Interpretation of the Ussing flux ratio from the fluctuation theorem

The fluctuation theorem gives a mathematical expression to quantify the probability of observing events violating the second law of thermodynamics in a small system over a short period of time. The theorem predicts the ratio of forward (entropy-producing) runs to the backward (entropy-consuming) runs for a nanometer-sized molecular machine in a nonequilibrium system. However, few experimental verifications of the theorem have been carried out. In this paper, I show that the Ussing flux ratio, the ratio of outward to inward unidirectional ion fluxes across a membrane channel, can be derived from the fluctuation theorem if we consider the ion channel and the contacting solutions as a small nonequilibrium system. The entropy change due to ion electrodiffusion is expressed from the fundamental equation for the entropy change. Thus, the empirical flux ratio equation can be interpreted from the more general fluctuation theorem, and serves as a verification of the theorem.     

Deviations from the flux ratio equation for chloride ions in ouabain- and acetazolamide-treated frog skin

In order to establish whether or not chloride ions behave as freely moving particles in “passive”, i.e. ouabain-and acetazolamide-treated, frog skin, tracer fluxes of ³⁶Cl⁻ have been measured while a voltage (generally +40 mV, serosal side positive) across the skin was applied. Ussing's flux ratio equation has been used as a criterion for this type of transport. One group of skin samples exhibited significant exchange diffusion phenomena. Most samples in a second group either behaved according to the flux ratio equation or showed significant and extreme exchange diffusion. From flux ratios obtained at two different voltages across various skin samples, showing extreme exchange diffusion, it appeared that the simple form of Kedem and Essig's law derived from irreversible thermodynamics, which is valid for homogeneous systems, does not apply to the type of exchange diffusion found. The system can, however, be described by a 1 : 1 exchange mechanism working in parallel with a diffusional pathway. The ratio exchange flux/observed efflux must then have a constant value (0.83) at the voltages applied, which implies that the exchange flux is voltage dependent. By comparison with iodide flux experiments as carried out by Ussing, it is shown that iodide exhibits the same type of exchange diffusion. A carrier, possibly responsibe for the observed behaviour, is described.     

Deviations from the flux ratio equation for chloride ions in ouabain- and acetazolamide-treated frog skin.

In order to establish whether or not chloride ions behave as freely moving particles in "passive", i.e. ouabain- and acetazolamide-treated, frog skin, tracer fluxes of 36Cl-have been measured while a voltage (generally +40 mV, serosal side positive) across the skin was applied. Ussing's flux ratio equation has been used as a criterion for this type of transport. One group of skin samples exhibited significant exchange diffusion phenomena. Most samples in a second group either behaved according to the flux ratio equation of showed significant and extreme exchange diffusion. From flux ratios obtained at two different voltages across various skin samples, showing extreme exchange diffusion, it appeared that the simple form of Kedem and Essig's law derived from irreversible thermodynamics, which is valid for homogeneous systems, does not apply to the type of exchange diffusion found. The system can, however, be described by a 1:1 exchange mechanism working in parallel with a diffusional pathway. The ratio exchange flux/observed efflux must then have a constant value (0.83) at the voltages appled, which implies that the exchange flux is voltage dependent. By comparison with iodide flux experiments as carried out by Ussing, it is shown that iodide exhibits the same type of exchange diffusion. A carrier, possibly responsible for the observed behaviour, is described.     

Single-file diffusion through K+ channels in frog skin epithelium

Summary The ratio between the unidirectional fluxes of K⁺ across the frog skin with K-permeable outer membranes was determined in the absence of Na⁺ in the apical solutions. The experiments were performed under presteady-state conditions to be able to separate the flux ratio for K⁺ through the cells from contributions to the fluxes through extracellular leaks. The cellular flux ratio deviated strongly from the value calculated from the flux ratio for electrodiffusion. The experiments can be explained if the passive K transport through the epithelial cells proceeds through specific channels by single-file diffusion with a flux ratio exponent of about 2.5.     

Ionic transfer across the isolated frog large intestine

The unidirectional fluxes of sodium, chloride, and of the bicarbonate and CO(2) pair were determined across the isolated large intestine of the bullfrog, Rana catesbiana. The isolated large intestine of the frog is characterized by a mean transmembrane potential of 45 mv., serosal surface positive with respect to mucosal. The unidirectional sodium flux from mucosal to serosal surface was found to be equal to the short-circuit current, thus the net flux was less than the simultaneous short-circuit current. This discrepancy between active sodium transport and short-circuit current can be attributed to the active transport of cation in the same direction as sodium and/or the active transport of anion in the opposite direction. The unidirectional fluxes of chloride and the bicarbonate and CO(2) pair revealed no evidence for active transport of either anion. A quantitative study of chloride fluxes at 45 mv. revealed a flux ratio of 1.8 which is considerably less than a ratio of 6 expected for free passive diffusion. It was concluded that a considerable proportion of the isotopic transfer of chloride could be attributed to "exchange diffusion." Study of the electrical properties of the isolated frog colon reveals that it can be treated as a simple D. C. resistance over the range of -20 to +95 mv.     

Single and Multicomponent Gas Phase Diffusion in a Porous Media: Modeling and Laboratory Measurements

Subsurface vapor migration of volatile chemicals may impact ambient and indoor air quality, increasing the importance to investigate the fate and transport of these chemicals. This project involved both modeling and experimental work to study the vapor phase transport behavior of single, binary, and tertiary component systems present in the gas phase. The experimental phase resulted in the development of a diffusion cell for measuring vapor phase transport. Three organic compounds (toluene, cumene, and isooctane) common to petroleum-based products were selected. The objective of this research project was to evaluate how the rate of a component diffusing alone in a stagnant gas mixture compares to the rate of the same component when diffusing in the presence of multiple diffusing species. The equipment was first validated by measuring the unobstructed gas phase diffusion fluxes for each organic compound. The diffusion coefficients were then calculated from the experimentally measured diffusive fluxes using Fick's Law at 20 and 25°C and compared to the respective literature values. The experimental/literature (E/L) ratio was calculated for toluene, cumene, and isooctane. The range of the average E/L ratio for the single component data sets is 0.93 to 1.05. The validation data provided the baseline for extending the research to multicomponent data. The multi-component systems research was characterized as either binary systems or a three-component system. The binary systems were either isooctane/tolu-ene or isooctane/cumene. The three-component system consisted of a mixture of all three compounds. For both temperatures and all compounds the flux rate decreased for any single component due to the dilution effect by incorporation into a mixture. Applying Fick's Law to calculate the effective diffusion coefficient for each compound that corresponded to the resulting concentration gradient by the mixture, an enhancement in the diffusive flux of each individual species was observed. This enhancement can be explained by a compositional coupling of each component to all others which results in a total vapor phase mass flux comprised of both diffusive and pseudo-advective mass transport. This pseudo-advective component is attributed to simultaneous diffusion of other species in the presence of the one of interest. Since this research project incorporated a mixture of toluene and cumene present in a background carrier solvent of isooctane, by calculating the ratios Dexp(3-component)/ Dexp(2-component) and Dexp(2-compo-nent)/Dexp(single component), an estimate is obtained of the enhancement effect due to the advective component of simultaneously diffusing chemicals. The diffusivity ratios for the three-component system compared to the dual component system ranged from 0.8 to 3.7. The diffusivity ratios for individual compounds were for 1.5-3.7 cumene, 0.8-1.2 for toluene, and 1.0-1.2 for isooctane. The diffusivity ratios for the dual component system to the single component systems ranged from 0.8 to 4.0. The range of diffusivity ratios for individual compounds were for 2.0-4.0 for cumene, 0.8-1.6 for toluene, and 1.1-1.4 for isooctane. A ratio greater than 1.0 indicated an enhancement effect on the molecular diffusion rate due to the presence of one or more additional diffusing chemical species present. The majority of fate and transport models are based on single component behavior modeled by Fick's Law using the pure gas phase diffusion coefficient. The enhancement of the individual diffusive flux in a multicomponent mixture observed in this study and accounted for by pseudo-advec-tive mass transport results in an under-prediction of the actual multicomponent diffusive fluxes. It is recommended that a more rigorous diffusion equation such as the Stefan-Maxwell equation be considered for incorporation into vapor phase transport models when modeling multicomponent/ contaminant systems.     

The Validity of the Ussing Flux Ratio Equation in a Three-Dimensionally Inhomogeneous Membrane

Derivations of the Ussing flux ratio equation have, until now, required the membrane to be both bounded by parallel planes and homogeneous, except in the transmembrane direction. These constraints have been necessary for the theoretical demonstration that the equation is independent of membrane parameters in the absence of carriers, coupling, solvent drag, or “single-file” diffusion. In a new derivation, the flux ratio equation is shown to be valid in this kind of diffusion regime without regard to the three-dimensional structure of the membrane. Thus the constraints on both membrane homogeneity and membrane geometry are shown to be unnecessary. The general use of this equation to differentiate between simple, uncoupled diffusion and other membrane transport phenomena is thus placed on a firmer base. However, as in earlier derivations, it is necessary that isopotential, isobaric, constant concentration surfaces exist sufficiently close to the membrane on both of its sides.     

Transcapillary exchange of strontium and sucrose in canine tibia.

The walls of haversian capillaries have been proposed as a physiologic membrane controlling flux of solute between blood and bone. In this study, capillary permeability to 85Sr and [14C]sucrose was estimated in the dog tibia by using indicator dilution techniques. Sucrose was chosen as a tracer because it is inert and has no known transport system. The mean (+/-SD) observed ratio of permeabilities of 85Sr and sucrose was 2.36 +/- 0.46 (N = 14) which is not substantially different from the ratio of their free diffusion coefficients, 2.55. This ratio was not influenced when the dogs were made hyperparathyroid by injection of parathyroid hormone (2.16 +/- 0.55; N = 11). This suggests that free diffusion is the principal mechanism for moving 85Sr across the bone capillary wall.     

Analysis of the Components of Ionic Flux across a Membrane

The unidirectional flux of an ionic species may occur because of several mechanisms such as active transport, passive diffusion, exchange diffusion, etc. The contribution of such mechanisms to the total unidirectional flux across a membrane cannot be determined by only measuring that flux. It is shown that if the pertinent experimental data (the opposite unidirectional fluxes and the composite phenomenological resistance coefficient of the ionic species for a given electrochemical potential difference) obey a certain inequality, then the parameters of a model consisting of parallel, independent, active transport, and passive processes may be determined. Although the existence of "additional" processes including exchange diffusion, single-file pore diffusion, isotope interaction, etc. is not disproved, their existence is unnecessary if the inequality is satisfied. Two types of violations of the inequality may occur: (a) if the upper limit is disobeyed the presence of another substance contributing to the measured resistance and/or a constant affinity of the active transport process may be indicated; (b) if the lower limit is disobeyed it is necessary to postulate the existence of an additional process. For the latter type of violation, exchange diffusion is chosen as an example. Methods are given for determining the contribution of exchange diffusion, active transport, and passive diffusion to the unidirectional flux for some special cases.     

Bound-State Diffusion due to Binding to Flexible Polymers in a Selective Biofilter

Selective biofilters are used by cells to control the transport of proteins, nucleic acids, and other macromolecules. Biological filters demonstrate both high specificity and rapid motion or high flux of proteins. In contrast, high flux comes at the expense of selectivity in many synthetic filters. Binding can lead to selective transport in systems in which the bound particle can diffuse, but the mechanisms that lead to bound diffusion remain unclear. Previous theory has proposed a molecular mechanism of bound-state mobility based only on transient binding to flexible polymers. However, this mechanism has not been directly tested in experiments. We demonstrate that bound mobility via tethered diffusion can be engineered into a synthetic gel using protein fragments derived from the nuclear pore complex. The resulting bound-state diffusion is quantitatively consistent with theory. Our results suggest that synthetic biological filters can be designed to take advantage of tethered diffusion to give rapid, selective transport.     

Permselectivity of the glomerular capillary wall to macromolecules. I. Theoretical considerations.

The transport of macromolecules across the renal glomerular capillary wall has been described theoretically using flux equations based on (a) restricted transport through small pores, and (b) the Kedem-Katchalsky formulation. The various assumptions and limitations inherent in these two approaches are discussed. To examine the coupling between macromolecular solute transport and the determinants of glomerular filtration rate, these flux equations were combined with mass balance relations which allow for variations in the transmembrane driving forces along a glomerular capillary. It was predicted, using both pore theory and the Kedem-Katchalsky equations, that fractional solute clearance should be strongly dependent on the determinants of glomerular filtration rate when convection and diffusion both contribute to solute transport. When convection becomes the sole mechanism for transcapillary solute transport, however, fractional solute clearance is essentially independent of changes in the determinants of glomerular filtration rate. Consequently, unless diffusion is absent, fractional solute clearances alone are insufficient to characterize the permselective properties of the glomerular capillary wall, since these values may be altered by changes in glomerular pressures and flows as well as changes in the properties of the capillary wall per se.     

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.     

Analysis of lateral diffusion from a spherical cell surface to a tubular projection.

Cell surfaces are often heterogeneous with respect to the lateral distribution and mobility of membrane components. Because lateral mobility is related to membrane structure, measurement of a particular component's local diffusion coefficient within a distinct surface region provides useful information about the formation and maintenance of that region. Many structurally interesting cell surface features can be described as narrow tubular projections from the body of the cell. In a companion paper, we consider the thin "tethers" that can be mechanically drawn from the red blood cell membrane, and we measure the transport of fluorescent integral proteins from the surface of the cell body onto the tether. In this paper we present an analysis to describe the surface diffusion of membrane particles from a spherical shell onto a thin cylindrical process. Provision is made for different rates of diffusion within the two morphologically distinct regions. The relative role of each region in controlling the diffusive flux between regions is determined primarily by a single dimensionless parameter. This parameter incorporates the ratio of the two diffusion coefficients as well as the dimensions of each region. The analysis can be applied to a fluorescence photobleaching experiment in which the extended process is bleached. If the dimensions of the spherical cell body and the cylindrical extension are known, then the diffusion coefficients of both regions can be determined from the experimental fluorescence recovery curve.     

The flux-force relations across complex membranes with active transport

Equations are derived for the total material flux, and the total electric current flux, across a complex membrane system with active transport. The equations describe the fluxes as linear functions of forces across the system, and specifically of electrical potential, hydrostatic pressure, chemical potentials, and active transport rates. The equations can be simplified for experimental studies by making one or more of the forces equal to zero. The osmotic pressure difference across a membrane system is shown to be a function of the electrical potential and chemical potential differences and of the active transport rates. The transmembrane potential is shown to be the sum of a diffusion potential and an active transport potential. A simple equation is derived describing the current across a membrane as a linear function of the electrical potential and the active transport rate. Specific examples of the application of the equations to nerve membrane potentials are considered.     

Particle transport modeling in pulmonary airways with high-order elements

Inhaled particles can be either harmful (e.g., smoke, exhaust, viruses) or beneficial (e.g., a therapeutic drug). The accurate and computationally efficient simulation of particle transport and deposition remains a challenge because it requires the simultaneous solution of the Navier-Stokes equations and multiple advection-diffusion mass transport equations when the particles are modeled as multiple mono-dispersed populations. The solution of these equations requires that multiple length scales be resolved since the ratio of advection to diffusion varies among the different equations. Here, the spectral element method is examined because the high-order approximation provides greater flexibility in resolving multiple length scales. The problem geometry is based on the Weibel model A of the human airway for convergence tests and the first three generations of a typical rat airway for experimental validation. Particles in the size range 1 to 100 nm are simulated for deposition results. The particle concentration and flux were determined using meshes of varying coarseness to represent the geometry along with basis polynomials of order 5 to 11. The higher-order elements accurately propagate the short wavelengths contained in the advection-diffusion solution without sacrificing efficiency for the more computationally expensive Navier-Stokes solution. As the diffusion coefficient in the advection-diffusion equation decreases (i.e., particle size increases) the advantages of the spectral elements become apparent for the coupled system.     

The Intracellular Transport and Distribution of Cysteamine Phosphate Derivatives

Radioautography and extractive techniques were used to analyze the transport of cysteamine phosphate and its derivatives in salamander oocytes. The quantitative relations among the processes involved — membrane permeation, enzymatic dephosphorylation, binding through mixed disulfide formation, and cytoplasmic diffusion — were elucidated. Within the detection limits, all of the intracellular material is present as dephosphorylated derivatives. Cytoplasmic diffusion is effectively slowed by binding (the “chromatographic” effect) and makes an appreciable contribution to cellular flux rates. As a consequence, one can observe by radioautography a cortical diffusion ring which spreads inward as a function of influx time, while also increasing in peak density because of the finite membrane permeability. Good agreement was found between the transport parameters determined by radioautography and those from influx data for the whole oocyte. The ratio of nuclear to cytoplasmic concentrations of the cysteamine phosphate derivatives at equilibrium is about 0.4. The nuclear membrane is, however, a negligible barrier to transport, and the asymmetry appears to arise primarily from the quantity and sulfhydryl content of the binding proteins in the two compartments.     

Sodium fluxes through the active transport pathway in toad bladder.

To assess the active components of sodium flux across toad bladder as a function of transepithelial potential, unidirectional sodium fluxes between identical media were measured before and after adding sufficient ouabain (1.89 X 10(-3)M) to eliminate active transport, while clamping transepithelial potential to 0, 100 or 150 mV. Evidence was adduced that ouabain does not alter passive fluxes, and that fluxes remain constant if ouabain is not added. Hence, the ouabain-inhibitable fluxes represent fluxes through the active path. Results were analyzed by a set of equations, previously shown to describe adequately passive fluxes under electrical gradients in this tissue, here modified by the insertion of E, the potential at which bidirectional sodium fluxes (beta E, and theta E) through the active pathway are equal. According to these equations, beta E and theta E are the logarithmic mean of bidirectional fluxes through the active path at any potential, and the flux ratio in this path is modified by a constant factor Qia, which represents the ratio of the bulk diffusion coefficient to the tracer diffusion coefficient in this pathway. The data are shown to conform closely to these equations. Qia averages 2.54. Hence, serosal-to-mucosal flux vanishes rapidly as potential falls below E. Mean E in these experiments was 158 +/- 1 mV. Thus, linear dependence of net flux in both active and passive pathways on potential is present, even though the sodium fluxes in both paths fail to conform to the Ussing flux ratio equation. Qip less than 1 in the passive path (qualitatively similar to exchange diffusion) and Qia greater than 1 in the active path (as in single file pore diffusion). Both of these features tend to reduce the change in serosal-to-mucosal sodium flux induced by depolarization from spontaneous potential to zero potential ("short-circuiting").     

Small-scale fluid motion mediates growth and nutrient uptake of Selenastrum capricornutum

1. A fluid‐flow reactor using submersible speakers was constructed to generate small‐scale fluid motion similar to conditions measured in open water environments; flow was quantified by particle image velocimetry. Additionally a Couette‐type rotating cylinder was used to generate shear flows; flow was quantified using an optical hotwire probe and torque measurements. Growth rates of the green alga Selenastrum capricornutum were determined from changes in cell counts and viability was tested using the fluorogenic probe fluoresceine diacetate. 2. Evidence that fluid motion directly affects growth rates was obtained as a significant difference between growth in a moving versus non‐moving fluid. A near 2‐fold increase in growth rate was achieved for an energy dissipation rate of ? = 10^?7 m² s^?3; a rate common in lakes and oceans. The onset of the viability equilibrium, identified as the day of the test period when the number of active cells equalled non‐active cells, was delayed by 2 days for moving fluid conditions as compared with a non‐moving fluid. 3. Nutrient uptake was determined by a decrease in the bulk fluid concentration and cellular phosphorus concentration was also estimated. The thickness of the diffusive sublayer surrounding a cell, a zone dominated by molecular diffusion, was estimated. Increasing fluid motion was found to decrease the thickness of this layer. The Sherwood number (ratio of total mass flux to molecular mass flux) showed that advective flux surrounding cells dominated molecular diffusion flux with regard to Péclet numbers (ratio of advective transport to molecular diffusion transport). Fluid motion facilitated uptake rates and resulted in increased growth rates, compared with no‐flow conditions. The rate‐of‐rotation and the rate‐of‐strain in a moving fluid equally mediated the diffusive sublayer thickness surrounding the cells. Our study demonstrates that small‐scale fluid motion mediates algal growth kinetics and therefore should be included in predictive models for algal blooms.     

Current noise around steady states in discrete transport systems

Subject of this paper is the transport noise in discrete systems. The transport systems are given by a number (n) of binding sites separated by energy barriers. These binding sites may be in contact with constant outer reservoirs. The state of the system is characterized by the occupation numbers of particles (current carriers) at these binding sites. The change in time of the occupation numbers is generated by individual “jumps” of particles over the energy barriers, building up the flux matter (for charged particles: the electric current). In the limit n → ∞ continuum processes as e.g. usual diffusion are included in the transport model. The fluctuations in occupation numbers and other quantities linearly coupled to the occupation numbers may be treated with the usual master equation approach. The treatment of the fluctuations in fluxes (current) makes necessary a different theoretical approach which is presented in this paper under the assumption of vanishing interactions between the particles. This approach may be applied to a number of different transport systems in biology and physics (ion transport through porous channels in membranes, carrier mediated ion transport through membranes, jump diffusion e.g. in superionic conductors). As in the master equation approach the calculation of correlations and noise spectra may be reduced to the solution of the macroscopic equations for the occupation numbers. This result may be regarded as a generalization to non-equilibrium current fluctuations of the usual Nyquist theorem relating the current (voltage) noise spectrum in thermal equilibrium to the macroscopic frequency dependent admittance.The validity of the general approach is demonstrated by the calculation of the autocorrelation function and spectrum of current noise for a number of special examples (e.g, pores in membrances, carrier mediated ion transport).     

Passive Asymmetric Transport through Biological Membranes

The magnitude of passive diffusional solute transfer through artificial membranes is usually considered to be independent of the direction of the concentration gradient driving force. It can be shown, however, that a composite membrane, having as one component a membrane with a chemical reaction-facilitated diffusion transport mechanism, can result in an asymmetrical flux. An asymmetric flux caused by this type of structural heterogeneity may be one mechanism contributing to the asymmetric properties of biological membranes. Similar vectorial fluxes can be generated in interfacial solute transfer through membranes if hydrodynamic boundary layers occur at the membrane interface and reversible chemical reactions with the permeant species are involved in either phase.