Theory and simulation of the time-dependent rate coefficients of diffusion-influenced reactions.

A general formalism is developed for calculating the time-dependent rate coefficient k(t) of an irreversible diffusion-influenced reaction. This formalism allows one to treat most factors that affect k(t), including rotational Brownian motion and conformational gating of reactant molecules and orientation constraint for product formation. At long times k(t) is shown to have the asymptotic expansion k(infinity)[1 + k(infinity) (pie Dt)-1/2 /4 pie D + ...], where D is the relative translational diffusion constant. An approximate analytical method for calculating k(t) is presented. This is based on the approximation that the probability density of the reactant pair in the reactive region keeps the equilibrium distribution but with a decreasing amplitude. The rate coefficient then is determined by the Green function in the absence of chemical reaction. Within the framework of this approximation, two general relations are obtained. The first relation allows the rate coefficient for an arbitrary amplitude of the reactivity to be found if the rate coefficient for one amplitude of the reactivity is known. The second relation allows the rate coefficient in the presence of conformational gating to be found from that in the absence of conformational gating. The ratio k(t)/k(0) is shown to be the survival probability of the reactant pair at time t starting from an initial distribution that is localized in the reactive region. This relation forms the basis of the calculation of k(t) through Brownian dynamics simulations. Two simulation procedures involving the propagation of nonreactive trajectories initiated only from the reactive region are described and illustrated on a model system. Both analytical and simulation results demonstrate the accuracy of the equilibrium-distribution approximation method.     

Theoretical analysis on the structure and properties of thiourea dioxide crystal

The equilibrium geometry, electronic structure and optical properties of thiourea S, S-dioxide crystal have been studied using DFT within generalized gradient approximation (GGA) and the local density approximation (LDA), implemented using ultrasoft pseudo-potentials. The optimum bulk geometry is in good agreement with crystallographic data. An analysis of electronic structure, charge and bond order is presented. The energy gap of thiourea dioxide with GGA and LDA calculation is 3.217 or 3.210 eV, respectively, indicating that the compound is an insulator. The calculated absorption spectrum shows a number of absorption peaks, which are believed to be associated with different exciton states, in the fundamental absorption region.     

Interval Estimation for Ratios in Bioequivalence Trials

The problem for assessment of equivalence in variability of bioavailability between two drug products is considered. An exact confidence region for the ratio between intrasubject variabilities is derived when the intersubject variance is known. When the intersubject variance is unknown, a large sample approximation is considered. The proposed method for assessing equivalence in variability of bioavailability appears to be asymptotically uncorrelated with the sample mean ratio for average bioavailabilty. As a result, the proposed method in conjunction with the sample mean ratio method can be utilized for assessing population bioequivalence. An example concerning a bioequivalence trial with 24 healthy volunteers is presented to illustrate the proposed method.     

A Confidence Region Approach for Assessing Equivalence in Variability of Bioavailability

The problem for assessment of equivalence in variability of bioavailability between two drug products is considered. Similar to the method for assessing bioequivalence in average bioavailability proposed by Chow and Shao (1990), an exact confidence region approach is derived when the intersubject variance is known. When the intersubject variance is unknown, a large sample approximation is considered. The proposed method for assessing equivalence of variability of bioavailability appears to be asymptotically uncorrelated with that of Chow and Shao (1990) for average bioavailability. As a result, the proposed method in conjunction with the method proposed by Chow and Shao (1990) constitutes a confidence region approach for assessing population bioequivalence. An example concerning a bioequivalence trial with 24 healthy volunteers is provided to illustrate the proposed method.     

A new look at the critical community size for childhood infections

Quasi-stationarity and time to extinction are studied for the classic endemic model. Attention is restricted to the transition region in parameter space where the quasi-stationary distribution is non-normal. A new approximation of the marginal distribution of infected individuals in quasi-stationarity is presented. It leads to a simple explicit expression for an approximation of the critical community size in terms of model parameters.     

Drift model of the cathode region of a glow discharge

A one-dimensional drift model of the cathode region of a glow discharge with allowance for both electron-impact ionization and charged particle loss is proposed. An exact solution to the model equations is obtained for the case of similar power-law dependences of the ion and electron drift velocities on the electric field strength. It is shown that, even in the drift approximation, a relatively wide transition layer in which the ion-to-electron current ratio approaches a constant value typical of the positive column of a glow discharge should occur between the thin space-charge sheath and the quasineutral plasma, the voltage drop across the space-charge sheath being comparable to that across the transition layer. The calculated parameters of the normal and anomalous glow discharges are in good agreement with available experimental data.     

A note on population growth under random dispersal

An approximation method using a sine function is used to solve the second degree growth equation for the case in which an organism may simultaneously become dispersed throughout a uniform region. The resulting expression for a special case is compared with the expression obtained by R. Barakat (1959,Bull. Math. Biophysics,21, 141–51), giving the first two terms, by an iterative, procedure. The agreement is satisfactory.     

An extension of the steady-state approximation of the kinetics of enzyme-containing systems

An extension of the steady-state approximation of multi-enzyme systems is obtained, which can also be applied when enzyme concentrations are of the same order of magnitude as substrate concentrations. This extension contains as a first order approximation the extension given by Vergonet and Berendsen. The mathematical procedure however is different and avoids difficulties inherent in their theory.     

A note on the use of the non-central t-distribution in setting numerical microbiological specifications for foods

An exact procedure is described and used to determine k values and probabilities of rejection used in the application of variables sampling plans to foods. Comparison of quantities so obtained to those calculated by Kilsby et al. (1979), who used an approximation procedure, reveals that for small samples the approximation can lead to probabilities of rejection which are larger than those obtained using the exact procedure. It is recommended that the exact procedure be used in the application of these plans.     

A note on approximating the cumulative distribution function of the time to tumor onset in multistage models

The general multistage theory of carcinogenesis is a very special type of interconnected birth and death process in which the final state is absorbing and all states except the first one are empty at time zero. An approximation proposed by Whittemore and Keller (1978, SIAM Review 20, 1-30) is assessed. It is shown that the adequacy of this approximation depends on the number of malignant cells resulting from a single normal cell. If more than one malignant cell is likely to occur, the approximation will fail.     

A possible role for placental lysosomes in the formation of villous syncytiotrophoblast

Summary An ultrastructural and ultrahistochemical study of first trimester human placentae confirms previous reports that the cytotrophoblastic cells show a spectrum of differentiation, that dissolution of the limiting membrane of the cytotrophoblastic cells occurs and that fragments of free membrane can be found in the syncytiotrophoblast. There is an aggregation of primary lysosomes in the region of approximation of the cytotrophoblast to the syncytiotrophoblast, free lysosomal enzymes are found in the space between the two trophoblastic components, secondary lysosomes have been noted in the vicinity of fragmenting cytotrophoblastic cell membrane and the incorporation of a segment of free membrane into a vesicular structure has been noted. It is suggested that placental lysosomes mediate the dissolution of the cytotrophoblastic cell membranes that is a necessary prerequisite for their full differentiation into syncytiotrophoblast and it is further suggested that one of the principal roles of placental lysosomes is in the structural refashioning of the organ that occurs during the first trimester.     

Estimating the Volume Density from Thin Sections

An approximation formula for the determination of the volume density from thin sections is given. It is applicable under very mild conditions on the structure studied and has a good accuracy.     

Circular dichroism of helical polynucleotides calculated by the linear response method

Linear response theory in the decorrelation or random-phase approximation is used to calculate the absorption and CD spectra of model helical polymers, including single-stranded polyadenylic acid. The method, which makes use of infinite polymer selection rules for the linear response tensor, has the advantages that (1) only a few three-dimensional matrices need be inverted; (2) spectral band shapes of the polymer arise naturally from those of the monomer, as well as from the geometry-dependent interactions in the helix; and (3) the spectral dependence on geometrical factors of the helix is made transparent. It is found that the structure of the polymer CD spectrum depends critically on monomer bandshape. An asymmetric CD spectrum, similar to some experimental spectra, arises from either a Gaussian or a composite monomer band. Single-stranded polyadenylic acid spectra are sensitive to helix geometry in the region 200–240 nm, in reasonable agreement with experimental spectra. This sensitivity arises from the 207-nm monomer transition, and the results suggest that this region of the spectrum should be more fully exploited as a tool for helix geometry studies.     

Stability of a stationary double layer

Electrostatic oscillations in a stationary beam double layer are studied in the hydrodynamic approximation. An analysis of the solutions to the linearized equations for perturbations shows that the double layer in an unbounded plasma may be subject only to convective instability, in which case the perturbations localized inside the region of the potential jump escape from the plasma without distorting the steady field structure. The double layer in a bounded plasma is investigated numerically by a particle-in-cell method. It is established that, as is the case in the Pierce system, the governing role in the stability of the double layer is played by the field of the charges induced on the surfaces of the conducting electrodes.     

An approximation to the likelihood for a pedigree with loops

This paper presents a new approximation to the likelihood for a pedigree with loops, based on cutting all loops and extending the pedigree at the cuts. An opimum loop-cutting strategy and an iterative extension technique are presented. The likelihood for a pedigree with loops is then approximated by the conditional likelihood for the entire cut-extended pedigree given the extended part. The approximate likelihoods are compared with the exact likelihoods obtained using the program MENDEL for several small pedigrees with loops. The approximation is efficient for large pedigrees with complex loops in terms of computing speed and memory requirements.     

A note on the transient stage of the random dispersal of logistic populations

The transient stage of the random dispersal of logistic populations is investigated, using a Sturm-Liouville series leading to an infinite system of non-linear integral equations. These equations are then solved via a successive approximation scheme. R. A. Fisher's (steady-state) velocity of advance paradox is discussed. An illustrative example is worked to the second order of approximation. Contribution No. 1020 of the Woods Hole Oceanographic Institution.     

Thermodynamic properties of two-dimensional nonideal structures with isotropic pair potential

An approximation is proposed for energy density in two-dimensional nonideal systems for a wide class of isotropic repulsive pair interparticle potentials. The approximation allows one to determine the main thermodynamic functions and characteristics of the system by using well-known thermodynamic formulas. The results obtained with the help of this approximation are compared with the data of numerical simulations of thermodynamic properties of the structures under study. The simulations were performed in a wide range of parameters typical of laboratory dusty plasmas. Main attention was paid to the screened Coulomb potential.     

Speed of invasion in lattice population models: pair-edge approximation

 We propose a simple approach to approximating the speed of invasion in lattice population models. Approximate critical parameter values for successful invasion are then found by solving for zero wave speed. The approximation is based on describing the occupied region by the ordinary pair approximation, and using quasi-steady-state pair approximations to describe the leading edge of the wave front. We illustrate this idea using the basic contact process on the 1 and 2 dimensional lattice (with and without nearest-neighbor migration), finding very good agreement between the approximation and simulation results. The approximate critical values obtained by our approximation are significantly more accurate than those obtained by the ordinary pair approximation. Received 4 September 1996     

Electrical conductivity of a partially ionized plasma with allowance for the effect of plasma screening on the electron scattering by neutral atoms

Using linear response theory, the effect of electron-atom scattering on the electrical conductivity of a partially ionized hydrogen plasma is studied in the relevant statistical operator approximation. A relationship is analyzed between the polarization potential, which is routinely used in the problem, and the adiabatic potential, which results from the separation of electrons into bound and unbound ones. An approximation accounting for the effect of unbound electrons on the interaction between neutral and charged plasma particles is constructed. It is found that, in a high-density plasma, the parameters of the interaction of an atom with charged particles can change significantly, so these changes should be taken into account in calculating kinetic plasma properties.     

An approximation method for solving the steady-state probability distribution of probabilistic Boolean networks

MOTIVATION: Probabilistic Boolean networks (PBNs) have been proposed to model genetic regulatory interactions. The steady-state probability distribution of a PBN gives important information about the captured genetic network. The computation of the steady-state probability distribution usually includes construction of the transition probability matrix and computation of the steady-state probability distribution. The size of the transition probability matrix is 2(n)-by-2(n) where n is the number of genes in the genetic network. Therefore, the computational costs of these two steps are very expensive and it is essential to develop a fast approximation method. RESULTS: In this article, we propose an approximation method for computing the steady-state probability distribution of a PBN based on neglecting some Boolean networks (BNs) with very small probabilities during the construction of the transition probability matrix. An error analysis of this approximation method is given and theoretical result on the distribution of BNs in a PBN with at most two Boolean functions for one gene is also presented. These give a foundation and support for the approximation method. Numerical experiments based on a genetic network are given to demonstrate the efficiency of the proposed method.