Quasi-stationary Approximations Research Articles

Stationary and Time-Dependent Molecular Distributions in Slow-Fast Feedback Circuits

Biochemical reaction systems, in particular those that govern the expression of genes in single cells, involve discrete numbers of molecules and are inherently stochastic. This study concerns reaction systems with two molecular species and an arbitrary number of reactions, the rates of which conform to a scaling of the classical type in the first species (but not the second). For limiting values of the scaling parameter, the first species is highly abundant, evolves slowly, and buffers the relatively fast fluctuation of the second, scarce, species. The scale separation facilitates the construction of asymptotic approximations to the molecular distributions: the large-time behavior is described by a WKB approximation, which is further resolved into a tractable mixture of metastable modes; the earlier-time dynamics are described by quasi-stationary and linear noise approximations. The paper presents the theory and the algebraic recipes that are required to implement it. The framework is applied on two reaction systems with positive feedback: a delayed feedback circuit and a gene autoregulation model with cooperative binding of the protein to the promoter. For the former, the approximation scheme results into a mixture of Gaussian/Poisson modes for the inactive/active protein distribution. For the latter, the analysis elucidates the effects of bursty gene expression, promoter responsivity, and protein sequestration on bimodality of protein distributions.

The influence of temperature on the intrinsic transport properties of water in inorganic and polymeric coatings

Barrier layers are often used to protect sensitive organic devices from the detrimental effects of oxygen and water vapor. We investigated the effect of temperature on the time-dependent behavior of water vapor permeation in multilayer barrier films, focusing on the water vapor solubility, diffusion and permeability coefficients of individual layers and layer sequences. Activation energy measurements helped to explain the permeation mechanisms and showed the possible interactions between water and silicon oxide (SiOx). The activation energy for permeation (EP) through a two-layer film of polyethylene terephthalate (PET) coated with SiOx was 5.7 kJmol−1, about twice the EP value of the uncoated PET substrate. An intermediate ORMOCER® layer, providing a smooth surface for a second SiOx layer in an alternating structure, had only a negligible effect on the EP. Temperature dependent quasi-stationary approximations were derived for permeation through films with at least one inorganic/polymer/inorganic triplet, and were validated according to experimentally determined values. These equations showed the temperature and pressure dependencies of steady-state permeation and lag time in multilayered structures. Accordingly, the temperature dependency of steady-state permeation is determined by the EP of SiOx, whereas the lag time was mainly influenced by the heat of sorption of the intermediate polymeric layer. Furthermore, the increase in water vapor transmission rate was dominated by the partial pressure change with increasing temperature. Using these equations and experimentally determined parameters, we can predict the influence of temperature and humidity on the barrier performance of multilayered barrier structures.

On the justification of the quasistationary approximation of several parabolic moving boundary problems – Part II

We rigorously justify the quasistationary approximations of two moving boundary problems. We work out a systematic procedure to derive a priori estimates that allow to pass to the singular limit. The problems under our consideration are a one-phase osmosis model and the one-phase Stefan problem with Gibbs–Thomson correction and kinetic undercooling.

Reflection and transmission of low-frequency spherical waves at gas-liquid interfaces

High-frequency asymptotic solutions of the canonical problem of a spherical wave reflection at and transmission through a plane interface are well known. These asymptotics are usually derived using the stationary phase method and its extensions assuming that the distance between the point sound source and a receiver is large compared to the wavelength. The opposite case, where the distance is either of the order of or smaller than the wavelength, is of interest in a number of problems such as radiation of infrasound into the atmosphere by underwater explosions and earthquakes. Available quasi-stationary approximations fail to describe correctly acoustic power flux through the interface. Here, a low-frequency asymptotics of the acoustic field is obtained for gas-liquid interfaces. It is found that the acoustic field transmitted into gas can be approximately represented as a field due to a virtual point source in an unbounded medium. Positions of the virtual sources of the low- and high-frequency transmitted waves are compared. It is found that the bulk of acoustic energy transmission through the interface occurs at epicentral distances of the order of the source depth, regardless of the wavelength.

Stochastic and deterministic simulations of a delayed genetic oscillation model: Investigating the validity of reductions

Quasi-stationary approximations are commonly used in order to simplify and reduce the number of equations of genetic circuit models. Protein/protein and protein/DNA binding reactions are considered to occur on much shorter time scale than protein production and degradation processes and often tacitly assumed at a quasi-equilibrium. Taking a biologically inspired, typical, small, abstract, negative feedback, genetic circuit model as study case, we investigate in this paper how different quasi-stationary approximations change the system behaviour both in deterministic and stochastic frameworks. We investigate the consistence between the deterministic and stochastic behaviours of our time-delayed negative feedback genetic circuit model with different implementations of quasi-stationary approximations. Quantitative and qualitative differences are observed among the various reduction schemes and with the underlying microscopic model, for biologically reasonable ranges and combinations of the microscopic model kinetic rates. The different reductions do not behave in the same way: correlations and amplitudes of the stochastic oscillations are not equally captured and the population behaviour is not always in consistence with the deterministic curves.

Different types of quasistationary formulations for time domain simulations

For many practical examples in electromagnetics, quasistationary approximations to the full set of the Maxwell equations are valid. Applying such approximations reduces the level of mutual coupling between the separate equations and leads to different sets of differential equations for the specific case. Based on the resulting subsets of the Maxwell equations, numerous formulations in terms of fields and potentials can be obtained. As the type of differential equation is changed due to the approximation in general, appropriate time‐integration schemes need to be selected for the numerical solution of the specific problem. An overview of existing quasistationary formulations is given while a simple test model is used to compare the different formulations.

Methods for refining the models of steady-state chemical kinetics on the basis of perturbation theory

The relation between Tikhonov’s theorem on the differential equations with a small parameter and the mathematical models of steady-state chemical kinetics is traced. A new algorithm for the construction of kinetic models with the use of the degrees of completeness of steps as the state variables is presented. Anomalous cases taking place in the formation of the subsystem of algebraic equations relative to the fast variables are considered, and procedures for refining the models of quasi-equilibrium and quasi-stationary approximations on the basis of perturbation theory are described.

Domains of validity of quasistatic and quasistationary field approximations

PurposeThe paper aims at proposing a uniform and demonstrative description of two well‐known and widely used approximations of slowly time‐varying electromagnetic fields, i.e. the electro‐quasistatic and the magneto‐quasistationary approximation to Maxwell's equations.Design/methodology/approachUnder both approximations, the orders of magnitude of the relative errors of the dominant fields are analyzed by using three characteristic time constants. These time constants are determined by considering the material properties, the characteristic length scale and the characteristic time scale.FindingsLimiting curves which show the domains of applicability of the two approximations are retrieved from the estimation of their relative errors. The relation between the domains of validity of the electro‐quasistatic and magneto‐quasistationary approximations was found and depicted in a combined diagram.Research limitations/implicationsThe study is restricted to slowly time‐varying electromagnetic fields. Heuristic and local estimates based on local material properties were used for the analysis. Rigorous estimations of the errors (e.g. also considering the field problem's topology) of the magneto‐quasistationary approximation are already known in the literature. A rigorous estimation of the error of the electro‐quasistatic approximation is, therefore, suggested for future research.Originality/valueThe combined diagram showing the domains of validity of both approximations considered here in a uniform way is novel. It gives rise to an intuitive and easily accessible understanding of their applicability.

Toward a Quantitative Modeling of the Synthesis of the Pectate Lyases, Essential Virulence Factors in Dickeya dadantii

A dynamic mathematical model has been developed and validated to describe the synthesis of pectate lyases (Pels), the major virulence factors in Dickeya dadantii. This work focuses on the simultaneous modeling of the metabolic degradation of pectin by Pel enzymes and the genetic regulation of pel genes by 2-keto-3-deoxygluconate (KDG), a catabolite product of pectin that inactivates KdgR, one of the main repressors of pel genes. This modeling scheme takes into account the fact that the system is composed of two time-varying compartments: the extracellular medium, where Pel enzymes cleave pectin into oligomers, and the bacterial cytoplasm where, after internalization, oligomers are converted to KDG. Using the quasi-stationary state approximations, the model consists of some nonlinear differential equations for which most of the parameters could be estimated from the literature or from independent experiments. The few remaining unknown parameters were obtained by fitting the model equations against a set of Pel activity data. Model predictions were verified by measuring the time courses of bacterial growth, Pel production, pel mRNA accumulation, and pectin consumption under various growth conditions. This work reveals that pectin is almost totally consumed before the burst of Pel production. This paradoxical behavior can be interpreted as an evolutionary strategy to control the diffusion process so that as soon as a small amount of pectin is detected by the bacteria in its surroundings, it anticipates more pectin to come. The model also predicts the possibility of bistable steady states in the presence of constant pectin compounds.

Exploring the connection between quasistationary and squared eigenfunction expansion techniques in soliton perturbation theory

We give a short review of the techniques that have been used to study soliton perturbations for nonlinear integrable systems in 1 + 1 dimensions. In particular, we discuss the connection between direct perturbation methods using squared eigenfunction expansions and quasistationary perturbation theory. We show for the KdV (Korteweg–de Vries) how the results for quasistationary approximations are related to those for the standard adiabatic perturbation theory.

Estimates for Solutions to Model Problems Connected with Quasistationary Approximation for the Stefan Problem

Model problems for elliptic equations are considered. The time-derivative of a solution can be contained in the boundary condition or in the conjugation condition. Suich problems appear, for example, in the study of free boundary problems for elliptic equations that can be considered as quasistationary approximations to free boundary problems for parabolic equations. Estimates for solutions to the model problems are obtained. Bibliography: 11 titles.

New Algorithm for Constructing Models of the Chemical Reaction Kinetics in the Quasistationary and Quasiequilibrium Approximations

An analysis is made regarding the possibility of using the concepts of routes, overall equations, and stationary rates in constructing kinetic models simultaneously in the quasiequilibrium and quasistationary approximations. A general algorithm is proposed to construct routs, formulate overall equations, and derive expressions for stationary rates. Features of the algorithm are revealed, which are related to some quantitative characteristics of the reaction mechanism.

Fluctuating Powder-Driven Pulsed Water Jet

A study is made of the internal ballistics of a powder-driven pulsed water jet which generates a fluctuating, high velocity stream of liquid. The velocity fluctuations are caused by the use of a light piston which, together with the gases produced by the powder, forms an oscillating system. Calculations of the water jet parameters in the nonstationary and quasistationary approximations are compared. It is shown that the period of the stream velocity fluctuations depends on the initial pressure of the gases produced by the powder, the masses of the piston and water, and the geometry of the apparatus. The period of the fluctuations estimated in a linear approximation for instantaneous powder burnup is in good agreement with the quasistationary calculations.

Acoustic oscillations and viscosity

Using a simple thermo-hydrodynamic model that respects relativistic causality, we reexamine the analysis of the qualitative features of acoustic oscillations in the photon-baryon fluid. The growing photon mean free path introduces transient effects that can be modelled by the causal generalization of relativistic Navier-Stokes-Fourier theory. Causal thermodynamics provides a more satisfactory hydrodynamic approximation to kinetic theory than the quasi-stationary (and non-causal) approximations arising from standard thermodynamics or from expanding the photon distribution to first order in Thomson scattering time. The causal approach introduces small corrections to the dispersion relation obtained in quasi-stationary treatments. A dissipative contribution to the speed of sound slightly increases the frequency of the oscillations. The diffusion damping scale is slightly increased by the causal corrections. Thus quasi-stationary approximations tend to over-estimate the spacing and under-estimate the damping of acoustic peaks. In our simple model, the fractional corrections at decoupling are ≳10-2.

Interior Ballistics of a Powder Water Cannon

The results of theoretical and experimental analyses of the interior ballistics of a powder water cannon are presented. A mathematical model of the process is constructed and the calculations for a specific installation are presented. The fluid flow and powder combustion are considered in the quasi-one-dimensional and quasi-stationary approximations, respectively. The pressure of the fluid and powder gases is measured in various sections of the installation. The theoretical and experimental results agree fairly closely.

Electromagnetic fields and the quasistationary approximation near a moving mirror

A detailed study of the electromagnetic fields generated by illuminating a plane mirror, in inertial motion, by a plane wave is described. A new and complete power series is obtained for the fields, and convergence conditions depending on the distance between the mirror and an observer are established. This series is shown to converge in bandpass systems provided the distance between the mirror and the observer is limited. The power series provides simple expressions for assessing the validity of quasi-stationary (QS) field approximations near the mirror. An example confirms that, with the exception of the immediate vicinity of wavefronts, QS approximations are valid under conditions arising in practical urban radio systems.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

On quasi-stationary approximations to non-stationary random vibration

Most engineering structures are subjected to the action of the environment, which is in reality a non-stationary random process. In calculating the induced structural response it is often assumed that the environmental conditions change slowly and that at any instant the environmental loading can be considered to be a stationary random process. This technique is referred to here as the quasi-stationary approximation. The aim of the present paper is to assess the validity of this approximation, which is done by considering the response covariances of a simple system. Physical insights into the approximation are given by expressing the covariances in the time and frequency domains and the results obtained are applied to a number of practical examples.

Quasi-Stationary Approximations for the Size Distribution of Aerosols

Abstract Quasi-stationary approximations for the size distribution of aerosols have been derived by Friedlander using similarity arguments and dimensional analysis. In this paper they are rederived using an approach based on explicit models of the coagulation mechanisms that determine the spectrum of particle sizes. The primary aim is to cast Friedlander's work into a form suitable for Author generalization, but the new approach also clarifies criticism of the “sedimentation subrange.” The balance between coagulation growth and sedimentation loss postulated by Friedlander is interpreted differently and a new solution for this range is given. In contrast to the other subranges, the solution is not unique (granted the assumption that it is quasi-stationary), because it depends on the form of the externally imposed vertical gradient in number density. The new approach is also applied to a condensation subrange for the smallest particles.

Theory of ostwald coagulation of microscopic particles in solid disperse systems

The problem of the evolution of a two-phase disperse system in which polydisperse spheroidal microscopic particles are distributed in the solid medium is solved in the nonstationary approximation. It is shown that for conservative and nonconservative systems there exist unified solutions from which, as special cases, the well-known results obtained in the stationary and quasistationary approximations follow. Questions relating to the quantitative analysis of the microscopic structure of real disperse systems are discussed.

Melt time and heat flux for a simple PCM body

Using data derived from computer calculations, exact solutions and quasi-stationary approximations, relations for the total melt time and average surface heat flux are given for a lump of phase change material (PCM) subject to a constant temperature at its boundary, and initially at critical temperature.