum heterogeneous biofilm model--a simple and accurate method for effectiveness factor determination

We present a novel analytical approach to describe biofilm processes considering continuum variation of both biofilm density and substrate effective diffusivity. A simple perturbation and matching technique was used to quantify biofilm activity using the steady-state diffusion-reaction equation with continuum variable substrate effective diffusivity and biofilm density, along the coordinate normal to the biofilm surface. The procedure allows prediction of an effectiveness factor, η, defined as the ratio between the observed rate of substrate utilization (reaction rate with diffusion resistance) and the rate of substrate utilization without diffusion limitation. Main assumptions are that (i) the biofilm is a continuum, (ii) substrate is transferred by diffusion only and is consumed only by microorganisms at a rate according to Monod kinetics, (iii) biofilm density and substrate effective diffusivity change in the x direction, (iv) the substrate concentration above the biofilm surface is known, and (v) the substratum is impermeable. With this approach one can evaluate, in a fast and efficient way, the effect of different parameters that characterize a heterogeneous biofilm and the kinetics of the rate of substrate consumption on the behavior of the biological system. Based on a comparison of η profiles the activity of a homogeneous biofilm could be as much as 47.8% higher than that of a heterogeneous biofilm, under the given conditions. A comparison of η values estimated for first order kinetics and η values obtained by numerical techniques showed a maximum deviation of 1.75% in a narrow range of modified Thiele modulus values. When external mass transfer resistance, is also considered, a global effectiveness factor, η(0) , can be calculated. The main advantage of the approach lies in the analytical expression for the calculation of the intrinsic effectiveness factor η and its implementation in a computer program. For the test cases studied convergence was achieved quickly after four or five iterations. Therefore, the simulation and scale-up of heterogeneous biofilm reactors can be easily carried out.     

Kinetic study of an enzyme-catalysed reaction in the presence of novel irreversible-type inhibitors that react with the product of enzymatic catalysis

In the present paper a kinetic study is made of the behaviour of a Michaelis-Menten enzyme-catalysed reaction in the presence of irreversible inhibitors rendered unstable in the medium by their reaction with the product of enzymatic catalysis. A general mechanism involving competitive, non-competitive, uncompetitive and mixed irreversible inhibition with one or two steps has been analysed. The differential equation that describes the kinetics of the reaction is non-linear and computer simulations of its dynamic behaviour are presented. The results obtained show that the systems studied here present kinetic co-operativity for a target enzyme that follows the simple Michaelis-Menten mechanism in its action on the substrate, except in the case of an uncompetitive-type inhibitor.     

Effects of conformational selectivity and of overlapping kinetically influential ionizations on the characteristics of pH-dependent enzyme kinetics. Implications of free-enzyme pKa variability in reactions of papain for its catalytic mechanism.

The effects of selection by a small molecule, when binding to a protein, of a particular conformation from an equilibrium stereopopulation on the characteristics of the pH-dependence of reaction with a reactivity probe or substrate were determined by analysis of an appropriate kinetic model. For reaction in one protonic state containing an equilibrium mixture of two conformational isomers, the pH-second-order rate constant (k) profile is of conventional sigmoidal form. The apparent pKa value is a composite of the pKa values of the two conformational states. The value of pKapp. for a given enzyme under given experimental conditions will always be the same (provided that the site-specificity assumed in the model is maintained) irrespective of whether only one conformation reacts or both react, with the same or with different rate constants. The experimentally determined pH-independent rate constant (kapp.) is an average of the reactivities of the two conformational states weighted in favour of the predominant form. The presence of an additional but unreactive conformational state also affects the value of kapp. The possibility that overlapping acid dissociations that affect the reactivity of the enzyme might provide pH-k profiles often indistinguishable in practice from simple sigmoidal dissociation curves and subject to variability in apparent pKa values was evaluated by a simulation study. If two reactive protonic states of the enzyme respond differently to changes in the structure of the substrate or site-specific reactivity probe, differences in apparent pKa values of up to approx. 1 unit can be exhibited without deviation from sigmoidal behaviour being reliably observed. Differences in apparent pKa values observed in some site-specific reactions of papain and their possible consequences for its catalytic mechanism are discussed.     

Prediction of methane yield at optimum pH for anaerobic digestion of organic fraction of municipal solid waste

A concept of methane yield at optimum pH was advanced and subsequently a mathematical model that simulates the optimal pH of a batch process for anaerobic digestion of organic fraction of municipal solid waste (MSW) was developed and validated. The model was developed on the basis of the microbial growth kinetics and was divided into three processes: hydrolysis of substrates by hydrolytic bacteria, consumption of soluble substrate by acidogenic bacteria, and finally consumption of acetate and methane generated by methanogenic bacteria. Material balance and liquid phase equilibrium chemistry were used in this study. A series of experiments were conducted to validate the model. The model simulation results agreed reasonably with experimental data in different temperatures and total solid (TS) concentrations under uncontrolled pH. A computer circulation program was used to predict the optimal pH in different conditions. Experiments in different temperatures and TS were run under optimal pH which predicted by the model. The model was succeeded in increasing the methane production and the cumulative methane production had an average increment about 35% in optimal pH of different temperatures and TS.     

A graphical method for determining inhibition parameters for partial and complete inhibitors.

A new simple graphical method is described for the determination of inhibition type and kinetic parameters of an enzyme reaction without any replot. The method consists of plotting experimental data as v/(vo--v) versus the reciprocal of the inhibitor concentration at different substrate concentrations, where v and vo represent the velocity in the presence and in the absence of the inhibitor respectively with a given concentration of the substrate. Partial inhibition gives straight lines that converge on the abscissa at a point away from the origin, whereas complete inhibition gives lines that go through the origin. The inhibition constants of enzymes and the reaction rate constant of the enzyme-substrate-inhibitor complex can be calculated from the abscissa and ordinate intercepts of the plot. The relationship between the slope of the plot and the substrate concentration shows characteristic features depending on the inhibition type: for partial competitive inhibition, the straight line converging on the abscissa at--Ks, the dissociation constant of the enzyme-substrate complex; for non-competitive inhibition, a constant slope independent of the substrate concentration; for uncompetitive inhibition, a hyperbola decreasing with the increase in the substrate concentration; for mixed-type inhibition, a hyperbola increasing with the increase in the substrate concentration. The properties of the replot are useful in confirmation of the inhibition mechanism.     

Parallel activation in the ATP supply-demand system lessens the impact of inborn enzyme deficiencies, inhibitors, poisons or substrate shortage on oxidative phosphorylation in vivo

A potential kinetic impact of parallel activation of different steps during an increased energy demand on the effect of inborn enzyme deficiencies, physiological inhibitors, external poisons and substrate shortage on oxidative phosphorylation was studied in the theoretical way. Numerical simulations were performed with the aid of the previously developed computer model of oxidative phosphorylation. It was demonstrated that the parallel activation mechanism diminishes significantly changes in fluxes and metabolite concentrations occurring at a given degree of inactivation of the system by one of the above-mentioned factors. It was also shown that parallel activation decreases greatly the threshold value of the relative activity of oxidative phosphorylation, below which the oxygen consumption flux and ATP turnover flux become significantly affected. Finally, computer simulations predicted that parallel activation leads to a considerable increase in the apparent affinity of oxidative phosphorylation to oxygen, which delays the effect of inhibitors and poisons competing with oxygen for the active centre of cytochrome oxidase. It is concluded that one of possible functions of parallel direct activation of different steps of oxidative phosphorylation is to increase the resistance of the system to a decrease in the concentration/activity of different oxidative phosphorylation complexes.     

Thiophosphorylation as a probe for subunit interactions in Escherichia coli succinyl coenzyme A synthetase. Further evidence for catalytic cooperativity and substrate synergism

Succinyl-CoA synthetase has an (alpha beta)2 subunit structure and shows half-of-the-sites reactivity with respect to the formation of the phosphohistidyl residues that acts as a catalytic intermediate. Adenosine 5'-O-(3-thio)triphosphate has been found to be a substrate, but the overall maximum velocity is 3 orders of magnitude lower than that seen with ATP. Moreover, steps of the reaction involving thiophosphoryl transfer are much slower than the corresponding phosphoryl transfers. These properties of adenosine 5'-O-(3-thio)triphosphate as a substrate have been exploited to test the concept of alternating sites catalytic cooperativity proposed earlier as a rationale for the subunit structure of succinyl-CoA synthetase. As predicted by this model for catalysis, the rate of discharge of thiophosphate from the enzyme in the presence of succinate and CoA is stimulated by ATP. Neither of two nonhydrolyzable analogs of ATP has an equivalent effect. The results indicate that the transfer of the thiophosphoryl group from the enzyme to succinate at one active site is not favored until the neighboring active site is phosphorylated by ATP, with accompanying reciprocal changes in the conformations of the two halves of the enzyme molecule.     

Theoretical studies of enzymic reactions: Dielectric,electrostatic and steric stabilization of the carbonium ion in the reaction of lysozyme

A general method for detailed study of enzymic reactions is presented. The method considers the complete enzyme-substrate complex together with the surrounding solvent and evaluates all the different quantum mechanical and classical energy factors that can affect the reaction pathway. These factors include the quantum mechanical energies associated with bond cleavage and charge redistribution of the substrate and the classical energies of steric and electrostatic interactions between the substrate and the enzyme. The electrostatic polarization of the enzyme atoms and the orientation of the dipoles of the surrounding water molecules is simulated by a microscopic dielectric model. The solvation energy resulting from this polarization is considerable and must be included in any realistic calculation of chemical reactions involving anything more than an isolated molecule in vacuo. Without it, acidic groups can never become ionized and the charge distribution on the substrate will not be reasonable. The same dielectric model can also be used to study the reaction of the substrate in solution. In this way the reaction in solution can be compared with the enzymic reaction.In this paper we study the stability of the carbonium ion intermediate formed in the cleavage of a glycosidic bond by lysozyme. It is found that electrostatic stabilization is an important factor in increasing the rate of the reaction step that leads to the formation of the carbonium ion intermediate. Steric factors, such as the strain of the substrate on binding to lysozyme, do not seem to contribute significantly.