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 binomial coefficients in representing central difference solution to a class of PDE arising in chemistry

The use of differential equations for modeling chemical systems and solving by numerical approaches (e.g. finite difference methods) are prevalent in chemistry-related problems. As an extension to the direct use of Pascal’s Triangle to obtain the forward and backward difference equations to partial differentials by Lim [Mathematical Medley 31 (2004) 2], this paper proposes the use of binomial coefficient to generate central difference equations to odd-ordered partial differentials in a single-step operation. All finite difference equations to partial differentials shown herein display finite series of palindromic coefficients with alternating signs     

非平衡性引起的对质量作用定律的偏离

在化学反应动力学中,质量作用定律占有重要的地位,通常认为,该定律仅适用于理想的反应体系,并假定,一旦反应在平衡条件下遵循质量作用定律,那么在非平衡条件下也应遵循该定律,但根据最近的研究结果,平衡条件下的理想反应体系,在非平衡条件下可能变成非理想体系,换言之,非平衡性可能引起非理想性,这表明,非平衡条件下的(基元)反应并不总是遵守质量作用定律,本文利用文献的一些结果,并结合1个具体反应模型,研究了山非平衡性引起对质量作用定律的偏离,结果表明,非平衡性本身对反应动力学规律有重要影响,对化学反应动力学的研究亦具有一定启发作用。     

Numerical Modeling of an Ar–H<Subscript>2</Subscript> Radio-Frequency Plasma Reactor under Thermal and Chemical Nonequilibrium Conditions

The species densities and the thermal and chemical nonequilibrium phenomena in an Ar–H₂ radio frequency inductively coupled plasma reactor used for hydrogenation of materials have been investigated through numerical simulation. The mathematical model consists of a two-temperature fluid dynamics model and a chemical kinetics model that takes into account the effect of local chemical nonequilibrium. Computations are carried out for the rf plasma running at 11.7 kW and 27 kPa for different Ar–H₂ mixtures and for pure argon. Predicted results for the electron and heavy-species temperatures, the species densities, as well as the degree of thermal and chemical nonequilibrium, are presented in detail. It is found that the electron and hydrogen atom densities in the reactor and in the near-wall region of the torch are strongly altered by nonequilibrium effects. The hydrogen atom density remains high in the reactor zone, and peaks in a region that has been found to be attractive for material processing. Deviations from thermal and chemical equilibrium are greatly reduced by the addition of hydrogen to an argon plasma.     

Numerical modeling of Joule heating-induced temperature gradient focusing in microfluidic channels

Temperature gradient focusing (TGF) is a recently developed technique for spatially focusing and separating ionic analytes in microchannels. The temperature gradient required for TGF can be generated either by an imposed temperature gradient or by Joule heating resulting from an applied electric field that also drives the flow. In this study, a comprehensive numerical model describing the Joule heating induced temperature development and TGF is developed. The model consists of a set of governing equations including the Poisson-Boltzmann equation, the Laplace equation, the Navier-Stokes equations, the energy equations and the mass transport equation. As the thermophysical and electrical properties including the liquid dielectric constant, viscosity, and electric conductivity are temperature-dependent, these governing equations are coupled, and therefore the coupled governing equations are solved numerically by using a CFD-based numerical method. The numerical simulations agree well with the experimental results, suggesting the valid mathematical model presented in this study.     

Solvent Reorganization Energy and Electronic Coupling for Intramolecular Electron Transfer in Biphenyl-Acceptor Anion Radicals

A novel algorithm was designed and implemented to realize the numerical calculation of the solvent reorganization energy for electron transfer reactions, on the basis of nonequilibrium solvation theory and the dielectric polarizable continuum model. Applying the procedure to the well-investigated intramolecular electron transfer in biphenyl-androstane-naphthyl and biphenyl-androstane-phenanthryl systems, the numerical results of solvent reorganization energy were determined to be around 60 kJ/mol, in good agreement with experimental datKoopman's theorem was adopted for the calculation of the electron transfer coupling element, associated with the linear reaction coordinate approximation. The values for this quantity obtained are acceptable when compared with experimental results.     

Some Aspects of the Solubility of Gases in Liquids

Summary.  Henry’s law constants may usually be used to calculate solubilities of gases at low pressures. If experimental measurements are unavailable values of Henry’s law constants may be estimated by various methods. Several of these methods depend upon quantitative structure-property relationships. A method developed by Hine and Mookerjee depends on the assumption that each bond of a particular type makes approximately the same contribution to the molar free energy change when different organic gases are dissolved in water. The solubility of gases and also the rate at which gases dissolve in cloud droplets is important for the understanding of processes which occur in the atmosphere. A simple model for the uptake of gases by water is based on an analogy with the behaviour of an electric circuit containing resistances in series and in parallel. This model is important for the interpretation of experimental measurements of rates of gas uptake. E-mail: pfogg@compuserve.com Dedicated to Prof. Heinz Gamsj?ger Received September 21, 2002; accepted September 30, 2002 Published online April 3, 2003     

Scoring Functions – the First 100 Years

The use of simple linear mathematical models to estimate chemical properties is not a new idea. Albert Einstein used very simple ‘gravity-like' forces to explain the capillarity of different liquids in 1900–1901. Today such models are used in more complicated situations, and a great many have been developed to analyse interactions between proteins and their ligands. This is not surprising, since proteins are too complicated to model accurately without lengthy numerical analysis, and simple models often do at least as good a job in predicting binding constants as much more computationally expensive methods. One hundred years after Einstein’s ‘miraculous year’ in which he transformed physics, it is instructive to recall some of his even earlier work. As approximations, ‘scoring functions’ are excellent, but it is dangerous to read too much into them. A few cautionary tales are presented for the beginner to the field of ligand affinity prediction by linear models.     

Maximum acceptable deviations from additivity in the photometric analysis of two-component mixtures by Firordt’s method

Photometric analysis of mixtures by Firordt’s method yields relative errors that do not exceed the specified limits only if the deviations from additivity at the analytical wavelengths do not exceed (in their numerical value) their own critical values (δA _crit). These deviations depend on the ratio of component concentrations. By virtue of the analytical geometry technique, we derived equations for the evaluation of δA _crit. These equations allow one to predict the possibility of the precise determination of two components in a mixture under investigation at one and the same set of wavelengths. The developed algorithm was verified on 40 model mixtures of organic compounds.     

Comparative analysis of linear and nonlinear models for ion transport problem in chronoamperometry

Ion transport problem related to controlled potential experiments in electrochemistry is studied. The problem is assumed to be superposition of diffusion and migration under the influence of an electric field. The comparative analysis are presented for three well-known models—pure diffusive (Cottrell’s), linear diffusion-migration, and nonlinear diffusion-migration (Cohn’s) models. The nonlinear model is derived by the identification problem for a nonlinear parabolic equation with nonlocal additional condition. This problem reduced to an initial-boundary value problem for nonlinear parabolic equation. The nonlinear finite difference approximation of this problem, with an appropriate iteration algorithm is derived. The comparative numerical analysis for all three models shows an influence of the nonlinear migration term, the valences of oxidized and reduced oxidized species, also diffusivity to the value of the total charge. The obtained results permits one to estimate bounds of linear and nonlinear ion transport models.     

Dynamic behavior of interfaces: Modeling with nonequilibrium thermodynamics

In multiphase systems the transfer of mass, heat, and momentum, both along and across phase interfaces, has an important impact on the overall dynamics of the system. Familiar examples are the effects of surface diffusion on foam drainage (Marangoni effect), or the effect of surface elasticities on the deformation of vesicles or red blood cells in an arterial flow. In this paper we will review recent work on modeling transfer processes associated with interfaces in the context of nonequilibrium thermodynamics (NET). The focus will be on NET frameworks employing the Gibbs dividing surface model, in which the interface is modeled as a two-dimensional plane. This plane has excess variables associated with it, such as a surface mass density, a surface momentum density, a surface energy density, and a surface entropy density. We will review a number of NET frameworks which can be used to derive balance equations and constitutive models for the time rate of change of these excess variables, as a result of in-plane (tangential) transfer processes, and exchange with the adjoining bulk phases. These balance equations must be solved together with mass, momentum, and energy balances for the bulk phases, and a set of boundary conditions coupling the set of bulk and interface equations. This entire set of equations constitutes a comprehensive continuum model for a multiphase system, and allows us to examine the role of the interfacial dynamics on the overall dynamics of the system. With respect to the constitutive equations we will focus primarily on equations for the surface extra stress tensor.     

Parametrical studies of electroosmotic transport characteristics in submicrometer channels

Spatially two-dimensional nonequilibrium mathematical model describing electroosmotic flow through a submicrometer channel with an electric charge fixed on the channel walls is presented. This system is governed by the hydrodynamic, electrostatic, and mass transport phenomena. The model is based on the coupled mass balances, Poisson, Navier-Stokes, and Nernst-Planck equations. Nonslip boundary conditions are employed. The effect of an imposed electric field on the system behavior is studied by means of a numerical analysis of the model equations. We have obtained the following findings. If the channel width is comparable to the thickness of the electric double layer, the system behaves as an ion-exchange membrane and the dependence of the electric current passing through the channel on the applied voltage is strongly nonlinear. In the case of negatively (positively) charged walls, a narrow region of very low conductivity (so-called ionic gate) is formed in the free electrolyte near the channel entry facing the anode (cathode) side. For a wide channel, the electric current is proportional to the applied voltage and the velocity of electrokinetic flow is linearly proportional to the electric field strength. Complex hydrodynamics (eddy formation and existence of ionic gates) is the most interesting characteristics of the studied system. Hence, current-voltage and velocity-voltage curves and the corresponding spatial distributions of the model variables at selected points are studied and described in detail.     

Mendeleev's Discovery of the Periodic Law: The Origin and the Reception

This paper addresses the conceptual as well as social origins of Mendeleev’s discovery of the periodic law and its reception by the chemical community by taking account of three factors: Mendeleev’s early research and its relevance to the discovery; his concepts of chemistry, especially that of the chemical elements; and the social context of the discovery and the reception in the chemical community. Mendeleev's clear distinction between abstract elements and simple bodies was a departure from Lavoisier’s famous definition of elements as an endpoint of analysis and originated from his research in indefinite compounds. As a comparison, the paper also analyzes Lothar Meyer’s approach to the classification of the elements. Mendeleev’s new concept of chemical elements and the existence of an audience in the form of the newly established Russian Chemical Society, and his ``German connection', helped Mendeleev in his discovery and its reception. This revised version was published online in July 2006 with corrections to the Cover Date.     

Ion Flow Effects on Negative Direct Current Corona in Air

Numerical simulations are presented for the ion flow effects on the negative direct current corona in air. One dimensional equivalent steady corona model containing the continuity equations for electrons and ions on the basis of Townsend theory coupled with Poisson’s equation is applied. The conductor radius and the electric field intensity on conductor surface keep constant under the standard atmosphere condition in this paper. The results suggest that the space charges within the plasma region have no influence on the electric field distribution throughout the interelectrode gap, which is only governed by the ion flow. The influences of the gap distance and the corona current on the corona discharge within the plasma region have been investigated. If the voltage-current characteristics of the negative corona discharge are measured, the ion flow region and the plasma region can be investigated individually. Based on Kaptzov’s hypothesis, the plasma region can be investigated solely.     

Hydration of nonelectrolytes in binary aqueous solutions

Literature data on the thermodynamic properties of binary aqueous solutions of nonelectrolytes that show negative deviations from Raoult’s law due largely to the contribution of the hydration of the solute are briefly surveyed. Attention is focused on simulating the thermodynamic properties of solutions using equations of the cluster model. It is shown that the model is based on the assumption that there exists a distribution of stoichiometric hydrates over hydration numbers. In terms of the theory of ideal associated solutions, the equations for activity coefficients, osmotic coefficients, vapor pressure, and excess thermodynamic functions (volume, Gibbs energy, enthalpy, entropy) are obtained in analytical form. Basic parameters in the equations are the hydration numbers of the nonelectrolyte (the mathematical expectation of the distribution of hydrates) and the dispersions of the distribution. It is concluded that the model equations adequately describe the thermodynamic properties of a wide range of nonelectrolytes partly or completely soluble in water.     

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.     

Model hysteresis dimer molecule I: Equilibrium properties

A Hamiltonian system describing hysteresis behavior in a dimeric chemical reaction is modeled in a MD simulation utilizing novel two-body potentials with switches that is particularly suitable for numerical thermodynamical investigations. It is surmized that such reaction mechanisms could exist in nature on the basis of recent experiments, which indicate that electromagnetic hysteresis behavior is exhibited at the molecular level, although experimental interpretations tend to construct models that avoid such mechanisms. Numerical results of various common equilibrium thermodynamical and kinetic properties are presented together with new algorithms that were implemented to compute these quantities, where no unusual thermodynamics was observed for the chemical reaction which might be interpreted as not being “time reversible invariant” and therefore susceptible to manifesting unusual thermodynamical phenomena, which might contradict any of the known laws of thermodynamics. A revision of the concept of “time reversibility” to accommodate the above results is suggested. The general design of the reaction mechanism also allows for the use of conventional potentials and by the utilization of switches, overcomes the bottleneck of computations which involves multi-body interactions. The initial theoretical, program and algorithm development was done at Norwegian University of Science and Technology-NTNU, Institute of Physical Chemistry, (Trondheim, Norway) during a sabbatical visit 2000–2001 financed by University of Malaya. The method of MD used here conforms to the periodic boundary conditions and thermostatting algorithms developed or refined by Ikeshoji and Hafskjold for standard, non-reacting particles. Their approach is non-synthetic using traditional Verlet integration of Newton’s equations of motion.     

Comparison of Accuracy and Convergence Rate between Equilibrium and Nonequilibrium Alchemical Transformations for Calculation of Relative Binding Free Energy

Estimation of protein-ligand binding affinity within chemical accuracy is one of the grand challenges in structure-based rational drug design. With the efforts over three decades, free energy methods based on equilibrium molecular dynamics (MD) simulations have become mature and are nowadays routinely applied in the community of computational chemistry. On the contrary, nonequilibrium MD simulation methods have attracted less attention, despite their underlying rigor in mathematics and potential advantage in efficiency. In this work, the equilibrium and nonequilibrium simulation methods are compared in terms of accuracy and convergence rate in the calculations of relative binding free energies. The proteins studied are T4-lysozyme mutant L99A and COX-2. For each protein, two ligands are studied. The results show that the nonequilibrium simulation method can be competitively as accurate as the equilibrium method, and the former is more efficient than the latter by considering the convergence rate with respect to the cost of wall clock time. In addition, Bennett acceptance ratio, which is a bidirectional post-processing method, converges faster than the unidirectional Jarzynski equality for the nonequilibrium simulations.