Nernst-planck analog equations and stationary state membrane electric potentials

A set of coupled nonlinear differential equations, determining the concentration profiles and electric potentials valid for isothermal transport of ions and molecules across a diffusion barrier are formulated, using a correction to the limiting expression for chemical potential gradients and the molecular expression for frictional force. These differential equations are similar to Nernst-Planck equations and reduce to these under appropriate approximations. Solutions of these equations valid under specified conditions are presented. Expressions for permeability, concentration profiles of many ion systems are included.     

Mathematical Modeling of a Carrier-Mediated Transport Process in a Liquid Membrane

An analysis of the reaction diffusion in a carrier-mediated transport process through a membrane is presented. A simple approximate analytical expression of concentration profiles is derived in terms of all dimensionless parameters. Furthermore, in this work we employ the homotopy perturbation method to solve the nonlinear reaction–diffusion equations. Moreover, the analytical results have been compared to the numerical simulation using the Matlab program. The simulated results are comparable with the appropriate theories. The results obtained in this work are valid for the entire solution domain.     

Permeation of ammonia across bilayer lipid membranes studied by ammonium ion selective microelectrodes.

Ammonium ion and proton concentration profiles near the surface of a planar bilayer lipid membrane (BLM) generated by an ammonium ion gradient across the BLM are studied by means of microelectrodes. If the concentration of the weak base is small compared with the buffer capacity of the medium, the experimental results are well described by the standard physiological model in which the transmembrane transport is assumed to be limited by diffusion across unstirred layers (USLs) adjacent to the membrane at basic pH values (pH > pKa) and by the permeation across the membrane itself at acidic pH values. In a poorly buffered medium, however, these predictions are not fulfilled. A pH gradient that develops within the USL must be taken into account under these conditions. From the concentration distribution of ammonium ions recorded at both sides of the BLM, the membrane permeability for ammonia is determined for BLMs of different lipid composition (48 x 10(-3) cm/s in the case of diphytanoyl phosphatidylcholine). A theoretical model of weak electrolyte transport that is based on the knowledge of reaction and diffusion rates is found to describe well the experimental profiles under any conditions. The microelectrode technique can be applied for the study of the membrane permeability of other weak acids or bases, even if no microsensor for the substance under study is available, because with the help of the theoretical model the membrane permeability values can be estimated from pH profiles alone. The accuracy of such measurements is limited, however, because small changes in the equilibrium constants, diffusion coefficients, or concentrations used for computations create a systematic error.     

Diffusion inside microtubules

Recent high-resolution analysis of tubulin's structure has led to the prediction that the taxol binding site and a tubulin acetylation site are on the interior of microtubules, suggesting that diffusion inside microtubules is potentially a biologically and clinically important process. To assess the rates of transport inside microtubules, predictions of diffusion time scales and concentration profiles were made using a model for diffusion with parameters estimated from experiments reported in the literature. Three specific cases were considered: 1) diffusion of αβ-tubulin dimer, 2) diffusion/binding of taxol, and 3) diffusion/binding of an antibody specific for an epitope on the microtubule's interior surface. In the first case tubulin is predicted to require only ∼1 min to reach half the equilibrium concentration in the center of a 40 μm microtubule open at both ends. This relatively rapid transport occurs because of a lack of appreciable affinity between tubulin and the microtubule inner surface and occurs in spite of a three-fold reduction in diffusivity due to hindrance. By contrast the transport of taxol is much slower, requiring days (at nm concentrations) to reach half the equilibrium concentration in the center of a 40 μm microtubule having both ends open. This slow transport is the result of fast, reversible taxol binding to the microtubule's interior surface and the large capacity for taxol (∼12 mm based on interior volume of the microtubule). An antibody directed toward an epitope in the microtubule's interior is predicted to require years to approach equilibrium. These results are difficult to reconcile with previous experimental results where substantial taxol and antibody binding is achieved in minutes, suggesting that these binding sites are on the microtubule exterior. The slow transport rates also suggest that microtubules might be able to serve as vehicles for controlled-release of drugs. Received: 2 March 1998 / Revised version: 3 May 1998 / Accepted: 3 May 1998     

Monte Carlo simulation of simultaneous gas flow and diffusion in an asymmetric distal pulmonary airway model

In order to evaluate the effect of anatomic asymmetries on the gas concentration distribution in the pulmonary airways, a Monte Carlo simulation of combined bulk flow and molecular diffusion was carried out in a realistic distal airway model (Parkeret al., 1971). This airway model, composed of branches distal to the 0.5-ram diameter airways, contained an upper symmetric segment consisting of four generations of conducting airways and a lower asymmetric segment of alveolar ducts and sacs arranged in five transport paths of varying lengths. In accounting for the volume increases of these ducts and sacs occurring during normal respiration, uniform alveolar filling rates and a fixed length-to-diameter ratio of all airways were assumed. For a pulse injection of inert tracer gas, the simulation was employed to determine the longitudinal concentration profiles in the conducting airways. In the alveolated airways, not only were the longitudinal profiles determined along each path, but radial transport from the core to the periphery of the airways was considered. The results of the simulations indicate that geometric asymmetries alone contribute substantially to regional concentration variations in the distal airways. For example, when a gas bolus is injected at mid*inspiration, there are concentration differences as great as 40% between two points along different transport paths located equi-distant from the proximal end of the model. As viewed from the terminal end of the model (acinus), average concentration differences as large as 6-to-1 exist between the longest and shortest transport paths respectively for gas boli introduced near the end of inspiration. The results further indicate because of large radial diffusion rates, no significant concentration differences exist between the periphery a-ld the central core of alveolated airways. Simulation of the expired concentration profiles indicate that boll injected very late during inspiration exhibit a sloping tail, unlike the earlier injected boll whose tails are virtually horizontal. Through the use of superposition teehniqnes, it was found that these sloping tails correspond to an alveolar slope of 1.5 vol% between 750 and 1250 ml expired for a continuous washing of tracer. This result is in disagreement with other transport analyses which did not directly account for the effect of geometric asymmetries.     

Monitoring of artificial enzyme membrane systems by electric currents: I. Analytical treatment with direct current and buffered conductivity

Continuous electric fields (E) modify the transport flows and the intramembrane concentration profiles of protons or of ionic substrates or cofactors (inhibitors). These ‘mediators’ induce variations in enzyme activity, quantifiable by a generalized Damköhler group II Ψ distinguishing electrotransport reactions from diffusion reactions. For three typical reaction schemas, using only one mediator, the steady-state equations have been established. Depending on boundary conditions, the direction of electric current (for asymmetrical systems) and the value of Ψ. activations, inhibitions or activations followed by inactivations have been found. With buffered conductivity (supporting electrolyte), the limiting concentration profiles (E → ∞) are uniformly equal to the boundary values; i.e., diffusion constraints are suppressed and the regime is controlled by the reaction. The calculations give the relative activity variations for partially suppressed transport controls.     

External and internal diffusion in heterogeneous enzymes systems

The intrusion of diffusion in heterogeneous enzyme reactions, which follow. Michaelis-Menten kinetics, is quantitatively characterized by dimensionless parameters that are independent of the substrate concentration. The effects of these parameters on the overall rate of reaction is illustrated on plots commonly employed in enzyme kinetics. The departure from Michaelis-Menten kinetics due to diffusion limitations can be best assessed by using Hofstee plots which are also suitable to distinguish between internal and external transport effects. A graphical method is described for the evaluation of the reaction rate as a function of the surface concentration of the substrate from measured data.     

Microbial response to environmental gradients in a ceramic-based diffusion system

A solid, porous matrix was used to establish steady-state concentration profiles upon which microbial responses to concentration gradients of nutrients or antimicrobial agents could be quantified. This technique relies on the development of spatially defined concentration gradients across a ceramic plate resulting from the diffusion of solutes through the porous ceramic matrix. A two-dimensional, finite-element numerical transport model was used to predict the establishment of concentration profiles, after which concentration profiles of conservative tracers were quantified fluorometrically and chemically at the solid-liquid interface to verify the simulated profiles. Microbial growth responses to nutrient, hypochloride, and antimicrobial concentration gradients were then quantified using epifluorescent or scanning confocal laser microscopy. The observed microbial response verified the establishment and maintenance of stable concentration gradients along the solid-liquid interface. These results indicate the ceramic diffusion system has potential for the isolation of heterogeneous microbial communities as well as for testing the efficacy of antimicrobial agents. In addition, the durability of the solid matrix allowed long-term investigations, making this approach preferable to conventional gel-stabilized systems that are impeded by erosion as well as expansion or shrinkage of the gel.     

Computer simulation of single-file transport

A stochastic model of single-file transport was developed as the Markov process in continuous time technique. The model was constructed using an EC-1060 computer. Unidirectional fluxes were investigated and populations of channels were correlated with flux fluctuations. The profiles of channel populations were shown to have nonlinear shapes even with the transport of nonelectrolyte (the classical diffusion approach gives linear profiles). The relationship between the paired correlation function F(AB) and the concentration of transported particles was examined. The F(AB) profile was shown to become flattened (or exponential for asymmetrical cases) at high concentrations. The concentration dependence jA/jA0 ratio were analyzed, where jA is a single-file unidirectional flux, jA0 is unidirectional flux for the case of free diffusion. An interesting "stack" phenomenon was observed for abnormal time correlations of single-file fluxes.     

Fallout radiocesium in an Antarctic region:

To improve the knowledge about the (137)Cs spatial distribution and vertical migration in soils of the Southern Hemisphere, the total areal activity density and the vertical transport parameters of this radionuclide were measured in an Antarctic region. For this purpose vegetation and incremental soil samples were collected at 21 representative sites located at 4 islands of the South Shetland Archipelago: King George, Robert, Greenwich and Snow (62-63 degrees S and 58-62 degrees W). The total (137)Cs activity density varied considerably from 118 to 662 Bq m(-2) (median 384 Bq m(-2), reference date 1995), with a high percentage of the total activity retained in the vegetation cover (5-98% in moss, 3-20% in lichen and 4-12% in grass). At most sites, the maximum activity density in soil was observed in the top layer from where it decreased continuously. To evaluate the transport parameters of (137)Cs from the activity-depth profiles, the classical convection-diffusion model was used based on the time-course of the annual deposition density of (137)Cs at the studied region. The values for the diffusion coefficient D(s) (median 0.043 cm(2) year(-1)) and the convection velocity v(s) (median -0.012 cm year(-1)) of radiocesium observed under a polar climate are small compared to the transport parameters determined in temperate zones. The data also indicate that at these sites the convectional transport of (137)Cs is almost negligible compared to the transport by diffusion. The high vulnerability of the Antarctic soils to (137)Cs deposition, as a consequence of its very slow transport due to the extreme climatic conditions at these latitudes, has been confirmed.     

Investigating sensitivity coefficients characterizing the response of a model of tau protein transport in an axon to model parameters

Evaluating the sensitivity of biological models to various model parameters is a critical step towards advancing our understanding of biological systems. In this paper, we investigated sensitivity coefficients for a model simulating transport of tau protein along the axon. This is an important problem due to the relevance of tau transport and agglomeration to Alzheimer’s disease and other tauopathies, such as some forms of parkinsonism. The sensitivity coefficients that we obtained characterize how strongly three observables (the tau concentration, average tau velocity, and the percentage of tau bound to microtubules) depend on model parameters. The fact that the observables strongly depend on a parameter characterizing tau transition from the retrograde to the anterograde kinetic states suggests the importance of motor-driven transport of tau. The observables are sensitive to kinetic constants characterizing tau concentration in the free (cytosolic) state only at small distances from the soma. Cytosolic tau can only be transported by diffusion, suggesting that diffusion-driven transport of tau only plays a role in the proximal axon. Our analysis also shows the location in the axon in which an observable has the greatest sensitivity to a certain parameter. For most parameters, this location is in the proximal axon. This could be useful for designing an experiment aimed at determining the value of this parameter. We also analyzed sensitivity of the average tau velocity, the total tau concentration, and the percentage of microtubule-bound tau to cytosolic diffusivity of tau and diffusivity of bound tau along the MT lattice. The model predicts that at small distances from the soma the effect of these two diffusion processes is comparable.     

Isoelectric focusing using non-amphoteric buffers in free solution: I. Determination of stable concentration profiles

For large-scale separations of proteins, the use of simple non-amphoteric buffers in free solution and in multicompartment electrolyzers seems promising for industrial applications. The stabilization of a pH profile with this type of buffer requires the strict observation of two conditions: choice of an adequate buffer; stationary profiles of concentrations. During electrolysis in free solution, the ions of the buffer are displaced across the compartments by migration and by diffusion. To keep a stationary composition, the inflow and outflow of all individual ionic species through each compartment must be identical. At high current, diffusion may be neglected against migration and the ionic flows will be identical if the transport number of each ion is constant at each location within the cell. In these conditions, stationary compositions will be independent of the electric current. This condition of constant transport numbers implies the use of profiles of buffer concentrations different from those published up to now. The new equations for these profiles of concentrations are given in the present paper. The constant migration of the ions must be compensated in the end compartments of the isoelectric focusing cell to provide a stable steady state. Two methods are proposed in the literature: the buffer renewal method and the external recycling method (rheoelectrolysis). Here modified buffer renewal method is proposed. Using stationary mass balances, analytical equations are given to calculate the flows and the composition of the solutions to be recycled or added. Using these equations and the profiles of concentrations to keep constant transport numbers, it is demonstrated that only a renewal of the buffers in the end compartments may lead to stable pH profiles and thus to valid conditions of separation.     

Facilitated transport of oxygen in the presence of membranes in the diffusion path.

Most of the experimental observations on facilitated transport have been done with millipore filters, and all the theoretical studies have assumed homogeneous spatial properties. In striated muscle there exist membranes that may impede the diffusion of the carrier myoglobin. In this paper a theoretical study is undertaken to analyze the transport in the presence of membranes in the diffusion path. For the numerical computations physiologically relevant values of the parameters were chosen. The numerical results indicate that the presence of membranes tends to decrease the facilitation. For the nonlinear chemical kinetics of the reaction of oxygen with the carrier, this decrement also depends on the location of the membranes. At the higher oxygen concentration side of each membrane the flow of combined oxygen is transferred to the flow of dissolved oxygen. The reverse process occurs at the lower concentration side. Jump discontinuities of the concentration of the oxygen-carrier compound at each membrane are associated with these transfers. The decrement of facilitation is due to the cumulative effect of these jump discontinuities.     

Measurement of mass transport and reaction parameters in bulk solution using photobleaching. Reaction limited binding regime.

Fluorescence recovery after photobleaching (FRAP) has been used previously to investigate the kinetics of binding to biological surfaces. The present study adapts and further develops this technique for the quantification of mass transport and reaction parameters in bulk media. The technique's ability to obtain the bulk diffusion coefficient, concentration of binding sites, and equilibrium binding constant for ligand/receptor interactions in the reaction limited binding regime is assessed using the B72.3/TAG-72 monoclonal antibody/tumor associated antigen interaction as a model in vitro system. Measurements were independently verified using fluorometry. The bulk diffusion coefficient, concentration of binding sites and equilibrium binding constant for the system investigated were 6.1 +/- 1.1 x 10(-7) cm2/s, 4.4 +/- 0.6 x 10(-7) M, and 2.5 +/- 1.6 x 10(7) M-1, respectively. Model robustness and the applicability of the technique for in vivo quantification of mass transport and reaction parameters are addressed. With a suitable animal model, it is believed that this technique is capable of quantifying mass transport and reaction parameters in vivo.     

Estimation of thickness of concentration boundary layers by osmotic volume flux determination

The estimation method of the concentration boundary layers thicknesses (δ) in a single-membrane system containing non-electrolytic binary or ternary solutions was devised using the Kedem-Katchalsky formalism. A square equation used in this method contains membrane transport (L(p), σ, ω) and solution (D, C) parameters as well as a volume osmotic flux (J(v)). These values can be determined in a series of independent experiments. Calculated values δ are nonlinearly dependent on the concentrations of investigated solutions and the membrane system configuration. These nonlinearities are the effect of a competition between spontaneously occurring diffusion and natural convection. The mathematical model based on Kedem-Katchalsky equations and a concentration Rayleigh number (R(C)) was presented. On the basis of this model we introduce the dimensionless parameter, called by us a Katchalsky number (Ka), modifies R(C) of membrane transport. The critical value of this number well describes a moment of transition from the state of diffusion into convective diffusion membrane transport.     

Slow potential changes due to transport number effects in cells with unstirred membrane invaginations or dendrites

Summary Many neurones are extremely invaginated and possess branching processes, axons and dendrites. In general, they are surrounded by a restricted diffusion space. Many of these cells exhibit large, slow potential changes during the passage of current across their membranes. Whenever currents cross membranes separating aqueous solutions, differences in transport numbers of the major permeant ions give rise to local concentration changes of these ions adjacent to the membranes, which will result in various electrical and osmotic effects. These transport number effects are expected to be enhanced by the presence of membrane invaginations. Dendrites are equivalent to reversed invaginations and there should be significant changes in concentrations of permeant ions within them. In general, the effects of such changes on the electrical response of a cell will be greater when the concentration of a major permeant ion is low. The effects have been modelled in terms of two nondimensional parameters: the invagination transport number parameter and the relative area occupied by the invaginations A. If these two parameters are known, the magnitudes and time course of the slow potential changes can immediately be estimated and the time course converted to real time, if the length of the invaginations (l) and ionic diffusion coefficient (D) within them are also known. Both analytical and numerical solutions have been given and predictions compared. It is shown that in the case of large currents and potentials the analytical solution predictions will underestimate the magnitudes and rates of onset of the voltage responses. The relative magnitude of the transport number effect within the invaginations (or dendrites) and other transport number contributions to slow potential changes have also been assessed and order-of-magnitude values of these are estimated for some biological data.     

Development of Steady-State Diffusion Gradients for the Cultivation of Degradative Microbial Consortia

A diffusion gradient plate was constructed and evaluated for its potential use in the isolation of degradative microbial consortia from natural habitats. In this model, a steady-state concentration gradient of diclofop methyl, established by diffusion through an agarose gel, provided the carbon for microbial growth. Colonization of the gel surface was observed with epifluorescence and scanning confocal laser microscopy to determine microbial responses to the diclofop gradient. A detectable gradient developed over a narrow band (<10 mm). Consequently, quantitative analyses of the microbial response to the gradient were difficult to obtain. A two-dimensional, finite-element numerical transport model for advective-diffusive transport was used to simulate concentration and flux profiles in the physical model. The simulated profiles were correlated with the measured concentration gradient (R² = 0.89) and the cell numbers on the gel surface (R² = 0.85). The numerical model was subsequently used to redesign the physical model. The detectable concentration gradient in the modified physical model extended over the length of the gel (38 mm). The simulated profile again showed a good correlation with the measured profile (R² = 0.96) and the microbial responses to the concentration gradient (R² = 0.99). It was concluded that these gradients provide the steady-state environments needed to sustain steady-state consortia. They also provide a physical pathway for the development of degradative biofilms from low to high concentrations of toxicants and simulate conditions under which low concentrations of toxicant are supplied at a constant flux over long periods of time, such as the conditions that could occur in natural environments.     

An Inverse Analysis Approach to the Characterization of Chemical Transport in Paints

The ability to directly characterize chemical transport and interactions that occur within a material (i.e., subsurface dynamics) is a vital component in understanding contaminant mass transport and the ability to decontaminate materials. If a material is contaminated, over time, the transport of highly toxic chemicals (such as chemical warfare agent species) out of the material can result in vapor exposure or transfer to the skin, which can result in percutaneous exposure to personnel who interact with the material. Due to the high toxicity of chemical warfare agents, the release of trace chemical quantities is of significant concern. Mapping subsurface concentration distribution and transport characteristics of absorbed agents enables exposure hazards to be assessed in untested conditions. Furthermore, these tools can be used to characterize subsurface reaction dynamics to ultimately design improved decontaminants or decontamination procedures. To achieve this goal, an inverse analysis mass transport modeling approach was developed that utilizes time-resolved mass spectroscopy measurements of vapor emission from contaminated paint coatings as the input parameter for calculation of subsurface concentration profiles. Details are provided on sample preparation, including contaminant and material handling, the application of mass spectrometry for the measurement of emitted contaminant vapor, and the implementation of inverse analysis using a physics-based diffusion model to determine transport properties of live chemical warfare agents including distilled mustard (HD) and the nerve agent VX.     

Diffusion with memory in two cases of biological interest

The complexity of a biological structure, such as membrane where the transport process may carry solid particles which may obstruct some of the pores, diminishing their size and making the permeability dependent on the local structure of the medium, suggests the introduction of a space-dependent diffusion constant. In this note, the profile concentration of diffusing solutes inside a cell membrane has been calculated on the basis of the Fick diffusion equation modified by introducing a memory formalism (diffusion with memory). This approach has been employed to describe the concentration profile inside the membrane when a sudden change of the concentration in the medium bathing one of its face is applied for a limited interval of time. A further application of the method concerns the so-called concentration boundary layer that occurs at the membrane-aqueous medium interface, where the solute concentration depends, even at considerable depth, on the local structure of the interface. These profiles are compared to some recent experiments concerning the diffusion of ethanol in a layer close to a nephrophane membrane. This approach generalizes the diffusion models based on the Fick equation to more complex systems, where a space-independent diffusion coefficient could be inappropriate to take into account the large variety of diffusion processes in biological systems.