Heterogeneity of membrane transport quantified by the analysis of a unidirectional flux transient of charged tracer

A planar mosaic membrane consists of patches, each with a given area, diffusion coefficient, and mobility of charged tracer; a common electric field, constant in space and time, lies across all the patches. Given the properties of the patches, the transient of the total unidirectional flux (summed over the patches) is predictable. Here we deal with the inverse problem: Given only the observed transient of the total unidirectional flux (as defined experimentally by Ussing), the unknown transport heterogeneity of the mosaic membrane is to be analyzed. Results obtained previously for uncharged tracers are generalized to include effects of the field. In particular, the ratio of the arithmetic and harmonic means (both area-weighted) of the diffusion coefficients, evaluated over the membrane, is expressed in terms of only the observed transient and the field strength and is used to characterize the heterogeneity; and the unique exact solution of the inverse problem for two kinds of patches is recovered at any field strength. If the mosaic consists of n distinct kinds of patches, a sweep of the field strength from low to high values reveals (at most) n steplike shapes in the time course of the total unidirectional flux (normalized to its final steady value), which permit an approximate analysis of the heterogeneity by elementary means.     

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.     

Dynamic patches of membrane proteins

We test here a previously proposed hypothesis about lateral heterogeneity of cell membranes, a model predicting that heterogeneity is maintained by a combination of delivery and intake of molecules with barriers to lateral free diffusion. To test the validity of the model, we observed green fluorescent protein tagged major histocompatibility complex class I patches on the plasma membrane of mouse fibroblasts, using total internal reflection fluorescence microscopy in real time. The dynamic characterization revealed the life course of these patches comprises delivery of molecules at a short instant, followed by a slow, exponential decay, corresponding to diffusion of the molecules over dynamic barriers to free lateral diffusion. The characteristic lifetime of the patches, extracted from the measurements, is approximately 30 s, in excellent agreement with the predictions of the model.     

Flux ratio theorems for nonstationary membrane transport with temporary capture of tracer

It has been shown recently that the ratio of unidirectional tracer fluxes, passing in opposite directions through a membrane which has transport properties varying arbitrarily with the distance from a boundary, is independent of time from the very first appearance of the two outfluxes from the membrane. This surprising proposition has been proved for boundary conditions defining standard unidirectional fluxes, and then generalized to classes of time-dependent boundary conditions. The operational meaning of all the resulting theorems is that when any of them appear to be refuted experimentally, the presence of more than one parallel transport pathway (that is, of membrane heterogeneity transverse to the direction of transport) can be inferred and analyzed. Recent experimental data have been interpreted accordingly. However, the proofs of the theorems given so far have not taken into account the possibility of temporary capture of tracer at sites fixed in the membrane (including also entrances to microscopic culs-de-sac). The possible presence of such a process, which would not affect fluxes in the steady state, left a fundamental gap in the aforementioned inferences. It is shown here that all the theorems previously proved for the flux ratio under unsteady conditions remain valid when temporary capture of tracer is admitted, no matter how the rate of capture, and the probability distribution of residence times of tracer at capture sites, may depend on the distance from a membrane boundary. The validity of the aforementioned inferences from observed time-dependence of the flux ratio is thereby extended to a much wider class of membrane transport processes.     

Flux ratio theorems for nonlinear membrane transport under nonstationary conditions

We extend flux ratio theorems concerning ratios of unidirectional flux transients passed (in complementary experiments) through a medium of spatially inhomogeneous transport properties pertaining to diffusion, migration and temporary trapping of the transported substance. Any nonlinearity in the transport equations leads to a breakdown of the Ussing flux ratio theorem pertaining to all times. An integrated flux ratio theorem is proved for the case when the nonlinearity is in the kinetics of trapping, as when trapping sites can be saturated. The new theorem is shown to fail when the nonlinearity is due to a concentration-dependence of the diffusion coefficient, as in facilitated transport. The nature of a nonlinearity in membrane transport can therefore be elucidated experimentally by the use of the integrated flux ratio.     

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.     

Effect of diuretics on intestinal transport of electrolytes, glucose, and amino acid.

The jejunal mucosal membrane of albino mice was used to study the electrical properties and ion transport. The membrane was bathed in Krebs-Ringer solution with or without glucose.When ethacrynic acid (EA), furosemide, or amiloride was added to the bathing fluid of both sides, a transient increase followed by a decrease of both potential difference (PD) and short circuit current (Isc) were observed. In glucose-containing bathing medium, EA inhibited both net Na and Cl flux and residual flux; however, EA had little effect on both Na and Cl flux in glucose-free bathing medium. Studies using everted intestinal sac technique showed that EA inhibited both glucose and L-tyrosine across the mucosal membrane against concentration gradients. Furosemide and amiloride were less potent than EA in inhibiting the Na and Cl flux when the bathing solution contained glucose. But these two compounds had no effect on glucose and L-tyrosine transport across the intestinal mucosa. Furthermore, they did inhibit Cl flux even in the condition of glucose-free bathing medium. It is postulated that all three diuretics act on the brush-border membrane of the intestine. EA probably inhibits the Na-glucose cotransporting system; furosemide and amiloride inhibit the simple diffusion process of Na entry of Cl exit by decreasing the conductance of the membrane.     

Single-file diffusion of uncharged particles

An equation for unidirectional fluxes of nonelectrolytes and for the total flux in the single-file transport through a narrow pore has been derived. The equation obtained accounts for the correlations of the population of the pore in the coordinate of transport. The problem has been solved using superposition approximation and unidirectional fluxes has been found. The population profile in the pore was shown to have nonlinear shape; this is principally different from the results of the classical diffusion approach.     

Periodic band pattern as a dissipative structure in ion transport systems with cylindrical shape

A theory is presented for appearance of periodic band patterns of ion concentration and electric potential associated with electric current surrounding a unicellular or multicellular system of a cylindrical shape. A flux continuity at the membrane (or the surface) is reduced to a nonlinear equation expressing passive and active fluxes across the membrane and intracellular diffusion flux. It is shown that, when an external parameter is varied from the sub-critical region, i.e. the homogeneous flux state, a symmetry breaking along a longitudinal axis usually appears prior to the one along a circumferential direction. The spectrum analysis shows that the correlation length is longer in the longitudinal direction. Growth of the band pattern from a patch-shaped pattern is demonstrated by the use of numerical calculations of proton concentration on the two-dimensional space of cylindrical surface. An experimental example of formative process of H⁺ banding is given for the internodal cell ofChara. It is shown that small patches on the surface decline or are sometimes gathered to the band surrounding the circle. The resulting pattern is suggested as a kind of dissipative structure appearing far from equilibrium.     

Hindered diffusion in excised membrane patches from retinal rod outer segments.

Excised inside-out membrane patches are useful for studying the cGMP-activated ion channels that generate the electrical response to light in retinal rod cells. We show that strong ionic current across a patch changes the driving force on the current by altering the ionic concentration near the surface membrane, an effect somewhat like that first described by Frankenhaeuser and Hodgkin (1956) in squid axons. The dominant concentration change occurs in the solution adjacent to the cytoplasmic (inner) surface of the membrane, where diffusion is impaired by intracellular material that adheres to the patch during excision. The magnitude and time course of the ionic changes are consistent with the expected volume of this material and with an effective diffusion coefficient about an order of magnitude less than that in free solution. Methods are described for correcting current transients observed in voltage clamp experiments, so that channel gating kinetics can be obtained without contamination by changes in driving force. We suggest that restricted diffusion may occur in patches excised from other types of cells and influence rapid kinetic measurements.     

Unidirectional sodium and potassium fluxes through the sodium channel of squid giant axons.

Unidirectional 22Na-traced sodium influx or 42K-traced potassium efflux across the membranes of voltage-clamped squid giant axons was measured at various membrane potentials under bi-ionic conditions. Tetrodotoxin almost entirely eliminated the extra K+ efflux induced by short repetitive depolarizations in the presence of tetraethylammonium or 3,4-diaminopyridine. A method of determining the voltage dependence of the unidirectional flux through voltage-gated channels is described. This technique was used to obtain the unidirectional flux-voltage relation for the sodium channel in bi-ionic and single-ion conditions. It allows the determination of the unidirectional flux at the zero-current potential which, for influx, was found to be approximately 20% of the value measured 80 mV negative to the zero-current potential. The unidirectional flux ratio under bi-ionic conditions was also measured and the flux ratio exponent found to average 1.15 with an external sodium and an internal potassium solution. A three-barrier, two-site, multi-occupancy model previously obtained for other conditions was found to predict a similar non-unity average for the flux ratio exponent. It is also shown that some single-occupancy models can predict non-unity values for the flux ratio exponent in bi-ionic conditions.     

The Gibbs paradox and unidirectional fluxes

The Gibbs paradox is introduced by means of a particular example: the interdiffusion of two molecular species through a membrane. In the limit when passing from the interdiffusion of two distinct species to that of identical species, the equation for dissipation (or, equivalently, entropy production) fails catastrophically. In the case of an equilibrium system, one finds the paradoxical situation of a zero net flux in conjunction with nonzero dissipation. A general analysis of the problem is given in terms of a metrical theory of irreversible thermodynamics. Its mathematical structure provides a criterion for identity sufficient to resolve the paradox. Metrical aspects of the Principle of Superposition are examined. The paper ends with a discussion of the conceptual and physical difficulties that the Gibbs paradox poses for the idea of unidirectional fluxes. In particular, it is demonstrated that the commonly held notion that net flux is a superposition of an inward and an outward flux violates the Second Law.     

Bipartite expressions for diffusional mass transport in biomembranes

Analytical expressions for solute diffusion through a membrane barrier for different initial and boundary conditions are available in the literature. The three commonest initial and boundary conditions are for a membrane without solute respectively immersed in a solution of constant concentration, immersed in such a solution for one side but with the other side isolated, and immersed in such a solution for one side and with the other side kept at zero concentration. The physical quantities for the first two initial and boundary conditions are concentration and average concentration (the total solute entering the membrane) with amperometric current (flux) and solute that permeates through the membrane (charge passed) for the third initial and boundary condition. Expressions for these methods in the literature are inconvenient for practical applications because of the infinite mathematical series required. An investigation of convergence of these expressions was therefore carried out. Simple but accurate bipartite expressions for these methods were constructed and provided theoretical support for studies on mass transport characterization of biomembranes. As a specific application, these expressions enabled a direct fit of the simulated observables to experimental values to obtain diffusion coefficients. For these initial and boundary conditions and corresponding physical quantities, simple one point methods for diffusion coefficient estimation are also suggested. These latter diffusion coefficients can be initial values for numerical fit methods.     

A direct substitution, equation error technique for solving the thermographic tomography problem

A new technique for solving the combined state and parameter estimation problem in thermographic tomography is presented. The technique involves the direct substitution of known skin temperatures into the finite difference form of the bio-heat transfer equation as formulated for solving an initial value problem with a convection boundary condition at the skin surface. These equations are then used to solve the inverse bio-heat transfer problem for the unknown subcutaneous tissue temperatures and physiological parameters. For a small number of nodal points, closed form algebraic solutions are obtained. For larger sets of equations, a hybrid technique is used in which the problem is initially posed as an unconstrained optimization problem in which the model equation error is minimized using the conjugate gradient descent technique to get close to a solution. Then a generalized Newton-Raphson technique was used to solve the equations. A numerical simulation of a one-dimensional problem is investigated both with and without noise superimposed on the input (transient) skin temperature data. The results show that the technique gives very accurate results if the skin temperature data contains little noise. It is also shown that if the physical properties of the tissue and the metabolism are known, that a given set of proper transient skin temperature inputs yields a unique solution for the unknown internal temperatures and blood perfusion rates. However, the similar problem with known blood perfusion rates and unknown metabolisms does not yield a unique solution for the internal temperatures and metabolisms.     

Activated membrane patches guide chemotactic cell motility

Many eukaryotic cells are able to crawl on surfaces and guide their motility based on environmental cues. These cues are interpreted by signaling systems which couple to cell mechanics; indeed membrane protrusions in crawling cells are often accompanied by activated membrane patches, which are localized areas of increased concentration of one or more signaling components. To determine how these patches are related to cell motion, we examine the spatial localization of RasGTP in chemotaxing Dictyostelium discoideum cells under conditions where the vertical extent of the cell was restricted. Quantitative analyses of the data reveal a high degree of spatial correlation between patches of activated Ras and membrane protrusions. Based on these findings, we formulate a model for amoeboid cell motion that consists of two coupled modules. The first module utilizes a recently developed two-component reaction diffusion model that generates transient and localized areas of elevated concentration of one of the components along the membrane. The activated patches determine the location of membrane protrusions (and overall cell motion) that are computed in the second module, which also takes into account the cortical tension and the availability of protrusion resources. We show that our model is able to produce realistic amoeboid-like motion and that our numerical results are consistent with experimentally observed pseudopod dynamics. Specifically, we show that the commonly observed splitting of pseudopods can result directly from the dynamics of the signaling patches.     

Lifetime of major histocompatibility complex class-I membrane clusters is controlled by the actin cytoskeleton

Lateral heterogeneity of cell membranes has been demonstrated in numerous studies showing anomalous diffusion of membrane proteins; it has been explained by models and experiments suggesting dynamic barriers to free diffusion, that temporarily confine membrane proteins into microscopic patches. This picture, however, comes short of explaining a steady-state patchy distribution of proteins, in face of the transient opening of the barriers. In our previous work we directly imaged persistent clusters of MHC-I, a type I transmembrane protein, and proposed a model of a dynamic equilibrium between proteins newly delivered to the cell surface by vesicle traffic, temporary confinement by dynamic barriers to lateral diffusion, and dispersion of the clusters by diffusion over the dynamic barriers. Our model predicted that the clusters are dynamic, appearing when an exocytic vesicle fuses with the plasma membrane and dispersing with a typical lifetime that depends on lateral diffusion and the dynamics of barriers. In a subsequent work, we showed this to be the case. Here we test another prediction of the model, and show that changing the stability of actin barriers to lateral diffusion changes cluster lifetimes. We also develop a model for the distribution of cluster lifetimes, consistent with the function of barriers to lateral diffusion in maintaining MHC-I clusters.     

A model for membrane patchiness: lateral diffusion in the presence of barriers and vesicle traffic

Patches (lateral heterogeneities) of cell surface membrane proteins and lipids have been imaged by a number of different microscopy techniques. This patchiness has been taken as evidence for the organization of membranes into domains whose composition differs from the average for the entire membrane. However, the mechanism and specificity of patch formation are not understood. Here we show how vesicle traffic to and from a cell surface membrane can create patches of molecules of the size observed experimentally. Our computer model takes into account lateral diffusion, barriers to lateral diffusion, and vesicle traffic to and from the plasma membrane. Neither barriers nor vesicle traffic alone create and maintain patches. Only the combination of these produces a dynamic but persistent patchiness of membrane proteins and lipids.     

Comparison of the enhanced steady-state diffusion of calcium by calbindin-D9K and calmodulin: possible importance in intestinal calcium absorption

The diffusion of calcium was measured using the unidirectional flux of 45Ca across an aqueous layer. The aqueous layer was bounded by two dialysis membranes and convection was eliminated by gelling the aqueous layer with agarose. The apparent self-diffusion coefficient was determined by the dependence of the tracer flux on the diffusion distance. The apparent self-diffusion coefficient increased linearly with the concentration of calbindin-D9K and calmodulin, but the effect of calmodulin was markedly less than that of calbindin-D9K. This difference is attributed to the lower association constant for calmodulin. The ion-exchange resin Chelex-100 also increased the steady-state of 45Ca, but the effect of Chelex-100 was much less efficient than the effect of calbindin-D9K. The mechanism of enhanced diffusion was attributed to an enhanced gradient of total 45Ca. These results indicate that the steady-state unidirectional calcium flux is a superposition of free calcium diffusion and bound calcium diffusion, with only a small contribution due to a 'bucket brigade' mechanism. We suggest that this phenomenon may be important in calcium absorption across the intestine.     

Far-field analysis of coupled bulk and boundary layer diffusion toward an ion channel entrance.

We present a far-field analysis of ion diffusion toward a channel embedded in a membrane with a fixed charge density. The Smoluchowski equation, which represents the 3D problem, is approximated by a system of coupled three- and two-dimensional diffusions. The 2D diffusion models the quasi-two-dimensional diffusion of ions in a boundary layer in which the electrical potential interaction with the membrane surface charge is important. The 3D diffusion models ion transport in the bulk region outside the boundary layer. Analytical expressions for concentration and flux are developed that are accurate far from the channel entrance. These provide boundary conditions for a numerical solution of the problem. Our results are used to calculate far-field ion flows corresponding to experiments of Bell and Miller (Biophys. J. 45:279, 1984).     

Role of diffusion potential on the flow of angiotensin II,bradykinin and related molecules through cellophane membranes

The diffusion of electrically charged peptides (angiotensin II, bradykinin and [Suc¹]angiotensin II) across tight cellophane membranes, obtained by different degrees of acetylation, shows a kinetic behaviour which was interpreted in the literature as indicative of the existence of different molecular conformations presenting slow interconversion velocities and different permeabilities across the membrane. A diffusion potential (Δψ) was found to be present across the membrane along diffusion experiments performed in low ionic strength. Upon annihilation of δψ by chemical voltage clamping (by equally increasing the ionic strength on both bathing solutions) the diffusion rate was decreased and the flow followed first order kinetics, indicating a major role of Δψ in the process. As the ionic strength increase could also affect molecular conformation, the role of Δψ on the diffusion of those molecules was tested by fitting flux and Δψ experimental results by an integrated form of Nernst-Planck flux equation. It is concluded that the deviation from first order diffusion kinetics, observed in low ionic strength, is solely due to the diffusion potential, and not to the existence of more than one molecular conformation in aqueous solution. This study was extended to amino acids and other related charged molecules.