Kinetics of adsorption of extended ligands on DNA at small fillings

In the present work, the adsorption kinetics of extended ligands on DNA duplexes at small fillings when molecules of DNA duplexes are on the underlayer within diffusion layer has been investigated. Both diffusion of ligands in solution (diffusion stage) and adsorption of ligands (kinetic stage) are taken into consideration at adsorption of ligands on DNA duplexes. Nonlinear system of differential equations describing adsorption of ligands where not only diffusion stage but also kinetic stage is taken into account, is obtained, moreover the equations allow localizing duplexes in arbitrary place within diffusion layer. Numeric solution of the equations makes possible to investigate the filling kinetics of DNA duplexes by ligands depending on parameters controlling adsorption process. It has been shown that depending on relation between adsorption parameters different kinetic regimes of adsorption – kinetic, complex, and diffusion regimes may be realized.     

Derivation of unstirred-layer transport number equations from the Nernst-Planck flux equations.

Since the late 1960s it has been known that the passage of current across a membrane can give rise to local changes in salt concentration in unstirred layers or regions adjacent to that membrane, which in turn give rise to the development of slow transient diffusion potentials and osmotic flows across those membranes. These effects have been successfully explained in terms of transport number discontinuities at the membrane-solution interface, the transport number of an ion reflecting the proportion of current carried by that ion. Using the standard definitions for transport numbers and the regular diffusion equations, these polarization or transport number effects have been analyzed and modeled in a number of papers. Recently, the validity of these equations has been questioned. This paper has demonstrated that, by going back to the Nernst-Planck flux equations, exactly the same resultant equations can be derived and therefore that the equations derived directly from the transport number definitions and standard diffusion equations are indeed valid.     

Linear and non-linear diffusion models applied to the behavior of a population of an intertidal snail

The flux and continuity equations are presented for a non-linear diffusion of populations. The generalized diffusion-reaction equations are presented for both random and biased diffusion, where a bias is introduced toward the existing gradient of the population density. An ecological field experiment is presented and analyzed in light of the non-linear diffusion models. The dispersion of the population of intertidal snails near the Red Sea has been measured over a period of 20 months. The dispersion can be described by a one-dimensional random motion and the obtained diffusion coefficient 0·1 m² day^?1 is in good agreement with velocity and free path measurements of individual snails.     

A Lagrangian particle method for reaction–diffusion systems on deforming surfaces

Reaction–diffusion processes on complex deforming surfaces are fundamental to a number of biological processes ranging from embryonic development to cancer tumor growth and angiogenesis. The simulation of these processes using continuum reaction–diffusion models requires computational methods capable of accurately tracking the geometric deformations and discretizing on them the governing equations. We employ a Lagrangian level-set formulation to capture the deformation of the geometry and use an embedding formulation and an adaptive particle method to discretize both the level-set equations and the corresponding reaction–diffusion. We validate the proposed method and discuss its advantages and drawbacks through simulations of reaction–diffusion equations on complex and deforming geometries.     

Solution of linear one-dimensional diffusion equations

The paper introduces a basic mathematical form, which is characteristic of a number of linear one-dimensional diffusion equations with coefficients being represented as simple polynomials in the spatial coordinate. A number of particular diffusion equations are introduced and their corresponding exact mathematical solutions are obtained.     

Rotating wave solutions of the FitzHugh–Nagumo equations

This paper will treat the bifurcation and numerical simulation of rotating wave (RW) solutions of the FitzHugh-Nagumo (FHN) equations. These equations are often used as a simple mathematical model of excitable media. The dependence of the solutions on a uniformly applied current, and also on the diffusion coefficients or domain size will be studied. Ranges of applied current and/or diffusion coefficients in which RW solutions are observed will be described using bifurcation theory and continuation methods. The bifurcation of time-periodic solutions of these FHN equations without diffusion is described first. Similar methods are then used to find RW solutions on a circular ring and numerical simulations are described. These results are then extended to investigate RW solutions on annular rings of finite cross-section. Scaling arguments are used to show how the existence of solutions depends on either the diffusion coefficient or on the size of the region.     

On bifurcations in complex ecological systems with diffusion and noise

We investigate here several methods for the qualitative investigation of complex ecological systems including diffusion and noise. Assuming that the systems are described by diffusion equations or Fokker–Planck equations, we formulate for gradient-type dynamical systems several statements about the number and type of possible bifurcations using theorems of catastrophe theory. We introduce stochastic potentials, present exact and approximate solutions. In the case of general dynamical systems we show that the case of strong noise may be appropriately described by expansions of the stochastic potential with respect to reciprocal noise. Finally we present examples for the bifurcations of the biomass.     

On the mechanics of viscous bodies and elongation in ellipsoidal cells

After deriving some auxiliary equations for the average elongation of a viscous body under the action of forces derived from a potential, the diffusion problem for an ellipsoidal cell with a constant rate of reaction is solved for the case of an infinite permeability. The equation of elongation of such a cell under the influence of diffusion forces is derived, and compared with the, approximate expression obtained by N. Rashevsky for any kind of oblong cell. The two equations are in fair agreement. Effects of constant and variable surface tension are studied.     

Homogenization of Large-Scale Movement Models in Ecology

A difficulty in using diffusion models to predict large scale animal population dispersal is that individuals move differently based on local information (as opposed to gradients) in differing habitat types. This can be accommodated by using ecological diffusion. However, real environments are often spatially complex, limiting application of a direct approach. Homogenization for partial differential equations has long been applied to Fickian diffusion (in which average individual movement is organized along gradients of habitat and population density). We derive a homogenization procedure for ecological diffusion and apply it to a simple model for chronic wasting disease in mule deer. Homogenization allows us to determine the impact of small scale (10–100 m) habitat variability on large scale (10–100 km) movement. The procedure generates asymptotic equations for solutions on the large scale with parameters defined by small-scale variation. The simplicity of this homogenization procedure is striking when compared to the multi-dimensional homogenization procedure for Fickian diffusion,and the method will be equally straightforward for more complex models.     

The diffusive Lotka-Volterra oscillating system

The Lotka-Volterra equations are coupled with diffusion processes in homogeneous systems. The inclusion of a negative cross diffusion coefficient can result in the appearance of a stationary wave-like dissipative structure. The cross diffusion coefficients represent a deceitful relationship between the two interacting populations.     

Growth with regulation in fluctuating environments

In this paper deterministic growth laws of a logistic-like type are initially introduced. The growth equations are expressed by first order differential equations containing a third order nonlinear term. Such equations are then parameterized in a way to allow for random fluctuations of the intrinsic fertility and of the environmental carrying capacity, thus leading to diffusion processes of new types. Their transition p.d.f. and asymptotic moments are then obtained and a detailed study of the extinction problem is performed within the framework of the first passage time problem through arbitrarily fixed threshold values. Some statistically significant quantities, such as the mean time necessary for the process to attain an assigned state, are obtained in closed form. The behavior of the diffusion processes here derived is finally compared with that of the well known diffusion processes obtained by parameterizing logistic and Gompertz growth equations.Work supported in part by the Group for Mathematical Information Sciences (GNIM) of the National Research Council and by Progetto Finalizzato Sofmat, Contract No. 82.00845.97     

Turing structures in an enzyme-induction system with gap junction-mediated non-linear diffusion

Two cells, each containing a reaction system modeling genetic induction, are coupled by diffusion. The substrate is moving through gap junctions, the number of which is regulated by the adjacent cells. This leads to a non-linear substrate diffusion term in the rate equations. Stability analysis reveals the conditions for the emergence of stable asymmetric solutions (dissipative structures). Due to non-linear diffusion rigid restrictions on the ratio of the two diffusion constants no longer exist. We demonstrate that substances operating as regulators of intercellular communication and participating in cellular metabolism may exhibit morphogenetic functions.     

A numerical model for studying the effect of plasma layer on the process of blood oxygenation in the pulmonary capillaries

A two layer model for the blood oxygenation in pulmonary capillaries is proposed. The model consists of a core of erythrocytes surrounded by a symmetrically placed plasma layer. The governing equations in the core describe the free molecular diffusion, convection, and facilitated diffusion due to the presence of haemoglobin. The corresponding equations in the plasma layer are based on the free molecular diffusion and the convective effect of the blood. According to the axial train model for the blood flow proposed by Whitmore (1967), the core will move with a uniform velocity whereas flow in the plasma layer will be fully developed. The resulting system of nonlinear partial differential equations is solved numerically. A fixed point iterative technique is used to deal with the nonlinearities. The distance traversed by the blood before getting fully oxygenated is computed. It is shown that the concentration of O2 increases continuously along the length of the capillary for a given ratio of core radius to capillary radius. It is found that the rate of oxygenation increases as the core to capillary ratio decreases. The equilibration length increases with a heterogeneous model in comparison to that in a homogeneous model. The effect of capillary diameters and core radii on the rate of oxygenation has also been examined.     

Diffusion in immobilized-cell gels

Summary The relationship between the diffusion coefficient, the effective diffusion coefficient and the partition coefficient for a solute in a cell-containing gel is discussed. The use of correlation equations that are based on some kind of physical model is recommended when the effect of cell concentration on diffusion is interpreted.     

Diffusion models for chemotaxis: a statistical analysis of noninteractive unicellular movement.

A program is developed for applying stochastic differential equations to models for chemotaxis. First a few of the experimental and theoretical models for chemotaxis both for swimming bacteria and for cells migrating along a substrate are reviewed. In physical and biological models of deterministic systems, finite difference equations are often replaced by a limiting differential equation in order to take advantage of the ease in the use of calculus. A similar but more intricate methodology is developed here for stochastic models for chemotaxis. This exposition is possible because recent work in probability theory gives ease in the use of the stochastic calculus for diffusions and broad applicability in the convergence of stochastic difference equations to a stochastic differential equation. Stochastic differential equations suggest useful data for the model and provide statistical tests. We begin with phenomenological considerations as we analyze a one-dimensional model proposed by Boyarsky, Noble, and Peterson in their study of human granulocytes. In this context, a theoretical model consists in identifying which diffusion best approximates a model for cell movement based upon theoretical considerations of cell physiology. Such a diffusion approximation theorem is presented along with discussion of the relationship between autocovariance and persistence. Both the stochastic calculus and the diffusion approximation theorem are described in one dimension. Finally, these tools are extended to multidimensional models and applied to a three-dimensional experimental setup of spherical symmetry.     

On the accuracy of a diffusion approximation to a discrete state–space Markovian model of a population

The traditional Kolmogorov equations treat the size of a population as a discrete random variable. A model is introduced that extends these equations to incorporate environmental variability. Difficulties with this discrete model motivate approximating the population size as a continuous random variable through the use of diffusion processes. The set of cumulants for both the population size and the environmental factors affecting the population size characterize the population–environmental system. The evolution of this set, as predicted by the diffusion approximation, closely matches the corresponding predictions for the discrete model. It is also noted that the simulation estimates of the cumulants against which the predictions of the diffusion model are checked can vary considerably between simulations — despite averaging over a large number of simulation runs. The precision of the simulation estimates–both over time and with differing cumulant order–is discussed.     

THE THEORY OF DIFFUSION IN CELL MODELS

The differential equations which describe the simultaneous diffusion of water and a salt in a cell model have been formulated and solved. The equations have been derived from the general laws which describe diffusion processes, thereby furnishing a physical interpretation for the constants which enter into the theory. The theoretical time curves for the two diffusing substances are in good agreement with the experimentally determined curves and accurately reproduce all of the essential characteristics of the experiment.     

On the accuracy of a diffusion approximation to a discrete state–space Markovian model of a population

The traditional Kolmogorov equations treat the size of a population as a discrete random variable. A model is introduced that extends these equations to incorporate environmental variability. Difficulties with this discrete model motivate approximating the population size as a continuous random variable through the use of diffusion processes. The set of cumulants for both the population size and the environmental factors affecting the population size characterize the population–environmental system. The evolution of this set, as predicted by the diffusion approximation, closely matches the corresponding predictions for the discrete model. It is also noted that the simulation estimates of the cumulants against which the predictions of the diffusion model are checked can vary considerably between simulations — despite averaging over a large number of simulation runs. The precision of the simulation estimates–both over time and with differing cumulant order–is discussed.     

Relevance of diffusion through bacterial spore coats/membranes and the associated concentration boundary layers in the initial lag phase of inactivation: a case study for Bacillus subtilis with ozone and monochloramine

Disinfectants are generally used to inactivate microorganisms in solutions. The process of inactivation involves the disinfectant in the liquid diffusing towards the bacteria sites and thereafter reacting with bacteria at rates determined by the respective reaction rates. Such processes have demonstrated an initial lag phase followed by an active depletion phase of bacteria. This paper attempts to study the importance of the combined effects of diffusion of the disinfectant through the outer membrane of the bacteria and transport through the associated concentration boundary layers (CBLs) during the initial lag phase. Mathematical equations are developed correlating the initial concentration of the disinfectant with time required for reaching a critical concentration (C*) at the inner side of the membrane of the cell based on diffusion of disinfectant through the outer membranes of the bacteria and the formation of concentration boundary layers on both sides of the membranes. Experimental data of the lag phases of inactivation already available in the literature for inactivation of Bacillus subtilis spores with ozone and monochloramine are tested with the equations. The results seem to be in good agreement with the theoretical equations indicating the importance of diffusion process across the outer cell membranes and the resulting CBL's during the lag phase of disinfection.