Use of ultrasound spectroscopy to examine the effect of cysteine on β-lactoglobulin interactions

We investigated the effects of cysteine on β-lactoglobulin interactions using ultrasound spectroscopy, rheological measurements, and differential scanning calorimetry. Changes in ultrasonic velocity and attenuation were monitored using ultrasound spectroscopy, and we discuss the effects of cysteine on gel formation together with the results obtained using other methods. A decrease in ultrasonic velocity occurred around 54 °C, suggesting that the compressibility of the system increases at approximately this temperature. An increase in ultrasonic attenuation was observed at approximately 54 °C, which is much lower than the commonly observed denaturation temperature of 75–80 °C. The temperature coincided with the onset of phase transition by differential scanning calorimetry and the initial rise in temperature of dynamic modulus for rheological measurements under heat treatment. We conclude that cysteine promotes the polymerization processes of denatured proteins during the initial stage of gelation. The ultrasonic spectroscopic analysis is a useful tool to monitor protein molecule interactions prior to gelation.     

Degradation of substrate and/or product: mathematical modeling of biosensor action

Mathematical model for evaluation of the multilayer heterogeneous biocatalytic system has been elaborated. The model consists of nonlinear system of partial differential equations with initial values and boundary conditions. An algorithm for computing the numerical solution of the mathematical model has been applied. Two cases: when product diffuses out of the biosensor and when the outer membrane is impermeable for product (product is trapped inside the biosensor) have been dealt with by adjusting boundary conditions in the mathematical model. Profiles of the impact of the substrate and product degradation rates on the biosensor response have been constructed in both cases. Value of the degradation impact has been analyzed as a function of the outer membrane thickness. The initial substrate concentration also affects influence of the degradation rates on the biosensor response. Analytical formulae, defining approximate values of relationships between the degradation rates and the outer membrane thickness or the initial substrate concentration, have been obtained. These formulae can be employed for monitoring of the biosensor response.     

Numerical solution of ordinary differential equations by Fluctuationlessness theorem

This paper presents a numerical method based on Fluctuationlessness Theorem for the solution of Ordinary Differential Equations over appropriately defined Hilbert Spaces. We focus on the linear differential equations in this work. The approximated solution is written in the form of an nth degree polynomial of the independent variable. The unknown coefficients are obtained by setting up a system of linear equations which satisfy the initial or boundary conditions and the differential equation at the grid points, which are constructed as the independent variable’s matrix representation restricted to an n dimensional subspace of the Hilbert Space. An error comparison of the numerical solution and the MacLaurin series with the analytical solution is performed. The results show that the numerical solution obtained here converges to the analytical solution without using too many mesh points.     

A Two‐Step Interval and Modified Rosenbrock Method for Kinetic Parameter Estimation in Copolymerization Systems: A Robust and Reliable Approach

Summary: The kinetics of solution free radical copolymerization of isobutyl methacrylate (i‐BMA) and lauryl methacrylate (LMA) in benzene, initiated with 2,2‐azoisobutyronitrile (AIBN) were studied at different monomer feed compositions at low conversion levels. In order to avoid the complications of copolymerization kinetics, the pseudo‐kinetic rate constant method was applied in constant and variable volume polymerization systems. A two‐step procedure based on interval analysis and the modified Rosenbrock method was used to estimate the kinetic parameters of copolymerization. In the first step, initiation, coupled propagation‐termination and transfer rate parameters were determined from steady state kinetic equations using interval analysis. Since the objective function is non‐linear, non‐convex and has multiple local optima, a robust computational technique, based on the Interval Newton/Generalized Bisection (IN/GB) algorithm, was developed to solve this set of non‐linear algebraic equations. This method was used with mathematical and computational guarantees of certainty to find the global optimum. In the second step, the system of mole balance, population balance and moment equations, which are highly stiff ordinary differential equations, were discritized and solved by the modified Rosenbrock method. The results of the first computational step were inserted as an active or equality constraint in the second step to calculate the individual elementary rate parameters of the reaction. Statistical analysis indicated that the copolymer composition is well described by the terminal unit model (TUM), but the implicit penultimate unit effect (IPUE) model of Fukuda and coworkers is more suitable for describing the rate data. In contrast to most previously studied systems, it was found that propagation and coupled rate parameters are greater than those predicted by the TUM.

Variation of number average molecular weight of the copolymer in polymerization system for various initial monomer feed compositions at different reaction times (solid lines are computed results).     

SULFONATION IN A FALLING FILM REACTOR: COMPARATION OF EXPERIMENTAL RESULTS WITH MATHEMATICAL MODEL PREDICTIONS

A mathematical model is developed to simulate a falling film reactor for sulfonation/sulfation. In the model, the reaction rate is considered to be controlled by the mass transfer in the gas phase or in the liquid phase. The gas phase mass and heat transfers are calculated by empiric equations; in the liquid phase, they are calculated by solving with numerical methods the partial differential equations which describe the system. In these equations, and eddy diffusion is considered, following the Levich's theories

The model results are compared with the experimental results obtained by the authors in a pilot plant, for the dodecylbenzene sulfonation.     

2D general nonlinear parabolic equation for unstable wave package envelope in structural macrokinetics

We reduce the mathematical model for a chemical reaction in a moving medium to the general nonlinear parabolic equation (GNPE) for the complex amplitude of envelope wave to analyse weakly nonlinear interactions in supercritical regions. We use the method of many scales, wave packages, modification of Mandelshtam method and take into account the group velocity of envelope wave that is typical for nonlinear dispersive medium. GNPE describes the long-term system behaviour after stability loss and its coefficients are explicitly expressed through parameters of the initial nonlinear partial differential equations. It is taking into account the location of wave packet centre out of harmonics with maximum increment.     

On finding the rate constants of linear and nonlinear systems

In lowest approximation, a certain chemical reaction is described by a system of first-order linear differential equations with unknown constant coefficients. One can therefore write down an expression for the state of the system at time t, and from this find the endpoint of the reaction in terms of the initial state and the rate constants. The relative values of some rate constants can then be estimated from experimental data. A better approximation in which the differential equations are nonlinear is also considered, and it turns out that because of symmetry in the reaction, the relationship between the final state and the ratios of the rate constants is unchanged. Although the differential equations now appear much less tractable, the problem of relating the rate constants to the endpoint of the reaction can be formulated and solved in terms of probabilities. The results illustrate an important property of reaction schemes in which some of the steps are reversible. More generally, this is a property of differential equations: provided that they continue to satisfy certain linear constraints, the parameters of a linear system of ordinary differential equations can vary without affecting the asymptotic solution.     

Chemical instabilities during oxidation of 1,4-naphthalenediol in a homogeneous medium: II. Thermodynamic analysis and mathematical modeling

A possible mechanism of 1,4-naphthalenediol oxidation in the oscillatory regime has been considered, and a mathematical model describing the kinetics has been developed. Based on a thermodynamic Lyapunov function, it has been shown that the source of chemical instabilities is in the existence of autocatalytic steps and dynamic feedbacks. Qualitative analysis and numerical solution of the set of differential equations that model the reaction kinetics have been performed. The character of the stationary state and the possibility of bifurcations have been determined. The mathematical model satisfactorily describes the processes occurring in the system.     

Effect of Internal Diffusional Restrictions on the Hydrolysis of Penicillin G: Reactor Performance and Specific Productivity of 6-APA with Immobilized Penicillin Acylase

A mathematical model that describes the heterogeneous reaction–diffusion process involved in penicillin G hydrolysis in a batch reactor with immobilized penicillin G acylase is presented. The reaction system includes the bulk liquid phase containing the dissolved substrate (and products) and the solid biocatalyst phase represented by glyoxyl-agarose spherical porous particles carrying the enzyme. The equations consider reaction and diffusion components that are presented in dimensionless form. This is a complex reaction system in which both products of reaction and the substrate itself are inhibitors. The simulation of a batch reactor performance with immobilized penicillin G acylase is presented and discussed for the internal diffusional restrictions impact on effectiveness and productivity. Increasing internal diffusional restrictions, through increasing catalyst particle size and enzyme loading, causes impaired catalyst efficiency expressed in a reduction of effectiveness factor and specific productivity. High penicillin G initial concentrations decrease the impact of internal diffusional restrictions by increasing the mass transfer towards porous catalyst until product inhibition becomes significant over approximately 50 mM of initial penicillin G, where a drop in conversion rate and a maximum in specific productivity are then obtained. Results highlight the relevance of considering internal diffusional restrictions, reactor performance, and productivity analysis for proper catalyst and reactor design.     

Kinetics of donor–acceptor complex polymerization. III. Assessment of some approximations in the kinetics

The mathematical solution of the equations derived from a kinetic scheme previously developed for donor–acceptor complex polymerization was based on steady-state conditions and the applicability of initial concentration conditions over a range of conversion. These assumptions are scrutinized and tested by computer simulation and by the exact differential equations utilizing Runge-Kutta method. The analysis shows that for case I conditions of low concentration of complexing-agent, the degree of conversion is not critical and that the previous approximate solutions are valid. The steady-state and non-steady-state conditions are compared, and the range of validity of the assumptions is established. The approximate solutions are found inapplicable in the instances of non-steady-state conditions coupled with a low concentration of either monomer.     

Mathematical modeling of kinetic behavior of the hydrogen electrode in carbonate melts under non-steady-state conditions: An analysis of various reaction schemes

An extended analysis of the kinetic behavior exhibited by the hydrogen electrode in a molten carbonate electrolyte is performed by a method of mathematical modeling of relaxation processes as applied to chronoamperometric and coulostatic experiments. A set of differential equations with corresponding initial and boundary conditions, which fixes the balance of charge and substance at the electrode/electrolyte interface, is presented for several particular versions of reaction schemes and the method used for bringing the system out of equilibrium. Numerical calculations are performed and their results are compared with the experimental time dependences of the current and overvoltage obtained in chronoamperometric and coulostatic conditions, respectively. As a result, possible intervals of variations in the kinetic, adsorption, and transport parameters of the system are evaluated. The deviations of their values when using different investigation procedures and different electrode materials (gold, nickel) are discussed. To differentiate assorted reaction schemes as applied to a real electrode process of the anodic oxidation of hydrogen in a carbonate electrolyte it is necessary to expand the base of experimental material to be analyzed and the circle of procedures to be used for investigation.     

Solid film at the anode influences the process of polyvalent metal electrorefining: theoretical bifurcation analysis of the problem

A theoretical study of polyvalent metal anodic electrorefining has been carried out, and the decisive influence of surface film on the long-term process has been shown. A mathematical model of the long-term continuous electrorefining process in the form of non-linear differential equations is proposed. The developed model was investigated by the methods of the bifurcation theory. The model predicts the following types of electrolytic system: 1. The low-valence compound of the anodic film does not form. 2. A steady-state process with a stationary thickness of the anodic deposit. 3. An unlimited accumulation of low-valence compounds at the anode. 4. An unstable steady state: a bifurcation with a transition to case 3 or to the regime of undamped oscillations. The model provides satisfactory explanations of the experimental results related to long-term continuous electrorefining processes in molten salts. Received: 19 February 1997 / Accepted: 18 August 1997     

Analysis of two-dimensional models for liquid chromatographic reactors of cylindrical geometry

This work presents the analytical solutions of two-dimensional isothermal reactive general rate models for liquid chromatographic reactors of cylindrical geometry. Both irreversible and reversible reactions are considered. The model equations form a linear system of convection-diffusion-reaction partial differential equations coupled with algebraic equations for isotherms. Analytical solutions are derived by integrated implementation of finite Hankel transform, Laplace transform, eigen-decomposition technique, and conventional ordinary differential equations solution technique. To verify the analytical results, a high-resolution finite volume scheme is also applied to numerically approximate the model equations. The current results can be very useful to optimize and upgrade the liquid chromatographic reactors.     

Emulsion copolymerization of styrene and methyl methacrylate. II. Molecular weights

A mathematical model for the time evolution of both number-average molecular weight and weight-average molecular weight is described. The model results in a set of ordinary differential equations and its application is not limited by the maximum number of radicals per particle. The model was used to analyze the effect of the monomer molar ratio in the initial charge on the weight-average molecular weight during the emulsion copolymerization of styrene and methyl methacrylate in a batch reactor.     

Biomedical simulation of heat transfer in a human heart

Theory and practical experiences from numerical simulations of heat transfer in the field of medicine are presented in this paper. The cooling of a human heart during surgery was taken as an illustrative example. All phases of the simulation process are described starting with the construction of an irregularly shaped 3-dimensional model. The mathematical model is based on diffusion and Navier-Stokes equations. The system of partial differential equations is solved by finite difference approximation using an explicit time-stepping scheme to obtain the time evolution of the solution for the complete simulated interval, which is typically 1 h. A typical domain is composed of several million voxels; therefore, the program was parallelized to speed up the simulation. A speed-up of 8.2 was obtained on 16 processors in a Linux cluster.     

Computing minimal entropy production trajectories: an approach to model reduction in chemical kinetics

Advanced experimental techniques in chemistry and physics provide increasing access to detailed deterministic mass action models for chemical reaction kinetics. Especially in complex technical or biochemical systems the huge amount of species and reaction pathways involved in a detailed modeling approach call for efficient methods of model reduction. These should be automatic and based on a firm mathematical analysis of the ordinary differential equations underlying the chemical kinetics in deterministic models. A main purpose of model reduction is to enable accurate numerical simulations of even high dimensional and spatially extended reaction systems. The latter include physical transport mechanisms and are modeled by partial differential equations. Their numerical solution for hundreds or thousands of species within a reasonable time will exceed computer capacities available now and in a foreseeable future. The central idea of model reduction is to replace the high dimensional dynamics by a low dimensional approximation with an appropriate degree of accuracy. Here I present a global approach to model reduction based on the concept of minimal entropy production and its numerical implementation. For given values of a single species concentration in a chemical system all other species concentrations are computed under the assumption that the system is as close as possible to its attractor, the thermodynamic equilibrium, in the sense that all modes of thermodynamic forces are maximally relaxed except the one, which drives the remaining system dynamics. This relaxation is expressed in terms of minimal entropy production for single reaction steps along phase space trajectories.     

Numerical approximation of a nonisothermal two-dimensional general rate model of reactive liquid chromatography

A nonlinear and nonisothermal two-dimensional general rate model is formulated and approximated numerically to allow quantitatively analyzing the effects of temperature variations on the separations and reactions in liquid chromatographic reactors of cylindrical geometry. The model equations form a nonlinear system of convection-diffusion-reaction partial differential equations coupled with algebraic equations for isotherms and reactions. A semidiscrete high-resolution finite volume method is modified to approximate the system of partial differential equations. The coupling between the thermal waves and concentration fronts is demonstrated through numerical simulations, and important parameters are pointed out that influence the reactor performance. To evaluate the precision of the model predictions, consistency checks are successfully carried out proving the accuracy of the predictions. The results allow to quantify the influence of thermal effects on the performance of the fixed beds for different typical values of enthalpies of adsorption and reaction and axial and radial Peclet numbers for mass and heat transfer. Furthermore, they provide useful insight into the sensitivity of nonisothermal chromatographic reactor operation.