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.     

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}.     

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.     

Transport and kinetics in electrocatalytic thin film biosensors: bounded diffusion with non-Michaelis-Menten reaction kinetics

In this paper, we describe the problem of describing the transport and catalytic kinetics at immobilized enzymes in an electronically conductive polymer thin film where substrate inhibition is important. Here, the enzyme kinetics are not well described by the Michaelis-Menten equation. We describe a mathematical procedure based on the recently developed Akbari-Ganji method (AGM) which facilitates a full analytical solution of the boundary value problem which is valid for all values of substrate concentration. Closed form expressions for both the substrate concentration in the film and the steady-state amperometric current response are presented. Limiting kinetic cases are identified and are expressed pictorially in parameter space using a kinetic case diagram.

     

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.     

Concentration profiles near an activated enzyme

When a resting enzyme is activated, substrate concentration profile evolves in its vicinity, ultimately tending to steady state. We use modern theories for many-body effects on diffusion-influenced reactions to derive approximate analytical expressions for the steady-state profile and the Laplace transform of the transient concentration profiles. These show excellent agreement with accurate many-particle Brownian-dynamics simulations for the Michaelis-Menten kinetics. The steady-state profile has a hyperbolic dependence on the distance of the substrate from the enzyme, albeit with a prefactor containing the complexity of the many-body effects. These are most conspicuous for the substrate concentration at the surface of the enzyme. It shows an interesting transition as a function of the enzyme turnover rate. When it is high, the contact concentration decays monotonically to steady state. However, for slow turnover it is nonmonotonic, showing a minimum due to reversible substrate binding, then a maximum due to diffusion of new substrate toward the enzyme, and finally decay to steady state. Under certain conditions one can obtain a good estimate for the critical value of the turnover rate constant at the transition.     

An integration method of enzyme analysis

The application of an integration method of kinetic analysis to first-order and second-order reactions is discussed with particular emphasis on enzyme analyses. Transducer signals or concentrations of products or substrates are integrated for a Fixed time and the net integral of the increased or decreased signal or concentration is related to the initial substrate or enzyme concentration. Equations are developed describing the dependence of the integrals on enzyme and substrate concentrations for first- and second-order reactions, and examples are presented illustrating different cases. The application of this method to complicated enzymatic systems is discussed.     

Time dependent diffusion in a disordered medium with partially absorbing walls: a perturbative approach

We present an analytical study of the time dependent diffusion coefficient in a dilute suspension of spheres with partially absorbing boundary condition. Following Kirkpatrick [J. Chem. Phys. 76, 4255 (1982)] we obtain a perturbative expansion for the time dependent particle density using volume fraction f of spheres as an expansion parameter. The exact single particle t operator for partially absorbing boundary condition is used to obtain a closed form time dependent diffusion coefficient D(t) accurate to first order in the volume fraction f. Short and long time limits of D(t) are checked against the known short time results for partially or fully absorbing boundary conditions and long time results for reflecting boundary conditions. For fully absorbing boundary condition the long time diffusion coefficient is found to be D(t)=5a(2)/(12fD(0)t)+O((D(0)t/a(2))(-2)) to the first order of perturbation theory. Here f is small but nonzero, D(0) the diffusion coefficient in the absence of spheres, and a the radius of the spheres. The validity of this perturbative result is discussed.     

Computer simulation of the steady state currents at enzyme doped carbon paste electrodes

The plate-gap model of porous enzyme doped electrode has been proposed and analyzed. It was suggested that reaction diffusion conditions in pores of bulk electrode resemble particular conditions in thin gap between parallel conducting plates. The model is based on the diffusion equations containing a nonlinear term related to the Michaelis–Menten kinetic of the enzymatic reaction inside gap. Steady state current was calculated for the wide range of given parameters and substrate concentrations. All dependences of current on substrate concentration were approximated by hyperbolas in order to obtain “apparent” parameters (maximal currents and apparent Michaelis constants) of modelled biosensors. Simple approximate relationships between given and apparent parameters were derived. The applicability of theoretical plate-gap model was tested for the case of carbon paste electrodes which were doped with PQQ – dependent glucose dehydrogenase. It was found, that soluble glucose dehydrogenase based biosensors exhibit characteristic features of the theoretical plate-gap biosensors.     

Approximate Analytical Expressions for the Steady‐State Concentration of Substrate and Cosubstrate over Amperometric Biosensors for Different Enzyme Kinetics

Mathematical models of amperometric biosensors at three basic types of enzyme kinetics in nonstationary diffusion conditions are discussed. The models are based on nonstationary diffusion equations containing a linear term related to the first‐order and nonlinear term related to the Michaelis–Menten and ping–pong of the enzymatic reaction mechanism. In this paper, we obtain approximate closed‐form analytical solutions for the nonlinear equations under steady‐state condition by using the homotopy analysis method. Analytical expressions for concentrations of substrate and cosubstrate and corresponding current response have been derived for all possible values of parameters. Furthermore, in this work, the numerical simulation of the problem is also reported using Scilab/MATLAB program. An agreement between analytical and numerical results is noted.     

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.

     

A mathematical model for stress-driven diffusion in polymers

A model for case II diffusion into polymers is presented. The addition of stress terms to the Fickian flux is used to produce the characteristics progressive front. The stress in turn obeys a concentration-dependent evolution equation. The model equations are analyzed in the limit of small diffusivity for the problem of penetration into a semiinfinite medium. Provided that the coefficient functions obey two monotonicity conditions, the solvent concentration profile is shown to have a steep front that progresses into the medium. The formulas governing the progression of the front are developed. After the front decays away, the long time behavior of the solution is shown to be a similarity solution as in Fickian diffusion. Two techniques for approximating the solvent concentration and the front position are presented. The first approximation method is a series expansion; formulas are given for the initial speed and deceleration of the front. The second approximation method uses a portion of the long time similarity solution to represent the short time solution behind the front.     

Reactant stationary approximation in enzyme kinetics

In the application of the quasi-steady-state approximation, it is generally assumed that there is an initial transient during which the substrate concentration remains approximately constant while the complex concentration builds up. In this paper, we address the assumption that the substrate concentration does not change significantly during this initial transient and name it the reactant stationary approximation. For the single enzyme, single substrate reaction, the reactant stationary approximation is generally considered a sufficient condition to apply the quasi-steady-state approximation. Studying the dynamic behavior of this reaction with endogenous substrate, we show that the quasi-steady-state approximation and reactant stationary approximation are two separate approximations. We discuss the consequence of this result for the determination of reaction parameters in enzyme catalyzed reactions.     

On the simple Michaelis-Menten mechanism for chemical reactions

Several mathematical properties associated with the simple Michaelis-Menten mechanism for enzymatic reactions are proven. In particular it is shown that the usual interpretation of the slope of the experimental Michaelis-Menten rate law in terms of the reaction constants of the mechanism can be obtained, in the approximation in which the total concentration of the enzyme is small compared with the Michaelis-Menten constant, independently of the ratio between the total initial concentrations of the enzyme and substrate. Furthermore, the ratio of the total concentration of the enzyme to the Michaelis-Menten constant allows for the elimination of a fast variable in a singular perturbation method, yielding the Michaelis-Menten rate law as a first order approximation.     

Mathematical treatment of concentration profiles and anodic current for amperometric enzyme electrodes

Formulae are presented for the exact solution of partial differential equations describing the transient behaviour of concentration profiles and the anodic current in amperometric enzyme electrodes. The mathematical treatment is based on a reaction/diffusion model in which the reaction rate depends linearly on the substrate concentrations. Numerical results are presented to demonstrate the feasibility of the given formulae.     

Immobilized capillary enzyme reactor based on layer-by-layer assembling acetylcholinesterase for inhibitor screening by CE

A method for creating an immobilized capillary acetylcholinesterase (AChE) reactor based on a layer-by-layer (LBL) assembly for inhibitor screening is described. The unique capillary AChE reactor was easily prepared by the instrument in three steps: first, a 0.5 cm long plug of a solution of the cationic polyelectrolyte polydiallyldimethylammonium (PDDA) was injected into the capillary to produce a positively charged coating on the surface of the capillary; subsequently, the enzyme solution with the same plug length was injected into the capillary and incubated for 10 min to immobilize the enzyme on the capillary wall via electrostatic interaction; third, PDDA solution with the same plug length was injected again into the capillary to cover the immobilized enzyme by forming PDDA-AChE-PDDA sandwich-like structure. The enzyme reactor can be easily renewed after removing the immobilized enzyme by flushing the column with 1 M NaCl solution. Activity of the immobilized enzyme can be assayed simply by carrying out an electrophoretic separation, i.e., the substrate solution was injected and incubated for a short time, followed by applying a voltage to separate the product from the unreacted substrate. The measured peak area of the product then represented the enzyme activity. For enzyme inhibitor screening, the mixture solution of the substrate and the inhibitor was injected and assayed the reduction of the enzyme activity. The immobilized enzyme could withstand 100 consecutive assays by only losing 10% activity. The reproducibility in terms of time-to-time, day-to-day, and batch-to-batch was measured with RSD% less than 4.7%. Furthermore, the screening system was validated by a known inhibitor. Finally, screening a small compound library containing four known AChE inhibitors and 42 natural extracts was demonstrated, and species with inhibition activity can be straightforwardly identified with the system.     

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.     

Semiclassical initial value series solution of the spin boson problem

A numerical solution for the quantum dynamics of the spin boson problem is obtained using the semiclassical initial value series representation approach to the quantum dynamics. The zeroth order term of the series is computed using the new forward-backward representation for correlation functions presented in the preceding adjacent paper. This leads to a rapid convergence of the Monte Carlo sampling, as compared to previous attempts. The zeroth order results are already quite accurate. The first order term of the series is small, demonstrating the rapid convergence of the semiclassical initial value representation series. This is the first time that the first order term in the semiclassical initial value representation series has been converged for systems with the order of 50 degrees of freedom.     

High-throughput screening of enzyme inhibition using an inhibitor gradient generated in a microchannel

A new rapid microfluidic method for measuring enzyme inhibition is presented. The assay relies upon the creation of a uniform concentration of substrate and a microfluidically generated concentration gradient of inhibitor using a single microchannel and a single initial inhibitor concentration. The IC(50) values of two enzyme inhibitors were determined using the new technique and validated using a conventional microtiter plate assay. Using both experimental and computational simulation techniques, the assay was shown to be sensitive to inhibitor potency and the distribution of inhibitor in the system. The method has the potential to be more accurate than conventional methods because of the comparatively large amount of data that may be collected. Recommendations for use of the assay are provided, including its use for high-throughput screening in drug discovery and general use in measurement of enzyme inhibition.     

Analytical expression of non steady-state concentration for the CE mechanism at a planar electrode

The analytical solutions of the non-steady-state concentrations of species at a planar microelectrode are discussed. The analytical expression of the kinetics of CE mechanism under first or pseudo-first order conditions with equal diffusion coefficients at planar electrode under non-steady-state conditions are obtained by using Homotopy perturbation method. These simple new approximate expressions are valid for all values of time and possible values of rate constants. Analytical equations are given to describe the current when the homogeneous equilibrium position lies heavily in favour of the electroinactive species. Working surfaces are presented for the variation of limiting current with a homogeneous kinetic parameter and equilibrium constant. In this work we employ the Homotopy perturbation method to solve the boundary value problem. Furthermore, in this work the numerical simulation of the problem is also reported using Scilab program. The analytical results are found to be in excellent agreement with the numerical results.