Derivation of general equations describing tracer diffusion in any two-compartment tissue with application to ionic diffusion in cylindrical muscle bundles

A set of generalized diffusion equations have been derived which describe radioactive tracer movement in any tissue that can be modeled as a distributed two-compartment system. These equations have been applied to ionic tracer movement in cylindrical muscle bundles, and the boundary conditions used correspond to experimental conditions during various ionic tracer diffusion experiments on cardiac papillary muscles. Specifically, solutions were obtained for extra- and intracellular tracer washout as well as for the extra- and intracellular steady-state tracer diffusion experiments of Weidmann (1966). These solutions are presented in series form as well as in graphical form and are compared with the corresponding experimental data. A comparison of these solutions with those obtained using simple exponential kinetics is presented, and it is shown that there is a marked discrepancy between these two methods of analysis for bundles of any appreciable diameter.     

Stochastic model of population growth and spread

A stochastic model for the population regulated by logistic growth and spreading in a given region of two-or three-dimensional space has been introduced. For many-species population the interactions among the species have also been icorporated in this model. From the random variables that describe stochastic processes of a Wiener type the space-dependent random population densities have been formed and shown to satisfy the Langevin equations. The Fokker-Planck equation corresponding to these Langevin equations has been approximately solved for the transition probability of the population spreading and it has been found that such approximate expressions of the transition probability depend on the solutions of the deterministic equations of the diffusion model with logistic growth and interactions. Also, the stationary or equilibrium solutions of the Fokker-Planck equation together with the special discussion on the pattern of single-species population spreading have been made.     

Denitrification and a numerical modelling approach for shallow waters

A numerical model (`DiffDeni') has been developed to describe the disappearance of nitrate from the water column of 10–200 cm deep waters. The disappearance is caused by bacterial denitrification in the sediments. The model employs the molecular diffusion constant, an acceleration factor describing eddy diffusion, and three bacterial growth constants, viz. the inoculum size, the maximum growth rate and the half saturation constant for the hyperbolic process. The values of these system-constants were varied over a wide range. The curves obtained were compared with the curves for well-defined situations, viz. in which diffusion takes place without any or with a complete, immediate reaction. These cases have analytical solutions, and were simulated closely by the model `DiffDeni', though this model is based on different assumptions. It is shown that, when the bacterial growth rate is above a critical value, a negative exponential curve describes the nitrate disappearance well. On the other hand, a more complicated negative exponential equation can be used to describe the first phase of this denitrification in which bacterial activity is low and nitrate behaves as a conservative compound. The change-over period from phase 1 (no reaction) to phase 2 (complete, immediate reaction) which may vary between <1 and 50 days cannot be described analytically (mathematically correctly). The influence of temperature on denitrification is assessed and it is shown that both bacterial activity and diffusion may influence the denitrification rate.     

Effects of Hydraulic Soil Properties on Vegetation Pattern Formation in Sloping Landscapes

Current models of vegetation pattern formation rely on a system of weakly nonlinear reaction–diffusion equations that are coupled by their source terms. While these equations, which are used to describe a spatiotemporal planar evolution of biomass and soil water, qualitatively capture the emergence of various types of vegetation patterns in arid environments, they are phenomenological and have a limited predictive power. We ameliorate these limitations by deriving the vertically averaged Richards’ equation to describe flow (as opposed to “diffusion”) of water in partially saturated soils. This establishes conditions under which this nonlinear equation reduces to its weakly nonlinear reaction–diffusion counterpart used in the previous models, thus relating their unphysical parameters (e.g., diffusion coefficient) to the measurable soil properties (e.g., hydraulic conductivity) used to parameterize the Richards equation. Our model is valid for both flat and sloping landscapes and can handle arbitrary topography and boundary conditions. The result is a model that relates the environmental conditions (e.g., precipitation rate, runoff and soil properties) to formation of multiple patterns observed in nature (such as stripes, labyrinth and spots).     

Analysis of growth factor gradients in hydra

Differential equations have been written and solved to describe the concentration gradients for the two hydra growth factors. The equations consider diffusion, catabolism and synthesis of the materials based primarily on the model of hydra growth described by A. L. Burnett. Concentration gradient profiles were obtained which correspond with the concentration gradients deduced from experimental evidence. It has also been possible to show that in areas of high mitotic rate, mitotic rate is related to the calculated concentration ratio of the two growth factors (stimulator/inhibitor). However, the correspondence of mitotic rate to growth factor concentration ratio does not hold for extreme values of the ratio indicating that very high and very low concentration ratios are not conductive to mitotic activity.     

Nonequilibrium-Facilitated Oxygen Transport in Hemoglobin Solution

We have used the quasi-linearization method to obtain numerical solutions to the equations which describe steady-state diffusion of oxygen through layers of hemoglobin solution. The numerical solutions show how the facilitated flux of oxygen depends upon the layer thickness, reaction-rate coefficients, and other parameters of the system. The results indicate that steady-state oxygen diffusion in layers of hemoglobin solution, similar to those studied by Scholander, should be adequately described by the models which assume chemical equilibrium exists throughout the layer, but for layers of concentrated hemoglobin solution about the thickness of a human erythrocyte, the facilitation of oxygen diffusion should be much less than the equilibrium models predict.     

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.     

Calculating thermodynamic data for transitions of any molecularity from equilibrium melting curves

In this paper, we derive the general forms of the equations required to extract thermodynamic data from equilibrium transition curves on oligomeric and polymeric nucleic acids of any molecularity. Significantly, since the equations and protocols are general, they also can be used to characterize thermodynamically equilibrium processes in systems other than nucleic acids. We briefly review how the reduced forms of the general equations have been used by many investigators to evaluate mono- and bimolecular transitions, and then explain how these equations can be generalized to calculate thermodynamic parameters from common experimental observables for transitions of higher molecularities. We emphasize the strengths and weaknesses of each method of data analysis so that investigators can select the approach most appropriate for their experimental circumstances. We also describe how to analyze calorimetric heat capacity curves and noncalorimetric differentiated melting curves so as to extract both model-independent and model-dependent thermodynamic data for transitions of any molecularity. The general equations and methods of analysis described in this paper should be of particular interest to laboratories that currently are investigating association and dissociation processes in nucleic acids that exhibit molecularities greater than two.     

Molecular mass determination by sedimentation velocity experiments and direct fitting of the concentration profiles.

A new method for the direct molecular mass determination from sedimentation velocity experiments is presented. It is based on a nonlinear least squares fitting procedure of the concentration profiles and simultaneous estimation of the sedimentation and diffusion coefficients using approximate solutions of the Lamm equation. A computer program, LAMM, was written by using five different model functions derived by Fujita (1962, 1975) to describe the sedimentation of macromolecules during centrifugation. To compare the usefulness of these equations for the analysis of hydrodynamic results, the approach was tested on data sets of Claverie simulations as well as experimental curves of some proteins. A modification for one of the model functions is suggested, leading to more reliable sedimentation and diffusion coefficients estimated by the fitting procedure. The method seems useful for the rapid molecular mass determination of proteins larger than 10 kDa. One of the equations of the Archibald type is also suitable for compounds of low molecular mass, probably less than 10 kDa, because this model function requires neither the plateau region nor a meniscus free of solute.     

Determination of Brain Interstitial Concentrations by Microdialysis

Microdialysis is an extensively used technique for the study of solutes in brain interstitial space. The method is based on collection of substances by diffusion across a dialysis membrane positioned in the brain. The outflow concentration reflects the interstitial concentration of the substance of interest, but the relationship between these two entities is at present unclear. So far, most evaluations have been based solely on calibrations in saline. This procedure is misleading, because the ease by which molecules in saline diffuse into the probe is different from that of tissue. We describe here a mathematical analysis of mass transport into the dialysis probe in tissue based on diffusion equations in complex media. The main finding is that diffusion characteristics of a given substance have to be included in the formula. These include the tortuosity factor (lambda) and the extracellular volume fraction (alpha). We have substantiated this by studies in a well-defined complex medium (red blood cell suspensions) as well as in brain. We conclude that the traditional calculation procedure results in interstitial concentrations that are too low by a factor of lambda 2/alpha for a given compound.     

A diffusion equation for tapered plastrons

An equation for the diffusion of oxygen along a plastron of uniform thickness was provided by Thorpe and Crisp in their classical work on plastrons. Since then it has been discovered that some plastrons are tapered, e.g. those of many spiracular gills, in which the thickness of the plastron becomes less as the distance from the spiracle increases. Here a differential equation is provided for calculating the efficiency of a uniformly tapered plastron. A series of curves is also given that show the efficiency of the tapered plastron as a function of the distance from the spiracle, thickness, and degree of taper. Using these curves the efficiency of the plastron can be estimated immediately without the necessity of solving the differential equations.     

The process of gas exchange in the pulmonary circulation incorporating the contribution of axial diffusion

A mathematical model is made to describe the process of gas exchange in the pulmonary circulation incorporating the contribution of axial diffusion. The model takes into account the transport mechanisms of molecular diffusion, convection and facilitated diffusion due to the presence of haemoglobin as a carrier of the gases. The mathematical formulation leads to a coupled system of non-linear elliptic partial differential equations. A numerical scheme is described to solve such a system. It is found that the axial diffusion does not have an appreciable effect on the transport of the species in the blood.     

Computer simulation of sedimentation in the ultracentrifuge: V. Ideal and non-ideal monomer-trimer systems

Treatments of velocity sedimentation which neglect diffusion (“asymptotic” solutions) predict that a minimum will appear in the gradient curves given by certain rapidly equilibrating monomer-trimer and monomer-dimer-trimer systems. The velocity sedimentation of a variety of such systems has been simulated by a computer model. The model takes into account the diffusion of the solutes. Both ideal and non-ideal sedimentation have been examined. Sedimentation is assumed to have been done in water at 60,000 rpm in a standard ultracentrifuge rotor.     

THE THEORY OF DIFFUSION IN CELL MODELS : II. SOLUTION OF THE STEADY STATE FOR THREE DIFFUSING SUBSTANCES

The differential equations describing diffusion in cell models have been extended to include the simultaneous penetration of water and two salts. These equations have been solved for the steady state. Values for the concentrations in the steady state which may be computed from the equations compare favorably with the experimental values obtained by Osterhout, Kamerling, and Stanley. Moreover, it has been shown elsewhere that the solution for the steady state is essential to a discussion of the volume change or "growth" of phase C in the models and, by analogy, in living cells.     

A Comparison of Computational Models for Eukaryotic Cell Shape and Motility

Eukaryotic cell motility involves complex interactions of signalling molecules, cytoskeleton, cell membrane, and mechanics interacting in space and time. Collectively, these components are used by the cell to interpret and respond to external stimuli, leading to polarization, protrusion, adhesion formation, and myosin-facilitated retraction. When these processes are choreographed correctly, shape change and motility results. A wealth of experimental data have identified numerous molecular constituents involved in these processes, but the complexity of their interactions and spatial organization make this a challenging problem to understand. This has motivated theoretical and computational approaches with simplified caricatures of cell structure and behaviour, each aiming to gain better understanding of certain kinds of cells and/or repertoire of behaviour. Reaction–diffusion (RD) equations as well as equations of viscoelastic flows have been used to describe the motility machinery. In this review, we describe some of the recent computational models for cell motility, concentrating on simulations of cell shape changes (mainly in two but also three dimensions). The problem is challenging not only due to the difficulty of abstracting and simplifying biological complexity but also because computing RD or fluid flow equations in deforming regions, known as a “free-boundary” problem, is an extremely challenging problem in applied mathematics. Here we describe the distinct approaches, comparing their strengths and weaknesses, and the kinds of biological questions that they have been able to address.     

A network model of successive partitioning-limited solute diffusion through the stratum corneum

As the most exposed point of contact with the external environment, the skin is an important barrier to many chemical exposures, including medications, potentially toxic chemicals and cosmetics. Traditional dermal absorption models treat the stratum corneum lipids as a homogenous medium through which solutes diffuse according to Fick's first law of diffusion. This approach does not explain non-linear absorption and irregular distribution patterns within the stratum corneum lipids as observed in experimental data. A network model, based on successive partitioning-limited solute diffusion through the stratum corneum, where the lipid structure is represented by a large, sparse, and regular network where nodes have variable characteristics, offers an alternative, efficient, and flexible approach to dermal absorption modeling that simulates non-linear absorption data patterns. Four model versions are presented: two linear models, which have unlimited node capacities, and two non-linear models, which have limited node capacities. The non-linear model outputs produce absorption to dose relationships that can be best characterized quantitatively by using power equations, similar to the equations used to describe non-linear experimental data.