Theoretical Analysis of Net Tracer Flux Due to Volume Circulation in a Membrane with Pores of Different Sizes : Relation to solute drag model

When osmotic pressure across an artificial membrane, produced by a permeable electrically neutral solute on one side of it, is balanced by an external pressure difference so that there is no net volume flow across the membrane, it has been found that there will be a net flux of a second electrically neutral tracer solute, present at equal concentrations on either side of the membrane, in the direction that the "osmotic" solute diffuses. This has been ascribed to solute-solute interaction or drag between the tracer and the osmotic solutes. An alternative model, presented here, considers the membrane to have pores of different sizes. Under general assumptions, this "heteroporous" model will account for both the direction of net tracer flux and the observed linear dependence of unidirectional tracer fluxes on the concentration of the osmotic solute. The expressions for the fluxes of solutes and solvent are mathematically identical under the two models. An inequality is derived which must be valid if the solute interaction model and/or the heteroporous model can account for the data. If the inequality does not hold, then the heteroporous model alone cannot explain the data. It was found that the inequality holds for most published observations except when dextran is the osmotic solute.     

A two-barrier compartment model for volume flow across amphibian skin

The amphibian skin has long been used as a model tissue for the study of ion transport and osmotic water movement across tight epithelia. To understand the mechanism of water uptake across amphibian skin, we model the skin as a well-stirred compartment bounded by an apical barrier and a tissue barrier. The compartment represents the lateral intercellular space between cells in the stratum granulosum. The apical barrier represents the stratum corneum, the principal/mitochondria-rich cells, and the junctional area between cells. This barrier is hypothesized to have the ability to actively transport solutes through Na+-K+-ATPase. The actively transported solute flux is assumed to satisfy the Michaelis-Menten relationship. The tissue barrier represents a composite barrier comprising the stratum spinosum, the stratum germinativum, the basal lamina, and the dermis. Our model shows that 1) the predicted rehydration rates from apical bathing solutions are in good agreement with the experiment results in Hillyard and Larsen (J Comp Physiol 171: 283-292, 2001); 2) under their experimental conditions, there is a substantial volume flux coupled to the active solute flux and this coupled volume flux is nearly constant when the osmolality of the apical bathing solution is >100 mosmol/kgH2O; 3) the molar ratio of the actively transported solute flux to the coupled water flux is about 1:160, which is the same as that reported in Nielsen (J Membr Biol 159: 61-69, 1997).     

Exact solutions of a two parameter flux model and cryobiological applications

Solute-solvent transmembrane flux models are used throughout biological sciences with applications in plant biology, cryobiology (transplantation and transfusion medicine), as well as circulatory and kidney physiology. Using a standard two parameter differential equation model of solute and solvent transmembrane flux described by Jacobs [The simultaneous measurement of cell permeability to water and to dissolved substances, J. Cell. Comp. Physiol. 2 (1932) 427-444], we determine the functions that describe the intracellular water volume and moles of intracellular solute for every time t and every set of initial conditions. Here, we provide several novel biophysical applications of this theory to important biological problems. These include using this result to calculate the value of cell volume excursion maxima and minima along with the time at which they occur, a novel result that is of significant relevance to the addition and removal of permeating solutes during cryopreservation. We also present a methodology that produces extremely accurate sum of squares estimates when fitting data for cellular permeability parameter values. Finally, we show that this theory allows a significant increase in both accuracy and speed of finite element methods for multicellular volume simulations, which has critical clinical biophysical applications in cryosurgical approaches to cancer treatment.     

Effects of Abscisic Acid on the Hydraulic Conductance of and the Total Ion Transport through Phaseolus Root Systems

The response of solute and volume fluxes of Phaseolus root systems to applied AbA was observed under conditions of applied pressure which were used to enhance the volume flow. The growth regulator elicited three separate responses: a transient release of solutes to the xylem which was responsible for an initial increase in volume flux; a long term increase in the total ion flux; and a long term decrease in the hydraulic conductance of the root systems. The exact response was highly dependent on the magnitude of the pre-AbA volume flux density, the relative contributions of osmotic and pressure-induced flow, and the applied dosage.     

A mixture theory analysis for passive transport in osmotic loading of cells

The theory of mixtures is applied to the analysis of the passive response of cells to osmotic loading with neutrally charged solutes. The formulation, which is derived for multiple solute species, incorporates partition coefficients for the solutes in the cytoplasm relative to the external solution, and accounts for cell membrane tension. The mixture formulation provides an explicit dependence of the hydraulic conductivity of the cell membrane on the concentration of permeating solutes. The resulting equations are shown to reduce to the classical equations of Kedem and Katchalsky in the limit when the membrane tension is equal to zero and the solute partition coefficient in the cytoplasm is equal to unity. Numerical simulations demonstrate that the concentration-dependence of the hydraulic conductivity is not negligible; the volume response to osmotic loading is very sensitive to the partition coefficient of the solute in the cytoplasm, which controls the magnitude of cell volume recovery; and the volume response is sensitive to the magnitude of cell membrane tension. Deviations of the Boyle-van't Hoff response from a straight line under hypo-osmotic loading may be indicative of cell membrane tension.     

A mixture theory model of fluid and solute transport in the microvasculature of normal and malignant tissues. I. Theory

In order to better understand the mechanisms governing transport of drugs, nanoparticle-based treatments, and therapeutic biomolecules, and the role of the various physiological parameters, a number of mathematical models have previously been proposed. The limitations of the existing transport models indicate the need for a comprehensive model that includes transport in the vessel lumen, the vessel wall, and the interstitial space and considers the effects of the solute concentration on fluid flow. In this study, a general model to describe the transient distribution of fluid and multiple solutes at the microvascular level was developed using mixture theory. The model captures the experimentally observed dependence of the hydraulic permeability coefficient of the capillary wall on the concentration of solutes present in the capillary wall and the surrounding tissue. Additionally, the model demonstrates that transport phenomena across the capillary wall and in the interstitium are related to the solute concentration as well as the hydrostatic pressure. The model is used in a companion paper to examine fluid and solute transport for the simplified case of an axisymmetric geometry with no solid deformation or interconversion of mass.     

Osmoregulation,solute distribution,and growth in soybean seedlings having low water potentials

Soybean (Glycine max (L.) Merr.) seedlings osmoregulate when the supply of water is limited around the roots. The osmoregulation involves solute accumulation (osmotic adjustment) by the elongating region of the hypocotyls. We investigated the relationship between growth, solute accumulation, and the partitioning of solutes during osmoregulation. Darkgrown seedlings were transplanted to vermiculite containing 1/8 (0.13 x) the water of the controls. Within 12–15 h, the osmotic potential of the elongating region had decreased to-12 bar, but it was-7 bar in the controls. This osmoregulation involved a true solute accumulation by the hypocotyls, since cell volume and turgor were virtually the same regardless of the water regime. The hypocotyls having low water potentials elongated slowly but, when deprived of their cotyledons, did not elongate or accumulate solute. This result indicated a cotyledonary origin for the solutes and a dependence of slow growth on osmotic adjustment. The translocation of nonrespired dry matter from the cotyledons to the seedling axis was unaffected by the availability of water, but partitioning was altered. In the first 12 h, dry matter accumulated in the elongating region of the 0.13 x hypocotyls, and osmotic adjustment occurred. The solutes involved were mostly free amino acids, glucose, fructose, and sucrose, and these accounted for most of the increased dry weight. After osmotic adjustment was complete, dry matter ceased to accumulate in the hypocotyls and bypassed them to accumulate in the roots, which grew faster than the control roots. The proliferation of the roots resulted in an increased root/shoot ratio, a common response of plants to dry conditions.Osmotic adjustment occurred in the elongating region of the hypocotyls because solute utilization for growth decreased while solute uptake continued. Adjustment was completed when solute uptake subsequently decreased, and uptake then balanced utilization. The control of osmotic adjustment was therefore the rate of solute utilization and, secondarily, the rate of solute uptake. Elongation was inhibited by unknown factors(s) despite the turgor and substrates associated with osmotic adjustment. The remaining slow elongation depended on osmotic adjustment and represented some optimum between the necessary inhibition for solute accumulation and the necessary growth for seedling establishment.     

Compatible solute biosynthesis in cyanobacteria

Compatible solutes are a functional group of small, highly soluble organic molecules that demonstrate compatibility in high amounts with cellular metabolism. The accumulation of compatible solutes is often observed during the acclimation of organisms to adverse environmental conditions, particularly to salt and drought stress. Among cyanobacteria, sucrose, trehalose, glucosylglycerol and glycine betaine are used as major compatible solutes. Interestingly, a close correlation has been discovered between the final salt tolerance limit and the primary compatible solute in these organisms. In addition to the dominant compatible solutes, many strains accumulate mixtures of these compounds, including minor compounds such as glucosylglycerate or proline as secondary or tertiary solutes. In particular, the accumulation of sucrose and trehalose results in an increase in tolerance to general stresses such as desiccation and high temperatures. During recent years, the biochemical and molecular basis of compatible solute accumulation has been characterized using cyanobacterial model strains that comprise different salt tolerance groups. Based on these data, the distribution of genes involved in compatible solute synthesis among sequenced cyanobacterial genomes is reviewed, and thereby, the major compatible solutes and potential salt tolerance of these strains can be predicted. Knowledge regarding cyanobacterial salt tolerance is not only useful to characterize strain-specific adaptations to ecological niches, but it can also be used to generate cells with increased tolerance to adverse environmental conditions for biotechnological purposes.     

Some compartmental models of the root: steady-state behavior

Plots of the pressure difference (DeltaP) applied to plant roots vs. the resulting volume flow rate (Q(v)) often exhibit an anomalous offset that has been difficult to explain. The present analysis suggests that solute build-up in two- and three-compartment models of the root cannot account for this offset. The Ginsburg-Newman three-compartment model explains the offset in terms of differing reflection coefficients for the membranes bounding the intermediate compartment. This model appears more promising, but it predicts a minimum in the plot of xylem-sap osmotic pressure vs. Q(v)which is not observed in practice. Fiscus hypothesized that an internal asymmetric distribution of non-mobile solutes is responsible for the offset. In the present study, this hypothesis is incorporated into a four-compartment model of the root that is conceptually related to the three-compartment model of Miller. But according to the four-compartment model, the asymmetric solute distribution does not arise because of solvent drag. Rather the anomalous offset is associated with a concentration gradient of photoassimilates (the non-mobile solutes) that exists in the absence of volume flow, and which drives the diffusive transport of these solutes from the stele to the cortex via endodermal plasmodesmata. This model is consistent with the existence of radial symplastic osmotic-pressure gradients, and it appears to have greater explanatory power than the Ginsburg-Newman model. In particular, it suggests explanations for diurnal variations in DeltaP-Q(v)curves, as well as the effects of changing external solute concentrations. It also shows how the overall root reflection coefficient can be less than unity, even when the cell membranes are effectively ideally semipermeable, and there is negligible extracellular transport of water and solutes. The model makes a number of experimentally testable predictions.     

The Interaction between Osmotic- and Pressure-induced Water Flow in Plant Roots

This paper presents a general model for coupled solute and water flow through plant roots based on the thermodynamics of irreversible processes. The model explains in a straight-forward manner such experimentally observed phenomena as changes in root resistance, increased solute flux, and apparent negative resistance, which have been reported for root systems under the influence of a hydrostatic pressure gradient. These apparent anomalies are explained on the basis of the interaction between the osmotic and hydrostatic driving forces and the well known “sweeping away” or dilution effect. We show that with a constant hydraulic conductivity the only features necessary to explain these phenomena are some type of membrane or membranelike structure and a mechanism for actively accumulating solutes.     

Dynamic loading of immature epiphyseal cartilage pumps nutrients out of vascular canals

The potential influence of mechanical loading on transvascular transport in vascularized soft tissues has not been explored extensively. This experimental investigation introduced and explored the hypothesis that dynamic mechanical loading can pump solutes out of blood vessels and into the surrounding tissue, leading to faster uptake and higher solute concentrations than could otherwise be achieved under unloaded conditions. Immature epiphyseal cartilage was used as a model tissue system, with fluorescein (332 Da), dextran (3, 10, and 70 kDa) and transferrin (80 kDa) as model solutes. Cartilage disks were either dynamically loaded (± 10% compression over a 10% static offset strain, at 0.2 Hz) or maintained unloaded in solution for up to 20 h. Results demonstrated statistically significant solute uptake in dynamically loaded (DL) explants relative to passive diffusion (PD) controls for all solutes except unbound fluorescein, as evidenced by the DL:PD concentration ratios after 20 h (1.0 ± 0.2, 2.4 ± 1.1, 6.1 ± 3.3, 9.0 ± 4.0, and 5.5 ± 1.6 for fluorescein, 3, 10, and 70 kDa dextran, and transferrin). Significant uptake enhancements were also observed within the first 30s of loading. Termination of dynamic loading produced dissipation of enhanced solute uptake back to PD control values. Confocal images confirmed that solute uptake occurred from cartilage canals into their surrounding extracellular matrix. The incidence of this loading-induced transvascular solute pumping mechanism may significantly alter our understanding of the interaction of mechanical loading and tissue metabolism.     

Osmoregulation, growth and sucrose accumulation in germinated Trigonella foenum-graecum (fenugreek) seeds treated with polyethylene glycol

Germinated seeds of Trigonella foenum-graecum L. (fenugreek) were grown in water or in polyethylene glycol (PEG) solutions. After endosperm removal, the water relations, growth, dry weight, sucrose and reducing sugar content of the embryo were determined. Under water sstress conditions, water content and osmotic potential (π₀) at saturation, growth and dry weight were lower than in non-stressed controls. The reduction in dry weight indicated a lower uptake of solutes from the endosperm and the decrease in π₀ was not accompanied by an increase in the amount of the accumulated solutes. It is suggested that embryos of stressed fenugreek seeds control osmotic potential by reduction of water uptake and that this results in reduction of growth. Embryos isolated from germinated seeds ("naked" embryos) were grown in water or in PEG solutions, with or without galactose (as an external solute source substituting for the endosperm). The results indicate that a decrease in the external solute did not account for growth reduction under conditions of water stress, and that decreased solute transport to the embryo may be important. The sucrose contents of "naked" embryos and of embryos from whole seeds were higher after PEG treatment, while reducing sugar contents were lower compared to non-stressed controls. The increased sucrose accumulation may be due to decreased sucrose hydrolysis.     

Coupling of Solute and Solvent Flows in Porous Lipid Bilayer Membranes

The present experiments were designed to evaluate coupling of water and nonelectrolyte flows in porous lipid bilayer membranes (i.e., in the presence of amphotericin B) in series with unstirred layers. Alterations in solute flux during osmosis, with respect to the flux in the absence of net water flow, could be related to two factors: first, changes in the diffusional component of solute flux referable to variations in solute concentrations at the membrane interfaces produced by osmotic flow through the unstirred layers; and second, coupling of solute and solvent flows within the membrane phase. Osmotic water flow in the same direction as solute flow increased substantially the net fluxes of glycerol and erythritol through the membranes, while osmotic flow in the opposite direction to glycerol flow reduced the net flux of that solute. The observed effects of osmotic water flow on the fluxes of these solutes were in reasonable agreement with predictions based on a model for coupling of solute and solvent flows within the membrane phase, and considerably in excess of the prediction for a diffusion process alone.     

Uptake of EDTA-complexed Pb, Cd and Fe by solution- and sand-cultured Brassica juncea

Direct plant uptake of metals bound to chelating agents has important implications for metal uptake and the free-ion activity model. Uptake of hydrophilic solutes such as metal–EDTA complexes is believed to occur via bypass apoplastic flow, but many questions remain about the relative importance and selectivity of this pathway. In this study, Brassica juncea (Indian mustard) plants grown in solution- and sand-culture conditions were exposed to metal–EDTA complexes and to PTS, a hydrophilic fluorescent dye previously used as a tracer of apoplastic flow. The results suggest that there are two general phases of solute uptake. Under normal conditions, xylem sap solute concentrations are relatively low (i.e., <0.5% of concentration in solution) and there is a high degree of selectivity among different solutes, while under conditions of stress, xylem sap concentrations are significantly higher (i.e., >3% of concentration in solution) and the selectivity among solutes is less. In healthy plants, xylem sap metal–EDTA concentrations were generally an order of magnitude higher than those of PTS and differences among complexes were observed, with CdEDTA²⁻ exhibiting slightly higher xylem sap concentrations than PbEDTA²⁻ or FeEDTA⁻. Metal–EDTA complexes were found to dominate xylem sap metal speciation and the fraction of metal in xylem sap present as metal–EDTA was greater for non-nutrient metals (Pb, Cd) than for the nutrient metal Fe. Despite differences in root morphology between plants grown under solution- and sand-culture conditions, uptake of solutes was similar under both sets of growth conditions.     

A model for high-pressure ultrafiltration of blood

A semi-analytic model to predict the permeate flux during high-pressure ultrafiltration of blood with highly permeable membranes is proposed. This model explicitly considers the hydraulic resistance of the retained particles that limits the flux. An empirically derived relationship between particle surface concentration and hydraulic resistance is used. This model incorporates the axial variations in blood cell and solute surface concentrations (or concentration polarization), shear-induced diffusion coefficient for the blood cells, effective diffusion coefficient for the blood solutes, hydraulic (lumen) pressure, and flow rate. This model agrees well with experimental results in the pressure-independent filtration flux region.     

Growth,Water Relations,and Accumulation of Organic and Inorganic Solutes in Roots of Maize Seedlings during Salt Stress

Seedlings of maize (Zea mays L. cv Pioneer 3906), hydroponically grown in the dark, were exposed to NaCl either gradually (salt acclimation) or in one step (salt shock). In the salt-acclimation treatment, root extension was indistinguishable from that of unsalinized controls for at least 6 d at concentrations up to 100 mM NaCl. By contrast, salt shock rapidly inhibited extension, followed by a gradual recovery, so that by 24 h extension rates were the same as for controls, even at 150 mM NaCl. Salt shock caused a rapid decrease in root water and solute potentials for the apical zones, and the estimated turgor potential showed only a small decline; similar but more gradual changes occurred with salt acclimation. The 5-bar decrease in root solute potential with salt shock (150 mM NaCl) during the initial 10 min of exposure could not be accounted for by dehydration, indicating that substantial osmotic adjustment occurred rapidly. Changes in concentration of inorganic solutes (Na+, K+, and Cl-) and organic solutes (proline, sucrose, fructose, and glucose) were measured during salt shock. The contribution of these solutes to changes in root solute potential with salinization was estimated.     

Effect of heteroporosity on flux equations for membranes

An investigation is made of the possible errors in simple integrated equations for solute flux across both non-pieving and sieving porous membranes that can result from variations in the membrane structure. Detailed structural models are used, beginning with a membrane consisting of a parallel array of pores and progressing to series-parallel combinations of pore segments of various lengths and cross-sectional areas, with internal cross connections among pore segments allowed. It is shown that there are both upper and lower mathematical bounds on the possible variations that can be produced in a curve of solute flux versus volume flow by arbitrary variation in the membrane structure, subject only to certain general conditions. In particular, the flux equation for a homoporous membrane is a lower bound- The maximum deviations from this lower bound for a membrane of arbitrary structure are only moderately large, and require rather extreme pore size distributions; most distributions introduce only small errors. Implications of these results in studies of real membrane structure and in the design of experiments are discussed.     

The osmotic migration of cells in a solute gradient.

The effect of a nonuniform solute concentration on the osmotic transport of water through the boundaries of a simple model cell is investigated. A system of two ordinary differential equations is derived for the motion of a single cell in the limit of a fast solute diffusion, and an analytic solution is obtained for one special case. A two-dimensional finite element model has been developed to simulate the more general case (finite diffusion rates, solute gradient induced by a solidification front). It is shown that the cell moves to regions of lower solute concentration due to the uneven flux of water through the cell boundaries. This mechanism has apparently not been discussed previously. The magnitude of this effect is small for red blood cells, the case in which all of the relevant parameters are known. We show, however, that it increases with cell size and membrane permeability, so this effect could be important for larger cells. The finite element model presented should also have other applications in the study of the response of cells to an osmotic stress and for the interaction of cells and solidification fronts. Such investigations are of major relevance for the optimization of cryopreservation processes.