Analytical solution of non-linear enzyme reaction equations arising in mathematical chemistry

The boundary value problem in basic enzyme reactions is formulated and approximate expressions for substrate and substrate-enzyme complex are presented. He’s Homotopy Perturbation method is used to give approximate and analytical solutions of non-linear reaction equations containing a non-linear term related to enzymatic reaction. The pertinent analytical solutions for the substrate, enzyme- substrate complex and free enzyme are discussed in terms of dimensionless parameters σ, ρ and e{\varepsilon} . The obtained concentration results are compared with the numerical solution acquired using Matlab program. They are found to be in satisfactory agreement.     

On approximate analytical solutions of differential equations in enzyme kinetics using homotopy perturbation method

Homotopy perturbation method is used to extend the approximate analytical solutions of non-linear reaction equations describing enzyme kinetics for combinations of parameters for which solutions obtained in previous works are not valid. Also, by constructing a new homotopy, alternative approximate analytical expressions for substrate, substrate-enzyme complex and product concentrations are found. These first-order approximate solutions give more accurate results than the second-order approximations derived in previous works.     

Analytical expression of the steady-state catalytic current of mediated bioelectrocatalysis and the application of He’s Homotopy perturbation method

A mathematical model of mediated bioelectrocatalysis is restudied in this paper. Here He’s Homotopy perturbation method is implemented to give approximate and analytical solutions of steady-state non-linear reaction diffusion equation containing a non-linear term related to Michaelis–Menten kinetics of the enzymatic reaction. Approximate analytical expressions for mediator concentration and current have been derived for all values of saturation parameter α and reaction diffusion parameter k for the various types of boundary conditions. The Homotopy perturbation method which produces the solutions in terms of convergent series, requiring no linearization or small perturbation. These analytical results are compared with numerical results (Matlab program) and are found to be in good agreement.     

Solving nonlinear reaction–diffusion problem in electrostatic interaction with reaction-generated pH change on the kinetics of immobilized enzyme systems using Taylor series method

A mathematical model of electrostatic interaction with reaction-generated pH change on the kinetics of immobilized enzyme is discussed. The model involves the coupled system of non-linear reaction–diffusion equations of substrate and hydrogen ion. The non-linear term in this model is related to the Michaelis–Menten reaction of the substrate and non-Michaelis–Menten kinetics of hydrogen ion. The approximate analytical expression of concentration of substrate and hydrogen ion has been derived by solving the non-linear reactions using Taylor’s series method. Reaction rate and effectiveness factor are also reported. A comparison between the analytical approximation and numerical solution is also presented. The effects of external mass transfer coefficient and the electrostatic potential on the overall reaction rate were also discussed.

     

Modeling of nonlinear boundary value problems in enzyme-catalyzed reaction diffusion processes

A mathematical model of steady state mono-layer potentiometric biosensor is developed. The model is based on non stationary diffusion equations containing a non linear term related to Michaelis-Menten kinetics of the enzymatic reaction. This paper presents a complex numerical method (He’s variational iteration method) to solve the non-linear differential equations that describe the diffusion coupled with a Michaelis-Menten kinetics law. Approximate analytical expressions for substrate concentration and corresponding current response have been derived for all values of saturation parameter α and reaction diffusion parameter K using variational iteration method. These results are compared with available limiting case results and are found to be in good agreement. The obtained results are valid for the whole solution domain.     

Application of He’s variational iteration method in nonlinear boundary value problems in enzyme– substrate reaction diffusion processes: part 1. The steady-state amperometric response

A mathematical model of amperometric biosensors has been developed. In this paper, He’s variational iteration method is implemented to give approximate and analytical solutions of non-linear reaction diffusion equations containing a non linear term related to Michaelis–Menten kinetic of the enzymatic reaction. The variational iteration method which produces the solutions in terms of convergent series, requiring no linearization or small perturbation. These analytical results are compared with available limiting case result and are found to be in good agreement.     

Computational Modelling of the Behaviour of Potentiometric Membrane Biosensors

This paper presents a mathematical model of a potentiometric biosensor based on a potentiometric electrode covered with an enzyme membrane. The model is based on the reaction–diffusion equations containing a non-linear term related to theMichaelis–Menten kinetics of the enzymatic reaction. Using computer simulation the influence of the thickness of the enzyme membrane on the biosensor response was investigated. The digital simulation was performed using the finite difference technique. Results of the numerical simulation were compared with known analytical solutions.      

Computational modelling of amperometric biosensors in the case of substrate and product inhibition

In this paper the response of an amperometric biosensor at mixed enzyme kinetics and diffusion limitations is modelled in the case of the substrate and the product inhibition. The model is based on non-stationary reaction–diffusion equations containing a non-linear term related to non-Michaelis–Menten kinetics of an enzymatic reaction. A numerical simulation was carried out using a finite difference technique. The complex enzyme kinetics produced different calibration curves for the response at the transition and the steady-state. The biosensor operation is analysed with a special emphasis to the conditions at which the biosensor response change shows a maximal value. The dependence of the biosensor sensitivity on the biosensor configuration is also investigated. Results of the simulation are compared with known analytical results and with previously conducted researches on the biosensors.     

Stochastic dynamics of complexation reaction in the limit of small numbers

We study stochastic dynamics of the non-linear bimolecular reaction A + B?AB. These reactions are common in several bio-molecular systems such as binding, complexation, protein multimerization to name a few. We use master equation to compute the full distribution of several stochastic equilibrium properties such as number of complexes formed (N(c)), equilibrium constant (K). We provide exact analytical and simpler approximate expression for equilibrium fluctuation quantities to quickly estimate the amount of noise as a function of reactant molecules and rates. We construct the phase diagram for a fluctuational quantity f, defined as the ratio of standard deviation to average (f=√(ΔN(c))(2)/N(c)), as a function of different number of reactant molecules and reaction rates. One of the striking result is, it is possible to have f as high as 45% or higher in significant regions of the phase diagram even when number of reactants involved are around 20-40, typical in biology. Our finding indicates studying averages alone using mass action law needs careful scrutiny. We also outline possible application of our findings in gene expression. Furthermore, we compute average and fluctuation properties of time dependent quantities and derive equations of motion for different moments such as N(c)(t) and N(c)(t)(2). While mean-field mass action law fails to reproduce the exact time dependence, approximate solutions of coupled equations of motions for different moments, capturing fluctuation, is in good agreement with exact results. This may be a way to compute time development of averages and fluctuations in such non-linear systems where mass action law breaks down. Moreover, for this reaction, we outline connection to variational principle of maximum caliber and other more traditional approaches such as chemical Langevin equation. We derive noise statistics for the equivalent Langevin equation and show possible departure from Gaussian white noise. We believe quantitative estimates of phase diagrams for noise, time dependent quantities, and simple analytical expression for equilibrium quantities will be particularly useful to guide experiments involving such non-linear reactions with small numbers of reactants that are often encountered in biology.     

Development of a pressure-driven nanofluidic control system and its application to an enzymatic reaction

A novel air-pressure-based nanofluidic control system was developed and its performance was examined. We found that the flow in a 100 nm scale nanochannel on a chip (called an extended nanospace channel) could be controlled within the pressure range of 0.003–0.4 MPa, flow rate range of 0.16–21.2 pL/min, and residence time range of 24 ms–32.4 s by using the developed nanofluidic control system. Furthermore, we successfully demonstrated an enzyme reaction in which the fluorogenic substrate TokyoGreen-β-galactoside (TG-β-gal) was hydrolyzed to the fluorescein derivative TokyoGreen (TG) and β-galactose by the action of β-galactosidase enzyme as a calalyst in a Y-shaped extended nanospace channel. The parameters for the reaction kinetics, such as K _m, V _max and k _cat, were estimated for the nanofluidic reaction, and these values were compared with the results of bulk and microfluidic reactions. A comparison showed that the enzyme reaction rate in the Y-shaped extended nanospace channel increased by a factor of about two compared with the rates in the bulk and micro spaces. We thought that this nanospatial property resulted from the activated protons of water molecules in the extended nanospace. This assumption was supported by the result that the pH dependence of the maximum enzyme activity in the Y-shaped extended nanospace channel was slightly different from that in the bulk and micro spaces. Electronic supplementary material  The online version of this article (doi:) contains supplementary material, which is available to authorized users.     

The interaction energy between identical particles

The improved Derjaguin’s method is used to derive the approximate expressions for the electrostatic interaction energy between spherical colloidal particles. The approximate results are in good agreement with the exact numerical solutions. The results from the product of the original Derjaguin approximation with curvature correction 2a/R can be used satisfactorily at different αa and κh, no matter high or low the potential of spherical particle is. The text was submitted by the authors in English.     

Multiplex ELISA in a single microfluidic channel

In this article, we demonstrate a novel approach to implementing multiplex enzyme-linked immunosorbent assay (ELISA) in a single microfluidic channel by exploiting the slow diffusion of the soluble enzyme reaction product across the different assay segments. The functionality of the reported device is realized by creating an array of ELISA regions within a straight conduit that are selectively patterned with chosen antibodies/antigens via a flow-based method. The different analytes are then captured in their respective assay segments by incubating a 5-μL aliquot of sample in the analysis channel for an hour under flow conditions. Once the ELISA surfaces have been prepared and the enzyme substrate introduced into the analysis channel, it is observed that the concentration of the soluble enzyme reaction product (resorufin) at the center of each assay region grows linearly with time. Further, the rate of resorufin generation at these locations is found to be proportional to the concentration of the analyte being assayed in that segment provided that the ELISA reaction time in the system (τ _R ) is kept much shorter than that required by the resorufin molecules to diffuse across an assay segment (τ _D ). Under the operating condition τ _R  << τ _D , the reported device has been shown to have a 35% lower limit of detection for the target analyte concentration compared with that on a commercial microtiter plate using only a twentieth of the sample volume.     

Mathematical modelling of steady-state concentration in immobilized glucose isomerase of packed-bed reactors

The theoretical model of the steady-state immobilized enzyme electrodes is discussed. This model is based on diffusion equation containing a non-linear term related to Michaelis–Menten kinetics of the enzymatic reaction. Homotopy perturbation method (HPM) is employed to solve the non-linear diffusion equation for the steady-state condition. Simple and approximate polynomial expression of concentration and flux are derived for all small values of parameters ${\phi_p}$ (Theiele modulus) and β (kinetic parameter). Furthermore, in this work the numerical solution of the problem is also reported using SCILAB/MATLAB program. The analytical results are compared with the numerical results and found to be in good agreement.     

Flow-injection determination of nitrites based on their reaction with thiocyanates

A technique for the flow-injection determination of nitrites with spectrophotometric signal detection was developed and studied. Additionally, a combination of the developed technique with the on-line pre-concentration of low concentrations of nitrites in surface and potable waters was investigated. The technique is based on the reaction of nitrites with thiocyanates. The product of the analytical reaction (nitroso-S-thiocyanate) and its adducts with strong acids are stable in aqueous solutions, which is supported by quantum chemical calculations. It is shown that the reaction chosen is attractive for analytical applications, because it is highly selective. A decrease in the detection limit down to 1.5 ng/mL (3s) is achieved by combining FIA with on-line ion-exchange preconcentration. The technique developed is used for monitoring nitrites in natural and waste water. The throughput capacity is 300 samples per hour.Translated from Zhurnal Analiticheskoi Khimii, Vol. 60, No. 3, 2005, pp. 323–330.Original Russian Text Copyright © 2005 by Kuznetsov, Zemyatova, Ermolenko.     

Autocatalytic reaction on low-dimensional substrates

We discuss a model for the autocatalytic reaction A + B→ 2A on substrates where the reactants perform a compact exploration of the space, i.e., on lattices whose spectral dimension d͂ is < 2. For finite systems, the total time τ for the reaction to end scales according to two different regimes, for high and low concentrations of reactants. The functional dependence of τ on the volume of the substrate and the concentration of reactants is discussed within a mean-field approximation. Possible applications are discussed.     

Influence of substrate and product concentrations on the production of cyclodextrins by CGTase of Bacillus firmus,strain no. 37

The influence of substrate or product level on the initial velocity of cyclodextrin (CD) production by cyclodextringlycosyltransferase from a Brazilian isolate of Bacillus firmus was studied. Our results indicate that the product γ-CD is a stronger inhibitor to the reaction than β-CD. Small saccharides could also inhibit CD production, although to a lesser extent than the products, and maltose was the strongest inhibitor among small saccharides. Increasing substrate concentration resulted in greater reduction on enzyme activity for the formation of β-CD than for γ-CD. We modeled the kinetics of CD production with a set of four reversible reactions including the cyclization/coupling reaction that forms/opens CDs, and three disproportionation reactions. Our model on the initial velocity data explained well the substrate inhibition phenomenon. Kinetic parameters were determined by fitting the initial velocity data into our model.     

Kinetics of peroxidase-catalyzed oxidation of quercetin in the presence of β-cyclodextrin

It is established that the rate of peroxidase-catalyzed oxidation of flavonoid quercetin is increased by 20% in the presence of macrocyclic complexing agent β-cyclodextrin. The comparison of the kinetic parameters of the indicated reaction in the presence and in the absence of β-cyclodextrin shows that its introduction does not significantly influence the specificity of the enzyme with respect to the reducing substrate (characterized as k _cat/K _M ratio), while the increase in the reaction rate does not depend on the duration of the incubation of quercetin with β-cyclodextrin. It is assumed that the increase of the reaction rate is associated with nonspecific interaction between β-cyclodextrin and quercetin oxidation product.     

Implementation of solvent reaction fields for electronic structure

 Methods are described to incorporate solvent reaction field effects into solute electronic structure calculations. Included are several old and new approaches based on approximate solutions of Poisson's equation through boundary element methods, wherein the solutions are represented in terms of certain apparent surface charge or apparent surface dipole distributions. Practical algorithms to set up and solve the requisite equations are described and implemented in a new general reaction field computer program. Illustrative computational results are presented to show the performance of the program. Received: 2 July 2001 / Accepted: 11 September 2001 / Published online: 19 December 2001     

Inhibitory analysis of the effect of polycyclic aromatic hydrocarbons on the activity of chitinase by means of liquid chromatography-mass spectrometry of chitin oligosaccharides

The analytical method of determining enzyme activity by liquid chromatography-mass spectrometry (LC/MS) was developed and applied for investigation of the effect of polycyclic aromatic hydrocarbons (PAHs) on the enzyme activity of chitinase. The measurement of chitinase activity by LC/MS is useful in order to use the nonderivatized substrate, which can show in vivo chitinase activity. Substrate consumption and product formation were monitored in order to determine chitinase activity. It was shown that, for the first time, in vitro addition of PAHs inhibited the activity of chitinase in a noncompetitive manner. The IC₅₀ value of benzo[a]pyrene was 1.4 μM, and PAHs containing four or more aromatic rings showed the same or higher inhibitory effect, whereas PAHs with a lower number of aromatic rings showed lower inhibition of the chitinase activity than benzo[a]pyrene.     

Effects of Substrate Loading on Enzymatic Hydrolysis and Viscosity of Pretreated Barley Straw

In this study, the applicability of a “fed-batch” strategy, that is, sequential loading of substrate or substrate plus enzymes during enzymatic hydrolysis was evaluated for hydrolysis of steam-pretreated barley straw. The specific aims were to achieve hydrolysis of high substrate levels, low viscosity during hydrolysis, and high glucose concentrations. An enzyme system comprising Celluclast and Novozyme 188, a commercial cellulase product derived from Trichoderma reesei and a β-glucosidase derived from Aspergillus niger, respectively, was used for the enzymatic hydrolysis. The highest final glucose concentration, 78 g/l, after 72 h of reaction, was obtained with an initial, full substrate loading of 15% dry matter weight/weight (w/w DM). Conversely, the glucose yields, in grams per gram of DM, were highest at lower substrate concentrations, with the highest glucose yield being 0.53 g/g DM for the reaction with a substrate loading of 5% w/w DM after 72 h. The reactions subjected to gradual loading of substrate or substrate plus enzymes to increase the substrate levels from 5 to 15% w/w DM, consistently provided lower concentrations of glucose after 72 h of reaction; however, the initial rates of conversion varied in the different reactions. Rapid cellulose degradation was accompanied by rapid decreases in viscosity before addition of extra substrate, but when extra substrate or substrate plus enzymes were added, the viscosities of the slurries increased and the hydrolytic efficiencies decreased temporarily.