Mathematical modelling of enzyme kinetics reaction mechanisms and analytical solutions of non-linear reaction equations

The boundary value problem in basic enzyme reactions is formulated and approximate expressions for substrate and product concentrations are presented. He’s variational iteration method is used to give approximate and analytical solutions of non-linear reaction equations containing a non-linear term related to enzymatic reaction. The relevant analytical solutions for the substrate, enzyme, substrate-enzyme and product concentration profiles are discussed in terms of dimensionless reaction diffusion parameters K, λ and e{\varepsilon}.     

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.     

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.     

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.      

Bioaffine methods for determining ajmaline using an amperometric DNA-sensor and an immunoenzyme spectrophotometric test system

Bioaffine methods are developed for determining indole-containing alkaloid ajmaline, which has a cytostatic effect and is used as a cardiac drug. These methods are based on the proposed amperometric DNA-sensor and on immunoenzyme test system with the spectrophotometric indication of the analytical signal. The complex formation between ajmaline and immobilized native DNA allows ajmaline to be efficiently preconcentrated on the biosensor from test solutions. Optimum conditions for preconcentrating ajmaline and those for reactivating the biosensor for its repeated use are found. The time of analysis is 25–30 min, the determination limit for ajmaline is 3.0 × 10⁻¹⁰ M (RSD = 33%). In the test system, the immunological reaction of ajmaline with its antibodies and the enzyme marker, horseradish peroxidase, are used. The determination limit is 4.0 × 10⁻⁹ M (RSD = 33%). Ajmaline is determined by the two methods in model solutions of blood serum and in tablets and solutions for injections.     

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.     

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.     

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.     

Cellulose Hydrolysis by Cellobiohydrolase Cel7A Shows Mixed Hyperbolic Product Inhibition

In order to establish which are the contribution of linear (total), hyperbolic (partial) or parabolic inhibitions by cellobiose, and also a special case of substrate inhibition, the kinetics of cellobiohydrolase Cel7A obtained from Trichoderma reesei was investigated. Values of kinetic parameters were estimated employing integrated forms of Michaelis–Menten equations through the use of non-linear regression, and criteria for selecting inhibition models are discussed. With cellobiose added at the beginning of the reaction, it was found that cellulose hydrolysis follows a kinetic model, which takes into account a mixed hyperbolic inhibition, by cellobiose with the following parameter values: K _m 5.0 mM, K _ic 0.029 mM, K _iu 1.1 mM, k _cat 3.6 h⁻¹ and k _cat′ 0.2 h⁻¹. Cellulose hydrolysis without initial cellobiose added also follows the same inhibition model with similar values (4.7, 0.029 and 1.5 mM and 3.2 and 0.2 h⁻¹, respectively). According to Akaike information criterion, more complex models that take into account substrate and parabolic inhibitions do not increase the modulation performance of cellulose hydrolysis.     

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.     

Extraction–thermal lens quantification of cobalt with Nitroso-R-Salt in two-phase aqueous systems with water-soluble polymer polyethylene glycol

A novel method is proposed for the extraction-thermal lens quantification of cobalt with Nitroso-R-Salt based on the distribution of the colored complex in a two-phase aqueous system on the basis of poly-ethylene glycol (PEG) and an ammonium sulfate solution followed by its thermal lens detection in the extract. The limit of detection is 0.3 μM (20 ng/mL); the lower limit of the analytical range is 0.7 μM (40 ng/mL); the relative standard deviation for the concentrations 1–50 μM makes 1–3% (n = 6, P = 0.95). In the determination of cobalt by spectrophotometry under the same conditions, the detection limit is 10 μM (0.6 μg/mL) and the lower limit of the analytical range is 40 μM (2.5 μg/mL). The precision of thermal lens measurements in PEG solutions is higher in comparison to that in aqueous ones because of the weaker interference of convection in aqueous solutions of PEG.     

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.     

Intensification of Biocatalytical Processes by Synergistic Substrate Conversion. Fungal Peroxidase Catalyzed <Emphasis Type="Italic">N</Emphasis>-Hydroxy Derivative Oxidation in Presence of 10-Propyl Sulfonic Acid Phenoxazine

Many industrial pollutants, xenobiotics, and industry-important compounds are known to be oxidized by peroxidases. It has been shown that highly efficient peroxidase substrates are able to enhance the oxidation of low reactive substrate by acting as mediators. To explore this effect, the oxidation of two N-hydroxy derivatives, i.e., N-hydroxy-N-phenyl-acetamide (HPA) and N-hydroxy-N-phenyl-carbamic acid methyl ester (HPCM) catalyzed by recombinant Coprinus cinereus (rCiP) peroxidase has been studied in presence of efficient substrate 3-(4a,10a-dihydro- phenoxazin-10-yl)-propane-1-sulfonic acid (PPSA) at pH 8.5. The bimolecular constant of PPSA cation radical reaction with HPA was estimated to be (2.5 ± 0.2)·10⁷ M⁻¹ s⁻¹ and for HPCM was even higher. The kinetic measurements show that rCiP-catalyzed oxidation of HPA and HPCM can increase up to 33,000 times and 5,500 times in the presence of equivalent concentration of high reactive substrate PPSA. The mathematical model of synergistic rCiP-catalyzed HPA–PPSA and HPCM–PPSA oxidation was proposed. Experimentally obtained rate constants were in good agreement with those calculated from the model confirming the synergistic scheme of the substrate oxidation. In order to explain the different reactivity of substrates, the docking of substrates in the active site of the enzyme was calculated. Molecular dynamic calculations show that the enzyme–substrate complexes are structurally stable. The high reactive PPSA exhibited higher affinity to enzyme active site than HPA and HPCM. Furthermore, the orientation of HPA and HPCM was not favorable for proton transfer to the distal histidine, and different substrate reactivity was explained by these diversities.     

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.     

Oxidation of Fursemide by Diperiodatocuprate(III) in Aqueous Alkaline Medium—a Kinetic Study

Fursemide is the chemical compound 4-chloro-2-(furan-2-ylmethylamino)-5-(aminosulfonyl) benzoic acid. It was oxidized by diperiodatocuprate(III) in alkali solutions, and the oxidation products were identified as furfuraldehyde and 2-amino-4-chloro-5-(aminosulfonyl) benzoic acid. The reaction kinetics were studied spectrophotometrically. The reaction was observed to be first order in [oxidant] and fractional order each in [fursemide] and [periodate], whereas added alkali retarded the rate of reaction. The reactive form of the oxidant was inferred to be [Cu(H₃IO₆)₂]⁻. A mechanism consistent with the experimental results was proposed, in which oxidant interacts with the substrate to give a complex as a pre-equilibrium state. This complex decomposed in a slow step to give a free radical that was further oxidized by reaction with another molecule of DPC to yield 2-amino-4-chloro-5-(aminosulfonyl) benzoic acid and furfuraldehyde in a fast step. This reaction was studied at 25, 30, 35, 40 and 45 °C, and the activation parameters E _a,ΔH ^#,ΔS ^# and ΔG ^# were determined to be 51 kJ⋅mol⁻¹,48.5 kJ⋅mol⁻¹,−63.5 J⋅K⁻¹⋅mol⁻¹ and 67 kJ⋅mol⁻¹, respectively. The value of log ₁₀ A was calculated to be 6.8.     

On the stability of the bioactive flavonoids quercetin and luteolin under oxygen-free conditions

The natural flavonoid compounds quercetin (3,3′,4′,5,7-pentahydroxyflavone) and luteolin (3′,4′,5,7-tetrahydroxyflavone) are important bioactive compounds with antioxidative, anti-allergic, and anti-inflammatory properties. However, both are unstable when exposed to atmospheric oxygen, which causes degradation and complicates their analytical determinations. The oxidative change of these flavonoids was observed and followed by UV–visible spectrophotometry, both in aqueous and ethanolic solutions. The distribution of the degradation products in aqueous media was monitored by LC–MS and LC–DAD analysis. The amounts of oxidative reaction products increase with the exposure time. The oxidative degradation reduces the pharmacological efficiency of these antioxidants and renders analytical determination inaccurate. The oxidative changes in flavonoid test solutions can explain the inconsistent dissociation constants reported in the literature. Dissociation constants of quercetin and luteolin were determined both by alkalimetric titration and by UV–visible spectrophotometry under deaerated conditions. The values pK ₁ = 5.87 ± 0.14 and pK ₂ = 8.48 ± 0.09 for quercetin, and pK ₁ = 5.99 ± 0.32 and pK ₂ = 8.40 ± 0.42 for luteolin were found.     

Inactivation Kinetics of Polyphenol Oxidase from Pupae of Blowfly (Sarcophaga bullata) in the Dimethyl Sulfoxide Solution

The effects of dimethyl sulfoxide (DMSO) on the activity of polyphenol oxidase (PPO, EC 1.14.18.1) from blowfly pupae for the oxidation of l-3,4-dihydroxyphenylalanine were studied. The results showed that low concentrations of DMSO could lead to reversible inactivation to the enzyme. The IC₅₀ value, the inactivator concentration leading to 50% activity lost, was estimated to be 2.35 M. Inactivation of the enzyme by DMSO was classified as mixed type. The kinetics of inactivation of PPO from blowfly pupae in the low concentrations of DMSO solution was studied using the kinetic method of the substrate reaction. The rate constants of inactivation were determined. The results show that k ₊₀ was much larger than k _{+ 0}￠ k_{ + 0}^\prime , indicating that the free enzyme molecule was more fragile than the enzyme–substrate complex in the DMSO solution. It was suggested that the presence of the substrate offers marked protection of this enzyme against inactivation by DMSO.