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PubMed2003

Phase transition classes in triplet and quadruplet reaction-diffusion models.

Phase transitions of reaction-diffusion systems with site occupation restriction and with particle creation that requires n=3,4 parents, whereas explicit diffusion of single particles (A) is present are investigated in low dimensions by the mean-field approximation and simulations. The mean-field approximation of general nA-->(n+k)A, mA-->(m-l)A type of lattice models is solved and a different kind of critical behavior is pointed out. In d=2 dimensions, the 3A-->4A, 3A-->2A model exhibits a continuous mean-field type of phase transition, that implies d(c)<2 upper critical dimension. For this model in d=1 extensive simulations support a mean-field type of phase transition with logarithmic corrections unlike the recent study of Park et al. [Phys. Rev E 66, 025101 (2002)]. On the other hand, the 4A-->5A, 4A-->3A quadruplet model exhibits a mean-field type of phase transition with logarithmic corrections in d=2, while quadruplet models in one-dimensional show robust, nontrivial transitions suggesting d(c)=2. Furthermore, I show that a parity conserving model 3A-->5A, 2A--> zero in d=1 has a continuous phase transition with different kinds of exponents. These results are in contradiction with the recently suggested implications of a phenomenological, multiplicative noise Langevin equation approach and with the simulations on suppressed bosonic systems by Kockelkoren and Chaté [Phys. Rev. Lett. 90, 125701 (2003)].

Journal Article↗
PubMed2003

The frequency spectrum of a mutation, and its age, in a general diffusion model.

General formulae are derived for the probability density and expected age of a mutation of frequency x in a population, and similarly for a mutation with b copies in a sample of n genes. A general formula is derived for the frequency spectrum of a mutation in a sample. Variable population size models are included. Results are derived in two frameworks: diffusion process models for the frequency of the mutation; and birth and death process models. The coalescent structure within the mutant gene group and the non-mutant group is considered.

Alleles↗
PubMed1984

A generalized reaction diffusion model for spatial structure formed by motile cells.

A non-linear stability analysis using a multi-scale perturbation procedure is carried out on a model of a generalized reaction diffusion mechanism which involves only a single equation but which nevertheless exhibits bifurcation to non-uniform states. The patterns generated by this model by variation in a parameter related to the scalar dimensions of domain of definition, indicate its capacity to represent certain key morphogenetic features of multicellular systems formed by motile cells.

Animals↗
PubMed1995

Analysis of saturation transfer electron paramagnetic resonance spectra of a spin-labeled integral membrane protein, band 3, in terms of the uniaxial rotational diffusion model.

Algorithms have been developed for the calculation of saturation transfer electron paramagnetic resonance (ST-EPR) spectra of a nitroxide spin-label assuming uniaxial rotational diffusion, a model that is frequently used to describe the global rotational dynamics of large integral membrane proteins. One algorithm explicitly includes terms describing Zeeman overmodulation effects, whereas the second more rapid algorithm treats these effects approximately using modified electron spin-lattice and spin-spin relaxation times. Simulations are presented to demonstrate the sensitivity of X-band ST-EPR spectra to the rate of uniaxial rotational diffusion and the orientation of the nitroxide probe with respect to the diffusion axis. Results obtained by using the algorithms presented, which are based on the transition-rate formalism, are in close agreement with those obtained by using an eigenfunction expansion approach. The effects of various approximations used in the simulation algorithms are considered in detail. Optimizing the transition-rate formalism to model uniaxial rotational diffusion results in over an order of magnitude reduction in computation time while allowing treatment of nonaxial A- and g-tensors. The algorithms presented here are used to perform nonlinear least-squares analyses of ST-EPR spectra of the anion exchange protein of the human erythrocyte membrane, band 3, which has been affinity spin-labeled with a recently developed dihydrostilbene disulfonate derivative, [15N,2H13]-SL-H2DADS-MAL. These results suggest that all copies of band 3 present in intact erythrocytes undergo rotational diffusion about the membrane normal axis at a rate consistent with a band 3 dimer.

4-Acetamido-4'-isothiocyanatostilbene-2,2'-disulfo↗
PubMed2000

Fluorescence correlation spectroscopy in small cytosolic compartments depends critically on the diffusion model used.

Fluorescence correlation spectroscopy (FCS) is a powerful technique for measuring low concentrations of fluorescent molecules and their diffusion constants. In the standard case, fluorescence fluctuations are measured in an open detection volume defined by the confocal optics. However, if FCS measurements are carried out in cellular processes that confine the detection volume, the standard FCS model leads to erroneous results. In this paper, we derive a modified FCS model that takes into account the confinement of the detection volume. Using this model, we have carried out the first FCS measurements in dendrites of cultured neurons. We further derive, for the case of confined diffusion, the limits within which the standard two- and three-dimensional diffusion models give reliable results.

Animals↗
PubMed1994

A convective-diffusion model for dissolution of two non-interacting drug mixtures from co-compressed slabs under laminar hydrodynamic conditions.

A numerical convective-diffusion dissolution model has been extended to describe dissolution of two neutral non-interacting drugs co-compressed in a slab geometry. The model predicted the experimental dissolution rates of naproxen/phenytoin mixtures and hydrocortisone/nitrofurantoin mixtures quite accurately, except for phenytoin in the naproxen/phenytoin mixture at low weight proportions. A non-linear dependence of dissolution rate on weight proportion with a positive deviation from linearity was observed. An increase in flow rate increased the dissolution rate and the cube-root dependency of dissolution rate on the flow rate for a given weight proportion of the component in the slab, as proposed earlier by Shah and Nelson for pure compounds, was also observed here, suggesting that the changes in dissolution profile were caused by changes in surface area only. As expected from the model an increase in particle size of the powders used to make the slab decreased the dissolution rate. This was explained by an increase in the average length of the component resulting in a bigger 'carryover' of material from one section of the component in the slab to the next section of the same component, due to convection, and hence lower flux.

Diffusion↗
PubMed1999

Diffusion models for population dynamics incorporating individual behavior at boundaries: applications to refuge design.

We construct models for dispersal of a population which incorporate the response of individuals to interfaces between habitat types. The models are based on random walks where there may be a bias in the direction an individual moves when it encounters an interface. This sort of dispersal process is called skew Brownian motion. Our models take the form of diffusion equations with matching conditions across the interface between regions for population densities and fluxes. We combine the dispersal models with linear population growth models which assume that the population growth rate differs between regions of different habitat types. We use those models to study issues of refuge design. We specifically consider how the effectiveness of buffer zones depends on their size, quality, and the population's response to the interface between the buffer zone and the refuge.

Animals↗
PubMed2003

A fully coupled binary biochemical reactive-diffusion model with analytic solution.

Coupled multicomponent biochemical reactive diffusion underlies a variety of biological signalling processes and pharmacokinetic applications, such as paracrine signalling involving "cocktails" comprised of growth promoter/inhibitor factors and proteases associated with tumor angiogenesis, invasion and metastasis, extravascular drug delivery, and polymeric controlled-release drug codelivery design. Here, we present a model and develop a new analytic solution to illustrate the spatiotemporal behavior associated with fully coupled binary biochemical reactive diffusion. The complete coupling renders the solution appreciably more complex in structure and behavior than solutions for unicomponent or partially coupled models. Concentration behavior is illustrated by the computational simulation of binary-species tumor angiogenesis factor reactive-diffusion in the extravascular tissue matrix. The computational results indicate that (a) steady-state concentration profiles are achieved within 1 h of a change in factor production; (b) in the steady state, the spatial profiles of the two components tend to be similar; (c) exceedingly steep concentration gradients, involving several orders-of-magnitude differences in concentration over a few tenths of a millimeter, can occur in the vicinity of boundary sources due to inter-species reaction; (d) the concentration profiles of the two species differ from unicomponent predictions due to the simultaneous mass interchange between the two species. The analytic solution predictions are also used to provide a first-ever validation of a time-dependent, binary-component Crank-Nicholson numerical solution. The ability to quantitatively model interacting and often strongly varying concentration levels as a function of time and position can serve as a powerful complementary tool to experimental analyses for assessing disease state and interventional pharmacological efficacy, especially when the spatial scales on which in vivo behavior occurs taxes the limits of imaging capabilities.

Angiogenesis Inducing Agents↗
PubMed2002

Pattern formation in reaction-diffusion models with nonuniform domain growth.

Recent examples of biological pattern formation where a pattern changes qualitatively as the underlying domain grows have given rise to renewed interest in the reaction-diffusion (Turing) model for pattern formation. Several authors have now reported studies showing that with the addition of domain growth the Turing model can generate sequences of patterns consistent with experimental observations. These studies demonstrate the tendency for the symmetrical splitting or insertion of concentration peaks in response to domain growth. This process has also been suggested as a mechanism for reliable pattern selection. However, thus far authors have only considered the restricted case where growth is uniform throughout the domain. In this paper we generalize our recent results for reaction-diffusion pattern formation on growing domains to consider the effects of spatially nonuniform growth. The purpose is twofold: firstly to demonstrate that the addition of weak spatial heterogeneity does not significantly alter pattern selection from the uniform case, but secondly that sufficiently strong nonuniformity, for example where only a restricted part of the domain is growing, can give rise to sequences of patterns not seen for the uniform case, giving a further mechanism for controlling pattern selection. A framework for modelling is presented in which domain expansion and boundary (apical) growth are unified in a consistent manner. The results have implications for all reaction-diffusion type models subject to underlying domain growth.

Animals↗
PubMed2006

A simple technique to characterize proximal and peripheral nitric oxide exchange using constant flow exhalations and an axial diffusion model.

The most common technique employed to describe pulmonary gas exchange of nitric oxide (NO) combines multiple constant flow exhalations with a two-compartment model (2CM) that neglects 1) the trumpet shape (increasing surface area per unit volume) of the airway tree and 2) gas phase axial diffusion of NO. However, recent evidence suggests that these features of the lungs are important determinants of NO exchange. The goal of this study is to present an algorithm that characterizes NO exchange using multiple constant flow exhalations and a model that considers the trumpet shape of the airway tree and axial diffusion (model TMAD). Solution of the diffusion equation for the TMAD for exhalation flows >100 ml/s can be reduced to the same linear relationship between the NO elimination rate and the flow; however, the interpretation of the slope and the intercept depend on the model. We tested the TMAD in healthy subjects (n = 8) using commonly used and easily performed exhalation flows (100, 150, 200, and 250 ml/s). Compared with the 2CM, estimates (mean +/- SD) from the TMAD for the maximum airway flux are statistically higher (J'aw(NO) = 770 +/- 470 compared with 440 +/- 270 pl/s), whereas estimates for the steady-state alveolar concentration are statistically lower (CA(NO) = 0.66 +/- 0.98 compared with 1.2 +/- 0.80 parts/billion). Furthermore, CA(NO) from the TMAD is not different from zero. We conclude that proximal (airways) NO production is larger than previously predicted with the 2CM and that peripheral (respiratory bronchioles and alveoli) NO is near zero in healthy subjects.

Adult↗
PubMed1980

Diffusion models for the squid axon Schwann cell layer.

The Schwann cell, basement membrane, and connective tissue layers that surround the squid giant axon and constitute barriers to diffusion, were modeled in a number of ways to analyze various experimental results. The experiments considered are (a) the time-course of the potassium concentration in the space between the Schwann cell and the axon membrane (from now on referred to as the F-H space) after an initial loading, (b) the time-course of sodium concentration in the F-H space after a sudden change in the sodium concentration in the external fluid; (c) the time-course of the concentration of tetrodotoxin (TTX) or saxitoxin (STX) in the F-H space after a sudden change in external concentration, including (or not) the effects of specific binding of TTX or STX to sites on the axon membrane and nonsaturable binding to sites in the F-H space or in the spaces (clefts) between Schwann cells; (d) the effects of the F-H space, clefts, and diffusion into the clefts from the outside (from now on referred to as convergence into the clefts) on the measured series resistance.The analysis shows that (1) in no case is it necessary to include the effects of the convergence into the clefts from the outside; (2) in case a, the basement membrane, connective tissue layers, and the unstirred layer may be neglected, i.e., the clefts are rate limiting; (3) in case b the clefts may be neglected, i.e., the unstirred layer is rate limiting; (4) in most cases the clefts may be replaced by an equivalent thin diffusion barrier.

Animals↗
PubMed1992

The effect of auxiliary conditions on intestinal unstirred layer diffusion modelled by numerical simulation.

Estimation of intestinal unstirred layer thickness usually involves inducing transmural potential difference changes by altering the content of the solution used to perfuse the small intestine. Osmotically active solutes, such as mannitol, when added to the luminal solution diffuse across the unstirred water layer (UWL) and induce osmotically dependent changes in potential difference. As an alternative procedure, the sodium ion in the luminal fluid can be replaced by another ion. As the sodium ion diffuses out of the UWL, the change in concentration next to the intestinal membrane alters the transmural potential difference. In both cases, UWL thickness is calculated from the time course of the potential difference changes, using a solution to the diffusion equation. The diffusion equation solution which allows the calculation of intestinal unstirred layer thickness was examined by simulation, using the method of numerical solutions. This process readily allows examination of the time course of diffusion under various imposed circumstances. The existing model for diffusion across the unstirred layer is based on auxiliary conditions which are unlikely to be fulfilled in the same intestine. The present simulation additionally incorporated the effects of membrane permeability, fluid absorption and less than instantaneous bulk phase concentration change. Simulation indicated that changes within the physiologically relevant range in the chosen auxiliary conditions (with the real unstirred layer length kept constant) can alter estimates of the apparent half-time. Consequently, changes in parameters unassociated with the unstirred layer would be misconstrued as alterations in unstirred layer thickness.

Gastrointestinal Contents↗
PubMed2006

Multiscale modeling of protein transport in silicon membrane nanochannels. Part 2. From molecular parameters to a predictive continuum diffusion model.

Transport and surface interactions of proteins in nanopore membranes play a key role in many processes of biomedical importance. Although the use of porous materials provides a large surface-to-volume ratio, the efficiency of the operations is often determined by transport behavior, and this is complicated by the fact that transport paths (i.e., the pores) are frequently of molecular dimensions. Under these conditions, a protein diffusion can be slower than predicted from Fick law. The main contribution of this paper is the development of a mathematical model of this phenomenon, whose parameters are computed via molecular modeling, as described Part 1. Our multiscale modeling methodology, validated by using experimental results related to the diffusion of lysozyme molecules, constitutes an "ab initio" recipe, for which no experimental data are needed to predict the protein release, and can be tailored in principle to match any different protein and any different surface, thus filling gap between the nano and the macroscale.

Computer Simulation↗
PubMed1998

A gravitational diffusion model without dark matter.

In this model, without dark matter, the flat rotation curves of galaxies and the mass-to-light ratios of clusters of galaxies are described quantitatively. The hypothesis is that the agent of gravitational force is propagated as if it were scattered with a mean free path of approximately 5 kiloparsecs. As a result, the force between moderately distant masses, separated by more than the mean free path, diminishes as the inverse first power of the distance, following diffusion equations, and describes the flat rotation curves of galaxies. The force between masses separated by <1 kiloparsec diminishes as the inverse square of distance. The excess gravitational force (ratio of 1/r:1/r2) increases with the scale of structures from galaxies to clusters of galaxies. However, there is reduced force at great distances because of the approximately 12 billion years that has been available for diffusion to occur. This model with a mean free path of approximately 5 kiloparsecs predicts a maximum excess force of a few hundredfold for objects the size of galactic clusters a few megaparsecs in size. With only a single free parameter, the predicted curve for excess gravitational force vs. size of structures fits reasonably well with observations from those for dwarf galaxies through galactic clusters. Under the diffusion model, no matter is proposed in addition to the observed baryons plus radiation and thus the proposed density of the universe is only a few percent of that required for closure.

Journal Article↗
PubMed1977

Application of a convective diffusion model to membrane transport.

Studies were carried out on the permeation rate of butambed through a dimethicone membrane. Under conditions of "aqueous diffusion layer control", the permeation rate was accurately described by a mathematical model based on convective diffusion theory. In accordance with model, the rate of permeation from a saturated donor phase was shown to be equal to the rate of dissolution from a pure solid.

Aminobenzoates↗
PubMed2006

Phase transition of triplet reaction-diffusion models.

The phase transitions classes of reaction-diffusion systems with multiparticle reactions are an open challenging problem. Large scale simulations are applied for the 3A --> 4A, 3A --> 2A and the 3A --> 4A, 3A --> [formula : see text] triplet reaction models with site occupation restriction in one dimension. Static and dynamic mean-field scaling are observed with signs of logarithmic corrections suggesting d(c) = 1 upper critical dimension for this family of models.

Journal Article↗