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Analytical Expression for the Concentration of Substrate and Product in Immobilized Enzyme System in Biofuel/Biosensor  [PDF]
R. Malini Devi, O. M. Kirthiga, L. Rajendran
Applied Mathematics (AM) , 2015, DOI: 10.4236/am.2015.67105
Abstract: In this paper, an approximate analytical method to solve the non-linear differential equations in an immobilized enzyme film is presented. Analytical expressions for concentrations of substrate and product have been derived for all values of dimensionless parameter. Dimensionless numbers that can be used to study the effects of enzyme loading, enzymatic gel thickness, and oxidation/ reduction kinetics at the electrode in biosensor/biofuel cell performance were identified. Using the dimensionless numbers identified in this paper, and the plots representing the effects of these dimensionless numbers on concentrations and current in biosensor/biofuel cell are discussed. Analytical results are compared with simulation results and satisfactory agreement is noted.
Analytical solution of a class of coupled second order differential-difference equations  [cached]
L. Jódar,J. A. Martin Alustiza
International Journal of Mathematics and Mathematical Sciences , 1993, DOI: 10.1155/s0161171293000468
Abstract: In this paper coupled systems of second order differential-difference equations are considered. By means of the concept of co-solution of certain algebraic equations associated to the problem, an analytical solution of initial value problems for coupled systems of second order differential-difference equations is constructed.
Intrinsic Noise Analyzer: A Software Package for the Exploration of Stochastic Biochemical Kinetics Using the System Size Expansion  [PDF]
Philipp Thomas, Hannes Matuschek, Ramon Grima
PLOS ONE , 2012, DOI: 10.1371/journal.pone.0038518
Abstract: The accepted stochastic descriptions of biochemical dynamics under well-mixed conditions are given by the Chemical Master Equation and the Stochastic Simulation Algorithm, which are equivalent. The latter is a Monte-Carlo method, which, despite enjoying broad availability in a large number of existing software packages, is computationally expensive due to the huge amounts of ensemble averaging required for obtaining accurate statistical information. The former is a set of coupled differential-difference equations for the probability of the system being in any one of the possible mesoscopic states; these equations are typically computationally intractable because of the inherently large state space. Here we introduce the software package intrinsic Noise Analyzer (iNA), which allows for systematic analysis of stochastic biochemical kinetics by means of van Kampen’s system size expansion of the Chemical Master Equation. iNA is platform independent and supports the popular SBML format natively. The present implementation is the first to adopt a complementary approach that combines state-of-the-art analysis tools using the computer algebra system Ginac with traditional methods of stochastic simulation. iNA integrates two approximation methods based on the system size expansion, the Linear Noise Approximation and effective mesoscopic rate equations, which to-date have not been available to non-expert users, into an easy-to-use graphical user interface. In particular, the present methods allow for quick approximate analysis of time-dependent mean concentrations, variances, covariances and correlations coefficients, which typically outperforms stochastic simulations. These analytical tools are complemented by automated multi-core stochastic simulations with direct statistical evaluation and visualization. We showcase iNA’s performance by using it to explore the stochastic properties of cooperative and non-cooperative enzyme kinetics and a gene network associated with circadian rhythms. The software iNA is freely available as executable binaries for Linux, MacOSX and Microsoft Windows, as well as the full source code under an open source license.
Mathematical Modeling and Analysis of the Kinetics of Thermal Inactivation of Enzyme  [PDF]
S. P. Ananthi,P. Manimozhi,T. Praveen,A. Eswari,L. Rajendran
International Journal of Engineering Mathematics , 2013, DOI: 10.1155/2013/132827
Abstract: A theoretical model of Illeova et al. (2003) thermal inactivation of urease is discussed. Analytical expressions pertaining to the molar concentrations of the native and denatured enzyme are obtained in terms of second-order reaction rate constant. Simple and closed form of theoretical expression pertains to the temperature are also derived. In this paper, homotopy analysis method (HAM) is used to obtain approximate solutions for a nonlinear ordinary differential equation. The obtained approximate result in comparison with the numerical ones is found to be in satisfactory agreement. 1. Introduction Urease is a good catalyst for the hydrolysis of urea. Several excellent techniques are available to assess urease activity [1, 2]. In 1926, urease was isolated by Summner from the seeds of jack bean as a pure, crystalline enzyme [3]. These crystals, the first obtained for a known enzyme, played a decisive role in proving the protein nature of enzymes. Approximately 50 years later, jack bean urease was identified as the first nickel metalloenzyme [4]. A method for the determination of mercury (II) ions at trace levels is described. The method is based on the profound inhibitory effect of mercury on the enzyme urease [5]. For unknown reasons some seeds are particularly rich sources of urease, and this enzyme has been extensively studied in seeds of various Leguminosae, Cucurbitaceae, Asteraceae and Pinaceae [6, 7]. Jack bean urease, which is the most widely used plant urease, is a nickel containing oligomeric enzyme exhibiting a high degree of specificity to urea [8]. Numerous papers have been published on the applications of urease in biotechnology, including the determination of urea for analytical and biomedical purposes and analysis of heavy metal content in natural drinking water and surface water [9]. Hirai et al. [10] study the structural change of jack bean urease induced by addition of synchrotron radiation. Lencki et al. [11] discuss the effect of subunit dissociation, denaturation, aggregation, coagulation, and decomposition on enzyme inactivation kinetics. Omar and Beauregard [12] investigate the unfolding of jack bean urease by fluorescence emission spectroscopy. To our knowledge, no rigorous analytical expressions of molar concentrations of the native enzyme, denatured enzyme, and temperature for thermal inactivation of urease for the parameters , , , , , , and have been reported. The purpose of this communication is to derive simple approximate analytical expression for the nonsteady-state concentrations for thermal inactivation of urease using
Enzyme Kinetics: A critique of the quasi-steady-state approximation  [PDF]
Kamal Bhattacharyya,Sharmistha Dhatt
Physics , 2013,
Abstract: The standard two-step model of homogeneous-catalyzed reactions had been theoretically analyzed at various levels of approximations from time to time. The primary aim was to check the validity of the quasi-steady-state approximation, and hence emergence of the Michaelis-Menten kinetics, with various substrate-enzyme ratios. But, conclusions vary. We solve here the desired set of coupled nonlinear differential equations by invoking a new set of dimensionless variables. Approximate solutions are obtained via the power-series method aided by Pade approximants. The scheme works very successfully in furnishing the initial dynamics at least up to the region where existence of any steady state can be checked. A few conditions for its validity are put forward and tested against the findings. Temporal profiles of the substrate and the product are analyzed in addition to that of the complex to gain further insights into legitimacy of the above approximation. Some recent observations like the reactant stationary approximation and the notions of different timescales are revisited. Signatures of the quasi-steady-state approximation are also nicely detected by following the various reduced concentration profiles in triangular plots. Conditions for the emergence of Michaelis-Menten kinetics are scrutinized and it is stressed how one can get the reaction constants even in the absence of any steady state.
Analytical Solutions of System of Non-Linear Differential Equations in the Single-Enzyme, Single-Substrate Reaction with Non-Mechanism-Based Enzyme Inactivation  [PDF]
Govindhan Varadharajan, Lakshmanan Rajendran
Applied Mathematics (AM) , 2011, DOI: 10.4236/am.2011.29158
Abstract: A closed form of an analytical expression of concentration in the single-enzyme, single-substrate system for the full range of enzyme activities has been derived. The time dependent analytical solution for substrate, enzyme-substrate complex and product concentrations are presented by solving system of non-linear differential equation. We employ He’s Homotopy perturbation method to solve the coupled non-linear differential equations containing a non-linear term related to basic enzymatic reaction. The time dependent simple analytical expressions for substrate, enzyme-substrate and free enzyme concentrations have been derived in terms of dimensionless reaction diffusion parameters ε, λ1, λ2 and λ3 using perturbation method. The numerical solution of the problem is also reported using SCILAB software program. The analytical results are compared with our numerical results. An excellent agreement with simulation data is noted. The obtained results are valid for the whole solution domain.
The Kinetics of Enzyme Mixtures
Simon Brown, Noorzaid Muhamad, Kevin C Pedley, David C Simcock
Molecular Biology Research Communications , 2014,
Abstract: Even purified enzyme preparations are often heterogeneous. For example, preparations of aspartate aminotransferase or cytochrome oxidase can consist of several different forms of the enzyme. For this reason we consider how different the kinetics of the reactions catalysed by a mixture of forms of an enzyme must be to provide some indication of the characteristics of the species present. Based on the standard Michaelis-Menten model, we show that if the Michaelis constants (Km) of two isoforms differ by a factor of at least 20 the steady-state kinetics can be used to characterise the mixture. However, even if heterogeneity is reflected in the kinetic data, the proportions of the different forms of the enzyme cannot be estimated from the kinetic data alone. Consequently, the heterogeneity of enzyme preparations is rarely reflected in measurements of their steady-state kinetics unless the species present have significantly different kinetic properties. This has two implications: (1) it is difficult, but not impossible, to detect molecular heterogeneity using kinetic data and (2) even when it is possible, a considerable quantity of high quality data is required.
Analytical solution for a class of coupled linear second-order differential equations limited by transformations

Yang Peng-Fei,

物理学报 , 2006,
Abstract: By using the function and equation transformations,a coupled linear second-order differential equations are reduced to a nonlinear first-order Elliptic equation. And the analytical solutions to coupled transformations that include first-order and second-order transformations are given.The special approximate solution to a superconductivity question derived from a reference is reformed by using the new result given in this paper, and the existence of electric field in the surface of superconductor is validated.
Investigating the robustness of the classical enzyme kinetic equations in small intracellular compartments
Ramon Grima
BMC Systems Biology , 2009, DOI: 10.1186/1752-0509-3-101
Abstract: Starting from a master equation description of enzyme reaction kinetics and assuming metabolic steady-state conditions, we derive novel mesoscopic rate equations which take into account (i) the intrinsic molecular noise due to the low copy number of molecules in intracellular compartments (ii) the physical nature of the substrate transport process, i.e. diffusion or vesicle-mediated transport. These equations replace the conventional macroscopic and deterministic equations in the context of intracellular kinetics. The latter are recovered in the limit of infinite compartment volumes. We find that deviations from the predictions of classical kinetics are pronounced (hundreds of percent in the estimate for the reaction velocity) for enzyme reactions occurring in compartments which are smaller than approximately 200 nm, for the case of substrate transport to the compartment being mediated principally by vesicle or granule transport and in the presence of competitive enzyme inhibitors.The derived mesoscopic rate equations describe subcellular enzyme reaction kinetics, taking into account, for the first time, the simultaneous influence of both intrinsic noise and the mode of transport. They clearly show the range of applicability of the conventional deterministic equation models, namely intracellular conditions compatible with diffusive transport and simple enzyme mechanisms in several hundred nanometre-sized compartments. An active transport mechanism coupled with large intrinsic noise in enzyme concentrations is shown to lead to huge deviations from the predictions of deterministic models. This has implications for the common approach of modeling large intracellular reaction networks using ordinary differential equations and also for the calculation of the effective dosage of competitive inhibitor drugs.The inside of a cell is a highly complex environment. In the past two decades, detailed measurements of the chemical and biophysical properties of the cytoplasm have es
Analytical solution of the concentration and current in the electoenzymatic processes involved in a PPO-rotating-disk-bioelectrode  [PDF]
Govindhan. Varadharajan, Lakshmanan. Rajendran
Natural Science (NS) , 2011, DOI: 10.4236/ns.2011.31001
Abstract: A mathematical model for electroenzymatic process of a rotating-disk-bioelectrode in which polyphenol oxidase occurs for all values of concentration of catechol substrate is presented. The model is based on system of reaction-diffusion equations containing a non-linear term related to Michaelis-Menten kinetics of the enzymatic reaction. Approximate analytical method (He’s Homotopy perturbation method) is used to solve the non-linear differential equations that describe the diffusion coupled with a Michaelis-Menten kinetics law. Closed analytical expressions for substrate concentration, product concentration and corresponding current response have been derived for all values of parameter using perturbation method. These results are compared with simulation results and are found to be in good agreement. The obtained results are valid for the whole solution domain.
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