The unidirectional flux transient as a tool for quantifying parallel diffusional pathways through membranes. Exact solution for two pathways

If a plane membrane consists of patches, each with a given area and a given diffusion coefficient, then the transient of the total unidirectional flux of a diffusing substance (as defined experimentally by Ussing) is predictable. Here the inverse problem is studied: given only the observed transient of the total unidirectional diffusion flux, the unknown membrane heterogeneity transverse to the flux is to be quantified. The ratio of the arithmetic and of the harmonic means (both area-weighted) of the diffusion coefficients, evaluated over the membrane, is expressed in terms of the observed transient alone and is used to characterize the heterogeneity. A unique exact solution of the inverse problem for two kinds of patches is obtained in closed form. A singular limit of this solution pertains to currently postulated models of endothelial membranes, for which a characteristically shaped transient is predicted.     

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 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.     

The patch mosaic of an old-growth warm-temperate forest: patch-level descriptions of 40-year gap-forming processes and community structures

Old-growth forests consist of various types of small patches that reflect their own gap-forming process, which includes changes in environmental conditions occurring over several decades. We reconstructed the gap-forming processes that had occurred during a 40-year period for eight representative patches of an old-growth evergreen broad-leaved forest in Japan, and examined the current community structure. The selected patches were based on (1) changes in canopy heights estimated from aerial photographs taken in four different years, (2) long-term ecological research (LTER) monitoring records, and (3) a recent field survey, so that they sufficiently covered characteristic gap-forming processes such as a new gap, an old gap and consistently closed canopy. Height and diameter at breast height (DBH) were measured on all living trees taller than 1.3 m. In their height distributions, currently almost closed patches that were open in 1966 show a rotated sigmoid, whereas their DBH distributions are an inverse J-shape. In contrast, patches that have been consistently under a closed canopy exhibit gentle inverse J-shapes for both distributions. For species composition, there are no clear contrasts associated with the past gap-forming processes except for the existence of fast-growing deciduous species in large currently open patches. Our results suggested that the variation in several decades of gap-forming processes played a central role in the high patch diversity and the complex patch mosaic of the forest. Diverse gap-forming processes created micro-environmental heterogeneity both vertically and horizontally, and contributed to the maintenance of the species-rich, warm-temperate old-growth forest.     

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.     

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.     

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.     

Predator's invasion into an isolated patch with spatially heterogeneous prey distribution

The invasion success of a diffusing predator which changes its diffusion coefficient depending on whether the prey exists or not is investigated. The prey is assumed to be immobile and distributed in an isolated patch. The isolated patch consists of two kinds of region: prey-existing zone and prey-vacant zone. We discuss what relation a heterogeneity of prey distribution has with the predator's invasion success into the patch. Its spatial heterogeneity appears to affect significantly the predator's invasion. In an Appendix we briefly treat an analogous problem involving two competing species.     

Studies on the Movement of Water through the Isolated Toad Bladder and Its Modification by Vasopressin

Measurements of diffusion permeability and of net transfer of water have been made across the isolated urinary bladder of the toad, Bufo marinus, and the effects thereon of mammalian neurohypophyseal hormone have been examined. In the absence of a transmembrane osmotic gradient, vasopressin increases the unidirectional flux of water from a mean of 340 to a mean of 570 µl per cm² per hour but the net water movement remains essentially zero. In the presence of an osmotic gradient but without hormone net transfer of water remains very small. On addition of hormone large net fluxes of water occur; the magnitude of which is linearly proportional to the osmotic gradient. The action of the hormone on movement of water is not dependent on the presence of sodium or on active transport of sodium. Comparison of the net transport of water and of unidirectional diffusion permeability of the membrane to water indicates that non-diffusional transport must predominate as the means by which net movement occurs in the presence of an osmotic gradient. An action of the hormone on the mucosal surface of the bladder wall is demonstrated. The effects of the hormone on water movement are most simply explained as an action to increase the permeability and porosity of the mucosal surface of the membrane.     

Hyaluronate diffusion in semidilute solutions

Analysis of the mutual diffusion coefficient of hyaluronate reveals that it rapidly increases with increasing concentration or decreasing ionic strength. The mutual diffusion coefficients analyzed by boundary relaxation in the analytical ultracentrifuge by either Raleigh interference optics or absorption optics (through the use of fluorescein-labeled hyaluronate) yielded similar values. The theoretical treatment of the mutual diffusion coefficient has been analyzed in terms of experimentally measured intradiffusion coefficients and thermodynamic virial coefficients. Only approximate agreement between theory and experiment was found. The concept of formation of transient statistical network structures in semidilute solutions of hyaluronate was applied to evaluate a critical concentration at which network formation occurs. This has been discussed in relation to the marked decrease in the intradiffusion coefficient of hyaluronate with concentration. The formation of network structures in hyaluronate was found not to preclude the hyaluronate undergoing extremely rapid rates of mutual diffusion (with diffusion coefficients ～30 × 10^?11 m² s¹) under conditions of relatively large initial chemical potential gradients. Measurements of the unidirectional flux of hyaluronate for nonzero gradients demonstrated their marked sensitivity to the magnitude of the concentration difference across the boundary. An experimental feature of the unidrectional diffusion coefficients of hyaluronate is that they may be analyzed purely in terms of mutual and intradiffusion processes. The backflux diffusion coefficient (describing the flux against the imposed concentration gradient) appeared identical with the intradiffusion coefficient. The analysis of the various sources of errors made in this study suggests that the magnitude of the diffusion coefficients measured may be regarded only as approximate.     

Patch dynamics integrate mechanisms for savanna tree–grass coexistence

Many mechanisms have been suggested to explain the coexistence of woody species and grasses in savannas. However, evidence from field studies and simulation models has been mixed. Patch dynamics is a potentially unifying mechanism explaining tree–grass coexistence and the natural occurrence of shrub encroachment in arid and semi-arid savannas. A patch-dynamic savanna consists of a spatial mosaic of patches. Each patch maintains a cyclical succession between dominance of woody species and grasses, and the succession of neighbouring patches is temporally asynchronous. Evidence from empirical field studies supports the patch dynamics view of savannas. As a basis for future tests of patch dynamics in savannas, several hypotheses are presented and one is exemplarily examined: at the patch scale, realistically parameterized simulation models have generated cyclical succession between woody and grass dominance. In semi-arid savannas, cyclical successions are driven by precipitation conditions that lead to mass recruitment of shrubs in favourable years and to simultaneous collapse of shrub cohorts in drought years. The spatiotemporal pattern of precipitation events determines the scale of the savanna vegetation mosaic in space and time. In a patch-dynamic savanna, shrub encroachment is a natural, transient phase corresponding to the shrub-dominated phase during the successional cycle. Hence, the most promising management strategy for encroached areas is a large-scale rotation system of rangelands. In conclusion, patch dynamics is a possible scale-explicit mechanism for the explanation of tree–grass coexistence in savannas that integrates most of the coexistence mechanisms proposed thus far for savannas.     

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.     

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.     

The development of patterned mosaic landscapes: an overview

The articles in this special issue review what is known about the development of wetlands in landscapes and of patterns in various kinds of wetlands around the world with an emphasis on wetlands with treed patches (tree islands) like the Okavango Delta and the Everglades. Wetlands with treed patches have much in common with their terrestrial counterparts in arid regions. Both are mosaic landscapes in which the patches of trees are distributed in a non-random fashion in a matrix of herbaceous vegetation. In these landscapes, the treed patches are both biogeochemical and biodiversity hotspots. They are biogeochemical hotspots because treed patches preferentially acquire nutrients and other limited resources, like water in arid regions. They are biodiversity hotspots because treed patches are habitats with environmental and structural characteristics very different from those of the herbaceous matrix in which they are found. Consequently, treed patches attract and support populations of many plant and animal species that otherwise would not be found in the area. Studies and simulation models of arid mosaic landscapes have generated a number of hypotheses about the structure and functions of mosaic landscapes, most of which have yet to be tested, especially in wetland mosaic landscapes.     

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.     

Direct effects on the membrane potential due to "pumps" that transfer no net charge

The effects of active ionic transport are included in the derivation of a general expression for the zero current membrane potential. It is demonstrated that an active transport system that transfers no net charge (nonrheogenic) may, nevertheless, directly alter the membrane potential. This effect depends upon the exchange of matter within the membrane between the active and passive diffusion regimes. Furthermore, in the presence of such exchange, the transmembrane active fluxes measured by the usual techniques and the local pumped fluxes are not identical. Several common uses of the term “electrogenic pump” are thus shown to be inconsistent with each other. These inconsistencies persist when the derivation is extended to produce a Goldman equation modified to account for active transport; however, that equation is shown to be limited by less narrow constraints on membrane heterogeneity and internal electric field than those previously required. In particular, it is applicable to idealized mosaic membranes limited by these requirements.     

Measurement of facilitated calcium diffusion by a soluble calcium-binding protein

The flux of calcium through an aqueous compartment was determined in a flow-dialysis cell in which two dialysis membranes separated the middle aqueous compartment from two outer compartments. The contribution of convection to the total calcium flux was large but could be removed by addition of 1% agar. The flux of calcium through the gelled aqueous compartment agreed with theoretical expectations. The self-diffusion coefficient for calcium from these results was calculated to be 0.81 X 10(-5) cm2 X s-1. Carp parvalbumin significantly enhanced the calcium flux at 2.3 X 10(-6)M free calcium. The calcium flux increased linearly with parvalbumin concentration. These observations are consistent with the hypothesis that the overall unidirectional calcium flux J is the sum of free calcium diffusion and protein-calcium diffusion: J = D[Ca] + D'[CaPr]. The value of D', the self-diffusion coefficient for parvalbumin, was calculated from the flux data to be 13.7 X 10(-7) cm2 X s-1.     

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.